Research In Finance, Vol. 23

LIST OF CONTRIBUTORS Timothy J. Brailsford UQ Business School, The University of Queensland, Brisbane, Australia Chuan...

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LIST OF CONTRIBUTORS Timothy J. Brailsford

UQ Business School, The University of Queensland, Brisbane, Australia

Chuang-Chang Chang

Department of Finance, National Central University, Taiwan, Republic of China

Mo Chaudhury

Faculty of Management, McGill University, Montreal, Quebec, Canada

O. Emre Ergungor

Federal Reserve Bank of Cleveland, Cleveland, OH

Wayne Ferson

Carroll School of Management, Boston College, MA

Tyler Henry

Terry College of Banking and Finance, Athens, GA

William Hillison

College of Business, Florida State University, Tallahassee, FL

Bradley K. Hobbs

College of Business, Florida Gulf Coast University, Ft. Myers, FL

Darren Kisgen

Carroll School of Management, Boston College, MA

Jeff Madura

Florida Atlantic University, College of Business, Boca Raton, FL

Patricia A. McGraw

School of Business Management, Ryerson University, Toronto, Ontario, Canada

Bruce L. McManis

College of Business, Nicholls State University, Thibodaux, LA

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LIST OF CONTRIBUTORS

Kamphol Panyagometh

Graduate School of Business Administration, National Institute of Development Administration, Bangkok, Thailand

Carl Pacini

Department of Accounting and Finance, Florida Gulf Coast University, Fort Myers, FL

Jack H. W. Penm

Faculty of Economics and Commerce, The Australian National University, Canberra, Australia

Nivine Richie

Sigmund Weis School of Business, Susquehanna University, Selinsgrove, PA

Gordon S. Roberts

Schulich School of Business, York University, Toronto, Canada

Mark Schaub

Northwestern State University, Natchitoches, LA

Nadeem A. Siddiqi

LaSalle Bank Corporation, Ann Arbor, MI

Richard D. Terrell

National Graduate School of Management, The Australian National University, Canberra, Australia

James B. Thomson

Federal Reserve Bank of Cleveland, Cleveland, OH

Yu Jih-Chieh

Department of Finance, National Central University, Taiwan, Republic of China

INTRODUCTION Since its first appearance in 1979, Research in Finance has continued to publish novel, theoretical, and empirical research papers that represent significant contributions to important areas in finance, and economics. A total of 10 papers in this volume constitute original research spanning the topical areas of investments, financing, and banking. In a framework that relies on stochastic discount factor (SDF) modeling, Ferson et al. investigate the performance of fixed income mutual funds. They show that in some – but not all – economic states, the returns of fixed income funds during the 1985–1999 period were less than those of passive benchmarks that did not pay expenses. Schaub and McManis employ cross-sectional regression analysis to identify key factors affecting the long-term excess performance of American depository receipts (ADRs) listed on the NYSE. Brailsford et al. apply a time-series model of variable factors with kernel smoothing to forecast Euro/US Dollar exchange rates and the monthly net asset value (NAV) of U.S. open-end mutual funds. Using transactions-level data, Richie and Madura provide empirical evidence on differing degrees of fragmentation in day and night markets. Pacini et al. empirically examine the market reactions of U.S.-listed foreign banks to the passage of the Gramm-Leach-Bliley (GLB) Act of 1999. The contributions to this volume also examine derivatives pricing, corporate borrowing, and banking crises. For example, by establishing the properties of required analytical bounds, Chaudhury derives a more complete characterization of analytical upper bounds for American options. Chang and Yu extend the model of Das and Sundaram to value credit derivatives with correlated defaults, and counterparty risks. They also illustrate the impact of term structure interest rate volatility on the value of credit derivatives. McGraw et al. conduct a statistical analysis related to Diamond’s Life-Cycle Hypothesis, and they present empirical evidence supporting it. Siddiqi’s paper shows that the firm’s debt choice exhibits a life cycle, and that firms’ preferences change over the course of the life cycle.

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INTRODUCTION

Finally, in their two-part paper, Ergungor and Thomson explore in detail the issue of systemic banking crises – Part I discusses the underlying causes of banking system collapse, and Part II describes time-consistent crisis resolution policies. Andrew H. Chen Series Editor

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FIXED INCOME FUND PERFORMANCE ACROSS ECONOMIC STATES Wayne Ferson, Darren Kisgen and Tyler Henry ABSTRACT We evaluate the performance of fixed income mutual funds using stochastic discount factors motivated by continuous-time term structure models. Time-aggregation of these models for discrete returns generates new empirical ‘‘factors,’’ and these factors contribute significant explanatory power to the models. We provide a conditional performance evaluation for US fixed income mutual funds, conditioning on a variety of discrete exante characterizations of the states of the economy. During 1985–1999 we find that fixed income funds return less on average than passive benchmarks that do not pay expenses, but not in all economic states. Fixed income funds typically do poorly when short-term interest rates or industrial capacity utilization rates are high, and offer higher returns when quality-related credit spreads are high. We find more heterogeneity across fund styles than across characteristics-based fund groups. Mortgage funds underperform a GNMA index in all economic states. These excess returns are reduced, and typically become insignificant, when we adjust for risk using the models.

Research in Finance, Volume 23, 1–62 Copyright r 2007 by Elsevier Ltd. All rights of reproduction in any form reserved ISSN: 0196-3821/doi:10.1016/S0196-3821(06)23001-6

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WAYNE FERSON ET AL.

1. INTRODUCTION Recent years have witnessed an explosion of research on the performance of mutual funds, pension funds and related investment vehicles. The vast majority of this research focusses on equity-style funds. The relatively small amount of research on fixed income fund performance seems curious, given the importance of fixed income funds and assets in the economy. As of June 2002 there were 2,057 bond funds in the US, representing 25% of all mutual funds. Total assets under management by these funds totalled just over $1 trillion, or 15% of the $6.6 trillion in mutual fund assets. (These figures exclude balanced funds, which hold a mix of bonds and stocks.) Thus, fixed income funds represent a substantial economic interest. Fixed income funds have also seen rapid growth over the last decade, with the number of funds and assets under management increasing 97% and 245% respectively, between 1990 and 2002.1 Perhaps, the relatively small amount of research on fixed income funds reflects differences in the available empirical models for fixed income and equity returns.2 Standard models for expected equity returns lend themselves naturally to measures of risk-adjusted ‘‘abnormal’’ returns. For example, an ‘‘alpha’’ is measured as the difference between the actual average return of a fund and the expected return that is predicted by the model on the basis of the fund’s beta risk. Fixed income models, in contrast, are typically directed at the problem of solving for the prices of derivative claims. If a portfolio is formed with unobserved weights, such as a mutual fund, the value of the portfolio of claims is difficult to model (Farnsworth, 1997). This paper measures the performance of fixed income mutual funds in a stochastic discount factor (SDF) framework. The approach has several advantages. Popular term structure models identify SDFs that are easily time aggregated for monthly returns. The resulting theoretically motivated factors are appealing, in contrast to recently popular asset pricing factors for equities that arise from empirical regularities (e.g., Fama & French, 1996).3 Given a SDF, a measure of abnormal return similar to the traditional alpha can be easily constructed (Chen & Knez, 1996). Given the returns generated by the fund, the ‘‘SDF alpha’’ measure of performance does not require knowledge of the portfolio weights. The SDF approach lends itself naturally to conditional performance evaluation, where funds’ alphas are conditioned on ex-ante economic states. Term structure models in particular, suggest what to condition on. This removes some of the ambiguity in instrument selection that is typical of the conditional asset pricing literature. Finally, using discrete representations of the economic state, we avoid the linear

Fixed Income Fund Performance Across Economic States

3

functional form assumptions that are common in the conditional asset pricing literature. We find that the additional empirical ‘‘factors’’ implied by time aggregation of the continuous-time models contribute to an improved performance in explaining discrete period returns. We evaluate the SDF alphas, using passive benchmarks. The returns and volatility of the benchmarks vary significantly with the economic states. Using the benchmarks, a two-factor affine model outperforms a single-factor model for fitting the expected excess returns conditional on the states. (The single-factor affine model includes the models of Vasicek, 1977 and Cox, Ingersoll, & Ross, 1985a as special cases.) A twofactor Brennan and Schwartz (1979) model performs similarly to the twofactor affine model. Adding a third convexity factor to the affine models adds relatively little explanatory power. Extended models with non-term structure factors perform better than the pure term structure models. During 1985–1999 fixed income funds returned less than passive benchmarks that do not pay expenses, but not in all economic states. Fixed income funds offer relatively low returns when short-term interest rates or industrial capacity utilization rates are high, and offer higher relative returns when quality-related credit spreads are high. We find little cross-sectional variation in performance when funds are grouped into thirds by asset size, expense ratio, turnover, income yield, lagged return or lagged new money flows. There is more heterogeneity across fixed income fund styles. Mortgage funds underperform a GNMA index in all economic states. These excess returns are reduced, and typically become insignificant, when we adjust for risk using the SDFs. The rest of the paper is organized as follows. Section 2 describes and motivates our empirical approach. Section 3 presents the models for the SDFs, following term structure theory, and describes how we operationalize them to handle monthly mutual fund data. We also describe how we incorporate factors in the empirical models, for default risk and other risks outside of the default-free term structure. Section 4 describes the data. Section 5 presents a comparison of linear factor models and results on the estimation of the SDF models with passive benchmarks. Section 6 evaluates performance in our sample of mutual funds, grouped by fund characteristics. Section 7 studies performance in relation to fund style. Section 8 offers concluding remarks.

2. EMPIRICAL METHODS Most asset pricing models, including models for the term structure of interest rates, posit the existence of a SDF, m(f)t+1, which is a scalar random

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WAYNE FERSON ET AL.

variable that depends on data observed up to time t+1 and parameters f, such that the following equation holds: E t ðmðfÞtþ1 Rtþ1 Þ ¼ 1

(1)

where Rt+1 is an N-vector of gross (i.e., one plus) ‘‘primitive’’ asset returns, 1 is an N-vector of ones and Et(.) denotes the conditional expectation, given the information in the model at time t. We say that the SDF ‘‘prices’’ the primitive assets if Eq. (1) is satisfied. Re-arranging Eq. (1) reveals that the expected return is determined by the SDF model as: E t ðRtþ1 Þ ¼ ½E t ðmtþ1 Þ1 þ Covt mtþ1 =E t ðmtþ1 Þ; Rtþ1 (2) where Covt(.,.) is the conditional covariance given the information at time t. Thus, expected performance differs across funds in proportion to their conditional covariances with the SDF. We allow that a mutual fund, with return Rp,t+1, may not be priced exactly by the SDF. Its SDF alpha is defined as aptEt(mt+1 Rp,t+11). This follows Chen and Knez (1996) and Farnsworth, Ferson, Jackson, and Todd (2002), who show that the measure is proportional to the traditional alpha in a beta pricing representation, when the SDF is linear in the factors. In the case of the capital asset pricing model (Sharpe, 1964), the SDF is linear in the market return and ap is proportional to Jensen’s (1968) alpha.4 We estimate the conditional performance of a fund and the parameters of the SDF model simultaneously using the following system of moment conditions and the generalized method of moments (GMM, see Hansen, 1982). E mðfÞtþ1 Rtþ1 1 Dt ¼ 0 (3a) E

nh

i o mðfÞtþ1 Rp;tþ1 RB;tþ1 a0p Dt Dt ¼ 0

(3b)

Eq. (3a) says that the SDF prices the primitive returns, Rt+1. In Eq. (3b), the abnormal performance of a fund is measured relative to that of a benchmark return, RB,t+1. The conditional alpha is ap,t ¼ ap0 Dt, where Dt is the conditioning dummy variable, which includes a constant and a vector of (0,1) variables for the discrete economic states. For example, we define a conditioning dummy variable indicating whether the term structure slope is steeper or flatter than normal, as described below. In this way, we measure the expected abnormal performance of the fund conditional on the slope of the term structure being either steep, flat or normal. Eq. (3a) follows from Eq. (1) by the law of iterated expectations. Eq. (1) implies E(mt+1Rt+1|Zt)1 ¼ 0 for any instrument Zt that is public


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information at time t. A typical empirical approach with standard lagged instruments is to note that this implies E([mt+1Rt+11]Zt) ¼ 0, and to estimate the unconditional expectations. However, it may not be optimal to use the instruments with a linear functional form.5 When the instrument is a conditioning dummy variable the performance measure is ‘‘nonparametric.’’ If the underlying economy had discrete states, the perfect conditional measure would condition the levels of risk, expected return and performance on each discrete state. In practice, by using conditioning dummy variables and a small number of states we obtain simplicity and interpretability, and we are able to avoid a functional form assumption, at the cost of a coarse representation of the conditioning information. Of course, one can define more dummies to refine information, relative to the examples we use here. However, given recent studies that question the predictive ability of standard lagged instruments, our coarse representation may not entail a large cost.6 Farnsworth et al. (2002) show that estimating a system like (3a, 3b) for one fund at a time produces the same point estimates and standard errors for alpha as a system that includes an arbitrary number of funds. This is convenient, as the number of available funds exceeds the number of time series, and joint estimation with all of the funds is therefore not feasible.7 Farnsworth et al. also find small biases in SDF alphas, and we find small biases for fixed income benchmarks using term structure models. These biases are typically much smaller for excess returns than for raw returns. To the extent that the biases are similar for the fund and the benchmark, we control it by using Rp,t+1RB,t+1 in Eq. (3b). This has the additional advantage of increasing precision of the fund’s alpha, because the variance of the excess return is smaller than the raw return. Of course, if the model correctly prices the benchmark return, the point estimate of the fund’s alpha is not changed by the introduction of the benchmark.

3. STOCHASTIC DISCOUNT FACTOR MODELS We first explain how continuous-time term structure models specify the form of m(f)t+1 appropriate for a discrete-period return such as our monthly mutual fund data. The appropriate SDF involves integrals of functions of the continuous-time process. We describe how we approximate the integrals using daily data on interest rates. Finally, we describe how to combine a term structure model, designed for default-free bond returns, with a factor model for broader economic risks.

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3.1. Term Structure Stochastic Discount Factor Models Term structure models often specify a continuous-time stochastic process for the underlying state variable(s). For example, let X be the state variable following a diffusion process: dX ¼ mðX t Þ dt þ sðX t Þ dW

(4)

where dw is the local change in a standard Weiner process. The state variable(s) may be the level of an interest rate, the slope of the term structure, etc. Term structure models may be based on ‘‘no arbitrage’’ principles or general equilibrium. In either case the model specifies the form of a market price of risk, q(X), associated with the state variable, representing the expected return in excess of the instantaneous interest rate per unit of state variable risk. The models we study are based on time-homogeneous diffusions; that is, the functions m( ) and s( ) in Eq. (4) depend on time only through the level of the state variable at a point in time. In contrast, interest rate models such as Hull and White (1990) allow time variation in the functions, choosing them to fit closely the term structure of spot or forward rates observed at time t. Such models are attractive for the practice of pricing interest-ratedependent derivative securities, among other reasons, because by fitting the current term structure at each date the models can avoid derivative prices that allow arbitrage opportunities at the current prices. Our goal does not require us to fit precisely the structure of derivatives prices at each date. As Eq. (2) suggests, we want good models for the covariances of portfolio returns with the SDFs. Term structure models based on Eq. (4) can be shown (using Girsanov’s Theorem, see Cox, Ingersoll, and Ross (1985b) or Farnsworth, 1997) to imply SDFs of the following form: t mtþ1

¼ expðAtþ1 Btþ1 C tþ1 Þ; where Z tþ1 Atþ1 ¼ t rs ds Z tþ1 Btþ1 ¼ t qðX s Þ dws Z tþ1 C tþ1 ¼ 1=2 t qðX s Þ2 ds;

ð5Þ

where rs is the instantaneous interest rate at time s. The notation tmt+1 is chosen to emphasize that the SDF refers to a discrete time interval, in our


7

case one month that begins at time t and ends at time t+1. When there are multiple state variables, there is a term like Bt+1 and Ct+1 for each state variable. Note that, unlike beta pricing models where the SDF is linear in the factors, the SDF in (5) is nonlinear. Dietz, Fogler, and Rivers (1981) find that bond returns are nonlinearly related to bond risk factors, and argue that tests of bond portfolio performance should allow for nonlinearity.

3.2. Discretizations To use the term structure models with monthly mutual fund data, we adopt a simple first-order Euler approximation scheme for Eq. (4): X ðt þ DÞ X ðtÞ mðX t ÞD þ sðX t Þ½wðt þ DÞ wðtÞ

(6)

The period between t and t+1 is divided into 1/D increments of length D. Empirically, the period is one month, to match the mutual fund returns, and it is divided into increments of one day. For a given model, we have daily data on X(t+D) and X(t), and the functions m(Xt) and s(Xt) are specified. We can therefore infer the approximate daily values of [w(t+D)w(t)] from Eq. (6). The terms At+1, Bt+1 and Ct+1 in Eq. (5) are then approximated using daily data by Atþ1

X

rðt þ ði 1ÞDÞD

i¼1; ... 1=D

Btþ1

X

q½X ðt þ ði 1ÞDÞ½wðt þ iDÞ wðt þ ði 1ÞDÞ

i¼1; ... 1=D

C tþ1 1=2

X

q½X ðt þ ði 1ÞDÞ2 D

ð7Þ

i¼1; ... 1=D

Farnsworth (1997) and Stanton (1997) evaluate the accuracy of similar first-order approximation schemes. Stanton concludes that with daily data, these approximations are almost indistinguishable from the true functions over a wide range of values, and the approximation errors should be small when the series being studied is observed monthly. He also evaluates higher order approximation schemes, and finds that with daily data they offer negligible improvements over the first-order approximations.

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3.3. Single-Factor Models We include a single-state variable model in the affine class, where the shortterm interest rate rt is the state variable at time t: dr ¼ K ðy rt Þ dt þ sðrÞ dw sðrÞ ¼ ðY þ drÞ1=2 qðrÞ ¼ lðY þ drÞ1=2

ð8Þ

Eq. (8) includes as special cases, the single-factor models of Vasicek (1977), where d ¼ 0, and of Cox, Ingersoll, and Ross (1985a), where Y ¼ 0. The Euler approximations in Eq. (7) specialize as follows: X Atþ1 rðt þ ði 1ÞDÞD i¼1; ... ;1=D

Btþ1 lðrtþ1 rt Ky þ KAtþ1 Þ C tþ1 l2 2 ðY þ dAtþ1 Þ

ð9Þ

In our one-factor model the SDF is given by Eq. (5), with the coefficients approximated by Eq. (9). The term structure literature has directed a lot of firepower at modelling continuous-time interest rate processes like Eq. (8) as accurately as possible. When the objective is to price interest-rate-dependent derivative securities, it is important to accurately fit the stochastic process followed by state variables such as the short rate. This is because the value of an interest rate derivative may depend on the behavior of interest rates from the current date until the maturity date of the claim. Often the relation is highly nonlinear. Studies following Chan, Karolyi, Longstaff, and Sanders (1992) debate whether the power in the diffusion for the spot rate in Eq. (8) is 0.5, 1.0, 1.5 or some other number. Other studies ask whether the drift of the shortrate process is linear as in Eq. (8), or nonlinear (see, e.g., Ait-Sahalia, 1996). Indeed, Ait-Sahalia rejects most of the parametric models for the spot rate that have been proposed in the literature, by comparing their implied density functions with those observed in interest rate data. Dai and Singleton (2002) study the ability of a class of term structure models to capture the conditional first moments of returns and yield changes for zero-coupon bonds. In our application the SDF models should align the first moments of portfolio returns with their covariances with the SDF. The portfolio returns may have different dynamics from those of zero-coupon bond returns, since fund managers change their portfolio weights over time.


9

Therefore, while the term structure literature contains a lot of information on the performance of models for pricing derivatives and capturing the fine structure of interest rate dynamics, less is known about how useful the models are for the important task of risk-adjusting managed bond portfolio returns.

3.4. Multiple-Factor Models The defining characteristic of affine term structure models is that the natural logarithms of bond prices are affine (i.e., linear with an intercept) functions of the state variables. Duffie (1996, Chapter 7) provides a general representation for affine models and discusses special cases. We include two versions of two-factor term structure models. The first is the two-factor affine model: dr ¼ K 1 ðy1 rt Þ dt þ q1 =l1 dw1 þ r q2 =l2 dw2 d‘ ¼ K 2 ðy2 ‘t Þ dt þ r q1 =l1 dw1 þ q2 =l2 dw2 1=2 q 1 ¼ l 1 a 1 þ b1 r t þ Y 1 ‘ t 1=2 q 2 ¼ l 2 a 2 þ b2 r t þ Y 2 ‘ t ð10Þ where {K1, y1, K2, y2, l1, l2, r, a1, b1, a2, b2, Y1, Y2} are constant parameters. In this model, ‘t is the level of a long-term interest rate at time t, and r the correlation of the two diffusions. Both the drift and the squared diffusion terms are affine functions of the two-state variables rt and ‘t : We implement this model in the same fashion as the one-factor affine model; the empirical model is given in Eq. (13b) below. Our second two-factor term structure model is the Brennan and Schwartz (1979) two-factor model, which falls outside of the affine class: dr ¼ rt ½a lnð‘t þ krt Þ dt þ rt s1 dw1 d‘ ¼ ‘2t rt ‘t þ ‘t s22 þ q2 ‘t s2 dt þ ‘t s2 dw2

ð11Þ

where q1 and q2 are constants, E (dw1 dw2) ¼ r dt and the fixed parameters are {a, k, s1, s2, q1, q2, r}. Essentially, the same procedures are applied to implement this model. The reduced-form solutions for the term structure models are presented in system (13) below. We also consider a three-factor affine model, described below, where a measure of convexity is the third factor. The motivation for the three-factor

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model is provided by studies such as Litterman and Sheinkman (1988), Kahn (1991), Longstaff and Schwartz (1992), Balduzzi and Foresi (1998), D’Antonio, Johnsen, and Hutton (1997), and Dai and Singleton (2000).

3.5. Incorporating General Economic Risk Factors Fixed income funds hold securities that are exposed to default risk, mortgage prepayment risk and other risks not typically incorporated in pure term structure models.8 Trading by fund managers may also introduce additional dynamic structure into the managed portfolio returns. Elton, Gruber, and Blake (1995) use linear factor models, with as many as six factors, to evaluate fixed income fund performance. Our problem is to form an SDF that combines term structure and extra-term structure factors. Assume that the default-free bonds held by fixed income funds can be priced by a SDF from the term structure, m1t ¼ exp(AtBtCt), driven by the term structure factors, F1t. The funds also hold other securities whose returns are sensitive to the term structure and a set of additional factors, F2t. Partition the primitive returns as Rt ¼ (R1t, R2t), where the R1t are the default-free bonds, priced by the factors F1. We assume that the factors (F1,F2) price the returns in R2. The factors F1 and F2 may be correlated. However, we assume that the default-free bond returns in R1 are conditionally independent of the extra-term-structure factors: Covt1(R1t;F2t|F1t) ¼ 0. This says that the term structure factors F1t are sufficient to capture the ‘‘systematic’’ risks of the default-free bonds. We derive a combined SDF based on the union of the two sets of factors. First, assume that the term structure SDF is linear in its factors: m1t ¼ d00+d010 F1t. (This follows with F1texp(AtBtCt), d01 ¼ 1 and d00 ¼ 0.) Then, m1t prices the pure default-free bonds if and only if the expected returns on the R1t are linear in their betas on the F1t factors (e.g., Ferson, 1995). Dropping the notation indicating the dependence of the expectations on information at time t1, there exists an expected risk premium, l1 such that: R1t ¼ b11 ½F 1t þ l1 E ðF 1t Þ þ 1t ; with E ð1t Þ ¼ E ð1t F 1t Þ ¼ 0 where b11 is the regression slope vector and the regression has a zero intercept. If the combined factors price R2, there is a value of l2 and a regression: R2t ¼ b21 ½F 1t þ l1 E ðF 1t Þ þ b22 ½F 2t þ l2 E ðF 2t Þ þ 2t


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with E(e2t) ¼ E(e2tF1t) ¼ E(e2tF2t) ¼ 0. The combined set of factors prices the returns in R1, since the coefficient b12 on F2t in the regression of R1t on F1t and F2t is zero, by the conditional independence assumption, and the intercept therefore is equal to zero as in the first regression. Since the beta pricing relation holds for both R1 and R2, using the factors F1 and F2, it follows (e.g., Ferson, 1995), that the combined SDF is linear in the combined set of factors: mt ¼ d0+d10 F1t+d20 F2t. In summary, the combined SDF models are: t mtþ1

¼ d0 þ d1 expðAtþ1 Btþ1 C tþ1 Þ þ d02 F 2tþ1

(12)

Given the large number of parameters in a combined model, we study the extra-term structure factors F2t one at a time.

3.6. The Empirical SDF Models The empirical SDF models are written in reduced form, as follows: Single-factor affine: mðfÞtþ1 ¼ exp a þ b Artþ1 þ c½rtþ1 rt

(13a)

Two-factor affine: mðfÞ ¼ exp a þ b Artþ1 þ c½rtþ1 rt þ dA‘tþ1 þ e½‘tþ1 ‘t

(13b)

Two-factor Brennan and Schwartz: mðfÞtþ1 ¼ exp a þ b Artþ1 þ c A‘tþ1 þ d Drtþ1 þ e D‘tþ1 þ g Dr‘ tþ1

(13c)

Extended affine: mðfÞtþ1 ¼ exp a þ b Artþ1 þ c½rtþ1 rt þ d A‘tþ1 þ e ‘‘tþ1 ‘t þ d2 F 2;tþ1 (13d) Extended Brennan and Schwartz: mðfÞtþ1 ¼ exp a þ b Artþ1 þ c A‘tþ1 þ d D‘tþ1 þ e D‘tþ1 þ g Dr‘ tþ1 þ d2 F 2;tþ1 (13e)

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WAYNE FERSON ET AL.

where: Artþ1 ¼

X

rðt þ ði 1ÞDÞD

i¼1; ... 1=D

A‘tþ1 ¼

X

i¼1; ... 1=D

Drtþ1 ¼

X

i¼1; ... 1=D

D‘tþ1

¼

X

i¼1; ... 1=D

Dr‘ tþ1

¼

X

‘ðt þ ði 1ÞDÞD

rðt þ iDÞ=rðt þ ði 1ÞDÞ 1

‘ðt þ iDÞ=‘ðt þ ði 1ÞDÞ 1

In rðt þ ði 1ÞDÞ=‘ðt þ ði 1ÞDÞ D

i¼1; ... 1=D

The coefficients {a,b,c, y }, differ across the models. For identification, the coefficient d1 in Eq. (12) is set equal to 1.0 in the reduced forms, and the coefficient d0 is set equal to zero. The single-factor affine model actually depends on two short rate ‘‘factors.’’ Because of the effects of time aggregation, there is both a discrete change in the spot rate, [rt+1rt], and an average of the daily short rate levels over the month. The single-factor affine model is nested in the two-factor affine model by setting d ¼ e ¼ 0. The two-factor affine model depends both on the monthly changes in the long and short rates and on the average long rate and short rate values. The Brennan and Schwartz two-factor model replaces the discrete rate changes with the averages of daily relative changes, via the terms Drtþ1 ; D‘tþ1 and introduces the average slope measure, Dr‘ tþ1 Thus, the time-aggregated Brennan and Schwartz two-factor model actually uses five measured ‘‘factors’’ in monthly data. (We still refer to the models according to the number of theoretical factors.) We also consider a three-factor affine model, including a convexity factor, Ct. After time aggregation, the empirical factors are the discrete change over the month, [ct+1ct], and the monthly average of the daily convexity. If c(i) is the daily convexity for day i, the monthly average is Act+1 ¼ Si ¼ 1, ... 1/D c(t+(i1)D) D. Even with the additional factors that arise from time aggregation, the number of parameters that can be identified in the reduced form models is always smaller than the number of underlying parameters in the theoretical models. For example, the one-factor affine model of Eq. (8) has five parameters (four, in the special cases of the Vasicek and Cox–Ingersoll–Ross models), while


13

only three parameters can be identified using (13a). It would be possible to incorporate additional moment conditions, derived from the interest rate process specifications behind these models, and thereby identify additional parameters.9 However, if the interest rate process is misspecified, then in the attempt to fit these equations the misspecification would spill over into the estimated performance measures. It is not our goal to maximize the fit to the underlying interest rate processes. To identify the covariances of funds’ discreteperiod returns with the factors motivated by the time-aggregated models, it is sufficient to work with the smaller number of parameters identified by (13).

4. THE DATA We use several different data sets in our study. First, we describe our sample of returns and attributes for US fixed income mutual funds. We then describe the conditioning dummy variables for the states of the term structure and the broader economy. Finally, we describe our measures of the risk factors, benchmarks and primitive asset returns.

4.1. Fixed Income Mutual Fund Data The fixed income fund data are from the Center for Research in Security Prices (CRSP) mutual fund database, and include the period from 1962 through 1999. We select funds whose objectives indicate that they are primarily US fixed income funds. We exclude municipal bond funds, money market funds and international funds.10 The number of funds with some monthly return data in a given year varies from 53 in 1961 to 153 in 1973, to a high of 2,357 in 1999. However, in our version of the database, none of the fund objective codes exist prior to 1985. Using the fund returns prior to the first code indicating a fixed income fund would present a potential lookahead bias in fund classification. Our results for funds are therefore based on the returns after the first objective codes are observed. In Table 1 the funds are grouped by style according to their objective codes on the CRSP files. The return for each style group in any month is an equally weighted average of the returns of all fund shares, with return data for that month, whose most recently available objective codes fit into the style group. Panel A of Table 1 summarizes four groups: Government, High-yield Corporate, High-quality Corporate and Mortgage funds. In addition, we break out load and no-load funds.11

14

Table 1.

Summary Statistics for the Fixed Income Funds, Lagged Instruments and Factors.

Panel A: Fund return – equally weighted portfolios Fund Group All Government High quality High yield Mortgage Load No load

Period 1985–1999 1985–1999 1988–1999 1987–1999 1989–1999 1986–1999 1986–1999

Nobs 180.0 180.0 140.0 127.0 132.0 168.0 168.0

Mean 0.006708 0.006531 0.005953 0.006762 0.005456 0.006510 0.006639

Minimum 0.04876 0.06392 0.02130 0.07232 0.01602 0.03294 0.03421

Panel C: Risk factors, January 1973–December 1999 (324 observations) Factor Mean Minimum Ar 0.0006609 0.0002770 rt+1rt 9.389E–09 0.0003905 A‘ 0.006614 0.003539

Maximum 0.001579 0.0002434 0.01234

Standard 0.01473 0.01726 0.009791 0.01947 0.00820 0.01279 0.01388

r1 0.1140 0.0756 0.2153 0.3332 0.2402 0.07276 0.05838

r1 0.9709 0.8791 0.7829 0.9244 0.9635 0.9903 0.6052 0.3797 0.9798 0.9890 0.7388 0.2120

r1(Dhi) 0.8242 0.4096 0.5621 0.6516 0.8596 0.8729 0.3773 0.0013 0.8573 0.9277 0.3424 0.0874

r1(Dlo) 0.8483 0.7298 0.5638 0.6838 0.7590 0.9052 0.3351 0.3407 0.8542 0.8386 0.0939 0.1439

Standard 0.0002600 6.238E–05 0.001866

r1 0.9747 0.1145 0.9845

WAYNE FERSON ET AL.

Panel B: Lagged instruments, to predict January 1968–December 1999 (384 observations) Instrument Mean Minimum Maximum Standard Short rate 6.925 2.78 16.71 2.727 Slope 0.9279 4.25 5.208 1.340 Convexity 0.1010 0.626 0.9035 0.2025 Volatility 0.6030 0.023 1.552 0.2447 Credit 1.090 0.550 2.690 0.4415 BS-spread 4.259 1.772 8.435 1.558 Inflation 5.013 5.412 21.47 3.917 IP growth 2.889 50.96 40.14 9.518 Cap. util. 81.95 71.10 89.20 3.529 Xchange 103.3 80.97 158.4 15.43 Corp. iliq. 0.0805 0.099 1.149 0.135 Stock liq. 0.03116 0.469 0.203 0.059

Maximum 0.06134 0.08138 0.03218 0.06947 0.02714 0.05916 0.06225

6.652E–07 0.002073 0.001028 2.228 0.0002022 0.001811 0.004245 5.454E–07 0.002262 0.01698 0.004143 0.000559 0.002771 1.948E05

0.001659 0.3593 0.1680 2.733 0.5975 0.0007027 0.00451 0.00046 0.04247 3.600 0.2838 0.06248 0.4627 0.4394

0.001634 0.2601 0.1695 1.842 0.8945 0.004515 0.01789 0.00055 0.03345 2.600 0.1392 0.07049 0.2809 0.4417

0.0003637 0.07080 0.04789 0.1848 0.1359 0.0007408 0.00346 0.00010 0.00791 0.6378 0.04567 0.02157 0.06954 0.07464

0.1330 0.07691 0.1321 0.9665 0.3745 0.9326 0.6335 0.1888 0.3875 0.3777 0.01172 0.3319 0.2611 0.5134

15

Notes: Nobs is the number of monthly observations, standard is the sample standard deviation and r1 the sample, first-order autocorrelation. The instruments are as follows. The short rate is the bid yield to maturity on a 90-day Treasury bill. Slope is the difference between and fiveyear and a one-month discount Treasury yield, y5y1. Convexity is y3(y5+y1)/2. Credit is the difference between a Baa and an Aaa corporate bond index yield. BS-spread is the difference between a lagging, 12 month moving average of monthly values of y5 and the annual dividend yield of the CRSP value-weighted stock index. Inflation is the percentage change in the consumer price index, CPI-U. IP growth is the monthly growth rate of the seasonally adjusted industrial production index. Cap. util is a measure of industrial capacity utilization and Xchange is a trade-weighted purchasing power index for the US dollar. Corp. illiq. is the percentage spread of prime commercial paper over three-month Treasury rates, a measure of short-term corporate illiquidity. Stock liq. is a measure of stock market liquidity based on price reversals in response to trading volume, from Pastor and Stambaugh (2003). The factors in Panel C are measured in continuously compounded monthly decimal fractions and are defined as follows. Ar and A‘ are the monthly averages of daily short-and long-term interest rates. rt+1rt and lt+1lt are the first differences of the end-of-month values. Ar is the daily approximation for the integral of the short rate over the month, A‘ the integral of the long rate, Dr‘ the integral of the log of their ratio. Dr the cumulative percentage change in the short rate and Dl the cumulative percentage change in the long rate. vol the monthly spot rate volatility, estimated from daily data within the month. dconvex is the first difference of the convexity measure. cpi and ipx are the monthly growth rates of the consumer price index and industrial production index. dqual is the first difference of the Baa less Aaa yield spread. dcap is the first difference in the capacity utilization measure, and ddollar is the growth rate in the relative purchasing power of the US dollar. dcliq is the change in short-term corporate illiquidity and dsliq is the change in stock market liquidity.


‘tþ1 ‘t Dr D‘ Dr‘ dconvex vol cpi dqual ipx dcap spxret ddollar dcliq dsliq

16

WAYNE FERSON ET AL.

The summary statistics for the fund returns cover the indicated subperiods in Table 1. The returns are based on the end-of-month net asset values of the funds. Investors can trade open-end mutual funds at their net asset values per share at the close of each trading day, regardless of when the underlying assets of the funds trade. Not surprisingly, the returns look very different from equity mutual fund returns. The mean returns are all between 0.6% and 0.7% per month. The standard deviations are all on the order of 1.0–1.5% per month, about 1/10 the values of equity style mutual funds. The minimum return for any style in any month since 1985 is 7.23%, suffered by the high-yield fund group in August of 1998. October of 1987 was a high return month, where the ‘‘All funds’ portfolio’’ earned 3.47%.

4.2. Conditioning the Models One innovation of our study is the use of conditioning dummy variables, Dt, to condition on discrete economic states. Consider the Dt for the monthly spot rate series, rt. We first convert the spot rate into a deviation from its average level over the last 60 months: xt ¼ rt(1/60)Sj ¼ 1, y, 60 rtj. We then use the last 60 months of spot rate data to estimate a rolling standard deviation, s(rt). The dummy variable Dt,hi for a ‘‘higher than normal’’ level of the spot rate is then defined as the indicator function: I{[xt/s(xt)]>1}. Similarly, the dummy variable Dt,lo for a ‘‘lower than normal’’ level of the spot rate is I{[xt/s(xt)]o1}. The vector conditioning dummy variable for time-t in Eq. (3) is then defined as: Dt ¼ (1,Dt,lo,Dt,hi). If the data are approximately gaussian, we should get about 2/3 of the observations in the ‘‘normal’’ category, and 1/6 each in the ‘‘high’’ and ‘‘low’’ categories. Dummy variables for the other state variables are similarly defined. Many studies of conditional performance use a common set of lagged instruments, consisting of dividend yields, Treasury bill yields and yield spreads, following Fama and French (1988, 1989), Campbell (1987) and others. The choice of instruments in these studies is essentially ad hoc. One of the appeals of using term structure models is that the models suggest the relevant state variables. In the Cox–Ingersoll–Ross and Vasicek models, the level of the short-term interest rate is the relevant conditioning information. We therefore use data for a short-term spot rate to construct the conditioning dummy variable in the single-state variable models.12 In the two-factor models the state variables are the short rate and a long rate or term spread. We use the short rate and a term spread, the difference between a five-year and a onemonth discount bond yield, in these models.13 We also measure performance


17

conditional on high versus low ‘‘convexity,’’ which we measure as y3(y5+y1)/2, where yj is the j-year discount bond yield from the CRSP FAMABY term structure files. The final state of the term structure for which we measure performance is spot rate volatility. To construct this series we use the daily spot rates within each month to compute a monthly standard deviation.14 Our combined models incorporate extra-term-structure risk factors together with the term structure factors. We include state variables related to the relative yields of bonds versus stocks, inflation, credit spreads, industrial production, capacity utilization, exchange rates, short-term corporate illiquidity and stock market liquidity. We measure the relative yields of bonds versus stocks as the difference between a five-year discount Treasury bond yield and the dividend yield of the CRSP value-weighted stock index.15 Inflation is measured as the continuously compounded growth rate of the consumer price index, CPI-U, from CRSP. Credit spreads are the difference between Aaa and Baa bond yields, from the Federal Reserve Database (FRED). Industrial production is the growth rate of the industrial production index (indpro.txt) and capacity utilization is a decimal fraction (tcu.txt), both from the FRED. The state of exchange rates is measured as the relative purchasing power of the dollar against the major trading partners for the US.16 Short-term corporate illiquidity is the percentage spread of three-month high-grade commercial paper rates over three-month Treasury rates, which follows Gatev and Strahan (2006). Stock market liquidity is the measure from Pastor and Stambaugh (2003), based on price reversals. Summary statistics for the lagged instrument data are presented in Panel B of Table 1. Perhaps the most significant feature is the high persistence of the raw instruments, as indicated by the first-order sample autocorrelations. Four are in excess of 95%. This high persistence raises concerns about finite sample bias (e.g., Stambaugh, 1999) and spurious regression problems (e.g., Ferson, Sarkissian, & Simin, 2003). One potential advantage of our conditioning dummy variable approach is that the autocorrelations of the variables are always smaller than those of the underlying instruments, often substantially so. The maximum first-order autocorrelation of a dummy variable in Table 1 is 93%, and all but two are below 87%.

4.3. Data for the Stochastic Discount Factors In the term structure models, the SDFs depend on both the monthly averages of simple functions of daily interest rates, as well as the changes in their end-of-month values. For example, in the single-factor, affine model,

18

WAYNE FERSON ET AL.

the required data are the monthly change, rt+1rt, and the daily average, Art+1, given by Eq. (9). Our daily short rate series is the three-month Treasury bill rate, which is used by Stanton (1997) and evaluated by Chapman, Long, and Pearson (2001). The latter finds that the errors induced by using the three-month rate to approximate an instantaneous short rate, is economically insignificant in affine term-structure models. For our goal of measuring portfolio return covariances, the accuracy of this approximation should not be a first-order issue. Our daily long rate series is the seven-year Treasury yield from the FRED database. We also examine empirical factors implied by a three-factor affine model, where convexity is the third factor. The daily measure of convexity is the difference between a one-year, constant maturity Treasury yield and a weighted average of the three-month and seven-year yields, from the FRED database.17 Finally, we consider the contemporaneous value of an interest rate volatility factor, formed from the daily spot rate series as described above. In Merton’s (1973) model the current values of the state variables are the conditioning information, and the shocks or innovations in those same state variables are the factors. We therefore measure the extra-term-structure risk factors as the growth rates or changes in the variables that serve as the lagged instruments. The additional risk factors include (1) the return of the Standard and Poors (S&P500) index, measured in excess of the one-month Treasury bill, (2) the rate of inflation, based on the CPI-U, (3) the changes in the Baa-Aaa yield spread, (4) the growth rate of the industrial production index, (5) the first differences of the capacity utilization measure, (6) the log growth rate of the relative purchasing power of the US dollar, (7) a measure of short-term corporate illiquidity and (8) a measure of stock market liquidity, as described above. Cornell and Green (1991) find that an equity market factor helps to price low-grade bonds.18 Chen, Roll, and Ross (1986) use risk factors similar to (1–4), which they ‘‘prewhiten,’’ or transform to innovations with time series models. Since the conditioning information in our models is explicit there is no need to prewhiten the variables in a separate step. Summary statistics of the risk factors are in Panel C of Table 1.

4.4. Benchmarks and Primitive Assets The primitive assets of the model are the returns Rt in Eq. (3a). They are included in order to estimate the parameters of the SDF models under the restriction that the models correctly price these assets. The primitive assets should be representative of the securities that fixed income funds hold.


19

Farnsworth et al. (2002) find that SDF models for equity returns produce smaller pricing errors when a small number of primitive assets is used. Based on this evidence, we choose a small number of primitive assets, sufficient to identify the models’ parameters. Our primitive asset returns are one-month returns on (1) a 90-day Treasury bill, (2) a 20-year Treasury bond and (3) a long-term Baa rated corporate bond. The first two series are from the CRSP mcti index files and the third is from Lehman Brothers. The final data series is the benchmark return, the RB,t+1 of Eq. (3b). When we study funds grouped by style we use style-based benchmarks from Lehman Brothers. These include a GNMA series for mortgage funds, an Aaa bond index return for high-quality funds, and a Baa return index for high-yield funds. When funds are grouped by characteristics, such as expense ratios, turnover, flows, etc., funds of different styles are combined. We group within each style and then combine the groups across styles, to avoid style concentrations. In these cases, we use a broad bond market aggregate as our benchmark return, the Lehman Brothers combined Government– Corporate bond return series. Elton et al. (1995) find a similar benchmark to be the most important single factor for controlling variance in their sample of fixed income fund returns. In some experiments, we also use the one-year Treasury bond return from the CRSP mcti files and the Ibbotson long-term government bond return to check robustness.

4.5. Benchmark Returns Across Economic States Table 2 shows the sample averages and standard deviations of the gross returns for the five primitive and benchmark assets, conditional on the high, low and normal term structure and economic states. The columns are the various asset returns, from low to high risk as we move from left to right across the table; the rows correspond to the state variables. The state variable dummies are correlated, but not extremely so. The highest correlations among the low-state dummies, 1973–1999, are 0.821 (short rate level and volatility), 0.635 (slope and convexity) and 0.542 (short rate level and bond– stock spread). The highest correlations among the high-state dummies are 0.737 (short rate level and volatility) followed by 0.585 (slope and convexity). The other correlations are typically much smaller. Starting with the term structure state variables, we find that high levels of short-term interest rates predict relatively high and volatile short-term bond returns and low stock returns. There is a gradual transition between these two patterns as you move to the right across the columns. The difference in

Asset Return

N

State

One-Year Bond

20-Year Bond

Govcorp Return

Baa Return

S&P500

Mean

Standard

Mean

Standard

Mean

Standard

Mean

Standard

Mean

Standard

Mean

Standard

55.00 190.0 79.00

1.009 1.006 1.005

0.003541 0.002145 0.001512

1.008 1.007 1.006

0.01101 0.005377 0.003600

1.002 1.008 1.009

0.03825 0.02732 0.03320

1.003 1.008 1.008

0.02655 0.01591 0.01567

0.997 1.011 1.010

0.04102 0.02584 0.01984

0.998 1.013 1.015

0.04800 0.04575 0.03899

43.00 195.0 86.00

1.006 1.006 1.007

0.002472 0.002516 0.003134

1.007 1.006 1.007

0.005510 0.005717 0.008112

1.019 1.007 1.003

0.02641 0.03085 0.03191

1.014 1.007 1.004

0.01480 0.01701 0.02102

1.022 1.009 0.999

0.02492 0.02599 0.03152

1.019 1.011 1.006

0.03612 0.04740 0.04258

36.00 209.0 79.00

1.007 1.006 1.007

0.004122 0.002235 0.002970

1.008 1.006 1.007

0.007910 0.005071 0.008441

1.017 1.006 1.007

0.03185 0.02897 0.03482

1.014 1.007 1.005

0.01938 0.01600 0.02201

1.021 1.008 1.002

0.03119 0.02537 0.03202

1.012 1.010 1.012

0.03915 0.04402 0.04964

60.00 190.0 74.00

1.009 1.006 1.004

0.003697 0.002052 0.001453

1.009 1.006 1.005

0.01108 0.004915 0.003648

1.007 1.008 1.007

0.04041 0.02723 0.03134

1.006 1.008 1.007

0.02789 0.01506 0.01533

1.001 1.010 1.010

0.04313 0.02427 0.02084

1.004 1.012 1.014

0.05238 0.04500 0.03717

62.00 94.00

1.008 1.005

0.003816 0.001861

1.008 1.006

0.008796 0.004083

1.006 1.011

0.03229 0.02621

1.006 1.008

0.01828 0.01473

1.006 1.009

0.02734 0.02246

1.007 1.011

0.04963 0.03336

71.00 165.0 88.00

1.008 1.006 1.005

0.004119 0.001692 0.001711

1.009 1.006 1.005

0.01089 0.004410 0.003899

1.009 1.007 1.008

0.04093 0.02627 0.02986

1.009 1.007 1.007

0.02690 0.01460 0.01523

1.010 1.008 1.007

0.04106 0.02315 0.02406

1.012 1.012 1.007

0.04578 0.03925 0.05344

47.00 225.0 52.00

1.007 1.006 1.006

0.003382 0.002551 0.002765

1.007 1.006 1.008

0.01072 0.004936 0.006768

1.005 1.006 1.015

0.04027 0.02801 0.03275

1.005 1.006 1.012

0.02752 0.01529 0.01844

1.000 1.008 1.017

0.04146 0.02471 0.02574

1.000 1.011 1.019

0.06256 0.04211 0.03521

WAYNE FERSON ET AL.

Short rate High Normal Low Slope High Normal Low Convexity High Normal Low Volatility High Normal Low Credit High Low BS-spread High Normal Low Inflation High Normal Low

90-Day Bill

20

Table 2. Primitive and Benchmark Return Statistics in Different Economic States. The Sample Period is January, 1973 through December, 1999 (N ¼ 324). Returns are One Plus the Rate of Return, in Monthly Decimal Fractions.

40.00 238.0 46.00

1.006 1.006 1.007

0.002459 0.002728 0.002948

1.006 1.006 1.009

0.004662 0.006276 0.007979

1.004 1.007 1.011

0.02233 0.03113 0.03593

1.005 1.007 1.010

0.01289 0.01807 0.02196

1.004 1.008 1.011

0.02061 0.02792 0.03474

1.004 1.011 1.018

0.03466 0.04613 0.04557

70.00 186.0 68.00

1.006 1.006 1.006

0.001486 0.002951 0.003185

1.006 1.006 1.008

0.004268 0.006485 0.007796

1.008 1.006 1.009

0.02701 0.03165 0.03283

1.007 1.006 1.010

0.01419 0.01840 0.02077

1.008 1.006 1.016

0.01985 0.02835 0.03357

1.008 1.009 1.020

0.04909 0.04144 0.04860

72.00 141.0 111.0

1.008 1.005 1.006

0.003367 0.002615 0.001817

1.009 1.006 1.005

0.007365 0.006108 0.005630

1.013 1.009 1.001

0.03619 0.02888 0.02885

1.011 1.008 1.003

0.02083 0.01616 0.01795

1.015 1.010 1.002

0.03321 0.02250 0.03018

1.010 1.014 1.008

0.04416 0.03937 0.05146

34.0 270.0 20.00

1.006 1.006 1.006

0.002910 0.002766 0.002403

1.007 1.007 1.004

0.006444 0.006464 0.004962

1.011 1.007 1.001

0.03135 0.03115 0.02681

1.010 1.007 1.002

0.02010 0.01798 0.01615

1.014 1.008 1.005

0.03278 0.02789 0.02370

1.024 1.009 1.012

0.05654 0.04282 0.04808

46.00 238.0 40.00

1.006 1.006 1.007

0.002525 0.002566 0.003885

1.006 1.007 1.008

0.005180 0.005856 0.009962

1.008 1.007 1.007

0.03230 0.02904 0.03968

1.007 1.007 1.007

0.01675 0.01732 0.02393

1.009 1.009 1.005

0.02683 0.02711 0.03562

1.011 1.011 1.010

0.03765 0.04588 0.04708

Note: For each state variable, high (low) values are defined to occur when the difference between the current level of the variable and a lagged, 60-month moving average is more than one 60-month moving standard deviation above (below) zero. Normal is defined as the values that are neither high nor low. The instruments are as follows. The short rate is the bid yield to maturity on a 90-day Treasury bill. Slope is the difference between a five-year and a one-month discount Treasury yield, y5y1. Convexity is y3(y5+y1)/2. Credit is the difference between a Baa and an Aaa corporate bond index yield. BS-spread is the difference between a lagging, 12 month moving average of monthly values of y5 and the annual dividend yield of the CRSP value-weighted stock index. Inflation is the percentage change in the consumer price index, CPI-U. IP growth is the monthly growth rate of the seasonally adjusted industrial production index. Cap. util. is a measure of industrial capacity utilization and Xchange is a trade-weighted purchasing power index for the US dollar. Corp. illiq. is the percentage spread of prime commercial paper over three-month Treasury rates, a measure of short-term corporate illiquidity. Stock liq. is a measure of stock market liquidity based on price reversals in response to trading volume, from Pastor and Stambaugh (2003).


IP growth High Normal Low Cap. util. High Normal Low Xchange High Normal Low Corp. iliq. High Normal Low Stock liq. High Normal Low

21

22

WAYNE FERSON ET AL.

the conditional mean stock return, for low versus high spot rates, is 1.7% per month and strongly statistically significant.19 These results are generally consistent with previous evidence such as Fama and Schwert (1977) and Ferson (1989). Table 2 suggests that a steeply sloped term structure has little information about next month’s short-term bill returns, but it predicts high expected and low-volatility long-term bond returns, and high stock returns. The former result reflects a failure of the constant-premium version of the expectations hypothesis of the term structure (e.g., Campbell & Shiller, 1991). The latter result is consistent with consumption-based model predictions such as Breeden (1986), which emphasize a positive relation between the slope of the term structure, expected economic growth and stock returns. Harvey (1989) also finds that a steep slope predicts high economic growth. Table 2 shows that higher convexity predicts higher returns on the longer term bonds, but bears no strong relation to the level of stock returns. The former result is consistent with the convexity/return relationship described in Grantier (1988), but seems to contradict the regression results described in Shyy and Lieu (1994). High spot rate volatility is associated with higher and more volatile short-term bond returns, and with lower returns on stocks and bonds exposed to default risks. The non-term-structure state variables are also associated with interesting return differences. High credit spreads predict high returns on stocks and lower-grade corporate bonds, consistent with Keim and Stambaugh (1986). High inflation is bad news for stocks and long-term bonds. When output growth is abnormally low, it predicts high returns, especially for the riskier assets. In the case of stocks, the difference between the low output state and the high output state is an average return of 1.4% per month. High capacity utilization predicts low returns on Baa bonds, consistent with Gudikunst and McCarthy (1997). When capacity utilization is low it predicts higher stock returns, but there is little information about short-term bond returns. These general patterns are consistent with the positive relation between expected economic growth and risky asset returns that most asset pricing models would predict if economic growth is mean reverting. The intuition is that when the real economy is performing poorly we expect it to get better, so expected growth and stock returns are high at such times. (See Chen, 1991, for related empirical evidence.) When the purchasing power of the dollar is high, it predicts high returns for the longer term, riskier bonds. When corporate illiquidity is high, it predicts high returns on the longer term bonds and stocks, and their volatility is slightly elevated as well. Finally, states defined by the level of stock


23

market liquidity, using the Pastor–Stambaugh measure, have little predictive ability for the future returns.

5. ESTIMATING THE STOCHASTIC DISCOUNT FACTOR MODELS ON PASSIVE BENCHMARKS In this section we evaluate the performance of the SDF models for pricing passive, benchmark returns. To this end, system (3) is modified as follows: E mðfÞtþ1 Rtþ1 1 Dt ¼ 0 ð14Þ E mðfÞtþ1 RB;tþ1 1 a0B Dt Dt ¼ 0 where aB0 Dt is the conditional alpha of the passive benchmark, RB,t+1. We conduct a series of experiments to evaluate the ability of the models to correctly price the returns of a one-year US Treasury bond and the Lehman Brothers Government–Corporate index. We evaluate the fit of the models informally by examining the coefficients and test statistics, paying special attention to the estimated alphas and their standard errors. A model with no bias produces a small alpha, and a model with high precision delivers a small standard error. We summarize here the results of this ‘‘prescreening’’ of the models, conducted before we use the models on actual mutual funds. First, we examine some linear regressions of empirical factor models.

5.1. A Comparison of Linear Factor Models The SDFs summarized in (13) are nonlinear functions of the term structure data. Blake, Elton, and Gruber (1993) and Elton et al. (1995) and most studies of equity funds use linear factor models, so it is interesting to compare the two approaches. The essential differences are three. First, with the linear beta pricing models the factors must be measured as excess returns to factor-mimicking portfolios, in order to get the right alpha. With SDF models this is not required, and our factors are typically not excess returns. Second, linear beta models imply SDFs that are linear in the factors, whereas the term structure models imply nonlinear functions. We perform some experiments to see how models that assume the SDF is linear in the various empirical factors perform, compared with the exponential function. We simply replace the exponential function for the SDF with a linear function, using the same factors, and estimate Eq. (14) on the passive

24

WAYNE FERSON ET AL.

benchmarks. The results are nearly identical to those using the exponential function. Any small differences seem to be in favor of the exponential specification. It does not appear that assuming the SDF to be an exponential versus a linear function of the same factors makes much of a difference. The third feature of the term structure model SDFs is the additional variables that appear due to time aggregation. We find that the additional variables provide additional explanatory power for discrete holding period returns. Table 3 compares time-series factor model regressions for three default-free bond returns. The regressors are measured over the same one-month period as the returns, and heteroskedasticity-consistent standard errors for the coefficients are shown on the second line. TB90 is the one-month return on a three-month Treasury bill, Tbond1 is the monthly return on a one-year Treasury bond and Tbond20 is the monthly return on a 20- year Treasury bond. Some of the averaged terms that arise from time aggregation are highly autocorrelated, as can be seen in Table 1. This raises concerns about bias in the regressions, due to persistent stochastic regressors. Stambaugh (1999) provides a first-order adjustment for bias, and we apply the adjustment to the regression coefficients and R2 in Table 3. We find that the effect of the adjustment is typically small, on the order of 1% of the coefficient, and never exceeding 10% of the coefficient.20 Unlike the examples for equity returns and dividend yields provided by Stambaugh, the effect of the bias adjustment here is to slightly increase the explanatory power. Campbell, Chan, and Viciera (2003) also find that Stambaugh bias adjustments increase the coefficient magnitudes when bond returns are regressed on bond yields. In affine term structure models the conditional expected returns of discount bonds are linear in the levels of the state variables, and such predictive regressions are explored by Dai and Singleton (2002). In Table 3 we ask the regressions to explain the ex post bond returns – both the expected and the unexpected parts – with contemporaneous values of the various factors. This can be motivated by recalling that if the SDF is approximately linear in the empirical factors there would be a linear factor model regression of bond returns on the factors, and the slope coefficients of this regression are the ‘‘betas’’ that describe the cross-section of the average returns. Here we describe how well the linear factor model regressions explain variance using the various empirical factors implied by continuous-time term structure models.21 Comparing the first two rows of Table 3 for each bond, we find that including the average short rate, as implied by the continuous-time theory, provides an improved fit relative to using only discrete spot rate changes. The adjusted R2 for TB90 jumps from 23% to 98% when the time-averaged

Bond Return TB90

Tbond1

A Comparison of Linear Factor Model Regressions for Default-Free Bond Returns. Dr

1.006 0.0001 1.000 0.0001 1.006 0.0001 1.006 0.0001 0.9998 0.0001 0.9998 0.0001 0.9995 0.0004 1.000 0.0003 1.007 0.0002 0.9997 0.0002 1.000 0.0003 0.9997 0.0004 1.007 0.0002 1.002 0.0006

20.77 3.82 20.97 0.78 22.22 4.347 28.69 6.344 20.34 0.937 23.94 1.086

20.48 7.825

D‘

Dc

Ar

Ac

Dr

D‘

Dr‘

R2 0.230

8.487 0.190 3.686 5.585 14.01 8.336 1.208 1.251 4.393 1.595

0.978 0.230 0.238

16.31 11.40

8.531 1.977

7.602 0.376 8.433 0.294 6.679 0.960 4.696 1.568

1.313 0.389 0.420 0.353 2.226 1.109 3.889 1.446

0.635 14.16 7.127 0.549 4.640 .554 5.469 2.265

80.07 7.468 80.24 6.203

A‘

1.722 0.627 3.927 1.438 3.415 2.050

0.984 4.750 1.299

0.989 0.706 1.681 0.126 0.164 0.542 1.669 0.126 1.722 0.127

0.110 0.11 0.146 1.080 0.131 0.118

1.661 0.267

0.180 0.089 0.818 0.049

0.929 0.576 0.932

0.184 0.089 0.103 0.122

0.928 0.785 0.639

7.168 0.930

0.736

25

Int.


Table 3.

26

Table 3. (Continued ) Bond Return

Dr

D‘

1.007 0.0002 1.007 0.0001 0.9996 0.0005 0.9996 0.0003 0.9985 0.0016 1.000 0.0009 1.008 0.0003 0.9994 0.0008 1.001 0.0011 0.9994 0.0013 1.007 0.0015 1.008 0.0041 1.007 0.0007 1.007 0.0007

50.80 6.702 91.25 6.658 48.82 5.001 89.08 3.488

74.33 7.073 9.689 9.076 78.25 5.702 13.85 4.395

47.91 10.54

Dc

Ar

Ac

Dr

D‘

Dr‘

R2 0.767 0.844

102.1 11.46

100.3 5.957

4.363 1.492 5.132 0.727 0.462 4.075 5.179 5.849

5.131 1.513 4.349 0.859 9.330 4.451 13.65 5.324

102.22 17.99 2.103 2.116 8.237 6.760 3.107 7.464

257.3 31.45 257.3 31.53 38.96 17.79 56.67 35.49

A‘

7.152 2.338 15.76 6.273 12.38 6.664

0.893 11.57 3.089

0.968 0.086 4.486 0.555 0.152 0.687 4.451 0.555 6.725 0.699

5.980 0.669 2.321 1.406 6.044 0.669

10.12 1.087

0.534 0.332 0.657 0.089

0.764 0.809 0.763

0.724 0.368 0.331 0.405

0.645 0.600 0.282

1.505 6.013 752.5 34.72 780.8 53.52

0.280 0.852

44.69 66.11

0.851

WAYNE FERSON ET AL.

Tbond20

Int.

40.80 17.85 60.81 36.78

21.31 52.86

753.4 34.07 784.6 56.46

47.88 68.99

4.219 6.834 7.870 7.876 28.67 16.84 25.09 22.17

12.88 7.041 16.80 7.518 31.48 20.60 32.49 20.29

600.5 88.39 13.66 7.787 56.74 37.09 25.13 22.30

22.30 9.210 54.47 35.55 32.55 20.36

0.854 0.854

22.07 21.94

0.009 0.200 1.807 0.968 4.378 0.255 1.803 22.96 3.151

61.88 3.027 13.80 6.719 61.98 3.034

61.69 2.843

0.900 1.362 0.013 0.359

0.796 0.854 0.796

2.837 2.177 0.897 1.369

0.272 0.796

Note: TB90 is the one-month gross return on a 90-day Treasury bill, Tbond1 is a one-year Treasury bond and Tbond20 is the monthly gross return on a 20-year Treasury bond. The regressors are measured for the same month as the return, and heteroskedasticity-consistent standard errors are shown on the second line. The intercepts are shown in the first column. The other regressors are indicated as follows: Dr is the change in the 90-day spot rate, D‘ the change in the seven-year Treasury yield, Dc the discrete change in the monthly convexity measure, Ar the daily average spot rate over the month, A‘ the daily average seven-year yield, Ac the daily average of the convexity measure, Dr the daily average change in the spot rate, D‘ the daily average change in the yield, and Drl a daily average slope, measured using the three-month and seven-year yields. Rsq is the adjusted R2. The sample period is January 1973 through December 1999 (N ¼ 324). Returns are one plus the rate of return, in monthly decimal fractions. The coefficients and standard errors, excepting the intercept, are multiplied by 100.


1.000 0.0027 0.9997 0.0026 0.9967 0.0077 1.000 0.00043 1.007 0.0010 0.9987 0.0037 1.004 0.0071 1.000 0.0043

27

28

WAYNE FERSON ET AL.

short rate enters the regression, and for the one-year bond the R2 increases from 64% to 79%. However, for the 20- bond return, introducing the average short rate slightly lowers the adjusted R2. It seems sensible that refined information about the path of the short rate is more important for explaining short term than for long-term bond returns. Given the short rate, adding the discrete second factor D‘ makes a large difference in the Tbond20 regression, bumping the R2 from 28% to 85%, while the D‘ factor does not improve the fit for TB90. Comparing the third versus fourth rows of Table 3 reveals the marginal contribution of the discrete change in convexity when discrete changes in both the short and the long rate are present. The contribution is insignificant for TB90 and the 20-year bond, but highly significant for the one-year bond, where the adjusted R2- increases from 77% to 84%. A comparison of the fifth and sixth rows examines the incremental explanatory power of the convexity factors, Dc and Ac, given the factors implied by the two-factor affine model. The convexity factors are significant for TB90, but the change in the adjusted R2 is modest. The convexity factors provide no marginal explanatory power for the 20-year bond return. The largest improvement is found for the intermediate, one-year maturity. In this case, the convexity factors increase the adjusted R2 from 89% to 97%.22 It makes sense that convexity should be more important for the intermediate maturities, controlling for long and short rates, than for the long and short-maturity bonds. The third versus fifth rows for each asset evaluate the time-aggregation terms suggested by a two-factor affine model. The time-aggregation terms markedly improve the two-factor regressions for the two shorter-term bonds, but are not significant for the 20-year return. The third versus ninth rows provide a similar comparison for the three time-aggregation terms introduced by the Brennan and Schwartz model, Dr, D‘ and Dr‘ : When these variables join the regressions featuring the discrete variables Dr and D‘ ; they are significant for TB90 and Tbond1, but less potent than the Ar, Al combination. For Tbond20 the Brennan and Schwartz variables remain significant, and the D‘ term has a t-ratio larger than three. The fifth versus eight rows of Table 3 provide a head-to-head comparisons of the variables suggested by the two-factor affine model (which are Dr, D‘ ; Ar and A‘ ) versus the Brennan–Schwartz model (which are Ar, A‘ ; Dr, D‘ ; and Dr‘ ). Based on the adjusted R2, the affine model’s variables win for TB90, by an R2 of 97% versus 93%. Similarly for Tbond1, the R2 are 89% versus 78%, in favor of the affine model. For Tbond20 the two sets of variables are closer in explanatory power, at 85% versus 83%.


29

When we examine the SDF models’ performance in pricing the passive benchmarks below, we find that the Brennan and Schwartz model is over parameterized when all of its time-aggregation terms are used.23 The last three rows of Table 3 for each asset explore the effects of dropping one of the D terms from the Brennan and Schwartz variable regressions. These experiments show that it is possible to drop one of the variables without degrading the explanatory power of the regressions, but give mixed signals on which one to drop. Regressions for the shorter-term bonds suggest that D‘ or Dr‘ can be dropped, while the 20- year bond suggests dropping either Dr or Dr‘ : We drop the Dr‘ term in the empirical models presented below. We draw two main conclusions from this section. First, for these data, 0 models where the SDF is eb f perform similarly to models where the SDF is 0 b f, using the same empirical factors, f. This does not say that term-structure motivated SDFs and linear factor models are equivalent. The additional factors that arise from the explicit time aggregation of the continuous-time term structure models improve the explanatory power of factor model regressions for the discrete period returns.

5.2. Term Structure Models Meet Passive Benchmarks At least two primitive assets are required in the first equation of (13) to identify the parameters of the SDF models. We use the 90-day Treasury bill and the 20-year Treasury bond return. The second equation identifies the SDF alphas for the benchmarks, which we take to be the one-year government bond and the Lehman Brothers Government–Corporate index. All of the models are overidentified, and we find that Hansen’s J-statistic typically rejects the models. The coefficients of the pure term structure SDF models are not estimated with very high precision. The typical t-ratio for a coefficient is about 1.2, but values between 0.11 and 3.2 are observed. Thus, for example, we could not reject the hypothesis that the long rate factors may be excluded from the two-factor affine model, reducing it to a one-factor model. Our main interest is the estimates of the conditional alphas and their precision. Table 4 presents estimated alphas and their standard errors, conditional on the high and low values of the state variables. The first two columns present the raw conditional mean returns without risk adjustment. The third and fourth columns present the conditional alphas after risk adjustment. The far right column presents the excess-return alphas for the one-year government bond in excess of the Lehmann Brothers government–corporate

30

WAYNE FERSON ET AL.

Table 4. State Variable

Conditional Alphas on Passive Benchmarks.

rb1

rgovcor

Panel A: One-Factor Affine Model Short rate High 0.00825 0.00329 0.00148 0.00358 Low 0.00555 0.00764 0.000405 0.00176 Panel B: Discrete Two-Factor Model Short rate High 0.00825 0.00329 0.00148 0.00358 Low 0.00555 0.00764 0.000405 0.00176 Slope High 0.00744 0.0143 0.000840 0.00226 Low 0.00686 0.00385 0.000875 0.00227 Panel C: Two-Factor Affine Model Short rate High 0.00825 0.00329 0.00148 0.00358 Low 0.00555 0.00764 0.000405 0.00176 Slope High 0.00744 0.0143 0.000840 0.00226 Low 0.00686 0.00385 0.000875 0.00227

arb1

argovcor

Excess Alpha

0.000267 0.000684 0.000660 0.000201

0.00149 0.00111 0.000941 0.000452

0.00175 0.000759 0.000281 0.000337

0.000608 0.000599 0.000904 0.000163

0.00123 0.000906 0.00143 0.000360

0.00184 0.000665 0.000522 0.000304

0.000250 0.000383 0.000514 0.000336

0.00160 0.000568 0.000778 0.000527

0.00135 0.000460 0.00129 0.000447

0.000796 0.000611 0.000698 0.000194

0.000807 0.00105 0.000876 0.000446

0.00160 0.000756 0.000177 0.000336

0.000324 0.000380 0.000600 0.000373

0.00154 0.000544 0.000508 0.000643

0.00121 0.000451 0.00111 0.000503

0.00114 0.00108 0.000807 0.000453

0.00166 0.000767 0.000142 0.000334

0.00152 0.000552 0.000600 0.000681

0.00124 0.000439 0.00112 0.000510

Panel D: Two-Factor Brennan and Schwartz model Short rate High 0.00825 0.00329 0.000520 0.00148 0.00358 0.000617 Low 0.00555 0.00764 0.000665 0.000405 0.00176 0.000202 Slope High 0.00744 0.0143 0.000281 0.00084 0.00226 0.000374 Low 0.00686 0.00385 0.000522 0.000875 0.00227 0.000370


31

Table 4. (Continued ) State Variable

rb1

rgovcor

arb1

Panel E: Extended Two-Factor Affine Models Short rate High 0.00825 0.00329 0.000516 0.00148 0.00358 0.000578 Low 0.00555 0.00764 0.000692 0.000405 0.00176 0.000190 Slope High 0.00744 0.0143 0.000270 0.00084 0.00226 0.000369 Low 0.00686 0.00385 0.000486 0.000875 0.00227 0.000352 Convexity High 0.00841 0.0137 0.000654 0.00132 0.00323 0.000530 Low 0.00698 0.00539 0.000346 0.000950 0.00248 0.000416 Volatility High 0.00895 0.00609 0.000721 0.00143 0.00360 0.000601 Low 0.00523 0.00685 0.000646 0.000424 0.00178 0.000208 Credit High 0.00832 0.00810 0.00125 0.00112 0.00281 0.000490 Low 0.00595 0.00801 0.000167 0.000421 0.00152 0.000210 S&P 500 High 0.00872 0.00859 0.000653 0.00129 0.00319 0.000526 Low 0.00545 0.00679 0.000583 0.000416 0.00162 0.000217 Inflation High 0.00743 0.00505 0.000841 0.00156 0.00401 0.000629 Low 0.00797 0.0119 0.000931 0.0009 0.00256 0.000439 IP growth High 0.00576 0.00508 0.000418 0.000737 0.00204 0.000320 Low 0.00853 0.00970 0.00103 0.00118 0.00324 0.000527 Cap. util. High 0.00596 0.00696 4.29E–05 0.000510 0.00170 0.000265

argovcor

Excess Alpha

0.00117 0.00104 0.000988 0.000441

0.00169 0.000760 0.000296 0.000330

0.00152 0.000551 0.000683 0.000666

0.00125 0.000439 0.00117 0.000506

0.00166 0.000829 0.000680 0.000667

0.00101 0.000571 0.00103 0.000451

0.00117 0.00103 0.000954 0.000471

0.00189 0.000730 0.000308 0.000344

0.00195 0.000843 0.000500 0.000489

0.000699 0.000601 0.000667 0.000374

0.000597 0.000817 0.000605 0.000642

5.61E–05 0.000510 2.20E–05 0.000530

0.000579 0.00132 0.00151 0.000799

0.00142 0.000907 0.000576 0.000510

0.000379 0.000834 0.000652 0.00107

3.91E–05 0.000745 0.000381 0.000743

2.25E–05 0.000474

2.03E–05 0.000355

32

WAYNE FERSON ET AL.

Table 4. (Continued ) State Variable Low Xchange High Low

rb1

rgovcor

0.00788 0.000945

0.00968 0.00252

0.00139 0.000363

0.00218 0.000705

0.000792 0.000509

0.00931 0.000868 0.00545 0.000534

0.0114 0.00245 0.00309 0.00170

0.000783 0.000341 0.000208 0.000313

0.000761 0.000526 1.92E–05 0.000656

2.26E–05 0.000352 0.000188 0.000473

arb1

argovcor

Excess Alpha

Note: GMM estimation of the various SDF models in Eq. (14). The term structure models use the conditioning dummy variable for the relevant state variable(s) as the instruments. The extended models use one extra factor at a time, and are estimated using the conditioning dummy variables for the spot rate, slope and the extra factor as instruments. The benchmark returns are a one-year Treasury bond, with the conditional means in the column labeled rb1, and the Lehman Brothers government corporate aggregate bond index, denoted rgovcor. Standard errors of means and alphas, denoted by a, are shown below the point estimates. The excess alpha is the alpha for the return difference. The sample period is January 1973 through December 1999 (324 observations).

benchmark. The table shows that the conditional models explain a large fraction of the returns in most states. Even the one-factor affine model, shown in Panel A, does a reasonable job. The raw returns of the one-year government bond are 83 basis points (bp) in the high spot rate state and 55 in the low spot rate state. The conditional alphas are 3 and 7 bp, respectively. The one-factor model does not do as well on the government–corporate return, however, leaving alphas of 15 and 9 bp, respectively, in the two states. Comparing Panels A and C shows that the two-factor affine model performs better than the one-factor model. For example, the two-factor model produces alphas for the government–corporate index of 8 and +9 bp per month, conditional on the spot rate states. The excess return alphas are smaller in every case.24 A comparison of Panel B with either Panels C or D in Table 4 illustrates that the continuous time term structure models that include the time aggregation terms perform better than models using only the discrete factors, [rt+1rt] and ½‘tþ1 ‘t : For example, the discrete model shown in Panel B delivers conditional alphas for the government–corporate index equal to 12 and +14 bp in the two spot rate states, compared to the affine model’s 8 and +9 bp. The excess alphas are smaller in each state when the models include the time aggregation terms. Panels C and D show that the twofactor affine and the two-factor Brennan and Schwartz models perform similarly.


33

5.3. Models with Extra Factors Panel E of Table 4 summarizes extended affine models with one extra factor at a time. The models are estimated using the dummies for the spot rate, slope and the extra factor as instruments. We find that with the extra factors, the affine models and the Brennan and Schwartz models perform similarly, so we do not report results for the Brennan and Schwartz models. The coefficients, d2, on the additional factors are usually statistically significant, with t-ratios that average about 2.5 across the models, and values between 1.4 and 3.5 are observed. In the presence of the additional factors, the coefficients on the term structure variables often become more precise, with many t-ratios now in excess of 2.0 and values as large as 6.7 observed. Hansen’s J-test still produces asymptotic p-values less than 0.05 in most cases. The performance of the extended models on the passive benchmarks, conditional on the spot rate and term structure slopes, are typically similar for different choices of the third factor. Panel E of the table shows results for the spot rate and slope states, only for the first model, where convexity is the third factor. (We use only the discrete change in convexity here.) For the remaining models we show only the results conditioned on the state variable corresponding to the extra factor. (For example, when volatility is the extra factor, we report only the results conditioned on high- and low-volatility states.) The extended models generally work well at explaining the passive benchmark returns in the various states. For example, the high versus low output and capacity utilization states imply differences in the conditional mean returns on the order of 30 bp per month, but the excess return alphas for these states are 8 bp or smaller. The model with a stock market factor produces conditional alphas that are numerically close to zero. Based on the figures in Panel E, the average absolute ratio of the conditional alpha to the unadjusted return is 8.2% for the one-year government and 1.25% for the government–corporate returns. The precision of the alphas is generally good. The average ratio, across the models and states, of the standard error of the alpha to the standard error of the unadjusted mean return, is 21%. In some states statistically significant biases remain after risk adjustment, but the maximum bias is less than 20 bp per month. The biases for excess returns are typically smaller than for the raw returns. In panel E, the average excess return alpha is only 2.0 bp per month. We conclude from this section that the models can explain large fractions of the conditional mean returns on the passive benchmarks for all of the state variables. Two-factor models perform better than one-factor models, and the extended models are better yet. The two-factor affine and Brennan–Schwartz

34

WAYNE FERSON ET AL.

models perform similarly on the passive benchmarks. Excess return alphas are typically smaller than raw return alphas. But there are some cases where statistically significant alphas are found on passive benchmarks. High spot rates, spot rate volatility, extreme term structure slopes and high inflation states challenge the models.25 Conditional on these states the biases tend to be about 10 bp per month, and never exceed 20 bp. 5.4. Additional Experiments We estimated some of the models using three primitive assets, introducing the long-term Baa-rated corporate bond index. We find that the raw return alphas of the benchmark assets can be sensitive to the primitive assets. The excess return alphas appear less sensitive. This reinforces the impression that it is useful to measure conditional alphas in a relative form. Overall, the models with three primitive assets do not perform better than the models with two primitive assets. Furthermore, with three primitive assets the models are not as stable numerically, and the algorithm is more prone to running off to regions of the parameter space where the gradient matrix becomes rank deficient. Overall, we prefer the models with two primitive assets. Farnsworth et al. (2002) also argue on empirical grounds that a smaller number of primitive assets is preferred. We estimated models in which the continuous instrument, Zt is used in Eq. (13a) instead of the discrete dummy variable. These models perform markedly worse than the pure dummy variable models. One intuition for this result is that the models with the continuous instruments implicitly assume a linear relation to the instrument, while the dummy variable is a nonparametric form. Alternatively, the GMM solution with a dummy picks the parameters to fit returns in each discrete state, while the solution with a continuous instrument minimizes a different objective.

6. FIXED INCOME FUND PERFORMANCE IN RELATION TO CHARACTERISTICS Table 5 presents the mean excess returns of the fixed income funds, grouped according to high versus low asset size, turnover, expense ratio, one-year lagged return, reported income yield and new money flow over the previous year. Funds are grouped by characteristics within each of the style categories shown in Table 1, in order to avoid style concentrations. For example,


35

Table 5. Fixed Income Fund Excess Returns. Fund Grouping

State Nobs Uncond. 168 High short rate 3 Low short rate 64 High slope 17 Low slope 46 High convexity 15 Lo convexity 43 Hi volatility 11 Low volatility 62 High credit 14 Low credit 78 High BS spread 16 Low BS spread 62 High inflation 16 Low inflation 29 High IP growth 22 Low IP growth 24 High cap. util. 46 Low cap. util. 32 High dollar 21

Asset

Turnover

Expense

High

Low

High

Low

High

Low

2.673E–05 0.0005427 0.003959 0.0009229 9.650E–6 0.001091 0.001849 0.001819 0.0002245 0.0005937 0.001468 0.001974 0.001356 0.0008715 0.0008522 0.001209 0.0001940 0.001045 0.005819 0.002557 0.001017 0.000542 0.001167 0.001016 0.001044 0.000792 0.000473 0.000933 0.001089 0.001372 0.000878 0.001518 0.002023 0.002187 0.001310 0.000505 0.002214 0.002041 0.001312 0.001343

0.0001190 0.0005818 0.0008446 0.001221 0.0004889 0.001121 0.002604 0.001919 0.0002477 0.0006734 0.001249 0.001283 0.0008181 0.0009511 0.0007195 0.001412 0.0007226 0.001046 0.005923 0.002601 0.0007691 0.0004814 0.001588 0.001157 0.001134 0.000916 0.001040 0.001520 0.0001742 0.001422 0.001439 0.001480 0.003017 0.002180 0.001796 0.000894 0.002086 0.002066 0.001675 0.001490

0.0002778 0.0005522 0.004803 0.001363 0.0002193 0.001104 0.003060 0.002323 0.0001115 0.0006907 0.0009150 0.001310 0.001438 0.0009648 0.001107 0.001370 0.0001091 0.0009121 0.005131 0.002309 0.001024 0.0004822 0.001541 0.001149 0.001518 0.000816 0.000695 0.000992 0.0007489 0.001327 0.0007248 0.001503 0.001567 0.002223 0.002342 0.0008686 0.002438 0.001999 0.001708 0.001535

3.228E–05 0.0005281 0.002016 0.0007958 0.0004227 0.001020 0.001943 0.001721 0.0003615 0.0006124 0.001351 0.001845 0.0007417 0.0007555 0.0006918 0.001350 7.038E–05 0.001018 0.006020 0.002633 0.001259 0.0005078 0.001502 0.001079 0.0003216 0.0007137 7.467E–06 0.0009608 0.0007833 0.001328 0.001147 0.001430 0.003150 0.002046 0.001477 0.0004804 0.001937 0.002055 0.001458 0.001295

0.0003624 0.0006179 0.003499 0.0006687 0.0004340 0.001167 0.002994 0.002093 0.0003707 0.0006591 0.001942 0.002174 0.001894 0.001085 0.001288 0.001300 0.0004580 0.001126 0.006491 0.003026 0.001293 0.0005669 0.001485 0.001214 0.001633 0.000758 0.0003605 0.0008412 0.001106 0.001528 0.001783 0.001704 0.001965 0.002713 0.002109 0.0005724 0.002261 0.002513 0.002373 0.001864

7.061E–05 0.0004838 0.005232 0.001980 0.0002644 0.0009213 0.002605 0.001548 0.0004686 0.0007309 0.000822 0.001155 0.001148 0.0008063 0.001104 0.001624 3.843E–05 0.0008534 0.005467 0.002067 0.001155 0.0004916 0.001636 0.001260 0.0005408 0.0006822 0.0006050 0.001123 0.0006646 0.001172 0.0006036 0.001340 0.002291 0.001769 0.001992 0.0006857 0.002225 0.001697 0.001080 0.001315

36

WAYNE FERSON ET AL.

Table 5. (Continued ) Fund Grouping

Asset High

Turnover Low

High

Low

Low dollar 0.001110 0.0001614 0.0003148 52 0.001210 0.001418 0.001295 High corp. iliq. 0.001939 0.0009614 4.437E–05 22 0.001487 0.002032 0.002218 Low corp. iliq. 0.000225 0.0004896 0.000355 8 0.001159 0.001275 0.001047 High stock liq. 0.0001127 0.0008162 0.0005099 21 0.0009522 0.001181 0.001012 Low stock liq. 0.0007968 0.001047 0.0007385 22 0.001270 0.001366 0.001216 Fund Grouping

Uncond. 168 High short rate 3 Low short rate 64 High slope 17 Low slope 46 High convexity 15 Low convexity 43 High volatility 11 Low volatility 62 High credit 14 Low credit 78 High BS spread 16 Low BS spread 62 High inflation 16

Lag Return

Expense High

Low

0.0008941 0.0006764 0.0002573 0.001211 0.001412 0.001087 0.001088 0.001056 0.000401 0.001464 0.001886 0.001637 0.000338 3.165E–05 0.000755 0.001478 0.001189 0.001465 1.491E–05 0.0005191 0.0005424 0.001109 0.0009659 0.001300 0.0008812 0.0008808 0.0006012 0.001104 0.001146 0.001170

Yield

Lag Flow

High

Low

High

Low

High

Low

0.0006268 0.0005424 0.0003681 0.001414 6.293E–05 0.001083 0.002013 0.003093 0.001135 0.000758 0.002179 0.003479 0.001590 0.001097 0.001062 0.002281 0.000201 0.001120 0.001978 0.000887 0.001401 0.000744 0.001912 0.001969 0.000674 0.001054 0.001099 0.001158

0.0001378 0.0007610 0.005447 0.002214 0.000535 0.001427 0.001239 0.001320 0.0005442 0.0009145 0.001271 0.000972 0.001156 0.001089 0.000607 0.001491 0.0007568 0.001340 0.009034 0.004207 0.0006883 0.0006606 0.0008837 0.0005311 0.001477 0.0008753 0.000809 0.001683

0.0004992 0.0003925 0.005014 0.001352 0.0001057 0.0006638 0.0004289 0.0007106 0.0002150 0.0006863 0.0007349 0.0007412 0.001639 0.001024 0.001358 0.001365 3.860E–05 0.0006092 0.002713 0.001194 0.0008111 0.0004447 0.001191 0.001109 0.0009357 0.0006346 0.001053 0.001026

1.541E–05 0.0008509 0.001841 0.001760 0.0002523 0.001759 0.005200 0.003188 0.0005787 0.0007086 0.002771 0.003176 0.001108 0.001004 0.000664 0.001540 0.000561 0.001692 0.008783 0.004128 0.001418 0.000752 0.001584 0.001354 0.001348 0.001191 0.000929 0.001553

0.001425 0.0005681 0.0003037 0.001630 0.002462 0.001277 0.0006703 0.001465 7.879E–05 0.0006943 0.0009922 0.001388 0.001644 0.001182 0.000544 0.001554 0.002194 0.001184 0.001629 0.0008930 0.001193 0.0005932 0.001935 0.001245 0.002913 0.001313 0.001122 0.001375

0.001177 0.000562 0.003974 0.001129 0.002024 0.001315 0.001773 0.001876 0.0001377 0.0005882 0.001428 0.002045 0.001880 0.001120 0.000690 0.001177 0.001699 0.001234 0.001763 0.000939 0.001276 0.000616 0.001114 0.000985 0.002712 0.001321 0.000487 0.000976


37


Low inflation 29 High IP growth 22 Low IP growth 24 High cap. util. 46 Low cap. util. 32 High dollar 21 Low dollar 52 High corp. iliq. 22 Low corp. iliq. 8 High stock liq. 21 Low stock liq. 22

Lag Return

Yield

Lag Flow

High

Low

High

Low

High

Low

0.000613 0.001149 0.002423 0.001163 0.000326 0.001808 0.001741 0.000869 1.280E–06 0.001463 0.002044 0.001815 0.000537 0.001187 0.001644 0.001746 0.0002381 0.001025 0.0004052 0.001299 0.001128 0.0009591

0.001156 0.002026 0.0003311 0.002190 0.003722 0.003217 0.001108 0.0007817 0.003585 0.003058 0.001138 0.001343 0.001629 0.001796 0.001783 0.002018 0.0008307 0.002498 0.0001276 0.001483 0.0001310 0.001760

0.001171 0.000838 0.0002223 0.0009179 0.0008296 0.001672 0.001402 0.0005294 0.000635 0.001211 0.001659 0.001619 3.584E–06 0.0005821 0.0005855 0.001097 5.504E–05 0.001033 0.0005798 0.001035 0.0007022 0.001288

0.000353 0.002056 0.002293 0.002451 0.004992 0.003145 0.002376 0.000917 0.003386 0.003389 0.001813 0.001656 0.000848 0.002149 0.0004965 0.002818 0.0006821 0.001873 1.638E–05 0.001342 0.0007927 0.001298

0.001851 0.001849 0.001348 0.0008949 0.001418 0.001948 0.001220 0.000777 0.0001420 0.0009886 0.001627 0.001392 0.002157 0.000991 0.0009585 0.001316 0.0005009 0.001178 8.152E–05 0.001576 0.001987 0.002156

0.002100 0.001795 0.001129 0.0009214 0.002039 0.001897 0.001155 0.000541 3.381E–06 0.0009944 0.001292 0.001323 0.001465 0.000927 0.001278 0.001238 0.0003007 0.001078 0.0001110 0.001472 0.002371 0.002066

Note: Returns in excess of the Government–Corporate bond index are shown for equalweighted portfolios of funds grouped on high versus low asset size, turnover and expense ratios. High-asset funds are those in the top third, while low asset funds are in the bottom third, etc. The excess returns are decimal fractions per month, for the 1986–1999 period (168 observations); the flow group has 12 fewer. The first two rows report the unconditional sample means and standard deviations of the mean excess returns. Subsequent rows report excess returns conditional on various economic states, as measured by dummy variables. Figures larger than two standard errors of the mean are in bold. Nobs is the number of monthly observations for the state.

government bond funds are likely to have lower expense ratios on average than high yield funds, and we do not want the low-expense group to consist disproportionately of government funds. Within each style group we rank the funds from high to low on their expense ratios, reported for the previous year. We take the top third from each style group, and an equally weighted portfolio of these defines the high-expense fund returns for the next 12 months. The low-expense group is formed from the bottom third, and the other characteristics are treated symmetrically.

38

WAYNE FERSON ET AL.

The monthly returns for the characteristics-based groups in Table 5 are measured in excess of the Government–Corporate index, and the data cover the 1986–1999 period (168 observations). The first row shows the unconditional mean excess returns with no risk adjustment. Most of the fund groups’ average returns are slightly below the benchmark, the differences ranging from zero to 14 bp per month across the groups. These results are reminiscent of Blake et al. (1993), who find negative fixed income fund unconditional alphas, similar in magnitude to expense ratios, averaging about 1% per year. Dahlquist, Engstrom, and Soderlind (2000) find similar magnitudes for Swedish bond funds. This makes sense if the funds have no unconditional performance, and if the benchmark is of the same average risk, since the funds pay expenses and trading costs while the benchmark does not. The second line shows the standard errors of the means, and reveals that none of the average return differences across characteristics groups are statistically significant. Subsequent rows of Table 5 present the mean excess returns, conditional on the high versus low-economic states. The excess return differences across the states tend to dwarf their differences across the fund groups, and some of the conditional mean excess returns exceed two standard errors. High short-term interest rate states predict low returns for almost all fund groups. The funds deliver relatively low returns when capacity utilization is high, with performance ranging from 11 to 24 bp off the benchmark. Table 2 showed that the Government–Corporate benchmark returns are unusually low in high capacity-utilization states. It appears that the fund returns respond even more negatively to these states than the benchmark. When capacity utilization is low, most of the fund excess returns are higher than average, the magnitudes similar to those of the underperformance in the high utilization state. However, in the low utilization state, the volatility of the funds’ returns is unusually high as well, so the mean excess returns do not attain two standard errors. A similar pattern is observed in the low industrial production growth state, and in the high corporate liquidity state: high fund relative returns for most groups, but also high volatility. There is only one state in which we observe significant positive fund returns in excess of the benchmark, and this is the high credit spread state. The funds beat the benchmarks in this state by 16–90 bp points per month. The ‘‘significance’’ of the results in Table 5 should be interpreted with caution. We discussed the extreme outcomes across the 10 state variables, so we should account for the multiple comparisons. There are 284 conditional mean excess returns in Table 5, so we would expect about 14 of the t-ratios to be larger than 2.0 if all the mean excess returns are really zero. We find


39

39 t-ratios larger than 2.0 in Table 5, and the cases are certainly not independent. Still, it seems reasonable to conclude that the expected excess returns of the funds probably differ across some of the economic states that we measure.

6.1. Performance with Term Structure Models The pure term structure models have the advantage that the relevant factors and state variables have clean support from theory, as opposed to relying on an empirical search process. Table 6 summarizes the performance results for funds grouped by characteristics, using the term structure models. We estimate the system (3) for each fund group separately. The Appendix shows that this gives the same alphas as joint estimation with all funds simultaneously. The returns are measured in excess of the Government–Corporate index, which is used as RB in Eq. (3b). We report the conditional excess alphas in the high and low states, with their heteroskedasticity-consistent t-ratios in parentheses. Table 6 includes results for the one-factor affine model, where a comparison to the unadjusted excess returns in Table 5 is interesting. The risk adjustment cuts the performance, conditional on the high spot rate state, to about 1/2 or less of the unadjusted excess return. But alphas as large as 27 bp per month are found, and some of the t-ratios are quite large. None of the signs are changed, relative to Table 5. This means that funds’ excess conditional covariances with the one-factor SDF are the right sign, but the magnitudes are too small to explain the excess returns. Not many of the differences between the high- and low-characteristics groups are statistically significant. Seven of the 12 alphas have t-ratios larger than two, but six of these are in the high spot rate state, where there are not many observations (only three in the 1986–1999 period). Table 6 also reports the results for the two-factor affine model. We found in Table 4 that the two-factor model did a better job of controlling passive benchmark excess returns than the one-factor model, and here the funds’ alphas are also smaller. Under the two-factor model the largest conditional alpha in the table is 27 bp; and most are much smaller. Some of the alphas have t-ratios larger than two, such as in the high spot rate states. In no case is the performance difference between the groups of funds – high versus low asset size, turnover, etc. – of any statistical significance. A few cases, however, border on potential economic significance. High turnover funds underperform low turnover funds in the high slope states by about 14 bp per

40

WAYNE FERSON ET AL.

Table 6.

Fixed Income Fund Conditional Performance Using Term Structure Models.

Fund Grouping:

Asset Size High

Turnover

Expense

Low

High

Low

High

Low

State: One-Factor Affine Model High short rate 0.00198 (12.1) Low short rate 2.27E–05 (0.229) Tdiff 2.00

0.000478 (0.488) 0.000321 (0.301) 0.108

0.0025 (29.3) 0.000235 (0.231) 2.23

0.00098 (1.42) 0.000249 (0.248) 1.01

0.00191 (7.23) 0.000377 (0.338) 1.34

0.00261 (5.11) 0.000107 (0.118) 2.61

Two-Factor Affine Model High short rate 0.00139 (2.37) Low short rate 0.000316 (0.316) Tdiff 1.40 High slope 0.000204 (0.104) Low slope 7.14E–05 (0.055) Tdiff 0.0461

0.000432 (0.562) 0.00101 (0.456) 0.246 0.00104 (0.181) 9.97E–05 (0.145) 0.185

0.00142 (2.69) 0.00101 (0.357) 0.155 0.00158 (0.175) 0.000157 (0.227) 0.167

0.000503 (0.856) 0.000517 (0.467) 0.809 0.000187 (0.0969) 8.59E–05 (0.0618) 0.0347

0.00128 (2.60) 0.000594 (0.386) 0.459 0.00108 (0.283) 0.000358 (0.253) 0.257

0.000903 (1.92) 0.000189 (0.123) 0.460 0.000948 (0.211) 0.000169 (0.119) 0.239

Fund Grouping

Lag Return High

Yield

Lag Flow

Low

High

Low

High

Low

One-Factor Affine Model High short rate 0.000492 (0.517) Low short rate 0.000298 (0.285) Tdiff 0.138

0.00262 (7.62) 0.000184 (0.137) 1.76

0.00273 (8.47) 0.000263 (0.645) 4.74

0.000888 (0.626) 0.000124 (0.0723) 0.343

0.000139 (0.130) 0.000372 (0.524) 0.181

0.00212 (13.5) 4.05E–05 (0.0579) 3.01

Two-Factor Affine Model High short rate 0.000853 (1.26) Low short rate 0.000115 (0.0932) Tdiff 0.540 High slope 0.000673 (0.229) Low slope 0.000746 (0.677) Tdiff 0.0229

0.000874 (2.19) 1.13E–05 (0.0085) 0.620 0.000138 (0.0436) 0.000293 (0.253) 0.0383

0.00265 (6.10) 0.000159 (0.330) 4.41 0.000478 (0.713) 0.000325 (0.719) 0.192

8.94E–06 (0.00677) 0.000792 (0.287) 0.253 0.00142 (0.209) 0.000239 (0.0958) 0.242

0.000450 (0.621) 0.000342 (0.388) 0.104 0.000611 (0.432) 0.000295 (0.209) 0.209

0.00191 (7.28) 0.000908 (0.942) 2.60 0.0010 (0.564) 0.000183 (0.520) 0.453

Note: Abnormal returns in excess of the Government–Corporate bond index are shown for equal-weighted portfolios of funds grouped on high versus low asset size, turnover and expense ratios. High-asset funds are those in the top third of all funds, while low asset funds are in the bottom third, etc. The alphas are decimal fractions per month, for the January 1986 through December 1999 period (168 observations), the lag flow group has 12 fewer observations and starts in January of 1987. Heteroskedasticity-consistent T-statistics are shown in parentheses. Tdiff is the t-ratio for the difference between the high- and low-state alphas.


41

month. High asset size funds underperform low asset size funds in high spot rate states by about 10 bp, funds with large flows of new money outperform low flow funds by almost 20 bp, and high income yield funds underperform low income yield funds by about 25 bp. Only the latter case is clearly larger in magnitude than the biases we observe in the excess returns of passive benchmarks. It is possible that high-income yield funds sacrifice some total return performance in order to report high-income yields. We do not report results for the Brennan–Schwartz model, but the results are similar to the two-factor affine model. We conclude that the term structure models explain a substantial portion of the variation in the conditional mean returns of funds grouped by characteristics, when we condition on the level of interest rates or the slope of the term structure. We trust the results of the two-factor model more than the one-factor model, based on their performance on passive benchmarks. In most cases, the magnitudes of the two-factor conditional alphas are within the range of the biases we observed for passive benchmarks, and are not statistically significant.

6.2. Fund Performance with Extra Factors Given the empirical importance of factors outside the pure term structure, such as inflation, credit spreads and such, it makes sense to examine fund performance with models that incorporate these factors. This gives us the opportunity to examine performance conditioned on a wider range of state variables. In Table 7 we present results using the extended two-factor affine model with one additional factor at a time. The additional factors are selected to match the state variable that we condition on, and these are reported as the rows of the table. The models use dummy variables for the level and slope of the term structure and for the additional factor as instruments. The risk-adjusted performance measures in Table 7 are typically small – closer to zero than 10 bp in most cases – and statistically insignificant. Most of the values are negative. This is consistent with the view that fixed income funds have essentially neutral risk-adjusted performance in most economic states, net of their expenses and trading costs.26 None of the alpha differences between high- and low-characteristics groups is statistically significant. Only five of the 144 combinations of states and fund groups generate conditional alphas with t-ratios larger than two. However, a few cases do suggest potential economic significance. In high credit spread states, the alphas are 25 bp or greater for eight of the 12 fund groups. This is the only state

42

WAYNE FERSON ET AL.

Table 7. Fund Grouping

Fixed Income Fund Conditional Performance: Models with Extra Factors. Asset Size High

High convexity

0.000786 (0.355) Low convexity 0.000725 (1.13) Tdiff 0.0276 High volatility 0.000418 (0.667) Low volatility 4.05E–05 (0.042) Tdiff 0.325 High credit 0.00349 (1.19) Low credit 0.000362 (0.598) Tdiff 1.00 High BS spread 3.08E–05 (0.0480) Low BS spread 0.00055 (0.707) Tdiff 0.505 High cap. util. 0.000132 (0.175) Low cap. util. 0.000604 Tdiff High dollar Low dollar Tdiff Fund Grouping

High convexity Low convexity Tdiff High volatility

Turnover

Expense

Low

High

Low

High

Low

0.000580 (0.384) 0.000911 (1.00) 0.177 0.000355 (0.480) 0.000549 (0.574) 0.161 0.00340 (1.14) 0.000158 (0.336) 1.07 0.000297 (0.392) 0.00059 (0.506) 0.234 0.000615 (0.475) 0.00071

0.000137 (0.0986) 0.000785 (1.15) 0.428 6.07E–05 (0.098) 0.000459 (0.604) 0.409 0.00249 (1.08) 0.00059 (1.84) 1.32 0.000564 (0.537) 0.00114 (1.21) 0.428 n.a.

0.000730 (0.337) 0.000498 (0.782) 0.102 0.000375 (0.570) 0.000673 (0.677) 0.247 0.00309 (1.18) 0.000565 (1.42) 1.37 0.000179 (0.380) 3.20E–05 (0.039) 0.219 n.a.

0.00126 (0.464) 0.00127 (1.42) 0.00565 0.00809 (1.08) 0.000364 (0.002) 0.335 0.00313 (1.04) 0.000773 (1.68) 1.28 n.a.

0.000156 (0.138) 0.000620 (1.07) 0.374 0.00534 (0.841) 1.06E–05 (0.013) 0.507 0.00276 (1.33) 0.000547 (1.88) 1.57 n.a.

n.a.

n.a.

n.a. 0.00157 (3.30) n.a. n.a. 0.00167 (0.744) 0.431 0.743 n.a. n.a. 1.41 0.000377 0.000631 0.000638 0.000519 0.00127 (0.526) (0.795) (0.777) (0.798) (1.07) 0.00132 0.000360 0.000644 0.000960 0.000878 (1.21) (0.280) (0.546) (0.866) (0.666) 1.30 0.180 0.892 1.15 1.21 Lag Return High

Low

0.00177 (0.421) 0.00106 (1.03) 0.159 0.000726 (0.443)

0.000289 (0.689) 0.000878 (0.938) 0.663 0.000220 (0.207)

Yield High

n.a. 0.00111 (2.34) 0.00175 (1.17) 1.81 2.50E–05 (0.049) 0.000468 (0.470) 0.441

Lag Flow Low

High

0.000591 0.00131 0.000763 (0.946) (0.439) (0.592) 0.000963 0.000497 0.000611 (1.33) (0.553) (0.739) 0.393 0.259 0.099 0.000894 0.000264 0.00049 (1.30) (0.293) (0.532)

Low 0.000842 (0.436) 0.000576 (0.931) 0.130 0.000465 (0.718)


43


Lag Return High

Low volatility

0.000308 (0.277) Tdiff 0.211 High credit 0.000711 (0.783) Low credit 0.000969 (1.40) Tdiff 1.46 High BS spread 0.000425 (0.363) Low BS spread 0.000208 (0.143) Tdiff 0.110 High cap.util n.a. Low cap.util Tdiff High dollar Low dollar Tdiff

Low 0.000323 (0.257) 0.063 0.00467 (1.12) 0.000206 (0.0417) 1.16 0.000772 (1.45) 0.00136 (1.20) 0.538 n.a.

Yield High

0.000124 (0.286) 0.944 0.00116 (0.986) 0.000603 (1.87) 1.44 0.000286 (0.330) 0.00106 (1.26) 0.759 0.00106 (2.57) n.a. n.a. 0.000521 (0.558) n.a. n.a. 1.55 0.000888 0.000276 0.000664 (0.785) (0.324) (0.666) 0.000314 0.00134 8.66E–05 (0.264) (0.823) (0.178) 0.353 0.877 0.676

Lag Flow Low

High

Low

0.000383 (0.236) 0.064 0.00488 (1.09) 0.000639 (0.978) 1.23 3.32E–05 (0.0393) 0.000170 (0.115) 0.121 0.00184 (2.19) 0.00245 (0.815) 1.37 0.000649 (0.726) 0.000941 (0.483) 0.743

0.000504 (0.730) 0.013 0.000558 (0.810) 0.000640 (1.56) 1.50 0.00121 (2.11) 0.000611 (0.852) 0.632 0.000932 (1.32) 0.000182 (0.253) 0.735 0.00103 (1.41) 0.000978 (1.26) 0.045

1.96E–05 (0.027) 0.455 0.000753 (0.781) 0.000539 (1.37) 1.36 0.000680 (1.33) 0.000308 (0.442) 0.430 0.000706 (1.81) 0.000128 (0.161) 0.650 0.000735 (0.924) 0.000389 (0.607) 0.339

Note: Abnormal returns in excess of the Government–Corporate bond index are shown for equal-weighted portfolios of funds grouped on various high versus characteristics. High-asset funds are those in the top third of all funds, while low asset funds are in the bottom third, etc. The models are the extended affine models with one additional factor. The additional factor corresponds to the state of the economy being examained, as explained in the text. The instruments are a constant and dummy variables for high or low values of the spot rate, term structure slope and the additional state variable. The alphas are decimal fractions per month, for the January 1986 through December 1999 period (168 observations), results for funds grouped by lagged flow have start in 1987 and have 12 fewer observations. Heteroskedasticityconsistent T-statistics are shown in parentheses. Tdiff is the t-ratio for the difference between the conditional alphas in the high and low economic states. n.a. indicates that the algorithm encountered a singularity.

with consistently positive risk-adjusted performance. However, the t-statistics are smaller than two, which is probably explained by the high volatility of fund returns in high credit spread states. Table 5 illustrates that the standard error of the mean returns is about five times as large in high credit spread

44

WAYNE FERSON ET AL.

states than in low spread states. Overall, it seems that the differences in the conditional mean returns across the various states are well explained by the extended SDF models, and there is little evidence of abnormal risk-adjusted performance.

7. FUND PERFORMANCE IN RELATION TO STYLE When we grouped the funds according to characteristics we used a broad benchmark, because the groups included all fund styles. However, fixed income funds may adhere more closely to style than equity funds, so controlling for style may be important. There may be more heterogeneity across fund styles than in relation to the fund characteristics examined earlier, so grouping by style may reveal performance differences obscured by the characteristics groups. This section studies fund performance by the style groups summarized in Table 1, with returns measured relative to a stylerelated passive benchmark. For mortgage funds we use the Lehmann Brothers GNMA index as a benchmark. For high yield funds we use the return on the Lehmann Brothers index of all Baa rated bonds. For highquality funds we use the all Aaa bond index return. For government securities funds, we use the Ibbotson Associates, 20-year bond return. For load funds, no-load funds and the aggregate of the styles (‘‘all’’) we use the Government–Corporate index as before. Table 8 presents the excess returns with no risk adjustments, similar to Table 5. The first three lines summarize the unconditional means and the number of monthly observations, which differ across the style groups, and the sample for each group ends in December of 1999. The range of excess returns across styles is more than twice the range we saw across the characteristics groups. In Table 5 we saw that all of the fund groups’ returns were below benchmark. There is only one exception here. Over the 1990– 1999 period, high yield funds beat the Baa benchmark by about 9 bp per month. The remaining rows of Table 8 summarize excess returns conditional on the high or low state variable dummies. There are a number of interesting results. Consistent with the unconditional means, there is more heterogeneity across the fund styles than we found with the characteristics groups. Mortgage funds return less than the GNMA benchmark unconditionally and in every state. The t-ratio for the difference is below 2.0 in 16 of the 24 cases shown and the magnitude of the underperformance ranges from 9 to 33 bp per month. Overall, significant positive excess returns are rare.

Fund Style State Uncond.

High short rate

Low short rate

High slope

Low slope

High convexity

Low convexity

High volatility

Low volatility

Fixed Income Fund Returns in Excess of Style Benchmarks. All

Load

No Load

High Quality

High Yield

Mortgage

0.002607 0.001424 180.0 0.01651 0.008535 3.000 0.002491 0.002616 74.00 0.008091 0.002592 19.00 0.001622 0.002407 46.00 0.002054 0.002510 16.00 0.003121 0.002833 43.00 0.002052 0.005885 11.00 0.001744 0.002448 70.00

0.0006659 0.0006209 180.0 0.003843 0.001075 3.000 0.0002784 0.0008671 74.00 0.002466 0.001215 19.00 0.001083 0.001068 46.00 0.001502 0.001231 16.00 0.0008589 0.001010 43.00 0.002226 0.002746 11.00 0.0003042 0.0008481 70.00

0.0002210 0.0005789 168.0 0.004949 0.001330 3.000 0.0001301 0.001074 64.00 0.002751 0.001958 17.00 0.0002295 0.0007180 46.00 0.001336 0.001820 15.00 0.001722 0.001098 43.00 0.001268 0.001434 11.00 0.0001045 0.001002 62.00

9.196E–05 0.0004927 168.0 0.002220 0.0009297 3.000 0.0001436 0.0009671 64.00 0.001765 0.001467 17.00 0.0003159 0.0005843 46.00 0.001291 0.001555 15.00 0.0009038 0.0006428 43.00 0.0006972 0.001340 11.00 0.0005426 0.0009439 62.00

0.0009400 0.0004399 132.0 0.004463 0.003648 3.000 0.0001000 0.0005456 38.00 0.001944 0.0008749 14.00 0.0005923 0.0009220 43.00 0.001493 0.0006260 14.00 0.001562 0.001209 33.00 0.001574 0.001962 11.00 0.0004254 0.0004461 40.00

0.0008908 0.001536 120.0 0.000 0.000 0.000 0.005369 0.002046 38.00 6.594E–05 0.003334 14.00 0.001565 0.002807 32.00 0.0009287 0.002285 14.00 0.002369 0.005094 22.00 0.0006444 0.004932 9.000 0.004636 0.002114 40.00

0.001456 0.0002716 132.0 0.002933 0.003037 3.000 0.001131 0.0003687 38.00 0.002613 0.0007099 14.00 0.001626 0.0005232 43.00 0.001931 0.0005504 14.00 0.001932 0.0006480 33.00 0.002315 0.001492 11.00 0.001087 0.0003523 40.00

45

Government


Table 8.

46

Table 8. (Continued ) Government

All

Load

No Load

High Quality

High Yield

Mortgage

High credit

0.01054 0.004198 14.00 0.005065 0.001832 83.00 0.008426 0.004065 16.00 0.003187 0.002613 66.00 0.006209 0.005762 16.00 0.004731 0.003838 30.00 0.0004109 0.002698 22.00 0.002097 0.004872 25.00 0.004933 0.002442 46.00 0.003248 0.003191 32.00

0.006034 0.002573 14.00 0.001316 0.0006586 83.00 0.001403 0.001157 16.00 0.001202 0.0005276 66.00 0.002644 0.002936 16.00 0.001228 0.001261 30.00 0.001034 0.001519 22.00 0.003916 0.002327 25.00 0.001781 0.0005025 46.00 0.002167 0.002155 32.00

0.006144 0.002715 14.00 0.001099 0.0005548 78.00 0.001404 0.001240 16.00 0.001301 0.0007473 62.00 0.0007968 0.001017 16.00 0.001110 0.001390 29.00 0.001068 0.001576 22.00 0.001614 0.002525 24.00 0.002027 0.0006440 46.00 0.002271 0.002255 32.00

0.005715 0.002491 14.00 0.001145 0.0004559 78.00 0.001547 0.001068 16.00 0.0008924 0.000569 62.00 0.000388 0.000893 16.00 0.000611 0.001264 29.00 0.001148 0.001482 22.00 0.003003 0.001874 24.00 0.001466 0.000469 46.00 0.002271 0.001998 32.00

0.002582 0.0009020 14.00 0.001531 0.0005506 68.00 0.002460 0.001535 16.00 0.0008702 0.0005939 32.00 0.0009813 0.001950 11.00 0.001539 0.0006860 24.00 0.0007877 0.001260 20.00 0.0006928 0.001465 20.00 0.002814 0.001045 28.00 0.0005841 0.0008365 32.00

0.009122 0.003942 14.00 0.0003204 0.001634 56.00 0.0004272 0.002997 16.00 0.0007325 0.001929 32.00 0.01079 0.009645 8.000 0.001773 0.002586 24.00 0.002018 0.003462 19.00 0.002908 0.007393 16.00 0.003631 0.002714 21.00 0.005472 0.003713 32.00

0.001253 0.0005326 14.00 0.001858 0.0004645 68.00 0.001455 0.0004843 16.00 0.001244 0.0005815 32.00 0.003346 0.001349 11.00 0.001336 0.0005378 24.00 0.001955 0.0006996 20.00 0.0008667 0.0004809 20.00 0.003143 0.0008989 28.00 0.0009207 0.0003598 32.00

Low credit

High BS spread

Low BS spread

High inflation

Low inflation

High IP growth

Low IP growth

High cap. util.

Low cap. util.

WAYNE FERSON ET AL.

Fund Style

Low dollar

High corp. iliq.

Low corp. iliq.

High stock liq.

Low stock liq.

0.004828 0.003719 28.00 0.0006032 0.002808 52.00 0.002079 0.003368 22.000 0.0003646 0.005362 8.000 0.001629 0.004693 24.00 0.002642 0.004582 22.00

0.001443 0.001282 28.00 0.0006483 0.001613 52.00 0.0008359 0.001517 22.00 0.0004182 0.001301 8.000 0.0001388 0.001215 24.00 0.0003146 0.0009644 22.00

0.002041 0.001817 21.00 0.00073 0.001279 52.00 0.001004 0.001854 22.00 0.0002915 0.001198 8.000 0.0003374 0.001063 21.00 0.0005840 0.001152 22.00

0.001035 0.001122 21.00 0.000476 0.001200 52.00 0.0006339 0.001414 22.00 0.0005091 0.001375 8.000 0.0003119 0.001040 21.00 0.0003060 0.0008062 22.00

0.001875 0.001521 21.00 0.001360 0.0008173 23.00 0.001474 0.001573 14.00 0.0002779 0.001685 8.000 0.001412 0.001446 17.00 0.0004471 0.001603 16.00

0.001453 0.004083 21.00 0.002348 0.004750 23.00 0.004039 0.005296 14.00 0.002592 0.002097 8.000 0.0005970 0.003193 16.00 0.0009286 0.004007 15.00

0.0009795 0.0003951 21.00 0.001327 0.0005387 23.00 0.0008824 0.0004834 14.00 0.0006262 0.002014 8.000 0.001578 0.0004949 17.00 0.001030 0.001008 16.00

Note: Returns in excess of style indexes are shown for equal-weighted portfolios of funds grouped by style. The excess returns are decimal fractions per month, for various subperiods ending in December 1999 (180 or fewer observations). The first group of three rows reports the unconditional sample means and standard deviations of the mean excess returns, followed by the number of months available. Subsequent rows report the same statistics conditional on various economic states, as measured by dummy variables. Figures larger than two standard errors of the mean are in bold.


High dollar

47

48

WAYNE FERSON ET AL.

Cases where the t-ratios for the mean excess return exceed 2.0 include the high credit spread states (for all fund styles); also, high yield funds in three states (low short-term rates, low volatility and high credit spread states). Like in Table 5, some states predict positive excess returns that may be economically significant, but the funds’ return volatilities are also higher in these states, so the means are not statistically significant. These case include the low capacity utilization and low industrial production states. The states where low excess returns are indicated in Table 8 include the high slope states, the low credit spread states, and especially the high capacity utilization states. High capacity utilization predicts underperformance ranging from 15 to 49 bp off the benchmark. High short rates also predict low excess returns, but with only three months in the high-rate regime, these figures may not be very meaningful. There are 168 cases in Table 8, so we would expect about eight of the t-ratios to be larger than 2.0 if all the mean excess returns are really zero. We find 45 absolute t-ratios larger than 2.0; a larger portion than in the characteristics-based groups. It seems reasonable to conclude that the expected excess returns of the funds differ across some of the economic states.

7.1. Risk-Adjusted Performance by Style Table 9 summarizes conditional SDF alphas for the funds’ returns in excess of style benchmarks. The table shows that the risk adjustments of the onefactor affine model explain part of the excess returns, but not as much as we saw for the characteristics-based groups. The two-factor affine and Brennan– Schwartz models produce similar results as the one-factor model. There is more heterogeneity in risk-adjusted performance across styles than across the characteristics groups. Six or eight of the 28 alphas have absolute t-ratios larger than two, depending on the model. Under the two-factor models the largest conditional alphas are 31 bp or less, with one exception, and most are less than 15 bp. Mortgage funds have significantly negative alphas across all the states. The significant alphas for the mortgage funds reflect their small standard errors as much as the economic magnitudes of the alphas, which range from 14 to 19 bp. However, these magnitudes are similar to the excess returns of the mortgage funds before risk adjustment. The risk adjusted performance measures of the extended two-factor affine models are shown in Panel D. Mortgage funds have negative alphas, with t-ratios in excess of two for 13 of the 20 cases. Again, this reflects the small standard errors of the mortgage alphas: The values range from 3 to

Fund Style

Fixed Income Fund Risk-Adjusted Returns in Excess of Style Benchmarks.

Government

Load

No Load

High Quality

High Yield

Mortgage

0.002197 0.0003285 3.000 0.0004258 0.0008134 74.00

0.002562 5.380E–05 3.000 0.0001507 0.001001 64.00

0.001139 0.0007026 3.000 0.0001086 0.0009514 64.00

0.002695 0.001742 3.000 0.0004883 0.0003291 38.00

n.a. n.a. n.a. n.a. n.a. n.a.

0.002155 0.002408 3.000 0.001282 0.0003516 38.00

Panel B: Two-Factor Brennan–Schwartz Model High short rate 0.005075 0.001459 0.0007636 0.0001371 3.000 3.000 Low short rate 0.0008610 0.0004529 0.002301 0.002171 74.00 74.00 High slope 0.001983 0.0006831 0.0008741 0.0009093 19.00 19.00 Low slope 0.001047 0.0005340 0.003425 0.001504 46.00 46.00

0.0009529 0.0009940 3.000 0.0001043 0.001217 64.00 0.0009922 0.0005265 17.00 0.0002284 0.005813 46.00

0.0003710 0.0007155 3.000 5.854E–05 0.001251 64.00 0.0002242 0.0003876 17.00 0.0002014 0.001569 46.00

0.001270 0.001260 3.000 0.0007459 0.0004399 38.00 0.001076 0.000629 14.00 0.0004080 0.001931 43.00

Panel C: Two-Factor Affine Model High short rate 0.005063 0.0006846 3.000 Low short rate 0.0006925 0.001509 74.00

0.001489 0.0006546 3.000 0.0003879 0.001630 64.00

0.0004680 0.0006172 3.000 0.0001308 0.0009820 64.00

0.0002917 0.0002024 3.000 0.0001483 0.0004777 38.00

State Panel A: One-Factor Affine Model High short rate 0.009274 0.002886 3.000 Low short rate 0.002324 0.001265 74.00

0.001443 0.0001184 3.000 0.0006745 0.001318 74.00

n.a. n.a. n.a. n.a. n.a. n.a. 0.0006196 14.00 0.001518 0.001115 32.00 n.a.

n.a.

0.001549 0.002497 3.000 0.001425 0.0003613 38.00 0.001914 0.000540 14.00 0.001771 0.0006683 43.00 0.001696 0.0007333 3.000 0.001423 0.000441 38.00

49

All


Table 9.

50

Table 9. (Continued ) Government

All

Load

No Load

High Quality

High Yield

Mortgage

High slope

0.001833 0.001037 19.00 0.0008797 0.003563 46.00

0.000509 0.001858 19.00 0.0006049 0.0009018 46.00

0.001047 0.001724 17.00 0.0001674 0.006357 46.00

0.0002991 0.0006335 17.00 0.0001472 0.002144 46.00

0.001235 0.0001478 14.00 0.0001076 0.009653 43.00

0.001227 14.00 0.0001076 0.0009162 32.00

0.001862 0.0006075 14.00 0.001798 0.0008568 43.00

Panel D: Extended Two-Factor Affine Model High convexity 0.003133 0.000499 0.001745 0.001156 16.00 16.00 Low convexity 0.003138 0.0008786 0.001165 0.0009473 43.00 43.00 High volatility 0.002341 0.000287 0.003950 0.002608 11.00 11.00 Low volatility 0.002188 0.0005823 0.001258 0.0008213 70.00 70.00 High credit 0.005889 0.002324 0.004717 0.003201 14.00 14.00 Low credit 0.002554 0.0005991 0.000874 0.0005932 83.00 83.00 High BS spread 0.001377 0.0005723 0.000344 0.0005845 16.00 16.00

0.000238 0.002185 15.00 0.0007433 0.0009675 43.00 7.045E–05 0.0007317 11.00 0.0004015 0.0009286 62.00 0.003819 0.003088 14.00 0.0001907 0.0005728 78.00 0.001060 0.0007692 16.00

0.000171 0.001897 15.00 0.0005465 0.0006181 43.00 0.0003977 0.0005908 11.00 0.0001628 0.0008755 62.00 0.003128 0.002876 14.00 2.203E–05 0.0004699 78.00 0.001144 0.000428 16.00

0.001227 0.0004005 14.00 0.0008465 0.0007080 33.00 0.0004843 0.0007153 11.00 0.0007360 0.0003119 40.00 0.008151 0.002142 14.00 0.0006820 0.0004134 68.00 1.011E–05 0.0005290 16.00

0.001223 0.002149 14.00 0.0003069 0.003622 22.00 0.0009061 0.002797 9.000 0.001750 0.001790 40.00 0.003009 0.006926 14.00 0.000536 0.002623 56.00 0.007511 0.005991 16.00

0.001732 0.000568 14.00 0.001711 0.000604 33.00 0.001236 0.001159 11.00 0.001380 0.000345 40.00 0.001797 0.0006305 14.00 0.001314 0.0005088 68.00 0.0009876 0.0005598 16.00

Low slope

WAYNE FERSON ET AL.

Fund Style

High inflation

Low inflation

High IP growth

Low IP growth

High cap. util.

Low cap. util.

High dollar

Low dollar

High corp. iliq.

Low corp. iliq.

0.0006430 0.0004419 66.00 0.002151 0.003224 16.00 0.0004645 0.001219 30.00 0.0006901 0.001471 22.00 0.000563 0.002962 25.00 0.0006527 0.0003951 46.00 0.0007619 0.002519 32.00 0.0004285 0.001741 28.00 0.000396 0.002368 52.00 0.0004975 0.001335 22.00 0.0006123 0.0006881 8.000

0.0007760 0.0006693 62.00 0.0004051 0.0008425 16.00 7.030E–05 0.001319 29.00 0.000283 0.001518 22.00 0.000210 0.002976 24.00 0.001317 0.0004910 46.00 0.001764 0.002001 32.00 0.000901 0.001100 21.00 0.000991 0.001179 52.00 0.000189 0.002860 22.00 0.0003389 0.0007441 8.000

0.0007339 0.0006215 62.00 0.0003573 0.0009201 16.00 0.0003273 0.001274 29.00 0.000177 0.001466 22.00 5.616E–06 0.002517 24.00 0.001279 0.000421 46.00 0.001698 0.001779 32.00 0.0001243 0.0004212 21.00 0.000396 0.001087 52.00 0.001406 0.000981 22.00 0.0002371 0.0004257 8.000

0.0003275 0.0002952 32.00 0.001098 0.0008575 11.00 0.0008504 0.0003518 24.00 0.0004229 0.0006885 20.00 0.001210 0.002297 20.00 0.002456 0.000620 28.00 0.001236 0.000501 32.00 0.001059 0.000502 21.00 0.0009460 0.0006621 23.00 0.0008085 0.0008061 14.00 0.0003198 0.0007601 8.000

0.001466 0.002264 32.00 0.006722 0.006949 8.000 0.002421 0.001990 24.00 0.001126 0.003049 19.00 0.002136 0.005683 16.00 0.001650 0.001994 21.00 0.005241 0.003190 32.00 0.0008441 0.003160 21.00 0.000852 0.004410 23.00 0.004059 0.004446 14.00 0.001961 0.001924 8.000

0.0008469 0.0006553 32.00 0.001138 0.001316 11.00 0.001483 0.0005145 24.00 0.001812 0.000746 20.00 0.000365 0.001040 20.00 0.002922 0.000701 28.00 0.0009018 0.0004019 32.00 0.0009396 0.0004005 21.00 0.001416 0.000577 23.00 0.001001 0.000468 14.00 0.0005783 0.001682 8.000

51

0.003289 0.000744 66.00 0.003338 0.003938 16.00 0.001076 0.001892 30.00 0.002505 0.002161 22.00 0.002370 0.003848 25.00 0.002629 0.000529 46.00 0.002785 0.003240 32.00 0.001958 0.000596 28.00 0.000874 0.002133 52.00 0.002317 0.001961 22.00 0.002884 0.000849 8.000


Low BS spread

52

Table 9. (Continued ) Fund Style

Government

All

Load

No Load

High Quality

High Yield

Mortgage

High stock liq.

0.01482 0.003832 24.00 0.001493 0.002154 22.00

0.0002165 0.0004293 24.00 0.002462 0.001660 22.00

6.279E–05 0.0007297 21.00 0.0001750 0.0007015 22.00

1.382E–05 0.0008971 21.00 0.0003469 0.0007010 22.00

0.0003520 0.0004994 17.00 0.001218 0.0008206 16.00

0.001565 0.001802 16.00 0.001527 0.002992 15.00

0.001547 0.0004862 17.00 0.0008159 0.0008587 16.00

Low stock liq.

Note: Risk-ajusted SDF alphas for returns in excess of style indexes are shown for equal-weighted portfolios of funds grouped by style. Asymptotic standard errors are on the second line. The units are decimal fractions per month, for various subperiods ending in December, 1999 (180 or fewer observations), with the number of observations in the high and low states shown below the standard errors. Figures larger than two standard errors are in bold. n.a. indicates that the algorithm encountered a singularity.

WAYNE FERSON ET AL.


53

18 bp. In high credit spread states, where the characteristics groups produced all positive alphas in excess of 25 bp, the style groups produce a range of conditional SDF alphas. Four of the seven groups have negative alphas, and Government bond funds deliver a whopping 58 bp in the high credit spread state. Overall, 28 of the 140 combinations of states and fund style groups generate conditional alphas in the extended models with t-ratios larger than two. The average absolute alpha across the style groups and states is 13.2 bp. This compares with an average absolute excess return, before risk adjustment, of 22.4 bp.

8. CONCLUDING REMARKS This paper evaluates the performance of fixed income mutual funds using SDFs. Conditioning the models on discrete representations of the state of the term structure and the economy, the returns and volatility of fixedincome funds and benchmarks vary significantly across the economic states. The models can explain large fractions of this variation. Additional empirical factors arise from time-aggregation of the continuous-time term structure models. These factors enhance explanatory power; both in linear regressions, and in the asset pricing models, for monthly average returns on passive benchmarks. Two-state-variable models perform better than one-state-variable models, and extended models with additional factors are better yet. The two-factor affine and Brennan–Schwartz models perform similarly. Excess return performance measures are less sensitive to the choice of benchmarks and are typically less biased than raw return measures. We find that fixed income funds return less than passive benchmarks that do not pay expenses, but not in all economic states. The funds typically do poorly when short-term interest rates are high, the slope of the term structure is steep and industrial capacity utilization is high. The largest positive excess returns are found when quality-related credit spreads are high, but the volatility of returns is also high in these states. We find little cross-sectional variation in performance when funds are grouped into thirds by asset size, expense ratio, turnover, income yield, lagged return or lagged new money flows. There is more heterogeneity across fixed income fund styles. Mortgage funds underperform a GNMA index in all of the economic states. The underperformance of mortgage-style funds survives risk adjustment, but most of the other excess returns become insignificant when we adjust for risk using the SDFs.

54

WAYNE FERSON ET AL.

NOTES 1. Sources: The Investment Company Institute, Trends in Mutual Fund Investing, June 2002, and 2002 Mutual Fund Handbook. 2. Ferson, Henry, and Kisgen (2006) evaluate government bond funds using stochastic discount factors from term structure models, and this paper is the pilot study to that article. Other studies that focus on US fixed income funds include Blake et al. (1993), Elton et al. (1995) and Kang (1995). Cornell and Green (1991) and Gudikunst and McCarthy (1992, 1997) examine low-grade bond funds, Stock (1982) and Kihn (1996b) examine municipal bond portfolios and Kihn (1996a) examines convertible bond funds. Duke, Papaloannou, and Brierley (1993), Schadt (1996), Gallo, Lockwood, and Swanson (1997), Fjelstad (1999), Detzler (1999) and Silva and Cortez (2002) study international fixed income fund performance. Dahlquist, Engstrom, and Soderlind (2000) include bond and money market funds in their sample of Swedish funds. Fung and Hsieh (2002) compare the styles of fixed income hedge funds and mutual funds. Additional studies include D’Antonio et al. (1997), Dietz et al. (1981), Fong, Pearson, and Vasicek (1983), Grantier (1988), Kahn (1991) and Shyy and Lieu (1994). 3. Critiques by Lo and MacKinlay (1990), MacKinlay (1995) and Ferson, Sarkissian, and Simin (1999) illustrate the pitfalls of asset pricing factors motivated by empirical regularities. 4. See Ferson (1995, 2002) and Cochrane (2001) for more discussion and interpretation of SDF alphas. 5. The condition E(mR19Z) ¼ 0 is equivalent to E{(mR1)f(Z)} ¼ 0 for all functions f(.). The typical linear specification assumes that f(Z) ¼ IZ. See Ferson and Siegel (2003) for a discussion of optimized functions f(.) in the context of mean variance efficiency bounds, and Ferson and Siegel (2006) for an approach to asset pricing tests based on optimized functions for mean variance portfolios. 6. See, for example Bossaerts and Hillion (1999), Goyal and Welch (2003), Simin (2002) and Cooper, Gutierez, and Marcum (2005). 7. Efficient GMM parameter estimates can be obtained using any subset of funds, and the individual standard errors are numerically equivalent to those in the full system. Farnsworth et al. (2002) provide the invariance result for the special case where there is only a constant in Dt, so the alpha is a constant. The appendix to this paper refines and extends the result for a time-varying alpha. 8. See Kahn (1991) for a decomposition of bond returns into term structure effects and other effects. For a recent study of term structure models incorporating additional economic risk factors, see Ang and Piazzesi (2001). 9. For example, in the Cox–Ingersoll–Ross model the first and second moments of the discrete changes, rt+1rt, conditional on the current value of the state variable rt, may be expressed as a function of rt and the parameters of the square root interest rate process. We could append these moment conditions to system (13) to identify all of the model’s parameters. See Farnsworth (1997) for an illustration. 10. The objectives are the union of the following: Weisenberger objective codes CBD, CHY, GOV, LTG or MTG; ICDI fund objective codes BQ, BY, GM or GS; Strategic Insight fund objective codes CGN, CHQ, CHY, CIM, CMQ, CSI, CSM, GGN, GIM, GMB, GMA, GSM or IMX.


55

11. Government funds include the ICDI_OBJ code GS, OBJ codes GOV or LTG, POLICY code of GS or SI_OBJ codes of GGN, GIM, or GSM. High quality funds include ICDI_OBJ code BQ, OBJ code CBD or SI_OBJ codes CGN, CIM, CSM, CMQ, CHQ, IMX or CSI. High yield funds include ICDI_OBJ code BY, OBJ code CHY or SI_OBJ code CHY. Mortgage funds include ICDI_OBJ code GM, OBJ code MTG or SI_OBJ codes GMB or GMA. All is an equal-weighted portfolio of the above. Load funds have a positive value in at least one of the following fields: FRNT_LD, DEF_LD or REAR_LD. No load funds have a value of zero in all three of these fields. 12. The end-of-month value of the daily short rate is the secondary market threemonth Treasury rate from the Federal Reserve H.15 release, obtained from the FRED database. 13. The five-year yield is from the CRSP FAMABY file and the one-month yield is from the CRSP RISKFREE file. Both are converted to continuously compounded rates. 14. One complication is that the daily three-month spot rates are highly autocorrelated. Since the interest rates refer to overlapping periods longer than one month, the data should follow a moving average process with more terms than the number of days in the month. This causes a bias in the sample variance. We approximately control this bias by modeling the autocorrelation as an AR(1) process. Let the AR(1) coefficient be r, let the number of daily observations in the month be T, and let s2(r) be the maximum likelihood estimator of the variance, ignoring the autocorrelation. It is easy to show that the expected value of s2(r) differs from s2(r), the true variance. An unbiased estimator, in the sense that its expected value under the AR(1) assumption is s2(r), may be constructed as: s ¼ s2 ðrÞ=½1 ð1=TÞ ð2=T 2 Þfr=ð1 rÞgfTð1 rT1 Þ ð1 rT1 Þ=ð1 rÞ þ ðT 1ÞrT1 g We use s as our estimate of the monthly variance, where T is the number of daily observations in the month and r ¼ 0.990, the estimated autocorrelation using all of the daily observations in the sample. 15. The dividend yield is computed from the with- and the without-dividend index levels and returns of the CRSP value-weighted index. It sums the preceding 12 months of dividend payments, divided by the level of the index. The Treasury yield is measured to match, as a lagging, 12-month moving average. 16. From January of 1999 this series is twexbmth, from the FRED. Before 1999 we use the series twexmthy, which is measured relative to the G10 countries, but is discontinued at the end of 1998. We splice the two series together by multiplying twexbmth by a constant, so that the levels of both series are the same in December of 1998. 17. We tried a three-year constant maturity yield in place of the one-year yield in the convexity measure, but it did not work as well in the regressions of Table 3. We also experimented with the 10-year and 30-year yield in place of the seven-year yield; see below. 18. Gudikunst and McCarthy (1997) also find that multiple economic factors are significant in pricing low-grade bond returns. See Kihn (1996b) for municipal bond funds and Kihn (1996a) for convertible bond funds.

56

WAYNE FERSON ET AL.

19. The standard errors of the mean differences between the returns conditioned on the high and low states in Table 2 is approximately 0.05s(hi) [1+(s(lo)/s(hi))2]1/2, where s(lo) is the standard deviation shown in the table for the low state. This assumes that the returns in the high and low states are uncorrelated. For the S&P500 return in the high and low spot rate states, the standard error of the mean difference is about 0.003. 20. Stambaugh (1999) considers a regression system: rtþ1 ¼ a þ b Zt þ utþ1 Z tþ1 ¼ d þ rZt þ vtþ1 2 2 with E(u t+1vt+1) ¼ suv and 2 E(vt ) ¼ sv . He shows that the OLS estimator has bias ^ rÞ ð1 þ 3rÞ=T: Our adjusted estimas ; where E ð r E b^ b ¼ E ðr^ rÞsuv v tor is b^ þ ð1 þ 3r Þsuv Ts2v ; where r ¼ ðT r^ þ 1Þ=ðT 3Þ: This approximation treats the slope coefficients as simple regression coefficients. We compute the regression R2 using the adjusted slopes, as the ratio of the variance of the fitted values to the variance of the dependent variable. 21. For any discount bond return there is an exact factor model regression that works tautologically. That model includes a yield change, a term structure slope and an interest rate level as the ‘‘factors.’’ However, in the tautological model all three factors are maturity specific, and thus are different for different bonds. A good empirical factor model should use a small number of market-wide factors to explain bonds of different maturities. 22. We experiment by replacing the one-year with a three-year yield in the convexity measure, and the explanatory power is lower. We also replaced the seven-year yield with a ten-year or a 30-year yield (the latter available starting in 1977). The longer series do not offer any marked improvements over the seven-year series. In some cases, the explanatory power with the ten-year and 30-year yield series is significantly worse. The 30-year yield, in particular, usually results in larger standard errors of the regressions. 23. The gradient matrix becomes rank deficient. 24. While not shown in the table, if we ask the one-factor model to explain returns conditional on the slope, it performs much worse than the two-factor model. 25. Duffee (2002) also observes that affine models have trouble fitting expected returns conditional on extreme term structure slopes. He advocates ‘‘essentially affine’’ models, an extension we are currently exploring. 26. See Berk and Green (2004) for a model in which fund flows ensure neutral performance net of expenses in equilibrium.

ACKNOWLEDGMENTS This report is based on a pilot study for our 2006 paper, ‘‘Evaluating Government Bond Fund Performance with Stochastic Discount Factors,’’ Review of Financial Studies 19, 423–455. We are grateful to Warren Bailey, Edie


57

Hotchkiss, Clifton Green, Eric Jacquier and Russ Wermers for suggestions. This paper has also benefited from workshops at Babson College, the Berkeley Program in Finance, Boston University, Brandeis, Cornell, the Federal Reserve Bank of Atlanta, New York University, Northwestern, the Pennsylvania State University, the University of Texas at Dallas, Utah and Wharton. The authors appreciate financial support from the Gutmann Center for Portfolio Management at the University of Vienna and a Q-group research grant.

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Ferson, W. E. (1995). Theory and empirical testing of asset pricing models. In: R. A. Jarrow, V. Maksimovic & W. T. Ziemba (Eds), Handbooks in operations research and management science. North Holland, UK: Elsevier. Ferson, W. E., Sarkissian, S., & Simin, T. (1999). The alpha factor asset pricing model: A parable. Journal of Financial Markets, 2, 49–68. Ferson, W. E., Sarkissian, S., & Simin, T. (2003). Spurious regressions in financial economics? Journal of Finance, 58, 1393–1414. Fjelstad, M. (1999). Modelling the performance of active managers in the Euroland bond market. Journal of Fixed Income, 9, 32–44. Fong, G., Pearson, C., & Vasicek, O. (1983). Bond performance: Analyzing sources of return. Journal of Portfolio Management, 9, 46–50. Fung, W., & Hsieh, D. A. (2002). The risks in hedge fund strategies: Alternative alphas and alternative betas. In: J. Lars (Ed.), The new generation of risk management for hedge funds and private equity funds. London: Euromoney Institutional Investors. Gallo, J. G., Lockwood, L. J., & Swanson, P. (1997). The performance of international bond funds. International Review of Economics and Finance, 6, 17–36. Gatev, E., & Strahan, P. (2006). Bank’s advantage in hedging liquidity risk: Theory and evidence from the commercial paper market. Journal of Finance, 61, 867–892. Goyal, A., & Welch, I. (2003). Predicting the equity premium with dividend ratios. Management Science, 49, 639–654. Grantier, B. J. (1988). Convexity and bond performance: The benter the better. Financial Analysts Journal, 44, 79–81. Gudikunst, A., & McCarthy, J. (1992). Determinants of bond mutual fund performance. Journal of Fixed Income, 2, 95–101. Gudikunst, A., & McCarthy, J. (1997). High-yield bond mutual funds: Performance, January effects and other surprises. Journal of Fixed Income, 7, 35–46. Hansen, L. P. (1982). Large sample properties of the generalized method of moments estimators. Econometrica, 50, 1029–1054. Hull, J., & White, A. (1990). Pricing interest rate derivative securities. Review of Financial Studies, 3, 573–592. Jensen, M. C. (1968). The performance of mutual funds in the period 1945–1964. Journal of Finance, 23, 389–416. Kahn, R. N. (1991). Bond performance analysis: A multifactor approach. Journal of Portfolio Management, 18, 40–47. Kang, J. (1995). Bond mutual fund performance evaluation: The numeraire portfolio appraoch. Working Paper, University of Rochester. Keim, D. B., & Stambaugh, R. F. (1986). Predicting returns in the bond and stock markets. Journal of Financial Economics, 17, 357–390. Kihn, J. (1996a). The effect of embedded options on the financial performance of convertible bond funds. Financial Analysts Journal, 52, 15–26. Kihn, J. (1996b). The financial performance of low-grade municipal bond funds. Financial Management, 25, 52–73. Litterman, R., & Sheinkman, J. (1988). Common factors affecting bond returns. Working Paper, Goldman Sachs, Financial Strategies Group, New York. Lo, A. W., & MacKinlay, A. C. (1990). Data snooping in tests of financial asset pricing models. Review of Financial Studies, 3, 431–467.

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Longstaff, F., & Schwartz, E. (1992). Interest rate volatility and the term structure: A twofactor general equilibrium model. Journal of Finance, 47, 1259–1282. MacKinlay, A. C. (1995). Multifactor models do not explain deviations from the CAPM. Journal of Financial Economics, 38, 3–28. Merton, R. C. (1973). An intertemporal capital asset pricing model. Econometrica, 41, 867–887. Pastor, L., & Stambaugh, R. (2003). Liquidity risk and expected stock returns. Journal of Political Economy, 111, 642–685. Schadt, R. (1996). Testing international asset pricing models with mutual fund data. Unpublished Ph.D. dissertation, Graduate School of Business, University of Chicago. Sharpe, W. F. (1964). Capital asset prices: A theory of market equilibrium under conditions of risk. Journal of Finance, 19, 425–442. Shyy, G., & Lieu, C. (1994). A note on convexity and bond portfolio performance. Financial Management, 23, 14. Silva, F., & Cortez, M. D. C. (2002). Conditioning information and European bond fund performance. Working paper, Universidade do Minho. Simin, T. (2002). The poor predictive performance of asset pricing models. Working paper, Penn State University. Stambaugh, R. S. (1999). Predictive regressions. Journal of Financial Economics, 54, 315–421. Stanton, R. (1997). A nonparametric model of term structure dynamics and the market price of interest rate risk. Journal of Finance, 52, 1973–2002. Stock, D. (1982). Empirical analysis of municipal bond portfolio structure and performance. Journal of Financial Research, 5, 171–180. Vasicek, O. A. (1977). An equilibrium characterization of the term structure. Journal of Financial Economics, 5, 177–188.

APPENDIX: INVARIANCE OF PERFORMANCE MEASURES TO THE NUMBER OF FUNDS We show that estimating system (3) for a single fund produces the same alphas and standard errors as joint estimation with all funds simultaneously. This may be considered as the GMM extension of the well-known result that a seemingly unrelated regression system with the same variables on the right-hand side of each equation may be estimated equation by equation. From Eqs. (3a, 3b) we form the error terms: u1 ¼ mðfÞtþ1 Rtþ1 1 Z t (A.1) u2t ¼ mðfÞtþ1 Rp;tþ1 RB;tþ1 þ Ap Dt Dt

(A.2)

Note that we allow the two equations to have different instruments; but the Zt in Eq. (A.1) can be set equal to Dt, or vice versa. The sample moment condition is g ¼ (1/T)St (u1t0 , u2t0 )0 . Partition g ¼ (g10 , g20 )0 where g1 ¼ (1/T)Stu1t is an (n1 L1)-vector, where n1 is the number of assets in Rt and


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L1 is the dimension of Zt. Note that only the parameters of the SDF enter g1. Let g2 ¼ (1/T)Stu2t. The vector g2 is of length (n2 L), where n2 is the number of funds in the system and L is the length of the vector Dt. Conformably partition the GMM weighting matrix W, where W11 is the upper left block, etc. The GMM estimator for the system chooses the parameter vector y ¼ (f0 ,vec(Ap)0 )0 to minimize g0 Wg, which implies: (A.3) g0 W @g=@y ¼ 00 where 00 is a dim(f)+n2L row vector of zeros. A partition of (qg/qy) according to g1 and g2 (the rows) and the parameters f and Vec(Ap) (the columns) is of the form: ( ) gd 11 0 @g=@y ¼ (A.4) gd 21 O where gd11 and gd21 are full matrixes and O ¼ In2(1/T)St (DtDt0 ), is an (n2 L)-square, invertible matrix. For any value of f, say f*, if we set X 1 mðf Þtþ1 Rp;tþ1 RB;tþ1 D0t 1=T St Ap Dt D0t Ap ðf Þ ¼ 1=T then g2 ¼ 0 at this value. The Zellner seemingly unrelated regression result holds for the point estimates of alpha, taking the value off* as given, since Ap(f*) is the OLS estimator at this value. Using this result, the first-order conditions (A.3) specialize as follows: P P 0 1 Ap 1=T mðfÞtþ1 Rp;tþ1 RB;tþ1 D0t 1=T ¼0 t A p Dt Dt g01 W 12 O ¼ 0

ðA:5Þ

g01 W 11 gd 11 þ g01 W 12 gd 21 ¼ 0

These conditions show that the optimal GMM estimator for f is not independent of the funds, unless W12 ¼ 0. Thus, a two-step approach that estimates f using (A.1) alone and then plugs this estimate into (A.2) is not the optimal GMM estimator. The asymptotic covariance matrix of the parameter estimates is: AcovðyÞ ¼

0 1 @g=@y W @g=@y

(A.6)

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Partition this expression in conformance with (A.4), letting the matrix to be inverted, V ¼ (qg/qy)0 W(qg/qy), be conformably partitioned. Using (A.4) and the fact that g2 ¼ 0, we have 0 0 V 11 ¼ @g1 =@f W 11 @g1 =@f þ 2 @g2 =@f W 21 @g1 =@f 0 þ @g2 =@f W 22 @g2 =@f 0 V 12 ¼ @g1 =@f W 12 þ @g2 =@f W 22 O ¼ QO V 21 ¼ V 120 V 22 ¼ OW 22 O

ðA:7Þ

The lower right block of (A.6) is the asymptotic variance of Vec(Ap). Using standard expressions for partitioned matrix inversion and (A.7), this may be expressed as 1 Acov Vec Ap ¼ OW 22 O V 21 V 1 11 V 12 1 1 ¼ O1 W 22 Q0 V 1 O ðA:8Þ 11 Q Since (A.8) is block diagonal, this establishes that the asymptotic variance of the alpha for any fund is invariant to the number of funds in the system, for a given f*. By inspecting the upper left block of (A.6), it follows that the asymptotic variance of f is AcovðfÞ ¼

0 0 1 @g1 =@f W 11 @g1 =@f @g1 =@f W 12 W 1 22 W 21 @g1 =@f (A.9)

This expression does depend on the funds, unless W12 ¼ 0. We have shown that, for a given estimate off; the point estimates and asymptotic standard errors of the GMM alphas are the same with one fund in the system as with any number of funds, n2>1. We now argue that the impact of the particular estimatef*on the alphas vanishes asymptotically. From the first equation of (A.5), the value of f*only affects the estimate of Ap through a second moment term. Since under standard assumptions this covariance is consistently estimated with any consistent estimator of f in place of the true value, it follows that the estimator of Ap is consistent and has the same asymptotic distribution using any consistent estimator of f. In practice at our sample sizes, the estimates of f are only very slightly changed by varying the number of funds in the system, and this variation has virtually no detectable impact on the alpha for a given fund.

DETERMINANTS OF THE LONG TERM EXCESS PERFORMANCE OF AMERICAN DEPOSITORY RECEIPTS LISTED ON THE NEW YORK STOCK EXCHANGE Mark Schaub and Bruce L. McManis ABSTRACT We utilize cross-sectional regression analysis to identify key variables affecting the initial three-year holding period returns of foreign equities traded as American Depository Receipts (ADRs) on the New York Stock Exchange (NYSE). Our results suggest that U.S. market index movements and foreign exchange rates are the main determinants of the initial three-year holding period returns for 285 ADRs listed from January 1990 through December 2002. The determinants vary once the sample is broken into subsets comparing ADRs issued before 1998 to those issued afterwards, ADRs issued as IPOs versus SEOs, and Asia Pacific ADRs versus European and Latin American ADRs. We also find that U.S. interest rate movements and type of ADR issue (IPO versus SEO) provide little explanatory power for ADR returns overall.


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MARK SCHAUB AND BRUCE L. MCMANIS

1. INTRODUCTION American Depository Receipts (ADRs), first issued in 1927 by the investment firm of J. P. Morgan, provide investors with the convenience of investing in foreign securities without having to trade on foreign exchanges or trade in foreign currency. They represent the second most popular method for individual investors to diversify globally, the first being via mutual funds. ADR studies have shown mixed results in ADR performance. Callaghan, Kleiman, and Sahu (1999) suggest that ADRs yield significant marketadjusted gains in the long-term investment horizon whereas Schaub (2003) and Foerster and Karolyi (2000) find ADRs tend to underperform comparable firms during the three-year period following the date of issuance. They suggest ADRs underperform in the long-run, much like domestic IPOs (Ritter, 1991). Other studies, including but not limited to Jiang (1998) and Officer and Hoffmeister (1988), imply ADRs provide international diversification benefits. Along those lines, Schaub (2004) finds a timing effect for Asia Pacific ADR performance when the sample is broken down into those issued before mid-1998 and those issued after. Schaub and Highfield (2004) also found timing effects that provide some evidence that ADRs perform better in a U.S. stock market decline than in a bull market. Because prior studies contain only limited attempts to identify the main determinants of ADR returns, we utilize regression analysis to identify variables providing the strongest explanatory power of ADR holding period returns for the first 36 months of trading. In our examination, we identify the effects of the corresponding S&P 500 market performance, the influence of foreign exchange rate fluctuations, the effects of changing U.S. interest rates, the impact of issue type (initial versus seasoned), and location effects on the ADR returns. The sample consists of the ADRs initially listed on the New York Stock Exchange (NYSE) between January 1, 1990 and December 31, 2002. In addition to presenting the model estimates for the total sample, we also present results for the following subsets: (1) ADRs issued before and after 1998 to capture differences in ADR performance for those trading through the bull market and those through the bear; (2) ADR first issues (IPOs) and subsequent issues (SEOs); (3) ADRs issued from the European, Latin American, and Asia Pacific regions; and (4) ADRs issued in specific counties.

Determinants of the Long Term Excess Performance of ADRs

65

2. LITERATURE REVIEW ADRs are traded on the U.S. markets in the same manner as domestic stocks; however, due to international factors, investing in ADRs subjects the investor to additional risks and opportunities. First, the foreign firms tend to have a higher degree of asymmetric information due to limited access to management by U.S. investors. Also, the price movements of ADRs may reflect both the economy of the foreign country and any currency fluctuations (Liang & Mougoue, 1996). Positive characteristics of ADRs traded on the NYSE stem from the issuers being large, well-established firms and the ability of the ADRs to provide international diversification benefits as suggested by Jiang (1998) and Officer and Hoffmeister (1988). 2.1. Studies on ADR Performance Most ADR studies have focused on the excess performance of the securities over a specific length of time following issuance. These tend to be univariate in nature, examining only the influence of one variable, item, or event on ADR returns. Callaghan et al. (1999) found a sample of 66 ADRs issued for companies from 18 different countries from 1986 to 1993 amassed cumulative market-adjusted returns of 19.6% for the first year of trading on the NYSE. Foerster and Karolyi (2000) examined ADR returns for three years from the issue date and found the 333 ADRs issued from 1982 through 1996 underperformed domestic companies. The three-year cumulative excess returns for the ADR portfolio amounted to 27.53% using a U.S. Datastream index and 7.17% relative to a matched sample of U.S. firms. This suggests ADRs underperform in the long-run, much like Ritter (1991) found for IPOs. Schaub (2003) reports 36-month cumulative excess returns for a sample of 179 ADRs initially listed on the NYSE from January 1987 through May 1998. The sample underperformed the S&P 500 index by nearly 20% during that three-year period. Also, ADR SEOs outperformed ADR IPOs relative to the market index and issues from developed markets outperformed those from emerging markets. Schaub (2004) found market-timing differences over the first 36 months of trading as well. In this study, ADRs from the Asia Pacific region performed much better relative to the S&P 500 index for issues that traded through the U.S. correction period as opposed to those with 3 years of trading through the U.S. bull market. No long run market-timing effect existed for European ADRs however.

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Schaub and Highfield (2004) examined both short-term (21 trading days) and long-term (three-year) performance of 242 ADRs representing 36 countries listed on the NYSE between January 1987 and September 2000. They found that both IPO and SEO ADRs underperformed during the U.S. bull market and both outperformed during the U.S. bear market. 2.2. ADR Risks and Return Determinants In a risk examination context, Liang and Mougoue (1996) found ADRs expose U.S. investors to foreign exchange risk based on exchange rate fluctuations. However, their study is limited to 110 ADRs traded on the NYSE or NASDAQ for companies based in the United Kingdom, Japan, and South Africa. Their study period covered trading from January 1976 to December 1990 and did not focus on new listings. They also found that some of the exchange rate risk they identified could be diversified away. Choi and Kim (2000) sought determinants of ADR returns by examining the effects of firm-specific factors, industry factors (local, U.S., and global), and market factors (also local, U.S., and global). Their sample was limited to 156 ADRs from 15 countries trading in the U.S. markets from 1990 through 1996. Their sample was drawn from the NYSE, AMEX, and NASDAQ and did not focus on newly listed ADRs. The authors found each firm’s local equity performance, the firm’s domestic market index, the U.S. market index, the world market index, and the firm’s industry index all significantly impacted the returns of ADRs. However, exchange rate changes were not found to be a significant determinant. Most ADR studies highlighting factors that influence returns have drawn on samples that include issues traded on major exchanges as well as those traded over-the-counter. A potential weak point is the variation in information asymmetry that this presents. AMEX and over-the-counter traded ADRs are likely to be derived from lesser companies that could not meet NYSE listing standards. The amount and quality of the information that reaches U.S. investors is potentially significantly different for these firms and could confound the return generation process.

2.3. Comparison to Related ADR Papers As compared to the previous seminal papers on ADRs by Foerster-Karolyi (2000) and Errunza-Miller (2000), our paper differs in several significant


67

elements. First, we focused only on ADRs traded on the New York Stock Exchange to eliminate the additional asymmetrical information effects associated with private placement (Rule 144A) issues (which was a main contribution of the Foerster-Karolyi paper). Less than 30% of Foerster’s sample and 20% of Miller’s sample consisted of New York Stock Exchange-listed ADRs. Next, both of those papers used local market benchmarks and a matchedpair sample for the U.S. benchmark. We use the S&P 500 because we care more how the ADR portfolio performance compares to the most popular and attainable U.S. portfolio. We focus on investor implications as opposed to issuer implications. As a final comparison, the previous samples were dated 1985–1994 and 1982–1996 respectively. Our sample consists of only NYSE ADRs listed from 1990 to 2002, which includes many more Level III observations than were available in the test periods of the other two papers (285 versus 24 and 99). Our paper also differs in that we place emphasis on regional performance, including country-specific performance. Our analysis not only tests for comovements with the S&P 500 of different regional and country-specific portfolios, but also includes a measure to capture ‘‘market timing’’ effects that distinguish between performance during bull and bear markets; exchange rate influences on returns; and the effects of changes in the prime interest rate in the U.S. on these ADR portfolio values. We report results from 30 different regressions estimated on sample subsets that are both regional and sub-regional. Our reported holding period returns show that there are major return variations at the country level that other studies have appeared to ignore. Finally, our use of the entire sample of NYSE-traded ADRs enables us to have a large sample for analysis while drastically reducing the volatility that Foerster and Karolyi proved exists among lower level issues due to informational asymmetry in different markets.

3. DATA AND METHODS 3.1. The Sample and Return Computations The sample of firms included in the study were obtained from the list of all the non-U.S. equities listed and traded on the New York Stock Exchange (NYSE) as shown on their website. Limiting the sample to NYSE-listed ADRs allows us to examine returns of only large, well-established firms with less informational asymmetry (NYSE trading rules require these ADR issuers to meet the same strict information and reporting requirements as the

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Table 1. Region of Issue

Sample Description by Region, Date, and Typea. Number of Observations

Date of Issue

Type of Issue

Before 1/1/1998

After 1/1/1998

IPO

SEO

European Latin American Asia Pacific Other

138 100 57 9

62 66 26 4

76 34 31 5

66 61 36 4

72 39 21 5

Totals

304

158

146

167

137

a

The total sample contains 304 ADRs listed on the NYSE from January 1, 1990 through December 31, 2002. The sample is divided between IPOs and SEOs based on NYSE reports. The sample is also divided between issues prior to January 1, 1998 and those after January 1, 1998.

typical large U.S. firm). Because the ADR sample consists of the largest foreign companies, the S&P 500 index provides an appropriate market benchmark. Also, the S&P 500 represents the opportunity set most popular and attainable for U.S. investors. During the period January 1, 1990 through December 31, 2002, there were a total of 310 ADRs listed on the New York Stock Exchange. A review of a histogram of the 3-year holding period returns (HPRs) revealed six potential outliers. Each had a HPR more than 50% greater than the next highest ADR. After identifying the specific ADRs it was determined that no two were from the same country and they were spread across all three regions. We believed that these were truly outliers and removed them from the sample. Further sample description is provided in Table 1, which breaks the sample down by date of issue, type of issue (IPO versus SEO), and region of issue (Asia Pacific, Europe, or Latin America). We employ standard event study methodology to compute the 36-month holding period returns for the ADRs used as the dependent variable in the regression analysis. These holding period returns covering the first 36 months of trading for each security were computed as follows: HPR ¼ where

P36 P0 P0

(1)

HPR is the 36-month holding period return, P36 is the ending price for the 36-month period, and P0 is the opening price for secondary trading. Eq. 2 computes the excess holding period returns relative to the market benchmark by subtracting the holding period return of the S&P 500 for the


69

same time period. Excess holding period returns are computed to indicate ADR portfolio performance relative to the most popular U.S. portfolio. ERADR ¼ HPRADR HPRS&P

(2)

where ERADR is the excess return for the 36-month holding period, HPRADR is the 36-month holding period return for the ADR, and HPRS&P is the 36-month holding period return for the S&P 500.

3.2. Cross-Sectional Regression Model We employ cross-sectional regression analysis to determine the impact of U.S. market index movements, foreign exchange rate fluctuations, changing U.S. interest rates, issue type (IPO versus SEO), and regional effects on the 36-month holding period returns of ADRs. The regression model is estimated as follows: HPRi ¼ a þ b1 xSP500 þ b2 xTYPE þ b3 xREGION þ b4 xFOREX þ b5 xPRIME þ

ð3Þ

where HPR is the 36-month holding period return of each ADR; SP500 is the 36-month holding period return of the market index; TYPE is a binary variable set to one for IPOs and zero for SEOs; REGION represents three binary variables, for Europe, Latin America, and Asia Pacific; FOREX is the 3-year percent change in the dollar versus the foreign currency; and PRIME is the 3-year change in the prime interest rate in percent terms.

3.3. Independent Variables and Expectations 3.3.1. The Market Index Choi and Kim (2000) found the U.S. market index significantly impacted ADR returns, particularly when the firms headquartered in developed countries listed the ADRs. Also, their results suggest the ADR returns were positively related to the U.S. market index. Accordingly, we use the

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coinciding 36-month holding period return of the S&P 500 index as the main predictor of ADR holding period returns. We may expect to find a similar relationship as Choi and Kim (2000); however, sample differences may impact the relationship. Our sample, which includes only ADRs listed on the NYSE, is larger because our sample period extends well beyond the 1996 cutoff used in their study, and is broader because more foreign countries are represented. 3.3.2. Type of Issue The NYSE considers ADRs listed for the first time an IPO and those subsequently listed by the same company SEOs. A dichotomous variable is used to distinguish between these two types of ADR issues. Based on Schaub’s (2003) findings that ADR SEOs tend to outperform ADR IPOs, we can expect a negative coefficient for this variable set to one for IPOs and zero for SEOs. 3.3.3. Region of Issue Three dichotomous variables set to one for yes and zero for no are used to identify regional effects on ADR performance. The regions involved are Asia Pacific, Europe, and Latin America. The regression coefficient is expected to vary based on region of issue as some regional equities move more similar to U.S. equities than others. 3.3.4. Change in Exchange Rates Liang and Mougoue (1996) find exchange rate fluctuations significantly affect ADR returns, but these effects can be diversified away. Choi and Kim (2000) find exchange rates were not a significant determinant in ADR performance. Intuitively, an inverse relationship is expected between changes in exchange rates and ADR returns because when the dollar strengthens, the dollar value of assets denominated in foreign currencies decreases and vice versa. 3.3.5. Change in the Prime Interest Rate An independent variable capturing the effects of changing interest rates in the U.S. on ADR excess returns is included. This variable is measured as the net change in percentage points of the prime interest rate for the 3-year holding period of the ADR. Normally, interest rate increases are met with stock sell-offs. For these reasons, a priori, a negative relationship should exist between interest rate changes and ADR returns, assuming U.S. investors do not discriminate in their sell-off.


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4. RESULTS AND IMPLICATIONS 4.1. Excess Return Analysis Table 2 presents excess holding period return characteristics for various subsets of the sample. The total sample encountered an average underperformance of 23.3% relative to the S&P 500 for the initial three years of ADR trading. Underperformance was higher during the stock market boom as seen in the before 1998 sample results. After 1998, the average 3-year excess returns were positive, suggesting the ADRs outperformed the market index during that particular U.S. stock market decline. The IPO sub-sample underperformed on average by more than the SEO sub-sample and Latin American ADRs underperformed by more than European and Asia Pacific ADRs. The sample of ADRs issued after January 1, 1998 reported the smallest variation in results based on sample highs and lows. Also, only the ‘‘after 1998’’ sub-sample had more positive observations than negative; all other sub-samples had more observations below zero than above. Table 3 presents the excess holding period returns for ADR issues on a country-by-country basis in each region. There are tremendous variations in each regional dataset where a few countries outperformed the S&P 500 while most others underperformed. Of the countries represented, four (Italy, Brazil, India and Japan) actually outperformed the S&P 500 index on average during

Table 2. Sample

Excess Three-Year Holding Period Return Characteristics by Samplea. Number of Observations

Before 1998 After 1998 IPO SEO European Latin American Asia Pacific

158 146 167 134 138 100 57

Total

304

a

Positive

Mean

Median

High

Low

(18%) 129 (82%) (51%) 71 (49%) (27%) 122 (73%) (42%) 78 (58%) (39%) 84 (61%) (26%) 74 (74%) (42%) 33 (58%)

56.9% 13.0% 36.6% 7.1% 8.1% 45.9% 14.8%

66.4% 2.8% 45.4% 19.2% 21.0% 63.8% 10.6%

182.2% 214.7% 214.7% 212.7% 182.2% 212.7% 214.7%

181.7% 96.7% 181.7% 178.1% 178.1% 181.7% 168.5%

104 (34%) 200 (66%)

23.3%

30.2%

214.7%

181.7%

29 75 45 56 54 26 24

Negative

The total sample contains 304 ADRs listed on the NYSE from January 1, 1990 through December 31, 2002. The three-year excess holding period returns are computed using Eq. 1 and 2 in the text.

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Table 3.

Average Excess Three-Year Holding Period Returns by Countrya.

Number of Observations

Positive

Negative

Mean

Median

High

Low

Panel A. European France Germany Italy Netherlands Switzerland UK Other Totals

23 16 8 13 12 34 32 138

8 7 4 2 5 12 16 54 (39%)

15 9 4 11 7 22 16 84 (61%)

21.8% 23.5% 19.8% 33.6% 0.6% 4.4% 6.2% 8.1%

27.6% 42.4% 0.7% 21.8% 8.0% 23.8% 3.0% 21.0%

122.1% 74.0% 182.2% 32.0% 59.4% 171.0% 176.3% 182.2%

171.1% 133.7% 161.6% 132.1% 86.4% 178.1% 127.8% 178.1%

Panel B. Latin American Argentina Brazil Chile Mexico Other Totals

11 31 24 27 7 100

1 15 6 4 0 26 (26%)

10 16 18 23 7 74 (74%)

70.2% 4.0% 61.7% 63.3% 107.4% 45.9%

79.5% 6.9% 78.1% 78.0% 115.8% 63.8%

11.0% 212.7% 84.4% 92.6% 57.3% 212.7%

129.2% 143.7% 181.7% 180.7% 142.4% 181.7%

7 14 7 10 5 14 57

2 6 5 6 1 4 24 (42%)

5 8 2 4 4 10 33 (58%)

29.6% 7.6% 43.2% 23.9% 74.2% 50.1% 14.8%

37.9% 6.2% 40.5% 24.1% 96.9% 55.8% 10.6%

92.85% 214.7% 132.9% 183.2% 14.6% 122.1% 214.7%

86.1% 168.5% 22.2% 103.6% 149.5% 167.2% 168.5%

Panel C. Asia Pacific Australia China India Japan Korea Other Totals a

The respective regional samples are based on the ADRs listed on the NYSE from January 1, 1990 through December 31, 2002. The threeyear excess holding period returns are computed using equations 1 and 2 in the text. Results for countries with less than 5 ADRs are reported in the other category.


Region or Country


Table 4. Sample

1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002

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Excess Three-Year Holding Period Return Characteristics by Year of Issuea.

Number of Observations

Positive

Negative

Mean

Median

High

Low

5 9 8 19 31 18 28 40 34 19 40 34 19

2 5 0 5 4 6 2 5 10 10 20 23 12

3 4 8 14 27 12 26 35 24 9 20 11 7

0.5% 25.0% 76.0% 25.0% 76.8% 54.6% 92.2% 51.5% 15.1% 13.8% 6.4% 40.7% 27.0%

9.2% 26.5% 55.8% 32.9% 83.7% 90.7% 97.2% 52.3% 21.0% 18.8% 0.1% 27.2% 18.9%

101.9% 92.6% 42.5% 182.2% 122.1% 105.6% 74.0% 122.1% 212.7% 115.3% 173.8% 214.7% 176.3%

62.5% 86.1% 120.2% 142.4% 181.7% 161.6% 178.1% 155.3% 96.7% 61.7% 81.0% 86.4% 57.7%

a

The total sample contains 304 ADRs listed on the NYSE from January 1, 1990 through December 31, 2002. The three-year excess holding period returns are computed using Eq. 1 and 2 in the text.

the 3-year holding period. One other returned roughly the same as the market index (Switzerland). Only India and Japan had more ADRs outperforming the market index than underperforming. The notable performance variations suggest cross-sectional regressions should enhance our understanding of the main contributors to the ADR returns. In addition, separate regressions were estimated on all country samples with at least 10 observations to further explain subset variations. Table 4 presents the excess three-year holding period returns of ADRs broken down by the year of issue. There were five years when ADR returns exceeded the S&P 500 returns (1991 and 1999–2002). In 1990 the ADRs and market index performed roughly the same; while the remaining seven years’ issues underperformed the market. Notice that during the stock market boom in the US, the ADR excess performance was at its worst. Then, for issues trading through the correction, ADRs performed much better relative to the S&P 500.

4.2. Correlation Analysis Table 5 presents a correlation matrix expressing the relationships among individual variables and sub-samples. The three-year holding period returns

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Table 5. 3 YR HPR ADR 3 YR HPR ADR 3 YR HPR S&P 500 Date of Issue Type of Issue European Latin American Asia Pacific Forex Prime

1 0.171

3 YR HPR S&P 500

Variables and Sample Correlationsa. Date of Issue

0.171 0.098 1 0.841

0.098 0.841 0.017 0.326 0.115* 0.125* 0.103 0.179 0.013 0.064 0.187 0.132* 0.131 0.663

Type of Issue

European

0.017 0.115 0.326 0.125

1 0.406 0.406 1 0.121* 0.130* 0.192 0.085 0.064 0.079 0.134* 0.034 0.641 0.211

0.121 0.130* 1 0.638 0.438 0.250 0.194

Latin American

Asia Pacific

Forex

0.187 0.132

Prime

0.131 0.663

0.103 0.179

0.013 0.064

0.192 0.085 0.638 1 0.336 0.383 0.204

0.064 0.134 0.641 0.079 0.034 0.211 0.438 0.250 0.194 0.336 0.383 0.204 1 0.155 0.007 0.155 1 0.109 0.007 0.109 1

a

This table reports the Pearson correlations among the variables included in the regression analysis and samples. Correlation significant at the 0.05 alpha level (2-tailed). Correlation significant at the 0.01 alpha level (2-tailed).

of the ADRs are significantly correlated with the S&P 500 returns, European region issues, and exchange rate fluctuations. The market returns appear correlated with all samples and variables except Asia Pacific issues. Also, there appears to be high correlations among regional issues. These significant correlations suggest the regression analysis will be fruitful in explaining determinants in ADR returns.

4.3. Regression Analysis on Total and Regional Sub-Samples Table 6 summarizes the regression results for the total sample and regional sub-samples. The total and regional samples are further divided into pre 1998 versus post 1998 issues and IPOs versus SEOs. Multicollinearity problems were identified via examination of variance inflation factors (VIFs). Whenever a variable had a VIF over 10, it was removed from the regression. The regression run on the total sample is highly significant and includes two significant contributors as presented in Panel A of Table 6. The results are intuitive as the holding period returns of the S&P 500 index and the variation in foreign exchange rates were the significant contributors to ADR returns. The positive coefficient of the index variable suggests ADR returns moved with the market, although weakly as indicated by the beta of 0.27. Also, the negative coefficient of the exchange rate variable indicates a stronger dollar translates into lower ADR returns and vice versa. This coefficient is also small (0.22) suggesting a weak effect.

Sample Size

3 YR HPR S&P 500

Panel A. Total Sample and Sub-Samples Before 1/1/1998 158 After 1/1/1998 146 IPO 167 SEO 137 Total 304

0.20 1.23 0.19 0.40 0.27

Panel B. European Sample and Sub-Samples Before 1/1/1998 62 0.59 After 1/1/1998 76 0.75 IPO 66 0.88 SEO 72 0.44 Total 138 0.67 Panel C. Latin American Sample and Sub-Samples Before 1/1/1998 66 0.14 After 1/1/1998 34 0.79 IPO 61 0.08 SEO 39 0.46 Total 100 0.05 Panel D. Asia Pacific Sample and Sub-Samples Before 1/1/1998 26 0.49 After 1/1/1998 31 2.13 IPO 36 0.56 SEO 21 0.51 Total 57 0.40

Cross-Sectional Regression Results by Samplea. Type of Issue

European

2.87 9.84

45.49 43.32

5.96

49.70 40.78

3.00 0.74

2.69

16.75 23.56

24.62

41.22 46.52

9.98

Latin American

Asia Pacific

Forex

Prime

Intercept

F-Value

R2

70.58 35.43 62.31 26.08

15.62 71.84 16.51 53.59 34.02

0.24* 0.25 0.30 0.21 0.22

0.25 1.53 3.90 0.57 1.55

2.84 41.42 29.97 43.09 21.21

6.03 5.12 4.39 3.09 4.23

0.20 0.21 0.12 0.13 0.09

0.27 0.28 0.38 0.22 0.27

0.16 2.38 2.05 5.24 1.63

11.98 8.84 4.32 14.76 4.94

1.09 3.30 5.95 7.14 10.02

0.07 0.16 0.23 0.24 0.23

0.33 0.23 0.29 0.32 0.27

5.14 6.05 4.17 10.48 0.46

12.07 17.28 10.48 7.20 28.93

2.19 0.59 2.04 1.50 2.43

0.13 0.08 0.10 0.11 0.09

0.07 0.75 0.24 0.17 0.22

20.85 6.34 7.72 3.54 5.45

67.54 2.19 43.54 8.36 25.19

1.60 2.89 1.85 0.21 0.94

0.24 0.31 0.15 0.04 0.07


Table 6.

a

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The total sample contains 304 ADRs listed on the NYSE from January 1, 1990 through December 31, 2002. The three-year holding period returns computed using Eq. 1 in the text represent the dependent variable in all regressions. S&P 500 is the three-year holding period return in percent of the index; IPO is set to one for IPOs and zero for SEOs; European, Latin America and Asia Pacific are set to one for the respective regions and zero otherwise; Forex is the three-year change in the foreign exchange rate in percent; and Prime is the three-year change in the prime interest rate in percent. Significant at the 0.10 alpha level, but not the 0.01 level. Significant at an alpha level of 0.01 or lower.

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The results presented in Panel A suggest the returns of the S&P 500 contributed significantly in two sub-samples (after 1/1/1998 issues and SEOs). The strongest market effects occurred in the sample of ADRs issued after 1/1/1998 as indicated by the beta of 1.23. The foreign exchange variable was consistently significant, negative, and small for all sub-samples. Regional effects are also relevant as indicated by the existence of significant large regression coefficients present in all sub-samples. Panel B of Table 6 indicates the regression estimated on the European sample of ADRs was highly significant at the 1% alpha level. Consistent with the total sample results, European ADR holding period returns were significantly impacted by market index performance and foreign exchange rate fluctuations. The sub-samples of European issues indicate similar effects with the market index significant in three of four sub-samples and exchange rate fluctuations also significant in three of four. Similar to the total sample and sub-samples, the type of issue and change in the prime interest rates were not significant determinants in European ADR returns. The total Latin American ADR sample results as shown in Panel C of Table 6 indicate these issues from mostly emerging market companies were not impacted by the returns of the U.S. market index. The regression on the full sample was significant, and registered a significant negative relationship, small in magnitude, with the changes in foreign exchange rates. Also significant in the regression were the intercept term and the type of issue (IPOs on average returned 24.6% less than SEOs). The sub-sample regressions further illustrate the lack of relation to the market index, with no significance reported. In three of the four sub-samples, the changes in foreign exchange rates were significant. The sub-sample results also suggest no effects on ADR performance due to changes in the U.S. interest rates. Finally, the type of issue played no important role in explaining Latin American ADR performance for the samples. Panel D of Table 6 presents the results from the Asia Pacific sample and sub-samples. The regression produced no significant explanations for the entire sample. However, the pre 1998 sub-sample was significantly impacted by the change in the U.S. prime rate with over a 20% decline in ADR performance for each 1% increase in interest rates. Also, the sample of ADRs listed after January 1, 1998 showed strong sensitivity to the U.S. stock market index with a beta of 2.13 significant at the 10% alpha level. Finally, the IPO subset of Asia Pacific ADRs was also significantly affected by the movements of the S&P 500 index, although the relationship was negative. Overall, results imply that the two main factors affecting ADR holding period returns for those listed on the New York Stock Exchange are the


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holding period return of the U.S. market index and the change in foreign exchange rates. These results are intuitive and expected. Sample subsets based on date of issue (before or after January 1, 1998) and type of issue (IPO versus SEO) reveal regional effects as well. Breaking the total sample into regions provided insights also, as only the European sample was significantly affected by the U.S. market index, while both European and Latin American ADR returns were impacted by foreign exchange rates. Asia Pacific regional ADRs stood alone, with none of the four main independent variables significantly affecting ADR returns. Recall also that the median excess returns for Asia Pacific ADRs were the best relative to the U.S. market index. 4.4. Regression Analysis on Country Samples Because of the variation in returns among regions and countries, regressions were estimated on country-specific samples. The results of these regressions are presented in Table 7. Only those countries with at least 10 observations were included. Table 7. Country Sample

Argentina Brazil Chile China France Germany Japan Mexico Netherlands Switzerland UK

Cross-Sectional Regression Results by Countrya.

3 YR HPR S&P 500

Type of Issue

Forex

0.72 0.15 0.48 0.90 0.90 0.89 1.84 0.27 1.34 0.17 0.25

40.97 30.80 19.91

0.02 0.27 2.27 0.37 0.45 0.60 4.00 0.26 0.51 0.39 0.33

9.67 11.13 183.99 4.42 15.50 13.50 8.65

Prime

Intercept

F-Value

R2

3.45 0.97 7.94 9.58 1.19

25.40 44.38 78.08 66.42* 21.36 17.87 60.36 25.83 41.20 10.35 31.93

1.84 0.43 4.14 1.04 2.01 2.59 1.60 1.65 2.65 0.03 2.36

0.55 0.06 0.47 0.24 0.31 0.39 0.56 0.23 0.57 0.02 0.25

39.23 12.44 6.30 2.61 10.54

Variables omitted were due to multicollinearity. See the note to Table 6 for variable explanations. a The respective country regressions were estimated on those with at least 10 ADRs listed on the NYSE from January 1, 1990 through December 31, 2002. The three-year holding period returns computed using Eq. 1 in the text represent the dependent variable in all regressions. Significant at the 0.10 alpha level, but not the 0.01 level. Significant at an alpha level of 0.01 or lower.

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Of the 11 countries analyzed in Table 7, only three regressions were significant. The low number of observations in the country samples required quite large F-values to obtain significance. For that reason, four of the insignificant regressions actually had a significant independent variable (excluding the intercept). Also, the regression estimated on the UK was significant at the 10% alpha level, but had no significant regressor. The variable most commonly significant for the country samples was the return of the U.S. market index. The ADR returns for countries headquartered in France, Germany, and the Netherlands had strong positive relationships with the U.S. market index movements (with betas of 0.90, 0.89, and 1.34 respectively). The Chilean ADR returns were significantly sensitive to exchange rate changes during the three-year holding period, with a coefficient of 2.27. Japanese ADRs were significantly sensitive to whether the issue was an IPO or SEO. Finally, the change in U.S. interest rates significantly affected only Mexican ADR returns with a negative coefficient.

5. SUMMARY The results of this study provide evidence that the initial three-year holding period returns of ADRs listed on the NYSE are mostly affected by movements of the U.S. market index as proxied by the S&P 500 and the change in value of the dollar relative to the foreign currencies. These results are based on regressing the three-year holding period returns of the 285 ADRs issued on the NYSE from January 1990 through December 2002 against variables capturing the effects of the U.S. market index, type of issue (IPO or SEO), region of issue (Latin American, European, or Asia Pacific), three-year change in the dollar’s value against the foreign currency, and the three-year change in the U.S. prime interest rate. Regressions estimated on subsets of the entire sample indicate region of issue also played a part in determining the ADR returns. Although the results are encouraging and intuitive, they are in no way offered as exhaustive. The main weakness of the study stems from a lack of variables that significantly explain variation in regional issues, as the regressions estimated on the Asia Pacific and Latin American samples and sub-samples were mostly insignificant. Perhaps further research will pinpoint other macro- and micro-economic variables with explanatory power of ADR performance.


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REFERENCES Callaghan, J., Kleiman, R., & Sahu, A. (1999). The market-adjusted investment performance of ADR IPOs and SEOs. Global Finance Journal, 10, 123–145. Choi, Y., & Kim, D. (2000). Determinants of American Depository Receipts and their underlying stock returns: Implications for international diversification. International Review of Financial Analysis, 9, 351–368. Errunza, V., & Miller, D. (2000). Market segmentation and the cost of capital in international equity markets. Journal of Financial and Quantitative Analysis, 35(4), 577–600. Foerster, S., & Karolyi, G. (2000). The long-run performance of global equity offerings. Journal of Financial and Quantitative Analysis, 35, 499–528. Jiang, C. (1998). Diversification with American Depository Receipts: The dynamics and the pricing factors. Journal of Business, Finance & Accounting, 25, 683–699. Liang, Y., & Mougoue, M. (1996). The pricing of foreign exchange risk: Evidence from ADRs. International Review of Economics and Finance, 5, 377–385. Officer, D., & Hoffmeister, R. (1988). ADRs: A substitute for the real thing? Journal of Portfolio Management, 13, 61–65. Ritter, J. (1991). The long-run performance of initial public offerings. Journal of Finance, 46, 3– 27. Schaub, M. (2003). Investment performance of American Depository Receipts listed on the New York Stock Exchange: Long and short. Journal of Business and Economic Studies, 9, 1–19. Schaub, M. (2004). Market timing wealth effects of Asia Pacific and European ADRs traded on the NYSE. Applied Financial Economics, 14, 1059–1066. Schaub, M., & Highfield, M. (2004). Short-term and long-term performance of IPOs and SEOs traded as American Depository Receipts: Does timing matter? Journal of Asset Management, 5(4), 263–271.

KERNEL BANDWIDTH APPLICATIONS TO THE EURO AND THE U.S. MUTUAL FUND MOVEMENTS Timothy J. Brailsford, Jack H. W. Penm and Richard D. Terrell ABSTRACT This paper applies the variable forgetting factor and the fixed forgetting factor to financial time-series analysis, and establishes the linkage for the first time between the variable forgetting factor approach and kernel smoothing. We then demonstrate the use of the proposed variable forgetting factor approach to undertake forecasting of the Euro’s exchange rates and the CRSP monthly net asset values (NAV). For both applications, the findings show that the kernel bandwidth so determined can improve the forecasting performance.

1. INTRODUCTION In recent years the application of kernel smoothing methods in non-parametric regression framework to financial time-series analysis has become widespread.

Research in Finance, Volume 23, 81–97 Copyright r 2007 by Elsevier Ltd. All rights of reproduction in any form reserved ISSN: 0196-3821/doi:10.1016/S0196-3821(06)23003-X

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TIMOTHY J. BRAILSFORD ET AL.

For instance, Renault and Scaillet (2004) estimate the recovery rate density nonparametrically using a beta kernel approach. Rosenberg and Engle (2002) estimate an empirical pricing kernel using S&P 500 index option data. Guo and Wu (1998) adopt a non-parametric approach with the standard normal kernel to examine the exchange-rate exposure of Taiwanese firms. Kernel smoothing methods have not been applied, however, to a wide range of problems arising in financial time-series simulations and forecasting. Most algorithms developed in these areas are commonly used with time-series modelling, in particular subset autoregressive (AR) modelling. Subset AR models (see Yu & Lin, 1991), including full-order models as a special case, are often desirable. This is especially so when measurements exhibit some form of periodic behaviour with a range of different natural periods, such as data measured monthly, weekly, and daily, from periodic digital signals. Most important, if the underlying true AR process has a subset structure, the suboptimal model specification (for instance, a full-order structure) can give rise to inefficient estimates and inferior projections (see Holmes & Hutton, 1989). Empirical research has shown that it is impractical to ignore the possibility of zero coefficients in AR models, particularly in the presence of periodic behaviour, and the estimation and forecasting results could be very different if the presence of zero coefficients is allowed. Recent experience (Penm, Brailsford, & Terrell, 2000) has shown that using the forgetting factor has the potential to improve forecasting performance. Specifically, the forgetting factor has been widely used in linear models. Such models, which work well in explaining the behaviour of a process over a specific sample, may have to be adapted to capture slow evolution over time due to economic, political or structural changes. Consequently, the forecasts obtained by allocating greater weight to more recent observations and ‘forgetting’ some of the past, are likely to outperform alternatives in which such an allocation is not adopted. It is desirable to incorporate this approach into kernel smoothing methods for improving the performance of this approach through the framework of kernel smoothing. Forgetting factors can be both fixed and variable. Gijbels, Pope, and Wand (1999) propose an understanding of fixed forgetting factors via kernel smoothing. However the variable forgetting factor approach is not mentioned. Our paper establishes the linkage for the first time between the variable forgetting factor approach and kernel smoothing. Also the selection approach of the variable forgetting factor proposed by Cho, Kim, and Powers (1991) is used to choose the kernel bandwidth for data smoothing, and then to conduct model building. To demonstrate the effectiveness of the proposed new approach, two illustrations are provided. The first investigates the

Kernel Bandwidth Applications to the Euro and the U.S. Mutual Fund

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prediction of the Euro’s exchange rate with the US Dollar. The second examines the prediction of the average aggregate net asset value (NAV). This average aggregate series is computed from the monthly mutual fund data, which come from the ‘‘CRSP Survivor-bias free US Mutual Fund Database’’. The introduction of the Euro has been a significant recent event in global financial markets. The Euro is intended to create broader, deeper, and more liquid financial markets in Europe, and thus its main purpose is to improve the price stability and productivity of European countries. Rather than experiencing constant fluctuations in the member exchange rates, there has emerged a more consistent and predictable environment for international trade. Another reason why the European Central Bank introduced the Euro is based on its belief that the new currency will foster low inflation. The Euro has already established itself as a credible and important currency in the world. To date the Euro/Dollar trading has been very active in the world’s foreign exchange markets through a wide range of instruments, offering significant hedging possibilities. Over the period January 1999 to September 2002 the relative weakness of the Euro was a significant feature in international foreign exchange markets. During this period the value of the Euro relative to the US Dollar, in general, fell below the original par value. The Euro’s weakness throughout this period confounded earlier general expectations that it would trend upwards relative to the US Dollar (see ECB, 2001a), and possibly reach a value higher than the initial rate existing at 1 January 1999, which was 1Euro: 1.16675 US Dollar. The ‘‘CRSP Survivor-bias free US Mutual Fund Database’’ contains open-end mutual fund data from 1961. The funds cover all investment instruments including equity funds, taxable, and municipal bond funds, international funds, and money market funds. The price data are recorded as monthly NAV, calculated as total net assets (at market value). Further information on the database is provided in Carhart (1997). The monthly mutual fund data come from this database over the period January 1998 through June 2004. In order to focus on analysing complete fund data, we omit incomplete funds, which contain missing or invalid data during the test period. Any bad funds, including those with a record, which indicates no change in price, are also omitted. This pre-filtering process has identified 7454 satisfactory and complete funds in the CRSP database for further examination. An average of the total NAV approach is then adopted to examine the performance of forecasting. The major area of interest in both illustrations is whether kernel estimation, using Cho’s approach for kernel bandwidth selection, can improve the

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forecasting performance of both the Euro’s exchange rate and the average NAV within the framework of subset AR modelling. The forecasting performance is compared with the performance of AR modelling without the use of the forgetting factor. If improved forecasting performance is achieved, this can increase the potential use of kernel smoothing methods in time-series forecasting. The remainder of this paper is structured as follows. Section 2 reviews the use of the forgetting factor in financial time-series modelling. Section 3 provides a description of the forgetting factor via kernel regression. Section 4 illustrates the proposed kernel bandwidth application, associated with the variable forgetting factor, for the predictions of the Euro’s exchange rate and the average NAV, and Section 5 provides some concluding remarks.

2. THE USE OF THE FORGETTING FACTOR IN FINANCIAL TIME-SERIES MODELLING The use of the forgetting factor in financial time-series modelling has attracted attention in recent years. The forgetting factor method assesses each incoming observation and applies appropriate weights to update the model structure and parameters. Brailsford, Penm, and Terrell (2002) report the use of the forgetting factor in modelling and simulation of financial time-series, while Guo and Wu (1998) use the kernel regression to examine the exchange rate exposure of Taiwanese firms. The effect of their kernel is equivalent to the effect of a forgetting factor. Azimi-Sadjadi, Sheedvash, and Trujillo (1993) suggest the recursive updating procedure for the training process of a multi-layer neural network involving a forgetting factor, and Goto, Nakamura, and Uosaki (1995) use the forgetting factor in the recursive least squares ladder algorithm for spectral estimation of a non-stationary process. This section utilises AR modelling to illustrate the use of the forgetting factor in financial time-series modelling. Let Y ¼ ½ yð1Þ; yð2Þ; :::; yðT 1Þ; yðTÞ0 be a time-series observed at equally spaced time points x1 ; x2 ; . . . xT1 ; xT : An AR (p) model of the following form results: yðxt Þ þ

p X

ai yðxt xti Þ þ b ¼ ðxt Þ

(1)

i¼1

where b is an intercept term, and e(x P t) is a zero mean Gaussian white noise disturbance term with a variance .


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The coefficients in (1) are obtained by minimising: p T x x X X t T K ai yðxt xti Þ þ b2 ½yðxt Þ þ h t¼1 i¼1

(2)

For the case of the variable forgetting factor, lj, 1ZljZ0, the forgetting profile, Kðxt xT=hÞ; is defined as: K

T x x Y t T lj ; ¼ h j¼t

t ¼ T; T 1; :::; 1

(3)

where lT ¼ 1: For the case of the fixed forgetting factor, l, 1ZlZ0, the forgetting profile, Kðxt xT=hÞ is defined as: x x t T K (4) ¼ lTt ; t ¼ T; T 1; :::; 1 h In general, the equation of (2) can be re-written as: " # p p P P yðxT Þ þ ai yðxT xTi Þ þ b yðxT1 Þ þ ai yðxT1 xT1i Þ þ b i¼1

i¼1

2 xT xT K h 6 6 6 0 6 4 2

0

0 K

x

xT h

T1

0 p P

0

3

7 7 0 7 7 5 .. . 3

yðxT Þ þ ai yðxT xTi Þ þ b 7 6 i¼1 7 6 7 6 p P 7 6 6 yðxT1 Þ þ ai yðxT1 xT1i Þ þ b 7 7 6 i¼1 5 4 .. . (5) which is a typical weighted least squares problem in kernel regression. Following Hannan and Deistler (1988), the time-update recursions for fitting of a full-order AR(p) model of (1) can be described as follows. Let yp,T denote the vector of coefficients estimated using data up to y(xT). We have the following relationship: ^ yp;T ¼ ½a^ 1 ; a^ 2 ; :::; a^ p ; b

(6)

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The time-update recursions for yp;T is shown by Carayannis, Manolakis, and Kalouptsidis (1986) as: y0p;T ¼ y0p;T1 gp;T ep;T

(7)

gp;T ¼ Pp;T1 Z p;T ½lT1 þ Z 0p;T Pp;T1 Z p;T 1 is the Kalman gain vector, and Pp;T ¼ 1=lT1 ½Pp;T1 gp;T Z 0p;T Pp;T1 is the inverse information matrix, where Z0p;i ¼ ½yðxi Þ; yðxi1 Þ; :::yðxip Þ; 1 and ep;T ¼ yðxT Þ þ yp;T1 Zp;T1 is the prediction error. It is interesting to note that modelling researchers often use the assumption that if a coefficient in the AR is nonzero, then all the lower-order ones will be nonzero too. For example in the AR model when p ¼ 8 for every entry ak where k ¼ 1, 2, 8 is assumed to be non-zero. That is, they neglect the AR (p) models with possible zero entries ak. However, there are 28 ¼ 256 possible models in this example. More importantly, applications of AR models to economic and financial time-series data have revealed that zero entries are indeed possible. In such cases the use of a full-order AR can produce inefficient estimation and inferior projections. Subset AR models are AR models with intermediate lag coefficients constrained to zero, and include full-order AR models. The subset AR with the deleted lags i1, i2, y, is of (1) has the representation: yðxt Þ þ

p X

ai ðI s Þyðxt xti Þ þ b ¼ ðxt Þ

(8)

i¼1

where Is represent an integer set with elements i1, i2, y, is, and ai ðI s Þ ¼ 0; as i 2 I s: In fitting the subset AR model of (8), the time-update recursions from T1 to T now have the form: y0p;T ðI s Þ ¼ y0p;T1 ðI s Þ gp;T ðI s Þep;T ðI s Þ where gp;T ðI s Þ ¼ Pp;T1 ðI s ÞZp;T ðI s Þ½lT1 þ Z0p;T ðI s ÞPp;T1 ðI s ÞZp;T ðI s Þ1 Pp;T ðI s Þ ¼

1 ½PP;T1 ðI s Þ gp;T ðI s ÞZ 0p;T ðI s ÞPP;T1 ðI s Þ lT1

and ep;T ðI s Þ ¼ yðxT Þ þ yp;T1 ðI s ÞZ p;T1 ðI s Þ

(9)


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In (9), Z p;T ðI s Þ; gp;T ðI s Þ; and y0p;T ðI s Þ are formed by removing the ði1 ; :::; is Þ2th rows of Z p;T ; gp;T ; and y0p;T : Pp;T ðI s Þ is formed by removing the (i1,yis)–th rows and the (i1,yis)–th columns of Pp;T : Furthermore an order selection criterion, as suggested by Hannan and Deistler (1988), could be modified at each time instant to select the optimal subset AR model. From now on we will use MHQC as an abbreviation for the modified criterion, which is defined by X d MHQC ¼ log þ ½2 log log f ðTÞ=f ðTÞN P where f(T) is the effective sample size, and is denoted by Tt¼1 Kðxt PxT =hÞ: Also, N is the number of functionally independent parameters, and c is the estimated residual variance. The optimal model selected is the one with the minimum value of MHQC. In the next section we establish the linkage between the forgetting factor approach and kernel smoothing

3. UNDERSTANDING THE FORGETTING FACTOR VIA KERNEL REGRESSION This section provides a method of describing the forgetting factor via kernel regression, which is the focus of the paper. The forgetting factor method uses a sample of data and estimates the value of the forgetting factor from the sample. This method will tend to fit the data better than a parametric approach, which uses some assumed parameters. Since the forgetting factor method is equivalent to kernel estimation – which is a non-parametric method – it is likely to give more accurate estimates and better forecasting performance in financial time-series than using an inappropriate parametric one.1 If a parametric form for estimation is adopted, the cost arises from possible mis-specification of the parametric form.

3.1. The Variable Forgetting Factor Case In the variable forgetting factor case as proposed in Cho et al. (1991), the forgetting profile KðxTi xT =hÞ is now expressed as KðxTi xT =hTi Þ: If the bandwidth h and the forgetting profile are defined as: xT xTi hTi ¼ (10) loge lTi

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xTi xT K ¼ expð0Þ ¼ 1 hTi xTi xT xTi xT xTiþ1 xT ¼ exp K K hTi hTi hTiþ1

as i ¼ 0 as T4i40

Then the following identities arise: xT xT as i ¼ 0 K ¼ Kð0Þ ¼ expð0Þ ¼ lT ¼ 1; hT xT1 xT K ¼ Kðloge lT1 Þ ¼ expðloge lT1 ÞKð0Þ ¼ lT1 hT1 xT2 xT K ¼ Kðloge lT2 Þ ¼ expðloge lT2 ÞKðloge lT1 Þ hT2

as i ¼ 2

¼ lT2 lT1 Consequently (5) becomes: p T Y X X T l ½yðx Þ þ b þ ai yðxt xti Þ2 j t j¼t t¼1

as i ¼ 1

(11)

i¼1

Since (5) is re-written from (2), and (10) establishes the linkage between the variable forgetting factor and the kernel bandwidth, the variable forgetting factor method for coefficient estimation in (1) is equivalent to kernel estimation. 3.2. The Fixed Forgetting Factor Case In the fixed forgetting factor case as proposed in Penm et al (2000), if the bandwidth h and the forgetting profile are defined as follows: x T x1 h¼ (12) ðT 1Þloge l x x Ti xT Ti xT K ¼ exp h h Then the following relations emerge: x x T T K ¼ Kð0Þ ¼ expð0Þ ¼ 1; as i ¼ 0 h x T1 xT ¼ Kðloge lÞ ¼ expðloge lÞ ¼ l; as i ¼ 1 K x h x T2 T K ¼ Kðloge l2 Þ ¼ l2 ; as i ¼ 2 h


89

As a result (5) becomes: T X t¼1

lTt ½yðxt Þ þ b þ

p X

ai yðxt xti Þ2

(13)

i¼1

This outcome provides the linkage between the fixed forgetting factor and the bandwidth of the kernel equivalent, Thus the fixed forgetting factor approach2 for coefficient estimation in (1) is also equivalent to kernel estimation. As shown above, it has been demonstrated that the use of the forgetting factor, both fixed and variable, for coefficient estimation in (1) is equivalent to kernel estimation.

4. PREDICTION OF THE EURO’S EXCHANGE RATE This section provides two illustrations which examine whether the kernel bandwidth selection, using Cho’s approach for the choice of variable forgetting factors, can improve both the forecasting performance of Euro and the average NAV within the framework of subset AR modelling, which includes full-order AR models. 3 Cho et al. (1991) proposes the following formula for choosing the variable forgetting factor: l¼1

1 Nt

(14)

P P where N t ¼ e N max =Qt and where e is the expected noise variance based on real knowledge of the process, the maximum memory N max ¼ PM1length 1=1 lmaxPthe extended prediction variance Qt ¼ 1=M i¼0 e2ti and Qt will approach e for a stationary process. Also the value of M should be smaller than the minimum memory length N min ¼ 1=1 lmin so that the nonstationarity of the series will not be obscured (see Cho et al. 1991). As indicated in Section 1, in the period January 1999 to September 2002 the relative weakness of the Euro was a significant feature in international foreign exchange markets, despite earlier expectations that it would trend upwards relative to the US Dollar. Therefore the prediction of the Euro’s exchange rate is used as the first illustration of the proposed kernel estimation in time-series forecasting. The Euro exchange rate series we use covers monthly sampling over the period January 1997 to August 2002,4 a total of 68 observations (Fig. 1).

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TIMOTHY J. BRAILSFORD ET AL. The value of the Euro relative to the US Dollar 1.2 1.1 1.0 0.9 0.8 1997M1

1999M7 1998M4

2002M1

2002M8

2000M10

Fig. 1. The Euro Exchange Rate Series. The Euro Exchange Rate Series Used Covers Monthly Sampling Over the Period January 1997 to August 2002, a total of 68 Observations. ∆log(Euro Exchange Rate) 0.04 0.02 0.00 -0.02 -0.04 1997M1

1998M4

1999M7

2000M1

2002M1

2002M8

Fig. 2. The First Differenced Euro Exchange Rate Series. The First Differenced Euro Exchange Rate Series is in Logarithms. This Differenced Series Exhibits Varying Periodic Behaviour.

The first differenced series is in logarithms (Fig. 2). Clearly, the differenced series exhibits varying periodic behaviour. Therefore it is best estimated by subset time-series models, including full-order models, selected sequentially (see Brailsford et al., 2002). To assess out-of-sample forecasting performance, we compute the root mean squared error (RMSE) for the Euro exchange rate series. We undertake one- to five-period-ahead forecasts outside the observed first differenced series respectively, generated by both AR models with the forgetting factor and AR models without the forgetting factor. The forecasts for the first differenced series are converted to forecasts for the observed level series. These forecasts for the level series are then used to calculate RMSEs. Using the RMSEs produced by the AR models without the forgetting factor as the baseline, we calculate the percentage of RMSE improvement (or deterioration) for each period-ahead forecast. The average percentages computed for


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Table 1. Percentage Improvements of RMSE Based on Out-Of-Sample Forecasts of the Subset AR Forgetting Factor Modelling, Which Include Full-Order Models, for the Period 2002(1)–2002(8). RMSE Improvements Period-Ahead Forecast One month Two months Three months Four months Five months

Percentage Improvement (%) 58.8 20.2 5.3 1.9 0.2

The RMSEs computed from out-of-sample forecasts of the full-order AR modelling without the forgetting factor are used as a baseline. The improvements are expressed as percentages of the RMSE computed from out-of-sample forecasts, using the subset AR forgetting factor modelling, which has lags 1, 2, and 4 on every occasion. One- to five-period ahead forecasts have been undertaken. An alternative approach would be to use forecasts of the full-order AR modelling without the forgetting factor. Comparing the AR forgetting factor models with the full-order AR models without the use of the forgetting factor, the former performs better than the latter for one- to five-period ahead forecasts. This superiority is partly attributable to the inclusion of the forgetting factor into the AR forgetting factor modelling.

the Euro exchange rate series, from one- to five-period-ahead forecasts covering the period from January 2002 to August 2002, are presented in Table 1. We first present the results of the estimation procedure proposed in Section 2 for the variable forgetting factor. To undertake this procedure an initial subset AR model with a forgetting factor is required. To determine this initial model, the Penm and Terrell (1984) procedure is utilised with a given fixed forgetting factor incorporated in each of the AR models. The procedure is applied to the first differenced series over the period January 1997 to December 2001. To cope with this small sample environment, a maximum order of 12, i.e,. P ¼ 12, is selected. We start this approach with l ¼ 1:0; and then repeat this process by considering values of l ranging from 0.750 to 0.999 in increments of 0.001.5 The results indicate that an optimal initial AR model with l ¼ 0:999; and with lags 1, 2, and 4, is selected by the MHGC. Given these results we then proceed with the recursive estimation as proposed in Section 2 with the new differenced data. The selection of the P variable forgetting factor depends on the values of, lmin, M, and e : As investigated in Cho et al. (1991), to prevent lðtÞ from becoming negative, lmin is set at 0.75. If the value of an updated l falls P below lmin, the value of lmin will be set to the updated l. The quantity e is set at 2.8882 102,

92


which is approximated by averaging the squared residuals of the initial model (see Toplis & Pasupathy, 1988). lmax is set at 0.999. As reported in Cho et al. (1991), the value of M is small enough not to obscure the non-stationarity of the signal. However if this value were too small, the effect of a spurious large additive prediction error would become significant. This adverse effect leads to a wild fluctuation of the variable forgetting factor to be used in the next recursion of parameter estimation. Also the larger that M is, the higher is the likelihood of over-averaging conducted by qt. Subsequently the non-stationarity of the signal is obscured. Therefore the value of M is set at 6 to achieve a smooth updating of the variable forgetting factor and to prevent a large spurious noise error from creating the calculation of a misleading variable forgetting factor (see Brailsford et al., 2002). The value of the updated l is calculated at 0.998. We note that this value is marginally different from that determined in the initial model. This variable forgetting factor is then incorporated into the proposed time-recursive algorithm to select the updated subset AR model. The specification of this updated model remains as (1, 2, 4). To examine the effects on forecasting, the forecasts for the differenced series are converted to the forecasts for the level series, and we then calculate the RMSE for five forward forecasts for the level series. Compared with the subset AR model, which does not include a forgetting factor, improvement in forecasting performance is found. For five-period-ahead forecasts in this exercise, five monthly forecasts are first produced for the period from January 2002 to May 2002, using data from January 1997 to December 2001. Root mean squared errors for each AR model with the forgetting factor, and for each AR model without the forgetting factor, over the forecasting periods, are calculated for the Euro exchange rate. The forecast period is rolled forward by one month producing a second set of five monthly forecasts, covering the period from February 2002 to June 2002. The process is then repeated, and so on. The last set of forecasts covers the period from April 2002 to August 2002. The rolling average of the RMSEs for each AR forgetting factor modelling, and for each AR without the forgetting factor modelling, is computed. The former is then subtracted from the latter to obtain a difference. Then we divide this difference by the latter to calculate the percentage of RMSE improvement (or deterioration), which is presented in Table 1. For the remaining one- to four-period-ahead forecasts, the first set of forecasts covers the beginning period of January 2002, the second set covers the beginning period of February 2002, and so on.


93

The proposed time-recursive algorithm to select the updated subset AR model is also undertaken for observations y(1), y(2), y, y(T), T ¼ 61,y,68. The values of the updated variable forgetting factors and the updated subset AR models are presented in Table 2. Interestingly, the selected subset AR forgetting factor modelling on every occasion has lags 1, 2, and 4. The results illustrate a stable monthly lag pattern. This indicates that the value of the forgetting factor is the main contributor to forecasting improvement. It is also observed that high values of l have been updated. Comparing the AR forgetting factor models with the full-order AR models, the former performs better than the latter for one- to five-period- ahead forecasts, as shown in Table 3. This superiority is partly attributable to the inclusion of the forgetting factor into the AR forgetting factor modelling. That is, the AR forgetting factor modelling possesses a higher degree of flexibility in terms of the ‘forgetting’ process, which should lead to enhanced modelling, and hence improved forecasts. However the major improvement of the AR forgetting factor modelling over the full-order AR modelling, judged in terms of the average of percentage RMSEs, is in undertaking oneto two-period-ahead forecasts. The performance improvement then diminishes as forecast periods increase. The difference in forecasting performance

Table 2.

Outcomes of the Time-Update Recursions for the Euro for the Period 2002(1)–2002(8).

Sample Size (T)

61 62 63 64 65 66 67 68

Time Lags of the Selected Subset AR Forgetting Factor Model (1 (1 (1 (1 (1 (1 (1 (1

2 2 2 2 2 2 2 2

4) 4) 4) 4) 4) 4) 4) 4)

Value of Forgetting Factor Updated

0.998 0.999 0.996 0.999 0.994 0.993 0.991 0.998

The proposed time-recursive algorithm to select the updated subset AR model is applied for observations y(1), y(2), y, y(T), T ¼ 61,y,68. The values of the updated variable forgetting factors, and the updated subset AR models, are presented. The selected subset AR forgetting factor modelling on every occasion has lags 1, 2, and 4. The results illustrate a stable monthly lag pattern. This indicates that the value of the forgetting factor is the main contributor to forecasting improvement shown in Table 1.

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Table 3. Percentage Improvements of RMSE Based on Out-Of-Sample Forecasts of the AR Subset Forgetting Factor Modelling Which Include Full-Order Models, for the Period 1998(1)–2004(6). RMSE Improvements Period-Ahead Forecast One month Two months Three months

Percentage Improvement (%) 63.5 38.1 10.2

The RMSEs computed from out-of-sample forecasts of the full-order AR modelling without the forgetting factor are used as a baseline. The improvements are expressed as percentages of the RMSE computed from out-of-sample forecasts, using the AR forgetting factor modelling, which has lag 1 on each occasion. One- to three-period ahead forecasts have been undertaken. An alternative approach would be to use forecasts of the full-order AR modelling without the forgetting factor. Comparing the AR forgetting factor models with the full-order AR models without the use of the forgetting factor, the former performs better than the latter for one- to three-period ahead forecasts. This superiority is partly attributable to the inclusion of the forgetting factor into the AR forgetting factor modelling.

is quite insignificant for five-period-ahead forecasts. This is because the timeupdate recursions, including the forgetting factor, are applied to individual incoming observations, but not developed to operate on a block of incoming observations. Nevertheless, these results indicate that there are gains in undertaking financial time-series forecasting in the framework of AR forgetting factor modelling, in particular in undertaking short-period ahead forecasting. The second illustration examines the average NAV series as described in Section 1. To demonstrate the usefulness of the proposed AR forgetting factor approach, we investigate this NAV series covering the period from January 1998 to June 2004. Following the approach utilised in the first illustration, we undertake one- to three-period-ahead forecasts outside the observed first differenced series in logarithms respectively, generated by both the AR forgetting factor models and the AR models without the forgetting factor. On all occasions an AR forgetting factor model, which has a lag 1, has been consistently selected. The forecasts for the first differenced series are converted to forecasts for the observed level series to calculate RMSEs. For three-period-ahead forecasts in this exercise, three monthly forecasts are first produced for the period from February 2004 to April 2004, using data from January 1998 to January 2004. The last set of forecasts covers the period from April 2004 to June 2004. The percentage


95

improvements of the rolling average RMSEs computed for the NAV series, from one- to three-period-ahead forecasts, are presented in Table 3. Compared with the AR model that does not include a forgetting factor, improvement in forecasting performance is also found. The outcome confirms that the value of the forgetting factor is the main contributor to forecasting improvement.

5. SUMMARY In this paper a linkage is established for the first time between the variable forgetting factor approach and kernel smoothing. The linkage provides a new insight to understanding the characteristics of the forgetting factor method. To demonstrate the effectiveness of this method, the forecasting performances of the Euro’s exchange rate and of the average NAV fund data are investigated. The selection approach of the variable forgetting factor proposed by Cho et al. (1991) is used to choose the kernel bandwidth for data smoothing, and a stable monthly lag pattern for the AR forgetting factor modelling is identified in both illustrations. The findings also show that the kernel bandwidth so determined can improve the forecasting performance.

NOTES 1. The purpose of introducing the forgetting factor is to provide an appropriate data weighting process. This process does not give equal weight to each observation, but rather gives more weight to recent observations and less weight to earlier data. Thus a more appropriate parameter estimation approach can be undertaken with reweighted data. After the forgetting factor approach is applied to data, the parameter estimation with the given forgetting factor becomes a parametric OLS estimation, and the properties of the OLS estimation will apply. This approach therefore deals with a widely used form of parameter estimation in a more appropriate context, i.e., with re-weighted data. 2. For the fixed forgetting factor method, an approach for choosing lambda is to consider lambda as a function of the coefficients of the autoregressive polynomial. However no mathematical proof has been developed to underpin this approach; although Porat (1985,) asserts that the coefficients of the autoregressive polynomial are non-linear functions of the data, thus a fixed forgetting factor is a function of the data, and therefore a function of the autoregressive coefficients. 3. For a given fixed or variable l at a time point, the parameter estimation becomes a parametric OLS estimation. Further, re-sampling methods such as bootstrap and Markov Chain Monte Carlo (MCMC) methods can be used to enhance coefficient

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estimation in this context. Further, when l is a function of time, there is no unique methodology. This is because we need to know what function has been proposed for l. We, however, choose a distribution free approach to describe l, and thus there is no need to specify a function to describe l. The proposed approach is robust to the distribution assumption, because this approach does not depend on a particular distribution assumption in the estimation. It selects models based on goodness of fit with penalties for over-parameterisation. Therefore it does not fit into a maximum likelihood framework, or a conditional maximum likelihood framework. 4. We use data from January 1997 in order to achieve a workable sample size of 68. A sample size less than 50 is considered to be insufficient. The Euro was at a test stage prior to 1 January 1999. In this sense the data pre-1999 are not true marketdetermined rates but rather indicative figures 5. In selecting the value of the fixed forgetting factor, as mentioned above, a grid search is utilised to determine the value of the fixed forgetting factor. The results were obtained on a SUN 7800 running Unix, and the range of the grid search covers all possible candidate values of l within the numerical accuracy of the SUN computer.

REFERENCES Azimi-Sadjadi, M. R., Sheedvash, S., & Trujillo, F. (1993). Recursive dynamic node creation in multilayer neural network. IEEE Transactions on Neural Networks, 4(2), 242–256. Brailsford, T. J., Penm, J. H. W., & Terrell, R. D. (2002). Selecting the forgetting factor in subset autoregressive modelling. Journal of Time-series Analysis, 23(6), 629–650. Carayannis, C., Manolakis, C. D., & Kalouptsidis, N. (1986). A unified view of parametric processing algorithms for prewindowed signals. Signal Processing, 10, 335–368. Carhart, M. M. (1997). On persistence in mutual fund performance. Journal of Finance, 52(March), 57–82. Cho, Y. S., Kim, S. B., & Powers, E. J. (1991). Time-varying spectral estimation using AR models with variable forgetting factors. IEEE Transactions on Signal Processing, 39, 1422–1426. European Central Bank (ECB) (2001a). Monthly Bulletin, February. Gijbels, I., Pope, A., & Wand, M. P. (1999). Understanding exponential smoothing via kernel regression. Journal of the Royal Statistical Society Series, B61, 39–50. Goto, S., Nakamura, M., & Uosaki, K. (1995). On-line spectral estimation of nonstationary time-series based on AR model parameter estimation and order selection with a forgetting factor. IEEE Transactions on Signal Processing, 43, 1519–1522. Guo, J.-T., & Wu, R.-C. (1998). Financial liberalization and the exchange-rate exposure of the Taiwanese firms: A nonparametric analysis. Multinational Finance Journal, 2(1), 37–61. Hannan, E. J., & Deistler, M. (1988). The statistical theory of linear systems. New York: Wiley. Holmes, J. M., & Hutton, P. A. (1989). ‘Optimal’ model selection when the true relationship is weak and occurs with a delay. Economics Letters, 30, 333–339. Penm, J. H. W., & Terrell, R. D. (1984). Multivariate subset autoregressive modelling with zero constraints for detecting causality. Journal of Econometrics, 3, 311–330. Penm, J. H. W., Brailsford, T. J., & Terrell, R. D. (2000). A robust algorithm in sequentially selecting subset time-series systems using neural networks. Journal of Time-series Analysis, 21, 389–412.


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Porat, B. (1985). Second-order equivalence of rectangular and exponential windows in leastsquares estimation of Gaussian autoregressive processes. IEEE Transactions on Acoustics, Speech, and Signal Processing, 33(4), 1209–1212. Renault, O., & Scaillet, O. (2004). On the way to recovery: A nonparametric bias free estimation of recovery rate densities. Journal of Banking & Finance, 28(12), 2915–2931. Rosenberg, J. V., & Engle, R. F. (2002). Empirical pricing kernels. Journal of Financial Economics, 64(3), 341–372. Toplis, B., & Pasupathy, S. (1988). Tracking improvements in fast RLS algorithms using a variable forgetting factor. IEEE Transactions on Acoustics, Speech, and Signal Processing, 36(2), 206–227. Yu, G.-H., & Lin, Y.-C. (1991). A methodology for selecting subset autoregressive time-series models. Journal of Time-series Analysis, 12, 363–373.

FRAGMENTATION OF DAY VERSUS NIGHT MARKETS Nivine Richie and Jeff Madura ABSTRACT Stock markets during the day are relatively centralized, while night markets, due to the dominance of electronic trading venues, are fragmented. Though electronic markets at night allow more competition for order flow, they may result in decreased order interaction and decreased transparency. Using transaction data for three exchange traded funds (ETFs), we find that bid–ask spreads are wider at night due to higher order processing costs, market maker rents, and inventory holding costs. Results show that night markets are informationally fragmented and are not able to impound information available in net order flow to the same degree as day markets.

In February 2000, the Securities and Exchange Commission (SEC) asked ‘‘To what extent is fragmentation of the buying and selling interest in individual securities among multiple market centers a problem in today’s markets?’’ (SEC, 2000b) In addition, the SEC sought to ‘‘make information on prices, volume, and quotes for securities in all markets available to all investors, so that buyers and sellers of securities, wherever located, can make informed investment decisions and not pay more than the lowest price at which someone is willing to sell, or not sell for less than the highest price a Research in Finance, Volume 23, 99–125 Copyright r 2007 by Elsevier Ltd. All rights of reproduction in any form reserved ISSN: 0196-3821/doi:10.1016/S0196-3821(06)23004-1

99

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NIVINE RICHIE AND JEFF MADURA

buyer is prepared to offer’’ (see SEC Order Handling Rules, 6 Sept 1996). Market fragmentation, or order splitting across different trading locations, has the potential to decrease liquidity and transparency by isolating orders. In contrast, centralized markets like the NYSE provide quote and trade transparency, but at the potential cost of decreased competition for order flow. Recent accounts of improper specialist behavior add fuel to the ongoing fragmentation versus centralization debate.1 As markets continue to evolve and expand, particularly with the advent of electronic trading venues and after-hours markets, a central question remains as to whether the structure of the market itself carries implications for market efficiency. Two key themes emerge from the debates surrounding market fragmentation. The first is the benefit of price competition among market centers. When market centers vie for order flow, it leads to a decrease in the effective spread (or the implicit transaction cost). The second theme deals with the interaction of the order flow. Market centers compete on several levels in addition to price, such as service, speed, payment for order flow, and internalization of order flow, among others. The internalization of order flow, is hotly debated and is defined as a market center acting as principal in a customer’s agent order rather than routing the order to another market center. It may reduce the interaction of orders across markets, and possibly prevents the client from receiving better execution than the National Best Bid or Offer (NBBO) (SEC, 2000a). Furthermore, practices like the internalization of orders interfere with transparency and price discovery. In an environment with multiple market centers, there is a trade-off between competition and interaction of order flow.2 The issue of market fragmentation is of particular concern in the afterhours market where traditional exchanges are often closed and electronic communications networks (ECNs) are the ‘‘only game in town’’ (Sloan, 2000). Investment professionals warn that price fluctuations and illiquidity may increase the risks associated with trading after-hours.3 To address concerns about market fragmentation, the SEC investigated the role of ECNs and the after-hours markets in a report to Congress. In it, the SEC ‘‘highlights the liquidity constraints and price volatility that investors continue to face in this market and outlines recent initiatives to improve transparency and extend essential investor protection and market integrity measures to this environment’’ (SEC, 2000b). As noted by Hasbrouck in an SEC (2002) roundtable discussion of market structure, the central notion is that of a trade-off between market center competition and price competition. Decentralized markets compete for order flow, while centralized markets bring buyers and sellers together and thereby allow for more transparency.

Fragmentation of Day versus Night Markets

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This study offers insight into the ongoing debate regarding night market transparency by comparing the cost of informational fragmentation between day and night trading sessions. The objectives are: (1) to determine whether higher transaction costs at night are due to costs associated with dispersed information, and (2) to identify the degree of market transparency at night relative to the day as proxied by the sensitivity of returns to the order flow. The results show that information revealed in transaction data during the night sessions is not incorporated into trades to the same degree as during the day, indicating less transparency at night. Bid–ask spreads are significantly wider at night and can be explained by higher fragmentation costs at night, even after controlling for illiquidity. Night markets face higher order processing costs and higher market maker rents as well as higher inventory holding costs. Furthermore, costs associated with increased market concentration at night cause spreads to increase significantly.

1. ELECTRONIC COMMUNICATIONS NETWORKS AND THE ROUTING OF ORDERS Brokers have several avenues available to them for executing clients’ orders. Orders in exchange-listed securities can be directed to their respective listing exchange as well as to regional exchanges or to other dealers known as third-market makers. In the case of exchange-traded funds (ETFs) such as the SPY, DIA, and the QQQ, the primary exchange is the American Stock Exchange, but trades can be directed to regional exchanges such as the Pacific Exchange or to third-market makers. Some regional exchanges and market makers will pay a broker for order flow. An alternate method of executing trades is the ECNs such as Archipelago or RediBook where buy and sell orders are automatically matched and executed against one another. These venues offer high-speed low-cost order execution, but the potential exists at night for no execution if a willing counterparty cannot be found. Twenty-eight percent of the trading of ETFs is executed by the Archipelago ECN alone with QQQ, SPY and DIA market shares reported at 32%, 24%, and 32.7%, respectively.4 A final method of order execution is for the broker to serve as counterparty to a trade. As long as brokers are not violating the duty of best execution, they may route a client’s trade to the firm’s own inventory to be filled internally at the best bid or offer currently available. Investors receive the best

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bid if they are selling and the best offer if they are buying, but they do not experience price improvement, or the opportunity to trade inside the quoted bid or ask. This practice of internalization as well as the other practices associated with fragmented markets can cause the interaction of buy orders and sell orders to decrease, thereby reducing transparency of the order flow.

1.1. Informational Fragmentation and Transparency One of our main objectives is to compare the informational fragmentation in night markets versus day markets. The fear that fragmented markets lead to inefficient price discovery has spurred a body of literature to test the extent of integration of securities markets and to weigh the associated costs and benefits. Lee (1993) defines market integration as ‘‘the extent that electronic linkages communicate the available trading opportunities at different physical locations. A fully integrated market is one in which all the pricerelevant trading information available at each location is communicated quickly to the entire market.’’ (p. 1009) In his study of NYSE-listed stocks trading on different exchanges, he finds that trades differ by location, which suggests that markets are fragmented (not fully integrated). At the heart of informational fragmentation is the notion of transparency, which reflects the informational, or predictive, quality of available transaction data and is categorized as either pre-trade (disclosure of quotes) or post-trade (disclosure of transactions). Hasbrouck (1995) examines securities simultaneously traded on several exchanges and determines each market’s contribution to price discovery using cointegration analysis. He finds that the NYSE accounts for over 92% of the information share. Masulis and Shivakumar (2002) examine the effect of market structure on the speed at which stock prices respond to information. They find that the specialist systems of the NYSE/AMEX cause prices to incorporate news more slowly than the electronic multi-dealer system of NASDAQ. Huang (2002) finds that in spite of ECN practices that might adversely affect quote quality such as internalization and payment for order flow, all quotes are informative. He suggests that the Island and Instinet ECNs are frequently the price leaders and do not ‘‘free-ride’’ off of NASDAQ. Hendershott and Jones (2003) find that in response to Island’s decision in September 2002 to ‘‘go dark’’ and stop displaying their limit order book, transparency decreased, and fragmentation increased. Our study is concerned not with fragmented versus centralized markets per se but, more specifically, with fragmented night markets versus relatively


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centralized day markets. Some studies address the price discovery of night markets. McInish, Van Ness, and Van Ness (2002) investigate the price discovery of NYSE-listed stocks in the after-hours markets of Chicago, Philadelphia, and Pacific exchanges. They find that most trades happen at or near the closing NYSE price, suggesting little or no contribution to price discovery after-hours. Yet, Barclay and Hendershott (2003b) find price discovery is significant, and that the pre-open session offers more informed trading and price discovery than the post-close session. Barclay and Hendershott (2003a) find that the after-hours sessions exhibit higher adverse selection costs, lower order-processing costs, and more order persistence than day sessions. This study differs from related research on night markets in that it directly compares the informational fragmentation of night markets to day markets. Specifically, we investigate the incremental effect of night markets on price competition to isolate the costs associated with after-hours trading. Following Evans and Lyons (2002), we define informational integration as the degree to which information that arrives through order flow is immediately and fully impounded into prices. Market frictions, such as limited transparency and limited liquidity, can inhibit the transfer of information from order flow to prices, leading to informationally fragmented markets. We suggest that night markets will be more informationally fragmented than day markets. A key contribution is the identification of the competition component of the bid–ask spread in the after-hours market which we attribute to the night market fragmentation.

1.2. Cost of Transacting The economic impact of fragmented markets relative to centralized markets is dependent on the cost of transacting. In the broadest sense, transaction costs are composed of order-processing costs and costs associated with illiquidity (Tinic, 1972). The cost of illiquidity (stated differently, the cost of supplying liquidity) is further identified as being composed of inventory holding costs, adverse selection costs, and competition costs. We decompose the cost of transacting in the day and at night, and we borrow from the following studies to compare the cost of transacting across trading sessions. Demsetz (1968) suggests that dealers must be compensated for their ‘‘immediacy’’ or supply of liquidity. He shows that the opportunity costs associated with holding inventory are an increasing function of the dollars tied up in the transactions (proxied by the stock price) and a decreasing function of the trade frequency (proxied by the number of transactions or the number

104


of shareholders). Garman (1976) suggests that dealers must actively set bid and ask prices to prevent inventory from straying too far in one direction or the other. Order processing costs are assumed to be fixed and decreasing with trading volume (Tinic, 1972; Tinic & West, 1972, 1974; Stoll, 1978b; Harris, 1994). Inventory holding costs represent a dealer’s opportunity cost of holding securities and are often found to increase with stock price (Tinic, 1972; Tinic & West, 1972, 1974; Demsetz, 1968; Harris, 1994), increase with volatility (Stoll, 1978b; Harris, 1994), and decrease with trade frequency (Demsetz, 1968; Tinic, 1972). Asymmetric information costs, or adverse selection costs, are defined as the ‘‘information costs which arise if investors trade on the basis of superior information’’ (Stoll, 1978a). The inclusion of a spread component to account for the effect of competition is first introduced by Tinic (1972) and further refined in subsequent studies. While some use the number of dealers as an estimate of the cost associated with competition (Tinic & West, 1972, 1974; Stoll, 1978b), others use measures such as the number of exchanges (Tinic, 1972), the proportion of total volume traded on the primary exchange (Tinic & West, 1972), a Herfindahl index (Tinic, 1972) and, most recently, a modified Herfindahl index (Bollen, Smith, & Whaley, 2004). The effect of different market structures on bid–ask spreads has been examined empirically in several studies. Affleck-Graves, Hedge, and Miller (1994) find that the centralized NYSE has lower-order processing costs but higher adverse selection and higher inventory costs than the NASDAQ. Easley, Kiefer, and O’Hara (1996) examine transaction data from the NYSE and the Cincinnati Stock Exchange and find that the NYSE is exposed to larger adverse selection costs, as the specialist is left to contend with informed traders. In contrast, Heidle and Huang (2002) find a higher probability of informed trading in an anonymous dealer market like NASDAQ. Barclay, Hendershott, and McCormick (2003) determine that ECNs attract informed traders rather than uninformed traders. While studies have examined the effect of market structures on market quality and the effect of trading session on market quality, none has compared market structures across trading sessions. This study differs from existing research in that it examines the effect of informational fragmentation in the night market. It extends the work of Barclay and Hendershott (2003a) to include the cost of fragmentation in the composition of the bid– ask spread beyond that associated with illiquidity. Given the trade-off associated with multi-dealer markets whereby increased competition for order flow is potentially offset by decreased interaction of the order flow, we


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hypothesize a significantly higher cost of transacting in the night market than the day market. Applying a new model developed by Bollen et al. (2004) we capture the effect of fragmentation through the competition component of the bid–ask spread. This study applies this cost associated with competition to the night market to identify the proportion of marketmaking costs associated with fragmentation while controlling for liquidity. Prior literature and anecdotal evidence suggest that night market will be informationally fragmented. We attempt to document this phenomenon, and identify the associated cost to investors.

2. DATA AND RESEARCH DESIGN To achieve our objectives, we use a sample of exchange traded funds which should experience little uncertainty regarding the liquidation value of the shares as their NAVs are available intraday. Furthermore, the arbitrage that is available to large investors guarantees that the liquidation value and market value never stray far from one another. Other studies have also assessed fund trading to assess market microstructure characteristics, including research by Neal and Wheatley (1998) and Chen, Jiang, Kim, and McInish (2003). 2.1. Data Description Intraday trade and quote data from August 2001 are gathered from the NYSE TAQ database for the three largest ETFs: SPDRs, Standard and Poor’s Depository Receipts (ticker symbol SPY), the DIAMONDs Trust (ticker symbol DIA) and the Nasdaq-100 Index Tracking Stock (ticker symbol QQQ). The average daily volume of these securities is more than 22 million shares compared to a typical AMEX listed stock whose average daily volume is over 125 thousand shares. The month of August 2001 is typical in that the total monthly volume traded on the AMEX is approximately 2.3 billion shares, compared with an average total monthly volume of approximately 3 billion shares in 2001 and 2002 combined. August 2001 allows us to capture any additional volume that arrived as a result of the NYSE granting unlisted trading privileges (UTP)5 to the SPY, DIA, and QQQ securities in July 2001. Daily trade statistics are gathered from the Center for Research in Securities Prices (CRSP) daily files. We include all trades and quotes that arrive after 4 p.m. and before 9:30 a.m. in our definition of the after-hours market. This naturally includes all activity in the post-close session and in the pre-open session. Often the

106


question of liquidity arises when discussing the after-hours market. Table 1 shows that average volume per-session for this total sample is approximately 22.5 million shares and 960,000 shares during the day and during the night, respectively. DIAs have the lowest night time volume at 60,000 shares on average and QQQ has the highest average night time volume at over 500,000 shares on average. This compares with an average daily volume in August 2001 of approximately 125,000 shares per stock listed on the AMEX. Thus, in spite of the reduced volume during the night sessions, our sample of three ETFs still provides us with a rich dataset of after-hours transactions to investigate. Following Huang and Stoll (1996) and Boehmer and Boehmer (2003), the intraday data are screened to eliminate any reporting errors, irregular settlements, and non-positive spreads, prices, volumes, and depths. Records with a quoted spread greater than $4 are also eliminated. The Lee and Ready (1991) algorithm classifies a trade as a buy (sell) if it occurs at or near the quoted ask (bid) price or the prior quoted ask (bid) price. In the absence of a bid or ask price, the ‘‘tick test’’ classifies a trade as a buy if it occurs on an uptick or a zero-uptick and a sell if it occurs on a downtick or a zerodowntick. Following Bessembinder (2003), contemporaneous trades and quotes are compared rather than the five-second delay in reported trade times used in earlier studies. Table 1 shows the distribution of the data by market center and by ETF.6 During the day, the average number of participating market centers is seven. Market centers report 1,728 trades and 8.5 million shares trading volume on average. The after-hours market is thinner with five participants on average. At night, the average number of trades per market center and average trading volume per market center are 54 and 362,700, respectively. Signed order flow is defined as the volume of buyer-initiated trades less the volume of seller-initiated trades. On average, the net order flow per session is larger during the day. Panel B of Table 1 provides additional insight into the nature of the day and after-hours markets. The average trade size is higher in the after-hours market which is consistent with Barclay and Hendershott’s (2003a) findings that informed traders enjoy the anonymity of ECNs in the after-hours markets. Day and night markets are further distinguished by different transaction costs. The spreads are defined as: Quoted spread ðQSÞ ¼ Ask price bid price

(1)

Effective spread ðESÞ ¼ 2 Trade price Bid ask midpoint

(2)

Minimum Day

Mean Night

Day

Maximun Night

Day

Night

Panel A: Descriptive Statistics of Order Flow and Market Centers (Full Sample) Number of MCs DIA QQQ SPY Average number of trades per MC DIA QQQ SPY Average volume per MC DIA QQQ SPY

6 6 7 6

3 3 5 4

7 6 9 6

5 4 7 5

9 6 9 7

8 6 8 6

427

1

1,728

54

3,770

105

427 2,363 1,083

1 42 45

667 3,029 1,488

12 68 81

1,364 3,770 2,274

23 103 105

630,500

400

8,558,588

362,720

26,278,200

1,248,400

630,500 11,175,400 2,695,500

400 147,000 185,400

1,636,122 18,047,565 5,992,078

60,526 545,217 482,417

3,180,400 26,278,200 8,420,100

223,000 1,248,400 864,200

1,313,400

15,700

22,513,651

962,317

74,237,200

3,101,100

1,313,400 29,712,600 6,134,000

15,700 734,200 308,300

2,948,548 53,897,117 10,695,287

133,757 1,856,222 896,974

5,548,600 74,237,200 15,276,600

724,700 3,101,100 1,869,500

NOF per session DIA QQQ SPY

(9,455,900) (757,500) (9,455,900) (2,671,700)

(932,700) (642,500) (932,700) (811,100)

(704,881) (71,096) (1,687,978) (355,570)

(59,607) (37,261) (81,157) (60,404)

6,699,600 551,700 6,699,600 1,123,100

1,374,700 179,200 1,374,700 582,400

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Average total volume across MCs DIA QQQ SPY


Liquidity Descriptive Statistics.

Table 1.

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Table 1. (Continued ) Minimum Day

Mean Night

Maximun

Day

Night

Day

Night

Panel B: Transaction Level Descriptive Statistics Average trade price DIA (Nday ¼ 31,572 Nnight ¼ 609) QQQ (Nday ¼ 477,465 Nnight ¼ 13,857) SPY (Nday ¼ 80,441 Nnight ¼ 2,965)

99.25

102.87

103.18

106.25

105.62

35.75

36.10

39.51

39.69

44.00

44.05

112.04

113.16

118.21

118.26

123.25

122.96

Average trade size DIA QQQ SPY

100 – –

100 100 100

Average transaction return DIA QQQ SPY

0.008889 0.024393 0.017626

0.002686 0.033266 0.009420

2,148 2,596 3,058 0.000002 0.000000 0.000001

5,052 3,081 6,958 0.000018 0.000003 0.000005

250,000 984,000 971,200 0.009097 0.024690 0.017942

635,000 700,000 750,000 0.004304 0.034410 0.008054

Note: Descriptive statistics are presented in Panel A on a per trading session basis unless otherwise noted. Net order flow is defined as buyerinitiated volume and less seller-initiated volume. Trade direction is determined using the Lee and Ready (1991) algorithm. Panel B presents transaction-level descriptive statistics. MC, market center; NOF net order flow.


98.83


109

Table 2 shows that the average quoted and average effective spreads are wider in the after-hours market, which is consistent with the findings of McInish et al. (2002) and Frino and Hill (2000) that spreads widen and depth decreases in the after-hours market. A multivariate cross-sectional model confirms that transaction costs are higher at night. The model shows that effective spread is explained by several control variables which represent liquidity (Tinic, 1972; Stoll, 1978a, 1978b) and a dichotomous variable representing the night market. The model to estimate is Spread ¼ a þ b1 ln VOL þ b2 ln STD þ b3 ln Pt1 þ b4 NDUM þ e

(3)

where SPREAD is the effective spread or quoted spread calculated above, ln VOL the log of total volume of 50 prior trades, ln STD the log of standard deviation of 50 prior transaction returns, ln Pt1 the log of the prior price, and NDUM the dummy variable with a value of 1 if the transaction is afterhours, 0 otherwise. The first three variables are included as control variables based on prior microstructure work by Demsetz (1968) and Tinic (1972) which establishes the cost of supplying liquidity. Table 3 shows the coefficients of the regression specified above. Consistent with the results from Boehmer and Boehmer (2003), Panel A shows that the control variables enter into the regression with the correct positive sign and are significant in all but two cases. This study focuses on the incremental spread during the after-hours market and so the NDUM variable is of particular interest. For the full sample, the NDUM variable is 0.18 and is significant at the 0.1% level. The same conclusion is drawn when each ETF subsample is regressed independently. Panel B shows the regression estimated using effective spreads. The coefficient of the NDUM variable is once again positive and significant in all cases. Taken together, the data show that after controlling for volatility, volume, and price, market participants face higher costs at night than during the day.

2.2. Research Design To identify the costs associated with fragmentation, we decompose the effective spread into more detailed components. Several models for decomposing the bid–ask spread have been proposed over time. George, Kaul, and Nimalendran (1991) account for two components of the spread: order processing/inventory and adverse selection. Huang and Stoll (1997) further

110

Table 2. Spread

Comparison of Bid–Ask Spreads across Trading Session. Day Session

After Hour Session

Mean

Variance

Mean

Variance

Difference in Means Test T-statistic

Variance Ratio Test F-statistic

0.0497 0.1005

0.0053 0.0202

0.1668 0.3167

0.1927 0.2984

30.243 44.828

0.03 0.07

0.0672 0.1516 0.0468 0.0824 0.0594 0.1873

0.0090 0.0266 0.0038 0.0078 0.0126 0.0802

0.0861 0.4496 0.1 808 0.2979 0.1068 0.3937

0.0206 0.2342 0.2214 0.3124 0.0648 0.2249

2.390 11.181 29.186 39.528 8.311 19.373

0.44 0.11 0.02 0.03 0.20 0.36

Panel A: Full Sample Effective Quoted

DIA QQQ SPY

Effective Quoted Effective Quoted Effective Quoted

Note: Average effective and quoted spreads are reported for day and after-hours markets. Quoted spread is defined as ask–bid price. Effective spread is defined as 2|Trade price–bid ask midpoint|. Difference in means test assumes unequal variances and the variance ratio tests the null hypothesis that the day variance/night variance ¼ 1. Significance at the 5% level using a 1-tailed test of significance. Significance at the 0.1% level using a 1-tailed test of significance.


Panel B: Subsamples by ETF

Intercept

Incremental Spread during After-Soars Session. ln STD

ln VOL

ln Pt-1

NDUM

Adjusted R2 (%)

20.04459 (17.23) 14.48919 (3.27) 20.42432 (16.90) 63.67934 (7.46)

0.000000015 (5.51) 0.000000007 (0.72) 0.000000013 (4.91) 0.000000011 (0.67)

0.091032 (111.51) 0.121073 (2.42) 0.058879 (17.21) 0.365537 (6.18)

0.181081 (48.13) 0.169845 (19.91) 0.181397 (42.03) 0.223373 (12.23)

8.73

31.23125 (27.78) 41.05184 (8.96) 29.28457 (25.19) 36.18027 (5.38)

0.000000001 (0.71) 0.000000013 (3.48) 0.000000006 (2.77) 0.000000010 (1.03)

0.0222 (46.17) 0.05089 (2.84) 0.034019 (12.77) 0.065473 (1.96)

0.077352 (26.95) 0.020694 (5.43) 0.092343 (26.36) 0.011808 (2.38)

3.49

Panel A: Quoted Spread Full Sample (N ¼ 605,184) SPY (N ¼ 82,831) QQQ (N ¼ 490,747) DIA (N ¼ 31,606)

0.26735 (79.41) 0.40189 (1.68) 0.14909 (11.82) 1.56675 (5.71)

1.25 6.77 3.57

Panel B: Effective Spread Full sample (N ¼ 605,184) SPY (N ¼ 82,831) QQQ (N ¼ 490,747) DIA (N ¼ 31,606)

0.05439 (23.62) 0.290128 93.39) 0.09768 (-9.91) 0.2502 (1.62)


Table 3.

0.90 4.31 0.23

111

Note: The following cross sectional model is estimated using intraday data: SPREAD ¼ a+b1 ln STD+b2 ln VOL+b3 ln Pt1+b4 NDUM+e, where SPREAD is either the quoted or effective spread, ln STD the log of standard deviation of 50 prior transaction returns, ln VOL the log of total volume of 50 prior trades, ln P t1 the log of the prior price, and NDUM a dummy variable representing the after hours market. Significance at the 10% level using a 2-tailed test of significance. Significance at the 5% level using a 2-tailed test of significance. Significance at the 1% level using a 2-tailed test of significance. Significance at the 0.1% level using a 2-tailed test of significance.

112


decompose the spread by estimating three components: order processing, adverse information, and inventory. Barclay and Hendershott (2003b) follow their model to estimate the probability of informed trading during the pre-open session versus the post-close trading session. Most recently, Bollen et al. (2004) offer a new model to decompose the bid–ask spread into four components: order-processing costs, inventoryholding costs, adverse selection costs, and competition. The first component is order-processing costs which are generally fixed costs incurred by dealers and passed through to investors. In centralized markets, more orders are crossed without dealers taking principle positions in the transactions. In fragmented markets, however, the internalization of orders discussed earlier leads to less order interaction and more dealer participation in trades. Consequently, we should expect that order-processing costs in fragmented markets increase. Inventory costs represent the compensation dealers must earn to bear the risk and carrying costs of inventory positions. The higher the turnover rate in a dealer’s inventory, the lower the inventory holding costs will be. Since less volume changes hands at night relative to the day, we can assume the length of time that a dealer must carry the inventory is longer than it would be during the day. Consequently, we expect that inventory holding costs should be higher at night. Adverse selection costs are the compensation a trader requires to accept the risk that the counterparty has private information. Barclay and Hendershott (2003b) define the adverse selection component as the loss of liquidity externalities since liquidity traders will seek trading environments (i.e. day sessions) where they do not face costs associated with asymmetric information. The model proposed by Bollen et al. (2004) combines inventory holding costs and adverse selection costs into one term called the inventory holding premium (IHP). The inventory holding premium (IHP) is modeled as an at-the-money call using the Black and Scholes (1973) and Merton (1973) model and reduces to: h i pffiffiffi IHPi ¼ Si 2N 0:5s ti 1 (4) pffiffiffi where ti is the average of the square root of the annualized time between trades, S the average of the ending share price, s the standard deviation of pffiffiffiffiffiffiffi ffi the past 60 days returns annualized by a factor of 252: According to this model, this IHP can be further decomposed into its two components: (1) adverse selection costs and (2) inventory holding costs. However, our sample consists of exchange traded funds which have been shown to have


113

minimal adverse selection costs due to the broad diversification inherent in such portfolios (Neal & Wheatley, 1998; Datar & Dubofsky, 1999). Consequently, we do not seek to decompose the inventory holding premium into the two components, but rather take the IHP to be dominated by the inventory holding costs incurred by the market maker. The competition component of the spread is inversely related to the number of market makers participating in the security. The centralized auction markets of the NYSE and the AMEX are absent at night, and, consequently, the fragmented markets of ECNs and the NASDAQ dominate. The model proposed by Bollen et al. (2004) captures the effect of competition on spreads and is applied to estimate the cost of fragmentation at night. Bollen et al. (2004) note that ‘‘as competition increases, the bid/ask spread approaches the expected marginal cost of supplying liquidity; that is, the sum of inventory-holding costs and adverse selection costs.’’ (p. 5) Following their methodology, we estimate the competition component using the modified Herfindahl index and interpret a value approaching zero as highly competitive and fragmented and a value approaching one as monopolistic and centralized. The modified Herfindahl index (MHIi) is specified as: HI i 1=NM i MHI i ¼ (5) 1 1=NM i P 2 where HI i ¼ NM and Vi the number of shares traded by market j¼1 V j TV center j, NM the number of market centers, and TV the total number of shares traded in all market centers. The bid–ask spread function is estimated as: SPRDi ¼ a0 þ a1 InvT V i þ a2 MHI i þ a3 IHPi þ i

(6)

where SPRDi is either the average quoted spread or the average effective spreads, InvTVi the inverse of the total number of shares traded across all market centers which represents the fixed order-processing cost, MHIi the modified Herfindahl index which represents the costs associated with competition, and IHPi the inventory holding premium defined above. Therefore, controlling for liquidity by including the IHP term, we can isolate the effects of fragmentation as seen in the fourth term of the above regression. In addition, all cross sectional regressions are corrected for heteroskedasticity using White’s (1981) correction. A second methodology to investigate the costs associated with night markets follows Boehmer and Boehmer (2003) and is similar to the decomposition model described by George et al. (1991). The first step is a cross-sectional

114


model to identify the proportion of the spread that is due to information and inventory costs. The model is specified as: DMPt ¼ a þ bHSt1 I t1 þ e

(7)

where DMPt is the change in the midpoint of the quoted bid/ask spread, HSt1 the one-half of the prior quoted spread, It1 the trade indicator which takes a value of +1 if buyer initiated and 1 if seller initiated, b the proportion of the spread due to information and inventory costs, and 1b the proportion of the spread due to order processing and market maker rent. This model is estimated again using an interaction term of HSt1It1 *NDUM. The coefficient of the interaction term gives us an estimate of the incremental information and inventory component associated with the after-hours market. Subtracting both b and the coefficient of the interaction term from 1 gives us an estimate of the order processing and market maker rent component after including the after-hours market. The second step of this methodology involves multiplying the regression coefficients by the average effective spread to arrive at an estimate of the dollar cost associated with each component of the bid–ask spread. Our final methodology seeks to identify whether night markets are informationally integrated. We follow Evans and Lyons’ (2002) model which relates daily changes in exchange rates to order flow. Stock market returns are modeled as a linear function of order flow, and the following model is estimated to test the degree of informational integration. Rit ¼ a þ bOF it þ e

(8)

where Rit is the % D in value of the ETF from the ending quote midpoint during session i1 to the ending quote midpoint during session i, OFit the order flow during session i and is defined as the net of buyer-initiated trades and less seller-initiated trades, and i represents either the day or the night session.

3. RESULTS 3.1. Effect of Competition on Spreads Table 4 shows the regression results of quoted spreads and effective spreads regressed on order-processing costs, inventory-holding costs, and competition. Models 1 and 2 show the regressions using the day and after-hours subsamples separately. Models 3–5 use the full sample of day and night observations but with the inclusion of interaction terms to capture the incremental effects of each component at night.

Day Sample (N ¼ 69)

Effect of Fragmentation on Spreads.

After Hours Sample (N ¼ 69)

Full Sample (N ¼ 138)

Panel A: Average Quoted Spreads Intercept OPC IHP COMP

0.065 (6.17) 1371294 (7.84) 42.094 (12.76) 0.094 (1.77)

0.188 (3.54) 267245 (0.42) 15.814 (2.69) 0.090 (0.76)

NOPC

0.074 (2.72) 1800403 (6.95) 29.744 (6.57) 0.125 (1.27) 1839747 (3.77)

NIHP

0.100 (3.30) 264074 (0.59) 5.089 (0.68) 0.132 (1.42)

23.050 (5.04)

NCOMP Adjusted R2

0.135 (4.07) 206339 (0.49) 21.652 (4.55) 0.207 (1.72)


Table 4.

74.25%

17.25%

56.41%

58.59%

0.371 (5.86) 62.29%

Panel B: Average Effective Spreads Intercept OPC IHP

0.188 (3.84) 267245 (0.58) 15.814 (1.31) 0.090

0.081 (4.29) 232763 (1.88) 3.881 (2.06) 0.060

0.093 (4.21) 88634 (0.53) 3.625 (0.98) 0.063

0.108 (4.18) 171852 (1.07) 0.354 (0.15) 0.172

115

COMP

0.162 (8.55) 153356 (0.37) 4.111 (2.11) 0.126

116

Table 4. (Continued ) Day Sample (N ¼ 69)

(0.85)

After Hours Sample (N ¼ 69) (1.75)

NOPC

Full Sample (N ¼ 138)

(1.10) 185180 (1.00)

NIHP

(1.19)

5.887 (2.45)

NCOMP Adjusted R2 (%)

(2.16)

31.86

17.25

0.43

3.06

0.117 (2.80) 6.61


Note: The following pooled regression is estimated to decompose average quoted and average effective spreads: SPRD ¼ a0+a1; OPC+a2 IHP+a3 COMP+a4 NOPC+a5 NIHP+a6 NCOMP+e, where OPC is the inverse of total volume, IHP estimated as an at-the-money call, COMP the modified Herfindahl index, and NOPC, NIHP, and NCOMP are the night dummy interaction with OPC, IHP, and COMP, respectively. Significance at the 10% level using a 2-tailed test of significance. Significance at the 5% level using a 2-tailed test of significance. Significance at the 1% level using a 2-tailed test of significance. Significance at the 0.1% level using a 2-tailed test of significance.


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Panel A shows that order-processing costs are negative and significantly related to spreads in two of the five models but not significantly different from zero in the other three models. This indicates that in the day subsample as well as in model 3, increasing order-processing costs are associated with decreasing transaction costs. Though we would expect order processing to be positively related to spreads in individual stocks, it is quite possible that market makers active in portfolios such as these ETFs with heavy competition from ECNs find order-processing costs approaching zero in perfect competition. Of interest is model 3 which shows that the interaction of order-processing costs and night markets leads to increased transaction costs. This supports the hypothesis that order-processing costs in fragmented markets increase due to internalization of orders and more dealer participation in transactions. Centralized markets see less internalization of orders and more interaction of customer orders; so as volume increases, less fixed costs are incurred by dealers. Though our coefficients are large, they are similar to the results reported by Bollen et al. (2004). In their study, they found this coefficient to fall between +700 and +2300. These values are driven by the estimation of order-processing costs, which is proxied by the inverse of total volume. The economic interpretation of their coefficients is derived by multiplying the coefficient by the mean of the variable value and using this product to determine the proportion of the spread attributed to each component. Additionally, their study is a cross-sectional regression, which uses total monthly volume as reported by the NASDAQ. In contrast, we perform a time series analysis using total daily volume captured from the TAQ database. A similar economic interpretation cannot be derived because of the negative coefficients generated by the regression analysis. Consequently, these results differ in magnitude. The next component of the bid–ask spread, the inventory holding premium, is positive and significant in four of the five models in panel A. These results are consistent with the results of Bollen et al. (2004), which find the inventory holding premium to be the dominant explanatory variable in their regressions. Model 4 shows the NIHP coefficient to be positive and significant at the 0.1% level, indicating that traders in night markets face higher inventory holding costs. This is consistent with the hypothesis that less volume at night leads to lower turnover, and higher inventory holding periods lead to higher inventory holding costs. The final, and perhaps most important, component of the bid–ask spread addressed by this model is the competition component, which is modeled as a modified Herfindahl index. The MHI coefficient is negative and weakly

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significant in two of the five models and insignificant in the other three models. Alone, this result suggests that as the MHI decreases, transaction costs increase. Based on the interpretation that MHI approach zero in perfect competition, we are led to believe that as competition increases, transaction costs increase as well. This finding lends preliminary support to the hypothesis that increased market fragmentation is associated with higher transaction costs for investors. Since we are interested primarily in isolating such costs in the after-hours markets, we look further at the NCOMP variable in model 5. The NCOMP variable captures the interaction between the modified Herfindahl index and the night dummy variable, and is positive and significant at the 0.1% level. This finding suggests that after-hours, an increase in the MHI is associated with increased transaction costs. This lends support to the hypothesis that in the absence of NYSE and AMEX activity, night markets face higher costs associated with market concentration, suggesting the existence of monopolistic rents for those ECNs which dominate this market. Panel B shows these same results using the effective spread as the dependent variable rather than quoted spreads. The results are qualitatively similar but the regressions have less explanatory power as seen in the adjusted R2 values.

3.2. Decomposition of Spread We test the robustness of our findings above by decomposing the spread following Boehmer and Boehmer (2003) and George et al. (1991). Table 5 shows the results of the decomposition of spread into (1) the information asymmetry and inventory holding costs and (2) order-processing and market maker rents. Panel A shows that approximately 15% of the average effective spread is due to information and inventory costs while approximately 85% is due to order processing costs and market maker rents, resulting in a dollar cost of $0.00744 and $0.04 during the day, respectively. These results are consistent with the findings of Boehmer and Boehmer (2003) in their investigation of spreads upon the NYSE’s decision to grant UTP privileges to exchange traded funds. The after-hours market portrays a different picture. The dollar costs for each of these components is higher at night but the relative proportions shift toward market maker rents and away from the information and inventory component. The results show the dollar cost of information and inventory component rises to $0.01072 while the dollar cost associated with market

Average Effective Spread

Information and Inventory Component

% of Spread

$ Cost

Interaction of Information and Inventory Component with Night Dummy

% of Spread

$ Cost

Order Processing and Market Maker Rent Component

Order Processing and Market Maker Rent Component after interaction of Night Dummy

% of Spread

$ Cost

% of Spread

$ Cost

0.84636 0.91922

0.04100 0.12200

0.84636 0.72207 0.82895 0.88042

0.04305 0.04803 0.03999 0.05308

0.91922 0.93788 0.91445 0.93906

0.04676 0.06238 0.04411 0.05662

Panel A: Full sample partitioned by day and after-hours Day After-hours

0.04845 0.13273

0.15364 0.08078

0.00744 0.01072


Table 5. Decomposition of Spread.

Panel B: Results including the interaction term Full sample DIA QQQ SPY

0.05087 0.06652 0.04824 0.06029

0.15364 0.27793 0.17105 0.11958

0.00781 0.01849 0.00825 0.00721

0.07286 0.21581 0.08550 0.05864

0.00371 0.01436 0.00412 0.00354

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Note: Results are presented for a two-stage methodology following Boehmer and Boehmer (2003). The first stage is a cross-sectional model specified as DMP ¼ a+b1 HStrltr+b2 HStrltr NDUM+e where DMP is the change in the bid–ask midpoint, HS is half the quoted spread, /is a trade indicator and NDUM is dummy variable representing the night market. The second stage involves multiplying b1 by the average effective spread to arrive at the &doller; cost associated with information and inventory costs and multiplying (1b1) by the average effective spread to arrive at the &doller; cost associated with order processing and market maker rent. Multiplying b2 by the average effective spread returns the incremental &doller; cost associated with night trading and the cost associated with market maker rent after the inclusion of the interaction term is calculated as (1b1b2)* average effective spread. The coefficients, b1 and b2, extracted from the regression are all significant at the 0.1 % level.

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maker rents increases to $0.122. Though these costs are higher, we see that only 8% of the effective spread is due to information and inventory costs while almost 92% of the spread is due to order processing and market maker rents. A longer holding period at night does, in fact, result in higher inventory holding costs, but these costs explain relatively less of the effective spreads at night. These results support the hypothesis that fragmentation leads to higher costs at night. As fragmentation interferes with transparency, market makers are able to increase their monopolistic rents, leading to a higher market maker rent component at night. Panel B shows the results of the analysis with the inclusion of an interaction term representing the after-hours market. The results for the full sample show that information and inventory costs are approximately 7% lower in the after-hours market and that these results are similar for each ETF subsample. A decline in the inventory and information component leads to an increase in the order processing and market maker rent component, causing this component to be higher in all cases after the inclusion of the night interaction term. These results support the hypothesis that market fragmentation leads to higher costs at night after controlling for inventory and information asymmetry. An interesting implication of the analyses above is that the presence of additional market centers during the day increases the competition to the centralized market centers of the NYSE and AMEX. The dark side to such a market structure is that at night, the absence of the transparency associated with the NYSE and AMEX allows the fragmented ECN markets to operate under increased monopoly rents thereby driving up the cost of transacting.

3.3. Informational Fragmentation Results To further explore the effects of market fragmentation at night, we examine the informational integration of day and night markets following the methodology proposed by Evans and Lyons (2002). Panel A of Table 6 shows that for the full sample, order flow is directly and significantly related to the return during both the day and the night trading sessions.7 However, the night session results are weaker with the lower t-statistic and a 4.12% adjusted R2. The OF coefficient for the day subsample is significant at the 0.1% level and the model has an adjusted R2 of 28.5%. These results support the conclusion that information contained in the order flow is related to


Table 6.

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Informational Integration of Day and Night Markets. Intercept

OF

Adjusted R2 (%)

0.00087 (1.13) 0.00182 (1.42) 0.00016 (0.19)

0.0000000029 (4.30) 0.0000000028 (4.16) 0.0000000045 (1.77)

25.25

0.00157 (0.93) 0.0000 (0.01)

0.0000000038 (3.86) 0.0000000000 (0.03)

20.36

0.0007 (0.22) 0.00002 (0.10)

0.0000000028 (4.05) 0.0000000067 (1.94)

33.29

0.00279 (1.54) 0.00006 (0.61)

0.0000000026 (0.34) 0.0000000033 (0.90)

3.75

Panel A: Full Sample Full sample (N ¼ 135) Day (N ¼ 69) Night (N ¼ 66)

28.50 4.12

Panel B: SPY Subsample Day (N ¼ 23) Night (N ¼ 22)

5.00

Panel C: QQQ Subsample Day (N ¼ 23) Night

5.88

Panel D: DIA Subsample Day (N ¼ 23) Night (N ¼ 22)

3.79

Note: The following regression is estimated for the full sample and for individual day and night subsamples: Ra ¼ abOFa+e where Ra is the % change of the average ending ask–price from the average beginning bid–price over the first and last 15 min of each trading session i, OFa the net of buyer-initiated trades and less seller-initiated trades, and i the day or night session. Significance at the 10% level using a 2-tailed test of significance. Significance at the 0.1% level using a 2-tailed test of significance.

returns during the day, but related to returns to a lesser degree during the after-hours market. This strongly supports the concept of higher informational fragmentation in night markets than day markets. Panels B through D show the same analysis partitioned for each of the three ETFs. The same patterns hold for the SPY and the QQQ subsamples where the day order flow is informationally integrated but the night order flow is to a lesser degree or not at all. Only the DIA regressions in panel D

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show no explanatory power. These results provide additional support for less informational integration in night markets relative to day markets.

4. SUMMARY In response to the SEC’s request for comments on the issue of market fragmentation, the NYSE has called for an end to the practice of internalization and payment for order flow. Others have applauded the SEC for approaching additional rule making cautiously. Market fragmentation is particularly of concern in the after-hours markets when the centralized exchanges of the NYSE and the AMEX are closed and the NASDAQ and ECNs are active. Thus, this study seeks to answer the question of whether higher transaction costs at night are due to market fragmentation. Costs associated with market concentration cause spreads at night to be significantly larger. Wider bid–ask spreads at night are due to higher orderprocessing costs and market maker rents and higher inventory-holding costs. Furthermore, the results show that night markets are not able to impound information available in net order flow to the same degree as day markets. Thus, night markets are informationally fragmented. Investors face a trade-off. Though competition for order flow should lead to tighter spreads and improved market quality, competition can also lead to fragmentation, which may result in decreased order interaction and decreased transparency. At night, fragmentation comes with higher transaction costs which are attributed to competition, after controlling for the other three generally accepted components of the bid–ask spread. With different degrees of fragmentation, day and night markets may experience shocks in different ways. Future research can explore the impact of shocks like September 11 on the two markets. Additionally, the creation and expansion of the ETF market may be a contributing factor to the differences in market structure of day versus night markets.

NOTES 1. See Clark, Kelley, and Dugan (2004) as well as the written statement of Putnam (2004), Chairman & Chief Executive Officer, Archipeligo Holdings, LLC. 2. The issue of market fragmentation resurfaced with the Corinthian Colleges shares that were halted in 2003 by NASDAQ but resumed trading on Archipeligo leading to cries of improper order handling and calls for an examination of fragmented markets (see Karmin & Kelly, 2003).


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3. See for example Der Hovanesian (2003). Also see http://www.schwab.com for investor information regarding extended hours trading. 4. See ‘‘ARCA Exchange-Traded Fund Activity: Weekly Top 25’’ for the week of February 16, 2004 through February 20, 2004 available at http://www.tradearca. com/data/pdr/eft_weekly.pdf 5. Unlisted trading privilege is defined as ‘‘a right, provided by the Securities Exchange Act of 1934, that permits securities listed on any national securities exchange to be traded by other such exchanges.’’ See http://www.nyse.com. 6. The market centers are the American Stock Exchange, the Boston Stock Exchange, the Cincinnati Stock Exchange, the Midwest Stock Exchange, the New York Stock Exchange, the Pacific Stock Exchange, NASDAQ, CBOE, the Philadelphia Stock Exchange. The ECNs are known as ATS or alternative trading systems and so report their volume through the exchanges like the Pacific Exchange or the NASDAQ. The following market makers had quotes recorded in this sample: Archipeligo (ARCA), NASDAQ (CAES), National Clearing Corp (JBOC), Bernard Madoff Investment Securities (MADF), MH Meyers & Co (MHMY), Peters Securities Company (PTRS), Citigroup Global Markets (SBSH), Southwest Securities (SWST), The Third Market Group (THRD), Knight Capital Markets, Inc. (TRIM), VFinance Investments (VFIN), William R. Hough & Co. (WRHC). 7. An alternate measure of return is also estimated which assumes that investors buy at the average of the asking prices quoted over the first 15 min of the trading session and sell at the average of the bid prices quoted over the last 15 min of the trading session following Fehle and Zdorovtsov (2002) to directly account for transaction costs. The results are qualitatively similar and, consequently, not reported.

REFERENCES Affleck-Graves, J., Hedge, S. P., & Miller, R. E. (1994). Trading mechanisms and the components of the bid–ask spread. Journal of Finance, 49(4), 1471–1488. Barclay, M., & Hendershott, T. (2003a). Liquidity externalities and adverse selection: Evidence from trading after hours. Journal of Finance, 57, 681–710. Barclay, M., & Hendershott, T. (2003b). Price discovery and trading after hours. Review of Financial Studies, 16, 1041–1073. Barclay, M., Hendershott, T., & McCormick, D. T. (2003). Competition among trading venues: Information and trading on electronic communications networks. Journal of Finance, 58, 2637–2666. Bessembinder, H. (2003). Issues in assessing trade execution costs. Journal of Financial Markets, 6, 233–257. Black, F., & Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81(3), 637–654. Boehmer, B., & Boehmer, E. (2003). Trading your neighbor’s ETFs: Competition or fragmentation? Journal of Banking and Finance, 27, 1667–1703. Bollen, N., Smith, T., & Whaley, R. (2004). Modeling the bid/ask spread: Measuring the inventory-holding premium. Journal of Financial Economics, 72, 97–141. Chen, J., Jiang, C., Kim, J., & McInish, T. (2003). Bid–ask spreads, information asymmetry, and abnormal investor sentiment: Evidence from closed-end funds. Review of Quantitative Finance and Accounting, 21, 303–321.

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Clark, S., Kelley, K., & Dugan, I. (2004). NYSE traders are subject of investigation: SEC, Big Board expand probe of ‘specialists’ to include about a two dozen individuals. Wall Street Journal, 4(March), C1. Datar, V., & Dubofsky, D. (1999). The reaction of closed end funds to stock distribution announcements. The Financial Review, 34(2), 73–88. Demsetz, H. (1968). The cost of transacting. Quarterly Journal of Economics, 82(1), 33–53. Der Hovanesian, M. (2003). The market’s closed–wake up; after-hours trades give you the jump on most investors. Business Week, 3(March), 132. Easley, D., Kiefer, N., & O’Hara, M. (1996). Cream-skimming or profit sharing? The curious role of purchased order flow. Journal of Finance, 51(3), 811–833. Evans, M., & Lyons, R. (2002). Informational integration and FX trading. Journal of International Money and Finance, 21, 807–831. Fehle, F., & V. Zdorovtsov (2002). Large price declines, news, liquidity, and trading strategies: An intraday analysis. University of South Carolina Working Paper. Frino, A., & Hill, A. (2000). Intranight trading behaviour. University of Sydney Working Paper. Garman, M. (1976). Market microstructure. Journal of Financial Economics, 3, 257–275. George, T., Kaul, G., & Nimalendran, M. (1991). Estimation of the bid–ask spread and its components: A new approach. Review of Financial Studies, 4(4), 623–656. Harris, L. (1994). Minimum price variations, discrete bid–ask spreads, and quotation sizes. Review of Financial Studies, 7(1), 149–178. Hasbrouck, J. (1995). One security, many markets: Determining the contribution to price discovery. Journal of Finance, 40(4), 1175–1199. Heidle, H., & Huang, R. (2002). Information-based trading in dealer and auction markets: An analysis of exchange listings. Journal of Financial and Quantitative Analysis, 37(3), 391–424. Hendershott, T., & Jones, C. (2003). Island goes dark: Transparency, fragmentation, and liquidity externalities. University of California Working Paper. Huang, R. (2002). The quality of ECN and NASDAQ market maker quotes. Journal of Finance, 57(3), 1285–1319. Huang, R., & Stoll, H. (1996). Dealer versus auction markets: A paired comparison of execution costs on NASDAQ and the NYSE. Journal of Financial Economics, 41(3), 313–358. Huang, R., & Stoll, H. (1997). The components of the bid–ask spread: A general approach. The Review of Financial Studies, 10(4), 995–1034. Karmin, C., & Kelly, K. (2003). SEC urged to address electronic market risk. Wall Street Journal, 10(December). Lee, C. (1993). Market integration and price execution for NYSE-listed securities. Journal of Finance, 48(3), 1009–1038. Lee, C., & Ready, M. (1991). Inferring trade direction from intraday data. Journal of Finance, 46(2), 733–746. Masulis, R., & Shivakumar, L. (2002). Does market structure affect the immediacy of stock price responses to news? Journal of Financial and Quantitative Analysis, 37(4), 617–648. McInish, T., Van Ness, B., & Van Ness, R. (2002). After-hours trading of NYSE listed stocks on the regional stock exchanges. Review of Financial Economics, 11, 287–297. Merton, R. (1973). Theory of rational option pricing. Bell Journal of Economics and Management Science, 4(1), 141–183. Neal, R., & Wheatley, S. (1998). Adverse selection and bid–ask spreads: Evidence from closedend funds. Journal of Financial Markets, 1, 121–149.


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Putnam, G. D. (2004). Market structure III: The role of the specialist in the evolving modern marketplace. Written statement before the Committee on Financial Services – Subcommittee on Capital Markets, Insurance and Government Sponsored Enterprises United States House of Representatives, One Hundred Eighth Congress, 20 February 2004. SEC. (2000a). Electronic communications networks and after-hours trading. Special Study Department of Market Regulation, June. SEC. (2000b). Notice of filing of proposed rule change to rescind exchange rule 390; Commission request for comment on issues relating to market fragmentation. Release No. 34-42450 File No. SR-NYSE-99-48, Feb 23. SEC. (2002). Roundtable market structure hearing proceedings, October 29. Sloan, P. (2000). Trading the night away. U.S. News and World Report 128(10), 38, March 13. Stoll, H. (1978a). The supply of dealer services in securities markets. Journal of Finance, 33, 1133–1151. Stoll, H. (1978b). The pricing of security dealer services: An empirical study of NASDAQ stocks. Journal of Finance, 33(4), 1153–1172. Tinic, S. (1972). The economics of liquidity services. Quarterly Journal of Economics, 86(1), 79–93. Tinic, S., & West, R. (1972). Competition and the pricing of dealer services in the overthe-counter market. Journal of Financial and Quantitative Analysis, 8, 1707–1727. Tinic, S., & West, R. (1974). Marketability and common stocks in Canada and the USA: A comparison of agent versus dealer dominated markets. Journal of Finance, 29, 729–746. White, H. (1981). A heterskedasticity-consistent covariance matrix estimator and a direct test for heteroskedasticity. Econometrica, 48, 817–838.

THE SHARE PRICE AND TRADING VOLUME REACTIONS OF U.S.-LISTED FOREIGN BANKS TO THE FINANCIAL SERVICES MODERNIZATION ACT OF 1999 Carl Pacini, William Hillison and Bradley K. Hobbs ABSTRACT Recent research has examined the effect of the Financial Services Modernization Act of 1999, more commonly known as the Gramm–Leach– Bliley Act (GLB), on the market value of U.S. commercial banks, life insurers, property-liability insurers, thrifts, finance companies, and securities firms. This study fills a gap in our understanding of the Act by measuring the price and trading volume effects of the GLB on U.S.-listed foreign banks. A primary contribution of this study is to examine the role, if any, of two corporate governance perspectives, the stakeholder (code law), and shareholder (common law) models, in a cross-sectional analysis of foreign bank market reaction to the GLB. Using a generalized least squares (GLS) portfolio approach, Corrado’s rank statistic, and confirmed by the traditional market model approach, we find significant negative share price reactions to certain legislative announcements surrounding the passage of the GLB. Trading volume reactions corroborate the significant share price responses. In general, our Research in Finance, Volume 23, 127–159 Copyright r 2007 by Elsevier Ltd. All rights of reproduction in any form reserved ISSN: 0196-3821/doi:10.1016/S0196-3821(06)23005-3

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results indicate that investors in foreign banks reacted negatively to key legislative action. In a cross-sectional analysis, younger, higher-risk foreign banks with less concentrated ownership and more subordinated debt from countries with higher quality accounting standards appear to have more positive (or less negative) share price reactions.

1. INTRODUCTION Recent deregulation of financial services brought about by the Financial Services Modernization Act of 1999, also known as the Gramm–Leach– Bliley Act (GLB),1 has reduced the barriers between insurance, commercial banking, and investment banking. Given the link between the financial services industry and the global economy, it is imperative that a better understanding be gained about the market effects of regulation on financial institutions. Existing research primarily considers the shareholder wealth effects of the GLB on domestic firms including banks, life insurers, property-liability insurers, savings and loans, securities brokers, and finance companies (Akhigbe & Whyte, 2001; Carow & Heron, 2002; Hendershott, Lee, & Tompkins, 2002). Carow and Heron (2002) consider the GLB’s share price effects on U.S.-listed foreign banks but use a sample size of just 10. Such a small sample size makes tenuous the validity of statistical inferences. Most of these researchers also do not evaluate other measures of impact such as trading volume effects. Foreign banks are an important component of the financial system as they hold nearly 50 percent of all U.S. commercial and industrial loans (Deyoung & Nolle, 1996). The GLB’s influence cannot be properly evaluated without measuring its effects on foreign banks whose shares are listed in the U.S. As no study has sufficiently measured these effects, the impact of the GLB remains unclear. This study fills the gap in our understanding by empirically measuring the effects of the GLB on both share prices and trading volume. Determination of the market effects of the GLB is important to policymakers because it provides direction in formulating future regulations and to capital market participants because their wealth is affected by these regulations. Passage of the GLB allows an analysis of investors’ evaluations concerning scale economies, information sharing powers, and risk reduction through diversification by foreign banks. Unlike domestic bank securities, the cash flows of U.S.-listed foreign bank securities are primarily from a foreign country in a foreign currency. It is not clear if U.S.-listed foreign

The Share Price and Trading Volume Reactions

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bank securities should react to the GLB like U.S. bank securities, local market securities from their home country, or a mixture of both. Various researchers demonstrate that foreign banks in the U.S. may have comparative operational advantages over U.S. banks due to low-cost technology for intermediation, higher market capitalization, a lower cost of funds, greater cost efficiencies, and superior marketing strategies (Deyoung & Nolle, 1996). Hence, this study examines foreign banks as essentially a separate industry group from domestic banks. Our study differs from and improves upon existing studies in a number of ways. First, we consider a larger set of events relating to the passage of the GLB than most prior researchers. Second, we analyze a sample size of 41 U.S.-listed foreign banks as opposed to 10 by Carow and Heron (2002). Third, we provide a more thorough evaluation of market response because our study examines both trading volume and share price reactions. Tests of market response based on both trading volume and share price reaction are more reliable and precise than tests based on either metric alone (Cready & Hurtt, 2002). Fourth, we examine firm-specific determinants of the differential impact of the GLB on U.S.-listed foreign banks not considered by other studies that include such financial institutions. Fifth, this study examines the role, if any, of two corporate governance perspectives, the ‘‘stakeholder’’ (code law) and ‘‘shareholder’’ (common law) models, in explaining the market reaction of U.S.-listed foreign banks to the GLB. Consideration of corporate governance issues is important because the GLB incorporates such a perspective by requiring that a bank be ‘‘well-managed’’ as a condition for engaging in expanded activities (Macey & O’Hara, 2003). A determination of whether differences across perspectives exist should prove useful to legislators and regulators considering the inclusion of corporate governance viewpoints in related or similar legislation.2 This study identifies eight key events, starting with the reintroduction of HR 10 in October 1998 and ending with the disclosure that President Clinton would sign the legislation in November 1999. The dates and a description of the events are provided in Table 1. We find significant or marginally significant negative share price reactions for foreign banks to five key events. Those share price reactions were corroborated by significant abnormal increases in trading volume for the five events. In a cross-sectional analysis, we find that banks from countries with higher quality accounting standards (i.e., those that promote financial transparency) experience more positive (or less negative) share price reactions to the GLB events. Other variables for which we report significant results are organizational age, equity concentration, bank risk, and subordinated debt level.

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Table 1. Event

Date October 26, 1998 (Monday)

D2

April 26, 1999 (Monday)

D3

May 6, 1999 (Thursday)

D4

July 1, 1999 (Thursday)

D5

October 13, 1999 (Wednesdy)

Description Several House Republicans, led by James Leach, re-introduced HR10, last weekend, in the 106th Congress (which starts in 1/99). HR10 is a financial services modernization bill. An array of leading financial services firms signed a joint statement promising to work together for enactment (National Underwriter) The NAIC indicated that it opposes the House Banking Committee version of HR10 as hostile to the nation’s insurance consumers. Serious flaws the NAIC wants corrected are: (1) HR10 flatly prohibits states from regulating the insurance activities of banks except for certain sales practices; (2) HR10’s total elimination of state consumer protection powers; (3) HR10 prohibits states from preventing banks from affiliating with traditional insurers or engaging in insurance activities other than sales; (4) HR10 uses an ‘‘adverse impact test’’ to determine if state laws are preempted because they discriminate against banks; and (5) HR10 does not guarantee that state regulators will always have equal standing in federal court in disputes with federal regulators (PR Newswire) The Senate passed Senator Gramm’s financial services modernization bill after defeating two amendments that would have addressed concerns of President Clinton. One amendment would have allowed bank holding companies to engage in insurance underwriting. The second amendment would have strengthened Community Reinvestment Act evaluations. The measure passed on a 54–44 vote and was supported by all three financial services industries (The Houston Chronicle) The House passed HR10 and it is now headed for a House and Senate conference committee, so differences between the two bills can be worked out. Various parties in the banking and insurance industries are voicing concerns over HR10. The NAIC blasted HR10 saying it would leave insurance consumers without protection (National Underwriter) Republican House and Senate committee chairmen hammered out a compromise version of financial services reform but the White House said the proposed legislation inadequately protected consumers and would result in a presidential veto. Democrats criticized the bill for what they said are inadequate protections for consumer privacy. The issue many congressional aides and lobbyists say is most contentious is whether the Federal Reserve or Treasury will be the top bank regulator (The Washington Post)

CARL PACINI ET AL.

D1

Financial Services Modernization Act Legislative Events.

October 22, 1999 (Friday)

D7

November 4, 1999 (Thursday) November 11, 1999 (Thursday)

D8

A deal crafted in the predawn hours yesterday between the White House and Congress appears solid enough to ensure that a landmark bill to overhaul banking law will pass both the House and Senate and be signed into law by President Clinton within two weeks. The legislation is a compromise version of bills that earlier this year passed the House and Senate. The bill repeals the Glass–Steagall Act and a 1956 law that separates commercial banking from insurers. The legislation allows the Federal Reserve and Treasury to split oversight over banks entering new financial activities (The Washington Post) The Financial Services Modernization Act of 1999 passed the House of Representatives by a vote of 362–57 and the Senate by a vote of 90–8 (The Washington Post) President Clinton will sign the Financial Services Modernization Act of 1999 on Friday, November 12 (U.S. Newswire)


D6

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In the next section, we provide an overview of the GLB with an emphasis on provisions applicable to foreign banks that do business in the U.S. A review of theory and literature is followed by the study’s hypotheses and a description of our sample and methodology. We then discuss the results and conclude.

2. OVERVIEW OF THE GLB The Glass–Steagall Act of 1933 segmented the financial services industry and led to the development of separate and unique banking, insurance, and securities sectors in the U.S. Additional legislation enacted after Glass– Steagall, namely the McCarran Ferguson Act, the Bank Holding Company Act of 1956, and the Garn–St. Germain Act, created an awkward system of regulation among financial services industries. Regulatory barriers restricting financial services integration have been challenged in the courts and in Congress for much of the past two decades. In 1998, the 105th Congress nearly succeeded in repealing Glass–Steagall when the House narrowly passed HR 10; however, the Senate was unable to negotiate a compromise before the session ended. In the following year, both the House and Senate were able to reach an agreement and pass the GLB. The GLB allows the creation of financial holding companies (FHCs) that can engage in commercial and merchant banking, underwrite and sell both insurance and securities, and engage in certain real estate activities. U.S.based bank holding companies and foreign banks that meet certain criteria can become FHCs. Depository institutions held by an FHC must be and must remain well-capitalized, well-managed, and if FDIC-insured, have a satisfactory or better rating under the Community Reinvestment Act of 1977. A foreign bank is considered ‘‘well-capitalized’’ if: (1) the foreign bank’s home country has adopted risk-based capital standards consistent with the Basel Accord and the foreign bank maintains a Tier 1 capital-to-total-riskbased assets ratio of 6 percent and a total-capital-to-risk-based assets ratio of 10 percent, as calculated under home country standards; (2) the foreign bank maintains a Tier 1 capital-to-total-assets leverage ratio of at least 3 percent; and (3) the foreign bank’s capital is comparable to the capital required for a U.S. bank owned by an FHC. A foreign bank is ‘‘well-managed’’ if: (1) each of the U.S. branches, agencies, and commercial lending subsidiaries of the foreign bank has received at least a satisfactory composite rating at its most recent assessment; (2) the home country supervisor of the foreign bank considers the overall operations of the foreign bank to be satisfactory or better;


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and (3) the management of the foreign bank meets standards comparable to those required of a U.S. bank owned by an FHC. Thus, effective corporate governance and management are necessary for a foreign bank to operate in the U.S. in the post-GLB environment.

3. THEORY AND LITERATURE REVIEW 3.1. Wealth Effects and New Regulations Previous research indicates that new regulation impacts the value of financial services firms, both domestic and foreign. Cornett and Tehranian (1989) find that the passage of the Depository Institutions Deregulation and Monetary Control Act (DIDMCA) of 1980 had positive wealth effects for large domestic commercial banks and a negative impact on savings and loans. Mahajan, Dubofsky, and Fraser (1991) report a negative shareholder wealth effect for 31 foreign banks operating in the U.S. during enactment of the International Banking Act of 1978 (IBA). Wagster (1996) finds significant wealth effects for banks in Canada, Japan, Germany, the Netherlands, Switzerland, and the United Kingdom upon implementation of the Basel Accord in 1988. Research also reveals that new laws have asymmetric effects across banks with different characteristics. Liang, Mohanty, and Song (1996), using a sample of 164 BHCs, find that shareholders of well-capitalized banks benefited from passage of the Federal Deposit Insurance Corporation Improvement Act of 1991 (FDICIA) while those of undercapitalized banks experienced significant losses. Brook, Hendershott, and Lee (1998) find that banks with lower managerial stock ownership, higher outside block ownership, and/or fewer inside directors tend to report higher abnormal returns during the passage of the Interstate Banking and Branch Efficiency Act of 1994 (IBBEA). Asymmetric effects are also reported among banks relative to the passage of the GLB. Hendershott et al. (2002) find that larger and more profitable banks experienced more positive abnormal returns around passage of the GLB. Akhigbe and Whyte (2001) document that GLB enactment is associated with more positive share price reactions for larger and bettercapitalized banks. Carow and Heron (2002) report that the stock prices of domestic banks, both large and small, were unaffected by GLB enactment. Existing event studies involving the GLB, however, only consider stock price effects and do not analyze trading volume reaction.

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3.2. Trading Volume and New Regulations New regulation impacts not only firm share prices but also trading volume. Trading volume reflects changes in the expectations of individual investors while price reflects changes in the expectations of the market as a whole. Since public disclosures, including regulatory ones, convey relevant information about a firm, they will cause investors to revise their expectations about those attributes (Lobo & Tung, 1997). Investors’ expectation revisions should be more diverse around public announcements of unanticipated information and, as a result, trading volume should increase (Bamber, Barron, & Stober, 1999). Tkac (1999) supports this finding by reporting a link between increased trading volume and the information content of events such as earnings announcements, dividend policy changes, inclusion into the Standard & Poor’s index, and corporate control events. Volume and return responses are complementary measures that capture different aspects of investor response to information events (Cready & Hurtt, 2002). Volume and return responses are not substitutes for each other because their relation is closer to independence than it is to a strong positive association (Bamber & Cheon, 1995; Cready & Hurtt, 2002). The fact that substantial differences can exist between price and volume reaction suggests that trading volume-based research has the potential to yield insights beyond those attainable through price-based research (Bamber & Cheon, 1995). A literature review suggests that the GLB’s passage likely had an impact on foreign bank share prices and trading volume. Differences in countryand bank-specific features may help explain differences in share price and volume reactions. We now formulate hypotheses regarding the impact of the GLB on share price and trading volume reactions of foreign banks and the relation of governance-oriented variables to share price reactions.

4. HYPOTHESES DEVELOPMENT If diversification and synergies from selling different financial service products represent a benefit to foreign banks, then foreign bank equity values should rise when information supporting passage of the GLB becomes available. On a relative basis, foreign banks that are better positioned to capture scale economies, efficiency gains, risk reduction through diversification, and enter new markets should benefit most from the GLB. Moreover, foreign banks subject to an increased likelihood of takeover as a result of the GLB should experience a more significant increase in share value


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since the majority of gains from acquisitions accrue to target firm shareholders (Jerrell, Brickley, & Netter, 1988). If the GLB increased the threat of additional competition from insurers and other financial service providers, then stock prices of less competitive foreign banks should decline. Although banks had already gained partial entry into insurance, the GLB increased the threat of additional competition from insurers entering banking by the removal of protective barriers. Moreover, foreign banks may have been put at a competitive disadvantage by passage of the GLB. Many foreign banks have been able to underwrite securities and offer insurance services for years in their home markets. Because the U.S. market is larger than other markets, the GLB may spur U.S. banks to compete more aggressively in both domestic and overseas markets. Thus, it is unclear whether U.S.-listed foreign banks would experience expected net gains or losses from the passage of the GLB. This reasoning leads to the first hypothesis, stated in the null: H1. The abnormal returns of U.S.-listed foreign banks during the legislative enactment process of the GLB were not significantly different from zero. We also examine trading volume to provide additional evidence about reaction to the GLB. A test of both volume and return responses may be more powerful than an examination of just volume or return as the two are only modestly correlated (Bamber & Cheon, 1995). Moreover, Cready and Hurtt (2002) report that volume-based metrics are a more powerful measure of investor response to information events than return-based metrics. Since unanticipated disclosures concerning regulatory changes often convey relevant information about a firm, they are expected to lead investors to revise their expectations and trading volume should increase. Thus, the second hypothesis, stated in the null, is: H2. The trading volume of U.S.-listed foreign banks on GLB legislativeevent announcement days was not significantly different from trading volume on non-announcement days. Passage of the GLB may have different market effects on U.S.-listed foreign banks depending on firm- and country-specific characteristics, including various institutional and legal arrangements in a given country. Examination of the relation between bank market reaction and such characteristics is important for two reasons. First, the GLB incorporates a corporate governance perspective by requiring that a foreign bank be ‘‘well-managed’’ to expand. Second, extant international accounting research documents the significant influence of institutional arrangements, such as the quality of accounting

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standards, on the value relevance of financial reporting (Jaggi & Low, 2000; Ali & Hwang, 2000). Value relevance refers to the explanatory power of accounting variables for security returns (Ali & Hwang, 2000). To examine any asymmetrical market effects of the GLB, we test the following hypothesis: H3. The GLB legislative enactment process had no differential effect on the abnormal returns of U.S.-listed foreign banks possessing different firm-specific and/or country-specific characteristics.

5. SAMPLE AND METHODS 5.1. Sample Selection We collect our sample of U.S.-listed foreign banks by first identifying all foreign firms on Research Insight that have SIC Codes of 6021, 6022, 6029, or 6712. We also examine NYSE, AMEX, and NASDAQ listings of foreign financial institutions. We select only those foreign banks that have complete financial data on Research Insight and other required data contained in a 1999 Form 20-F or annual report and Standard & Poor’s Stock Reports. Information collected from these sources includes number of outstanding common shares, number of common shareholders, and age of the bank. This process resulted in an initial sample of 78 U.S.-listed foreign banks. The next step entailed searching the Lexis–Nexis Academic Universe database and other databases for confounding events on days –1, 0, and +1 related to any of the eight legislative events noted in Table 1.3 Earnings announcements, acquisitions, tender offers, bankruptcy filings, and income tax-related events in the U.S. and the foreign bank’s home country were those included as potential confounding events. Twenty banks with a confounding event were eliminated from the sample. Also, banks were excluded from the sample if daily stock return data were not available on the University of Chicago’s Center for Research on Security Prices (CRSP) database for at least 400 of the 476 trading days covered by this study.4 Seventeen banks were dropped for having insufficient returns leaving the final sample containing 41 U.S.-listed foreign banks from 19 countries. Nations represented in the sample include Argentina, Australia, Bermuda, Brazil, Canada, Chile, Columbia, Germany, Greece, Ireland, Italy, Japan, Luxembourg, the Netherlands, Panama, Peru, Portugal, Spain, and the United Kingdom. Banks from these nations were classified into the appropriate stakeholder (code law) and stockholder (common law) categories. Panels A and B of Table 2 provide sample information.


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Table 2. Sample Analysis. Panel A. Sample Size U.S.-listed foreign banks with complete data for cross-sectional model Less: banks with confounding events Less: banks with CRSP data unavailable (insufficient number of returns) Final sample size (37 NYSE and 4 NASDAQ)

78 20 17 41

Panel B. Sample Firms by Legal Environment Legal Environment

Number of Banks

Stakeholder (code law)

21

Stockholder (common law)

20

Countries Argentina, Brazil, Chile, Columbia, Germany, Greece, Italy, Japan, Luxembourg, Netherlands, Panama, Peru, Portugal, and Spain Australia, Bermuda, Canada, Ireland, and United Kingdom

Panel C. Descriptive Statistics for Cross-Sectional Independent Variables

Accounting standards (ACGSTD) (index score from 0 to 100 from La Porta et al. (1998) where higher scores indicate higher quality accounting standards) Organizational age (AGE) (number of years a bank has been in business) Ownership concentration (HOLD) (average number of shares per shareholder) Risk (RISK) (variance of abnormal returns) Size (SIZE) (market value of equity in millions of $) Subordinated Debt (SUBDEBT) (subordinated debt as a percent of total assets)

Mean

Median

63.05

71.00

109.35

115.00

37,238

13,309

0.000594 1,499,917

0.000487 318,373

0.0169

0.0187

5.2. Methodology for Analyzing Share Price Reactions We employ three methods to measure the share price reactions associated with GLB legislative event disclosures: a generalized least squares (GLS) portfolio approach, a non-parametric technique termed Corrado’s rank

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statistic (Corrado, 1989), and the traditional parametric approach. Since sample firms share common event dates and are members of the same industry, their stock returns may be subject to cross-sectional correlation (Bernard, 1987). The failure to compensate for cross-sectional dependence leads to downward-biased estimates in the standard errors of regression coefficients and excessive rejection of the null of no abnormal performance in event studies (Bernard, 1987). The first two methods used here are robust to cross-sectional correlation (Corrado & Zivney, 1992; Bernard, 1987). The traditional market model approach, a parametric procedure, is used for comparison purposes and as a sensitivity check. Following Baber, Kumar, and Verghese (1995), we employ a GLS portfolio approach (rather than SUR)5 that involves an expanded version of the market model with a zero-one dummy variable to reflect the occurrence or non-occurrence of each event: Rpt ¼ B0 þ B1 Rmt þ B2 Rmt1 þ B3 Rit þ Sj gj Djt þ ept

(1)

where Rpt is the equally weighted portfolio return for day t; B0 is the model intercept; B1 is the systematic risk of the portfolio; Rmt is the market return for day t, computed as the return for an equally weighted portfolio of NYSE, AMEX, and NASDAQ stocks;6 B2 is the coefficient on the lagged CRSP equally weighted market index; Rmt1 is the lagged return on the CRSP equally weighted market index on day t-1; Rit is the daily change in the interest rate on the 30-year treasury bond;7 gj is the coefficient measuring the abnormal return for event j; Djt assumes the value 1 for day t if it is the jth event day and 0 otherwise; and ept is a disturbance term. Dummy variable Djt distinguishes days involving legislative event disclosures. We set the dummy variable Djt equal to one for the announcement day (day 0), the preceding day (day –1), and the day after the announcement day (day +1) (Cornett & Tehranian, 1990).8 Thus, the model captures significant changes in market expectations across the eight three-day event windows. The parameters represented by gj are estimates of average abnormal portfolio returns. The forecast error of each gj considers the contemporaneous correlation between the residuals. For several compelling reasons, we also utilize a non-parametric technique (Corrado’s rank statistic). First, normality of abnormal returns identified from the traditional market model is a key assumption in event studies (Campbell & Wasley, 1993). A series of abnormal return distributions was tested for normality and found to be non-normal.9 Second, cross-sectional dependence exists because all sample firms share common event dates and belong to a common industry (Bernard, 1987; Corrado, 1989). Third,


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foreign firms listed on U.S. exchanges may be susceptible to thin trading and thin trading may cause parametric t-tests to be misspecified (Campbell & Wasley, 1993). Fourth, parametric tests on abnormal or standardized abnormal returns in traditional event study approaches are vulnerable to misspecification caused by an increase in the variance of event-day abnormal return distributions (Corrado, 1989). Finally, parametric t-tests are based on the assumption that market model residuals are not serially correlated. In sum, we conclude that the assumptions necessary for the use of traditional parametric t-tests are sufficiently violated to preclude using them other than for comparison purposes and as a sensitivity check. In the application of Corrado’s rank statistic, each sample firm’s series of abnormal returns from the standard market model is converted into ranks (from 1 to 476). The ranking procedure transforms each abnormal return distribution into a uniform distribution regardless of asymmetry in the original distribution (Corrado, 1989). Ranks are then standardized by dividing each abnormal return by one plus the number of non-missing returns in each bank’s return series (Corrado & Zivney, 1992). Standardization prevents the rank statistic from becoming misspecified in the presence of missing returns and serves as a cross-sectional variance adjustment to improve specification in tests for abnormal performance (Corrado & Zivney, 1992). The rank test statistic is the ratio of the mean deviation of the securities’ event day ranks to the estimated standard deviation of the portfolio mean abnormal return rank.10 Campbell and Wasley (1993) demonstrate that Corrado’s rank statistic is robust to cross-sectional dependence, multi-day event periods, combined samples of NYSE, AMEX, and NASDAQ securities, increases in the variance of abnormal returns on event dates, overlapping sample periods, alternative ways of estimating beta, and applies regardless of how serial dependence in abnormal returns is considered.11 As used in our analysis, Corrado’s rank statistic has the power and specification of the Wilcoxon two-sample rank test (Corrado, 1989).

5.3. Methodology for Analyzing Trading Volume Unlike measuring abnormal returns, there is no generally accepted method of measuring unexpected trading volume (Bamber et al., 1999). Current accounting and finance literature includes numerous unexpected volume measures (Tkac, 1999). Consistent with Bamber et al. (1999) and Lobo and Tung (1997), we use the percentage of a foreign bank’s outstanding shares traded on a given day in our analysis of trading volume. Abnormal volume

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for combined days –1, 0, and +1 for each of the eight event disclosures is computed as the deviation of a foreign bank’s event day volume from its mean daily volume for non-announcement days (Lobo & Tung, 1997). We use one-tail t-tests to assess whether daily abnormal volume on combined days –1, 0, and +1 for each event is significantly positively different from that on non-announcement days. 5.4. Methodology for Cross-Sectional Analysis A key element of this research is to investigate the relation between market returns and firm-specific, institutional, and corporate governance variables. We use a GLS rank regression model to test whether abnormal returns are related to a set of diverse variables: quality of accounting standards, organizational age, ownership concentration, bank size, bank risk, and subordinated debt. Descriptive data on these variables are contained in panel C, Table 2. A GLS approach is used to compensate for any cross-sectional correlation in cumulative abnormal return ranks recognizing that our sample involves event date clustering and intra-industry correlation (Bernard, 1987). We use a three-day event window (days –1, 0, and +1) because its use is consistent with prior research and is a conservative approach for rejection of the null hypothesis of no abnormal performance. Ranks rather than actual data values are used because ranks generalize the functional form of the model and minimize heteroskedasticity that can result from using a linear function to represent a non-linear relation (Cheng, Hopwood, & McKeown, 1992; Bamber & Cheon, 1995). Moreover, ranks are standardized by the number of observations plus 1 so that the ranked variable has a maximum value of N/(N+1) and a minimum value of 1/(N+1) with N equaling the number of data values. The standardization yields coefficients that are independent of the number of observations (Cheng et al., 1992). We derive the following cross-sectional rank regression model: CARRi ¼ b0 þ b1 ACGSTDi þ b2 AGEi þ b3 HOLDi þ b4 SIZEi þ b5 RISKi þ b6 SUBDEBTi þ ei

ð2Þ

where CARRi is the cumulative abnormal return ranks for bank i for respective event days; ACGSTDi the standardized rank of the foreign bank’s home country’s accounting standards quality index (from La Porta, Lopezde-Silanes, & Shleifer, 1998); AGEi the standardized rank of the number of years that foreign bank i has been in business; HOLDi the standardized rank of the average number of shares per shareholder; SIZEi the standardized


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rank of bank i’s market value of equity as of 10/26/98; RISKi the standardized rank of bank i’s variance of abnormal returns for the 476-day sample period; SUBDEBTi the standardized rank of subordinated debt as a percent of total assets of foreign bank i; and ei a disturbance term. Our cross-sectional model includes governance-related variables, such as quality of accounting standards, organizational age, ownership concentration, and subordinated debt level because corporate governance problems pose a greater risk for banks than other business enterprises. Higher risk stems from two sources: a virtual absence of hostile takeovers as a form of management discipline (Booth, Cornett, & Tehranian, 2002) and a higher degree of leverage in foreign banks relative to other firms and U.S. banks thus magnifying the impact of managerial actions on shareholder wealth (Deyoung & Nolle, 1996; Macey & O’Hara, 2003). Each of the independent variables is discussed next. 5.4.1. Accounting Standards (ACGSTD) Accounting information plays a crucial role in corporate governance. Financial statements and their footnotes provide information about economic transactions to investors and creditors while auditing serves as a monitoring mechanism to check on the fairness of reported information and to deter financial fraud. Contracts between managers and investors typically rely on the verifiability in court of measures of firms’ income, assets, and owners’ equity (La Porta et al., 1998). Accounting and disclosure standards may be necessary for financial contracting especially if investor rights are weak. Following Ball, Kothari, and Robin (2000) and Jaggi and Low (2000), we consider differences in countries’ financial accounting and disclosure standards as reflecting underlying differences in institutional influences on accounting. One common proxy for institutional influences is a classification of countries into code law systems with high institutional influence and common law systems in which accounting standards are determined mostly in the private sector (Ball et al., 2000). Differences between code law and common law countries in the extent of institutional influences on accounting are reflected in what Ball et al. (2000) call the ‘‘stakeholder’’ (code law) and ‘‘shareholder’’ (common law) models of corporate governance. The degree to which accounting rules are legislated can impact the nature of the accounting system. In code law countries, laws stipulate minimum requirements and accounting rules tend to be highly prescriptive and procedural (Jaggi & Low, 2000). The demand for information concerning accounting income is influenced more by the payout preferences of agents for labor, capital, and government and less by the demand for public disclosure.

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These stakeholders have incentives to reduce the volatility of accounting income. It has been observed that code law accounting provides managers more latitude in smoothing income (Jaggi & Low, 2000; Ball et al., 2000). In common law nations, laws establish limits beyond which it is illegal to venture and within those limits experimentation is encouraged and judgment is required (Jaggi & Low, 2000). Under the shareholder governance model typical of common law countries, accounting standards are developed more in the private sector rather than by the government. Payments to various groups are less closely connected to current period accounting income and third parties have less impact on corporate governance and less access to inside information (Guenther & Young, 2000). Hence, shareholders, creditors, and others demand accurate and timely information. Thus, managers in common law nations have less flexibility to smooth reported earnings. Jaggi and Low (2000) report that firms from common law countries are associated with a higher level of financial disclosure than firms from code law countries. Guenther and Young (2000) find that accounting earnings in common law nations are more closely related to underlying economic activity than accounting earnings in code law countries. Ball et al. (2000) show that common law accounting income is timelier than code law accounting income. In sum, accounting standards in common law nations tend to promote more transparent financial reporting than those in code law countries. The measure of the quality of accounting standards we use is from La Porta et al. (1998). It is an index based on an examination of annual corporate reports from 44 countries and consistent in the classification of code versus common law countries. The index ranges from 0 to 100 with a higher score representing more transparent accounting standards. We expect a positive relationship between ACGSTD and foreign bank CARRs.

5.4.2. Organizational Age (AGE) Organizational age has proven to be a powerful construct in institutional literature (Judge & Zeithaml, 1992). Because organizations change slowly, those founded earlier than others in different environmental conditions should yield different behaviors than those started later. It has been posited that older organizations have more difficulty overcoming momentum and will be less likely to respond quickly to change (Judge & Zeithaml 1992; Eisenhardt, 1988). In one study, Eisenhardt (1988) documented that organizational age was a reliable predictor of compensation practices in the retail industry. She attributed the different compensation practices of older and younger organizations to the varying conditions in the industry’s life cycle.


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Since older banks or other organizations were formed at a time when external pressures for board involvement and active outside directors were weaker than now, these banks may offer more resistance to increased board involvement in key policy decisions (Judge & Zeithaml, 1992). Expanded board involvement to facilitate strategic policy changes may be necessary in response to the increased competitive environment engendered by the passage of the GLB. In contrast, younger foreign banks may have more flexible boards of directors that adapt smoother and faster to the regulatory changes brought about by the GLB. In sum, the institutional perspective predicts that organizational age will be negatively associated with CARRs because inertia prevents older banks from adapting to the GLB’s new competitive environment. 5.4.3. Ownership Concentration (HOLD) Research suggests that large shareholders are often active in corporate governance. Thus, large shareholders such as institutions may affect firm value through their impact on managerial and board decisions. The relation between firm value and large holdings of stock is addressed by various theories. According to the efficient monitoring hypothesis, large shareholders often support managerial and board decisions enhancing firm value but oppose decisions detrimental to shareholder interests (Pound, 1988). Concentrated share ownership facilitates coordinated shareholder action to demand information from managers with which to assess their performance. Significant owners have more expertise and can monitor management at lower cost than individual shareholders. Large equity owners may inhibit managerial and director tendencies to reduce shareholder value through the adoption of risk-reducing strategies. The efficient monitoring hypothesis predicts a positive relation between ownership concentration and CARRs (McConnell & Servaes, 1990). The strategic alignment hypothesis suggests that large equity owners and management cooperate for their mutual benefit. For instance, in acquisition contests significant owners act with target management to defeat a takeover bid (Pound, 1988). Such cooperation offsets the positive effects on firm value from institutional monitoring (McConnell & Servaes, 1990). A third argument, the conflict-of-interests hypothesis, contends that large equity owners, in some situations, may negatively affect firm value (Pound, 1988). Owing to financially lucrative relationships with a firm, significant equity owners may be forced to vote with management on issues that are harmful to other shareholders.

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A fourth theory, the short-term investment hypothesis, maintains that the presence of large shareholders negatively impacts firm value because such owners are driven by short-term profit considerations (Graves, 1988). The latter three hypotheses predict a negative relation between foreign bank CARRs and ownership concentration (McConnell & Servaes, 1990). Empirical evidence is mixed on the relation between concentrated ownership and firm value. Holderness and Sheehan (1988) find no difference in Tobin’s q and accounting rates of return for a sample of 114 firms in which one shareholder owns more than one half of a firm’s stock and another sample of companies in which no shareholder holds more than 20 percent of the stock. Pound (1988) finds that some value-increasing proxy bids do not occur because significant owners are more likely to support management. Despite inconsistent empirical findings, the weight of theory leads us to predict a negative relation between CARRs and large shareholdings. We proxy large concentrated ownership with the average number of shares per shareholder. 5.4.4. Bank Size (SIZE) Research results are inconsistent regarding the relation between foreign bank size and the likely benefit from the passage of the GLB. Larger foreign banks may benefit if consolidation, due to deregulation, results in increased market power or leads to improved economies of scale. The advantages of a larger diversified organization may be offset by the increased costs of operating a more complex organization. Other studies suggest that smaller banks may benefit more from the passage of the GLB. For example, the majority of gains from acquisitions accrue to the shareholders of target firms (Jerrell et al., 1988) and smaller banks are more likely takeover targets than larger banks. In addition, the differential information hypothesis indicates that security price reactions to disclosures of unanticipated information are usually more substantial for smaller firms. Given the weight of theory, we predict a negative relationship between foreign bank size and CARRs. 5.4.5. Bank Risk (RISK) Foreign banks, like domestic banks, face numerous risks. Total risk is composed of several components: (1) the risk that loans will not be paid back in a timely fashion (credit risk); (2) the risk of financial instability associated with higher leverage (leverage risk); (3) the risk associated with the sensitivity of earning assets to interest rate changes (interest rate risk); (4) the risk from excess geographic or industry concentration (concentration risk); (5) the risk associated with the covariance of the bank’s cash flows with that of a market portfolio (systematic risk); and (6) the risk associated


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with poor or fraudulent management (management risk). We proxy total risk with the variance of abnormal returns, a market measure less subject to manipulation than an accounting measure (Demsetz & Strahan, 1997; Allen & Jagtiani, 2000). One important public policy issue surrounding the GLB is the effect of expansion by foreign banks into U.S. non-banking activities on bank risk. The ‘‘earnings diversification’’ hypothesis posits that expanding banks seek earnings diversification in an effort to generate greater cash flow for the same levels of total risk (Benston, Hunter, & Wall, 1995). Expanding foreign banks may chose to move along the risk-expected return frontier and take the benefits of diversification as higher returns by shifting their portfolios toward higher risk-expected return investments (Berger, Demsetz, & Strahan, 1999). However, diversification can also create costs. Diversified firms may invest more in negative net present value projects. Foreign bank expansion into U.S. non-bank activities may impose costs on or increase the risk of the U.S. financial system by expanding the safety net provided by deposit insurance to non-bank subsidiaries (Berger et al., 1999). Ultimately, the effect of the GLB on U.S.-listed foreign bank risk is an empirical question. Demsetz and Strahan (1997) find that through diversification large banks are able to operate with higher leverage and engage more in risky lending (with higher returns) without increasing bank risk. Allen and Jagtiani (2000) conclude that permitting banks to underwrite securities and insurance will likely lower the overall risk of banks but raise banks’ systematic risk depending upon the intensity of securities and insurance activities. Lown, Osler, Strahan, and Sufi (2000) find that mergers of banks and life insurers lower risk but mergers of banks with propertyliability insurers, securities firms, and real estate developers increase risk. Thus, we predict a positive relation between foreign bank risk and CARRs. 5.4.6. Subordinated Debt (SUBDEBT) Subordinated debt possesses characteristics that make it an attractive means by which to increase the market discipline of banks. Subordinated debt is not insured and is likely to decrease in value in the event of bank failure (Chen, Robinson, & Siems, 2004). If investors price the effects of changes in bank risk into securities then bank owners and managers are disciplined in the sense that they must take into account the full impact of their business decisions (Flannery & Sorescu, 1996). Moreover, some empirical research indicates that subordinated debt yields are risk-sensitive. Bank holding company subordinated debt spreads have been used for bank supervisory surveillance purposes (Hancock &

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Kwast, 2001). The GLB directs the Federal Reserve to report to Congress concerning the feasibility of requiring certain bank holding companies to maintain some portion of their capital as subordinated debt (Chen et al., 2004). Improved market discipline of both domestic and U.S.-listed foreign banks may be viewed in a positive light by shareholders. A regulatory requirement of inclusion of subordinated debt in a bank’s capital structure could result in greater disclosure and transparency in financial statements. In the long run, investor uncertainty diminishes with greater disclosure and more transparency. Chen et al. (2004) find that passage of the GLB is associated with positive share price reactions for domestic banks with relatively high percentages of subordinated debt in their capital structures. Hence, we predict a positive relation between the relative level of subordinated debt and share price response of U.S.-listed foreign banks.

6. EMPIRICAL RESULTS 6.1. Share Price and Trading Volume Results Table 3 reports the results for Corrado’s rank statistic, the GLS portfolio approach, and the traditional market model for share price reactions. Table 4 highlights trading volume results. The introduction of HR 10 on October 26, 1998 (D1) produced marginally significant negative share price reactions measured by Corrado’s rank statistic (t ¼ 1.68, p ¼ 0.093) and the traditional market model parametric test (t ¼ 1.72, p ¼ 0.085) but not using GLS. The marginally significant share price reaction is supported by significant trading volume for event D1 (t ¼ 1.66, p ¼ 0.050). The marginal nature of the share price response for D1 may be attributable to the uncertainty surrounding ultimate passage of HR 10 given that Congress had already failed to pass the same bill in the previous session. The announcement that the National Association of Insurance Commissioners (NAIC) opposed HR 10 in April 1999 (D2) is associated with marginally significant negative abnormal returns (GLS, t ¼ 1.73, p ¼ 0.084; Corrado, t ¼ 1.64, p ¼ 0.100; parametric, t ¼ 2.11, p ¼ 0.035). The share price reaction is corroborated by a significant rise in trading volume (t ¼ 1.74, p ¼ 0.041). Passage of Senator Gramm’s financial services modernization bill by the Senate (D3) generated a significant negative share price reaction under all three statistical approaches (Corrado, t ¼ 1.89, p ¼ 0.059; GLS, t ¼ 2.10,

Event

D1 Oct. 26, 1998

D2 April 26, 1999

D3 May 6, 1999

D4 July 1, 1999

D5 Oct. 13, 1999

Corrado’s Rank (T)

p-valuea

p-valuea

Parametric t-statistics

p-valuea

Reintroduced HR10 last weekend in the 106th Congress (which starts in 1/99) The NAIC indicated that it opposed the House Banking Committee’s version of HR10 The Senate passed Senator Gramm’s financial services modernization bill The House passed HR10 and it is headed for a Senate/House conference committee A compromise of the bill was adopted but the White House threatened a veto

1.68

0.093

0.00418

1.50

0.134

1.72

0.085

1.64

0.100

0.00502

1.73

0.084

2.11

0.035

1.89

0.059

0.00695

2.10

0.036

2.42

0.016

0.12

0.904

0.00102

0.18

0.855

0.01

0.995

0.69

0.490

0.00538

1.34

0.181

1.61

0.108

GLS Est’d GLS Coefficient t-statistics (portfolio average abnormal return)

147

Description


Table 3. Results for Corrado’s Rank Statistic, GLS Portfolio Approach, and Parametric Approach.

148

Table 3. (Continued ) Description

Corrado’s Rank (T)

p-valuea

The White House and Congress compromise appears solid enough to enact the bill The Financial Services Modernization Act passed both the House and Senate President Clinton will sign the Financial Services Modernization Act of 1999

1.68

0.093

0.00807

1.77

0.076

0.23

0.819

Event

D6 Oct. 22, 1999

D7 Nov. 4, 1999

D8 Nov. 11, 1999

a

p-valuea

Parametric t-statistics

p-valuea

1.78

0.075

1.87

0.062

0.00848

1.92

0.055

2.14

0.033

0.00254

0.63

0.527

0.37

0.712

GLS Est’d GLS Coefficient t-statistics (portfolio average abnormal return)

Two-tailed values.

CARL PACINI ET AL.

p-value of 0.05 or less. p-value of 0.10 or less.

Trading Volume Analysis.

Event D1 Oct. 26, 1998

D2 April 26, 1999

D3 May 6, 1999 D4 July 1, 1999

D5 Oct. 13, 1999

D6 Oct. 22, 1999

D7 Nov. 4, 1999 D8 Nov. 11, 1999

Several House Republicans led by James Leach, reintroduced HR10 last weekend in the 106th Congress (which starts in 1/99) The NAIC indicated that it opposed the House Banking Committee’s version of HR10 The Senate passed Senator Gramm’s financial services modernization bill The House passed HR10 and it is headed for a Senate/House conference committee A compromise version of financial services reform but the White House threatened a veto The White House and Congress appears solid enough that the banking law will be enacted The Financial Services Modernization Act passed both the House and Senate President Clinton will sign the Financial Services Modernization Act

p-valuea

Prediction

t-statistics

+

1.66

0.050

+

1.74

0.041

+

1.72

0.043

+

0.20

0.422

+

0.42

0.337

+

3.19

+

2.26

0.012

+

0.83

0.203


Table 4.

o0.01

a

149

One-tailed p-values. Trading volume reaction tests are one-tailed as significant abnormal trading volume occurs only in one direction (i.e., positive direction). p-value of 0.05 or less.

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CARL PACINI ET AL.

p ¼ 0.036; parametric, t ¼ 2.42, p ¼ 0.016). A significant increase in trading volume corroborated the significant negative share price reaction (t ¼ 1.72, p ¼ 0.043). One explanation for the negative share price response to the event D3 may be the GLB’s capitalization requirements for conducting business as an FHC in the U.S. Since many foreign countries impose lower capital requirements than the U.S., the GLB’s capitalization standards impose new costs and burdens on U.S.-listed foreign banks. Another reason is that the potential diversification and expansion opportunities made available to U.S. banks may represent a competitive threat to foreign banks. Analysis of events D4 and D5 did not reveal any significant shareholder reaction. These events apparently did not convey significant new information to the market. We now focus attention upon D6 and D7 – the two legislative events that resolved uncertainty about the form of regulatory change. Event D6, the compromise between Congress and the White House, essentially assured the repeal of the Glass–Steagall barriers allowing commercial banks to expand beyond traditional banking activities into insurance, merchant banking, real estate, and securities underwriting. Analysis of event D6 indicates marginally significant negative shareholder reaction for all three measures: Corrado’s rank statistic (t ¼ 1.68, p ¼ 0.093), GLS approach (t ¼ 1.78, p ¼ 0.075), and the traditional market model (t ¼ 1.87, p ¼ 0.062). This reaction is in stark contrast to the significant positive share price response of U.S. domestic banks reported by Akhigbe and Whyte (2001) and Hendershott et al. (2002). And although Carow and Heron (2002) also report a significant negative share price reaction for U.S.-listed foreign banks, their sample size was only 10 banks. Our results also indicate that trading volume increased significantly for event D6 (t ¼ 3.19, p o 0.01). Apparently event D6 led investors to revise their expectations and beliefs about the probability of enactment of the GLB with perceived losses for U.S.-listed foreign banks. Results related to event D7 extend this conclusion. All three measures, Corrado’s rank statistic (t ¼ 1.77, p ¼ 0.076), the GLS portfolio approach (t ¼ 1.92, p ¼ 0.055), and the traditional market model (t ¼ 2.14, p ¼ 0.033) indicate a marginally significant negative shareholder reaction to formal passage of the GLB by the House and Senate. The significant rise in trading volume corroborates that investors realigned their portfolios in response to D7 (t ¼ 2.26, p ¼ 0.012). These findings suggest that event D7 reaffirmed investor reactions associated with D6 by raising the probability of GLB passage to almost unity. However, the formality that the President supported the Act noted by event D8 did not result in significant shareholder reaction.


151

Overall, we conclude that the results shown in Tables 3 and 4 lead to the rejection of the first and second hypotheses. Most GLB legislative event disclosures are associated with negative share price reactions and increases in trading volume. Given these negative shareholder reactions, it is likely that institutional factors and firm-specific characteristics play a significant role in shareholder expectations. The next section addresses those factors and characteristics. 6.2. Cross-Sectional Results We analyze the cross-sectional variation of the stock price impact of all eight legislative events in the aggregate by estimating GLS rank regression Eq. (2). Given that some events resulted in no significant reactions, we view this as a conservative approach. CARRs were regressed on quality of accounting standards, organizational age, ownership concentration, size, bank risk, and subordinated debt level. All independent variables, except size, exhibit a significant or marginally significant relation with CARRs. Results in Table 5 demonstrate that quality of accounting standards (ACGSTD) is significant in the predicted direction (t ¼ 1.98, p ¼ 0.048). The findings indicate that foreign banks from countries with more transparent financial reporting experienced greater (or less negative) CARRs during passage of the GLB. Shareholders, creditors, and others have less access to inside information in common law countries that follow the shareholder governance model, so they likely demand more accurate and timely information. Information asymmetry and transaction/monitoring costs are both reduced by more transparent financial reporting. In sum, investor uncertainty is less with higher quality reporting and disclosure. The organizational age (AGE) variable carries a significant coefficient in the predicted negative direction (t ¼ 3.32, p o 0.01). This result implies that older foreign banks experienced more negative abnormal returns upon passage of the GLB. Our results are consistent with Judge and Zeithaml (1992) who contend older organizations offer more resistance when strategic change is necessary to respond to a changed competitive environment. It is likely that newer foreign banks have more flexible boards of directors and management who can adapt to the deregulation brought about by the GLB. We find that foreign banks with more concentrated ownership have greater negative abnormal returns (t ¼ 1.87, p ¼ 0.062). Our results are consistent with the strategic alignment hypothesis, the conflict-of-interests hypothesis, and the short-term investment hypothesis. The strategic alignment hypothesis predicts that significant owners act with target management

152

Table 5. Variable Intercept ACGSTD AGE HOLD SIZE RISK SUBDEBT Adjusted R2 F-Value F-Probability

CARL PACINI ET AL.

GLS Cross-Sectional Rank Regression Model Results (All Event Days Aggregated). Predicted Sign

Estimated Coefficient

t-statistics

p-valuea

n/a + + + 0.334 3.58 0.010

9.45 2.03 2.86 0.87 0.89 1.13 1.32

9.95 1.98 3.32 1.87 1.49 1.68 1.85

0.064 0.048 o0.01 0.062 0.136 0.094 0.064

Note: For evaluation purposes, additional tests of the general model were performed. The OLS regression model was tested for multicollinearity using partial correlation coefficients, variance inflation factors, and conditional indices. No pair of independent variables had a partial correlation coefficient greater than 0.7 or less than 0.7. Additionally, each independent variable had a VIF o3 and a condition index o12. Multicollinearity is a problem when the VIF exceeds 10, a condition exceeds 30, or a partial correlation coefficient is >0.7 or o0.7 (Kennedy, 1992). a Two-tailed p-values. Statistical significance at the 0.05 level. Statistical significance at the 0.10 level.

to defeat any takeover bids made for foreign banks. The conflict-of-interests argument suggests that significant equity owners who have financially beneficial relationships with foreign banks may side with management on issues that might decrease firm value. The short-term investment hypothesis speculates that large equity owners driven only by short-term profit considerations may reduce equity holdings in response to the GLB. As reported in Table 5, greater bank risk (RISK) is associated with marginally significant positive share price reactions of U.S.-listed foreign banks (t ¼ 1.68, p ¼ 0.094). Foreign banks with a higher variance of abnormal returns tend to have a more positive share price reaction to the GLB. These results provide support for the earnings diversification hypothesis which predicts that expanding foreign banks may choose to move along the riskexpected return frontier and reap benefits of higher returns by shifting their portfolios toward higher risk investments (Berger et al., 1999). Investors may believe that foreign banks, although already diversified, can further diversify by acquisition of insurers, U.S. banks, securities firms, or finance companies to pursue riskier more profitable investments (Demsetz &


153

Strahan, 1997). Moreover, the results for RISK are consistent with some foreign banks being potential takeover targets. Acquired higher risk foreign banks may experience greater financial gains from lower funding costs, improved credit standing, and additional contributed capital. Higher levels of subordinated debt are associated with marginally significant positive share price responses of U.S.-listed foreign banks (t ¼ 1.85, p ¼ 0.064). This result is consistent with the finding of Chen et al. (2004) that passage of the GLB is associated with positive wealth effects for shareholders of domestic banks with higher levels of subordinated debt. U.S.-listed foreign bank shareholders seem to both understand the greater reliance placed by the GLB on effective corporate governance (or market discipline) and the reduction of investor uncertainty that may accompany the required inclusion of subordinated debt in bank capital structure. From a public policy perspective, mandatory subordinated debt issuance may improve bank corporate governance, institutional monitoring, and mitigate the risk-taking incentives from deposit insurance. The cross-sectional model is significant with an F-value of 3.580. The adjusted R2 of 0.334 suggests that more than a modicum of return rank variance is explained by the independent variables in our cross-sectional model. In sum, the results permit us to reject H3 and conclude that quality of accounting standards, organizational age, ownership concentration, bank risk, and level of subordinated debt help explain significant variance in foreign bank shareholder reaction to the passage of the GLB. Younger, higher risk banks with more subordinated debt and less concentrated ownership from countries with higher quality accounting standards experienced more positive share price reactions upon passage of the GLB.

7. CONCLUSION In 1999, Congress passed the GLB Act to permit financial services integration. The GLB incorporates a corporate governance perspective because it requires that U.S.-listed foreign banks be ‘‘well-managed’’ as a prerequisite for engaging in expanded activities. The GLB legislative process provided new information to the market concerning the future of the financial services industry. We document significant negative share price reactions and increases in trading volume for U.S.-listed foreign banks to certain legislative events leading up to GLB passage. The significant negative market responses of U.S.-listed foreign banks are in contrast to positive share price reactions of domestic banks examined in other studies.

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This study evaluates the role of two corporate governance perspectives, the ‘‘stakeholder’’ (code law) and ‘‘shareholder’’ (common law) models, in explaining the cross-sectional variation of U.S.-listed foreign bank share price reactions to GLB passage. Results indicate that foreign banks from countries with more transparent financial reporting (‘‘shareholder’’ model) experienced more positive (or less negative) share price reactions than those from countries with lower quality accounting standards (‘‘stakeholder’’ model). Also, evidence shows that older foreign banks had more negative share price reactions than younger foreign banks. Older organizations offer more resistance when strategic change is required to respond to a changing competitive environment. Foreign banks with more concentrated ownership had more negative share price responses than foreign banks with greater ownership dispersion. Moreover, foreign bank risk levels appear to be related to stock returns. In addition, higher levels of subordinated debt are associated with positive share price responses. Investors were apparently able to discriminate between the impact of GLB enactment on U.S.-listed domestic and foreign banks on an individual event and industry basis.

NOTES 1. For detailed information on the GLB, see ‘‘Overview of the Gramm–Leach– Bliley Act’’ from the Federal Reserve Bank of San Francisco at http://www. frbsf.org/publications/banking/gramm/grammpgl.html 2. For example, the Sarbanes–Oxley Act of 2002 contains numerous references to the importance of corporate governance issues to stakeholders. Issues such as independent directors and directors with financial expertise are addressed by the Act. 3. The event dates used in our study differ slightly from those utilized in Carow and Heron (2002), Akhigbe and Whyte (2001), and Hendershott et al. (2002). The explanation for the differences in event dates involves the original sources from which the dates were obtained. Akhigbe and Whyte (2001) and Carow and Heron (2002) note that they obtain event dates from the Wall Street Journal. The event dates used here were taken from the source identified in the Lexis–Nexis database that first reported the event. The Wall Street Journal is not always the first source that publicly discloses legislative events. Also, one of our events, D6, was first disclosed on a Saturday when the market was closed. We coded the event days as the three market days surrounding the event. 4. The start date for the statistical analysis of share price reaction is 200 days before the announcement date of event D1 (the introduction of HR 10). The end date is 10 days after the announcement date of event D8 (the President’s signing of the GLB). 5. Akhigbe and Whyte (2001) and Carow and Heron (2002) use seemingly unrelated regression (SUR) while this study uses GLS. The GLS procedure used here (Parks method) is equivalent to Zellner’s two-stage SUR methodology (Zellner, 1962; Kennedy, 1992; see the appendix).


155

6. Chan, Cheung, and Wong (2002) compare various event study methods and stock indices for foreign stocks listed on U.S. exchanges. Their results indicate the CRSP equal- and value-weighted indices are as effective as the MSCI world index and MSCI country indices. The findings also show that the standard market, marketadjusted, and mean-adjusted models perform equally as well as the two-index model. 7. Banks and savings and loan stock returns appear not responsive to short-term rates but are sensitive to long-term rates (Unal & Kane, 1987). 8. Cornett and Tehranian (1990) use a two-day event window including the announcement day (day 0) and the preceding day (day 1). The use of a three-day event window allows for the possibility of information leakage prior to an event as well as post-information announcement drift. 9. The skewness and kurtosis coefficients and the Shapiro–Wilk statistic were calculated for a random sample of 25 days from the 476-day sample period. A perfectly symmetrical distribution has a kurtosis coefficient of three. The mean kurtosis coefficient across the 25 days is 2.75. Large kurtosis values indicate leptokurtic distributions or ones with ‘‘heavy tails.’’ Kurtosis has been shown in both univariate and multivariate analyses to have an effect on power. The abnormal returns are also positively skewed (mean skewness of 0.406). The Shapiro–Wilk statistic can assume a value between 0 and 1. The statistic must be extremely close to 1 (e.g., 0.99) for a distribution to be considered normal. The abnormal return distributions tested have a mean S–W statistic of 0.935. 10. The rank test statistic, T, substitutes (Uit1/2) for the abnormal return: N U it 12 1 X T ¼ pffiffiffiffiffi N i¼1 sðU Þ where Uit is the standardized abnormal return rank of bank i on day t during the 476-day sample period. Uit can assume any value between 0 and 1; N the number of firms; and s(U) the standard deviation of the portfolio mean abnormal return rank for the sample period.The denominator of T, s(U) is computed as follows: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u ! Nt þ10 u 1 X 1 X 1 2 t pffiffiffiffiffiffi U it 476 t¼200 2 N t i¼1 where Nt is the number of non-missing returns in the cross-section of N-firms on day t in the sample period (Corrado & Zivney, 1992). 11. Brockett, Chen, and Garven (1999) note that classical event study techniques may lead to incorrect statistical inferences by not allowing for shifts in the beta coefficient during the estimation and/or event periods and changes in the variance of abnormal returns during the event period. The methodologies employed in this study and a test we performed address these concerns. Corrado’s rank statistic is robust to increases in the variance of abnormal returns (or time-varying conditional variance) during the event period (Campbell & Wasley, 1993). The GLS approach allows for variance shifts. We also tested for beta shifts and found no statistically significant shifts in beta during the 476-day sample period.

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REFERENCES Akhigbe, A., & Whyte, A. M. (2001). The market’s assessment of the Financial Services Modernization Act of 1999. The Financial Review, 36, 119–138. Ali, A., & Hwang, L. S. (2000). Country-specific factors related to financial reporting and the value relevance of accounting data. Journal of Accounting Research, 38(Spring), 1–21. Allen, L., & Jagtiani, J. (2000). The risk effects of combining banking, securities, and insurance activities. Journal of Economics and Business, 52, 485–497. Baber, W., Kumar, K., & Verghese, T. (1995). Client security price reactions to the Laventhol and Horwath bankruptcy. Journal of Accounting Research, 33(Autumn), 385–395. Ball, R., Kothari, S. P., & Robin, A. (2000). The effect of international institutional factors on properties of accounting earnings. Journal of Accounting and Economics, 29, 1–51. Bamber, L., Barron, O. E., & Stober, T. (1999). Differential interpretations and trading volume. Journal of Financial and Quantitative Analysis, 34(September), 369–386. Bamber, L., & Cheon, S. (1995). Differential price and volume reactions to accounting earnings announcements. The Accounting Review, 70(July), 417–441. Benston, G., Hunter, W., & Wall, L. (1995). Motivations for bank mergers and acquisitions: Enhancing the deposit insurance put option versus earnings diversification. Journal of Money, Credit, and Banking, 27(August), 777–788. Berger, A. N., Demsetz, R., & Strahan, P. (1999). The consolidation of the financial services industry: Causes, consequences, and implications for the future. Journal of Banking and Finance, 23, 135–194. Bernard, V. (1987). Cross-sectional dependence and problems in inference in market-based accounting research. Journal of Accounting Research, 25(Spring), 1–48. Booth, J., Cornett, M. M., & Tehranian, H. (2002). Boards of directors, ownership, and regulation. Journal of Banking and Finance, 26, 1973–1996. Brockett, P., Chen, H. M., & Garven, J. R. (1999). A new stochastically flexible event study methodology with application to proposition 103. Insurance: Mathematics and Economics, 25(November), 197–216. Brook, Y., Hendershott, R., & Lee, D. (1998). The gains from takeover deregulation: Evidence from the end of interstate banking restrictions. Journal of Finance, 53(December), 2185–2204. Campbell, C. J., & Wasley, C. E. (1993). Measuring security price performance using daily NASDAQ returns. Journal of Financial Economics, 33(June), 73–92. Carow, K., & Heron, R. (2002). Capital market reactions of the passage of the Financial Services Modernization Act of 1999. Quarterly Review of Economics and Finance, 42(Summer), 463–485. Chan, K. C., Cheung, J., & Wong, H. (2002). A comparison of event study methods for foreign firms listed on U.S. stock exchanges. Journal of International Accounting Research, 1, 75–90. Chen, A. H., Robinson, K., & Siems, T. (2004). The wealth effects from a subordinated debt policy: Evidence from the passage of the Gramm–Leach–Bliley Act. Review of Financial Economics, 13, 103–119. Cheng, C. S. A., Hopwood, W., & McKeown, J. (1992). Non-linearity and specification problems in unexpected earnings response regression model. The Accounting Review 67(Summer), 579–598.


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APPENDIX Parks (1967) considered the model in which the random errors U ij l ¼ 1; 2; . . . N; j ¼ 1; 2; . . . T have the structure E U 2ij ¼ sii ðheteroscedasticityÞ E U ij U kj ¼ sik ðcontemporary correlatedÞ U it ¼ ri ui; t1 þ it ðautoregressionÞ where E(eij) ¼ 0, E(Ui,j1ekj) ¼ 0, E(eijekj) ¼ Fik, E(eijekl) ¼ 0(j6¼1), E(Uio) ¼ 0, E(UioUjo) ¼ Fij/(1rirj). The model assumed is first-order autoregressive with contemporaneous correlation between cross-sections.


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The covariance matrix, V, is estimated by a two-step procedure, leaving b to be estimated by the usual estimated GLS. The first step in estimating V involves the use of ordinary least squares to estimate b and obtain the fitted residuals U ¼ Y X bOLS The autoregressive characteristic of the data can be removed by a transformation of taking weighted differences. The second step in estimating the covariance matrix V is to apply ordinary least squares to the transformed model, obtaining U ¼ Y X bOLS from which S ij ¼

Fij 1 ri r j

where Fij ¼

1 T p

X T

U ik U jk

K¼1

provides a consistent estimator of Fij. Estimated GLS then proceeds in the usual manner as b ¼ ðX 0 V^

1

X Þ1 X 0 V^

1

Y

where V^ is the consistent estimator of V. The preceding set of steps is equivalent to Zellner’s two-stage methodology (Zellner, 1962).

UPPER BOUNDS FOR AMERICAN OPTIONS Mo Chaudhury ABSTRACT This paper provides a fuller characterization of the analytical upper bounds for American options than has been available to date. We establish properties required of analytical upper bounds without any direct reliance on the exercise boundary. A class of generalized European claims on the same underlying asset is then proposed as upper bounds. This set contains the existing closed form bounds of Margrabe (1978) and Chen and Yeh (2002) as special cases and allows randomization of the maturity payoff. Owing to the European nature of the bounds, across-strike arbitrage conditions on option prices seem to carry over to the bounds. Among other things, European option spreads may be viewed as ratio positions on the early exercise option. To tighten the upper bound, we propose a quasi-bound that holds as an upper bound for most situations of interest and seems to offer considerable improvement over the currently available closed form bounds. As an approximation, the discounted value of Chen and Yeh’s (2002) bound holds some promise. We also discuss implications for parametric and nonparametric empirical option pricing. Sample option quotes for the European (XEO) and the American (OEX) options on the S&P 100 Index appear well behaved with respect to the upper bound properties but the bid–ask spreads are too wide to permit a synthetic short position in the early exercise option.


161

162

MO CHAUDHURY

1. INTRODUCTION Analytical bounds for American option prices are interesting from both theoretical and practical perspectives. They provide theoretical restrictions for arbitrage-free pricing and optimal early exercise of American options. As most American option valuation problems require simultaneous determination of the early exercise boundary, their practical implementation involves numerical methods that may become computationally burdensome, in particular when there are multiple state variables.1 Bounds, especially if they are analytical and closed form, can be useful in such circumstances in providing valuation guidelines,2 developing approximations,3 implying information from the observed American option prices,4 setting trading restrictions such as dollar margin requirements on written options, managing the market risk of American option portfolios, and determining capital adequacy rules for institutional portfolios. The aim of this paper is twofold. First, we specify some general properties of the analytical upper bounds for American options where the bounds themselves are construed as contingent claims on the same asset. Second, we propose as upper bounds a class of generalized European claims that are not specific to preferences and are also independent of the exercise boundary. Together they provide a rigorous economic characterization of the analytical upper bounds for American options that is intuitively appealing. At the same time these bounds are easier to compute and invert, and hence should be useful for valuation guidance and information extraction purposes. There are three distinct and parallel lines of existing research on option bounds. Bounds based on the physical distribution or moments (e.g., Perrakis & Ryan, 1984; Lo, 1987; Grundy, 1991) are primarily limited to European options.5 Bounds relying on the exercise policy/boundary of American options (e.g., Broadie & Detemple, 1996, Rogers, 2002, Andersen & Broadie, 2004) tend to produce tighter bounds. These bounds are most useful in situations where optimal exercise and accurate option valuation is the main focus (like executive and employee options) and/or the options do not have a liquid secondary market (like many OTC and structured products). However, typically these bounds are not in closed form even when they are analytic, i.e., they require iterative optimization or regression. Accordingly, their use is highly restrictive in dealing with large datasets and in the presence of multiple state variables, especially in implying information from the observed American option prices. For example, the vast majority of exchange-traded equity options are American options, and so are some widely popular index (e.g., S&P 100) or exchange traded fund (e.g., S&P 500

Upper Bounds for American Options

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Depository Receipts known as SPIDERS, NASDAQ 100 Trust known as Cubes) options. Implying information from these option prices, using say a stochastic volatility model, can be a formidable task if one takes the early exercise boundary route. The third type of bounds are neither preference-dependent, nor do they have any direct reliance on exercise policies/boundaries (e.g., Margrabe, 1978, Chen & Yeh, 2002, Chaudhury & Wei, 1994, Chung & Chang, 2005). Instead this line of bounds research looks for analytic functions in closed form to bound American options. Obviously these bounds may not be as suitable as the early exercise based bounds when the highest level of accuracy in valuation and exercise are of primary importance. However, the analytic closed form bounds can be quite useful in dealing with large datasets and especially in implying information from observed American option prices. In prior research on closed form analytical bounds, Chaudhury and Wei (1994) and Melick and Thomas (1997) offer bounds for American futures options, but they do not apply to American put options on spot assets such as stocks, bonds, and foreign currency. Most recently, Chen and Yeh (2002) have provided closed form6 upper bounds that are applicable to these options as well and are quite fast computationally.7 Chung and Chang (2005) further generalize Chen and Yeh’s bounds and extend the approach to the case of options on multiple assets. Margrabe (1978) first noticed that the value of a European put option with the strike price compounded at the risk-free rate is an upper bound for the American put option. Chen and Yeh (2002)’s upper bound, on the other hand, is the expected maturity payoff or the pure (futures-style margining) option value of a European put option on a fraction of the asset, where the fraction adjusts for the net growth of the asset. We shall henceforth refer to these bounds as the Adjusted Strike European option (AKE) and the Adjusted Asset Pure European option (ASPE) bounds, respectively. The importance of these bounds is that they do not require the knowledge of the early exercise boundary and are as easy to calculate as the European option value. This paper builds on prior works on analytical upper bounds for American spot options in several ways. First, while an impressive literature exists on the characterization of American option upper bounds in terms of the exercise boundary, an independent characterization of the analytical upper bounds themselves is lacking. An important objective of this paper is to fill this gap. We develop fundamental properties required of an analytical upper bound that is a contingent claim on the same underlying asset and share the same maturity as the American option. Since our characterization is

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completely in terms of the value of claims and not their boundaries, this should enhance our understanding of the economic nature of American options and their bounds. Second, we propose a set of generalized European claims as upper bounds for an American spot option that contains the AKE and ASPE bounds as special cases. An important benefit of a European claim as an upper bound is that its value is considerably easier to calculate than the target American option while the specific option valuation setup remains largely in tact. Since Chen and Yeh’s (2002) bounds are pure option values or expected maturity payoffs and not European options, the bounding European options of this paper also help to tighten the bounds. In a closely related work and citing an earlier version of this paper, Chung and Chang (2005) generalizes Chen and Yeh’s bounds to analytic functions that amount to adjusting both the strike price and the units of assets of standard European options. As discussed later (Footnote 15), their generalized bounds are in fact special cases of the generalized European claims in this paper. Also, they do not provide full economic characterization of these bounds. It is to be noted that the analytical bounds of Chen and Yeh (2002), Chaudhury and Wei (1994), and this paper are bounds on model option values under the hypothesized distribution. While there is no restriction on the nature of the distribution,8 as mentioned earlier the closed form analytical bounds are more useful when there are multiple state variables, or when the exercise boundary of neither the target option nor the bounding claim is of interest. This is a special appeal of the European claims proposed in this paper as bounding claims. Third, although the primary role of an upper bound is to provide a ceiling on the American option’s value and guidance in regards to its exercise policy, other potentially important implications of an upper bound has not drawn much attention in the literature. In this paper, we discuss several of these implications. In the context of arbitrage conditions on option prices across various strikes, an interesting question is whether the respective upper bounds also satisfy similar conditions. This seems like a desirable property of a set of upper bounds as the credibility of the upper bounds in tracking the American options is enhanced. Another interesting implication of the generalized European claims of this paper as upper bounds is that European option spreads across different strikes essentially allow trading of early exercise options without ever trading the American options. Implications like this along with the traditional role of a price ceiling make upper bounds quite relevant for empirical option pricing.


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Although a detailed empirical study is beyond the scope of this paper, we examine sample option quotes for the European (XEO) and the American (OEX) options on the S&P 100 Index. These quotes appear well behaved with respect to the upper bound properties. A synthetic long position in the early exercise option seems quite expensive and the bid–ask spreads are too wide to permit a synthetic short position in the early exercise option. This could explain why the XEO contracts are not as popular as the OEX contracts and would suggest redesigning the OEX contracts purely as early exercise options. Lastly, despite their many benefits, one weakness of the analytical upper bounds that do not rely on the exercise boundary is that the bounds themselves are not quite accurate in approximating the American option value. To this end, we propose a quasi-bound that leads to significant improvement in pricing accuracy over the AKE and ASPE bounds. While the quasi-bound is truly an upper bound for most situations of interest, there still remains a set of circumstances where it is not meaningful. To summarize, the contribution of this paper lies in providing a thorough economic characterization of the analytical upper bounds for American options using (generalized) European claims that are more tractable both intuitively and computationally. An analytical and closed form quasi-bound is also proposed that is tighter and covers most practical situations. Further, the paper discusses novel implications for empirical pricing of options, spreads, and the early exercise option. As the dividend yield or leakage for most spot options is less than the riskfree rate, the pure option upper bound applies to American call options on these spot assets. Accordingly, we focus on spot put options in this paper. Section 2 specifies fundamental requirements for upper bounds on American options. In Section 3, a set of generalized and possibly randomized European claims (and a set of generalized but nonrandomized American claims) is proposed as upper bounds for the standard American options. We then discuss in Section 4 some interesting implications of our characterization of American option upper bounds. In section 5, we propose a quasi-bound that is truly an upper bound in most situations of interest. We present some numerical results to show the improvement in pricing accuracy offered by the quasi-bound. While an empirical study is beyond the scope of this paper, we briefly examine in Section 6 the applicability of bounds using sample CBOE option quotes for the S&P 100 European (XEO) and American (OEX) option contracts. Lastly, Section 7 summarizes and concludes the paper.

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2. THE FUNDAMENTAL UPPER BOUND REQUIREMENTS Let St denote the current price of the underlying spot asset with continuous and possibly stochastic leakage rate dt, and let Vt (vt) stand for the current value of an American (European) option with strike price K and maturity time (time to maturity) T (t ¼ Tt). The intrinsic value or immediate exercise proceeds, Xt, of Vt is (KSt)+ for a put option and (StK)+ for a call option, where (Y)+ indicates Max (0, Y). The fact that the European option value vt may fall below this intrinsic value Xt, and that the American option must satisfy the moving boundary condition VtZXt, is at the heart of the American option valuation problem. In the rest of the paper, we assume that a risk-neutral or equivalent martingale measure exists and all expectations and moments are under this measure. The instantaneous risk-free rate at time t is rt and it is allowed to vary stochastically over time unless mentioned otherwise. The discount factorR for valuing at time t the j-period hence cash flows is thus Rt,j ¼ exp(– tt+jrsds). Our objective here is to bind Vt from above by the value Gt of another contingent claim on the same asset and having the same maturity time T. Also, for obvious reason, we restrict our analysis to contingent claims with convex payoffs. Let us define such claims as the generalized claims, G. The intrinsic or immediate exercise value of G, if it is an American claim, is to be denoted as XG,t. The foremost requirement for an upper bound is stated in the following lemma. Lemma 1. To qualify as an upper bound for the American option value Vt, the value Gt of the generalized claim must never fall below the intrinsic value Xt of Vt . Proof: The American option value Vt is the greater of its continuation value and its immediate exercise or intrinsic value, where the continuation value itself represents the value of capturing potentially higher intrinsic value at a future time. Therefore, if Gt falls below Xt over any range of the asset price St, then Gt cannot be an upper bound for Vt. QED The converse of Lemma 1 is an important result on bounding the American option.


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Lemma 2. If the value Gt of the generalized claim never falls below the intrinsic value Xt of the American option, then Gt is an upper bound for Vt. Proof: Starting at maturity time T, we have VT ¼ XT and GT ¼ XG,T. At time TD, VTD ¼ Max[RTD,DETD(VT),XTD], and GTD ¼ RTD,DETD(GT) if G is European and GTD ¼ Max[RTD,DETD(GT), XG,TD] if it is American. Now, suppose GTZVT. Then, at time TD, the discounted value of the generalized claim is greater than the continuation value of the American option, i.e., RTD,DETD(GT)>RTD,DETD(XT). If, in addition, GTDZXTD, then we have GTDZVTD, as VTD ¼ Max[RTD,DETD(XT), XTD]. Continuing backward, the discounted value of G would be no less than the continuation value of V, and if in addition GT2DZ XT2D prevails, then once again GT2DZVT2D, where VT2D ¼ Max[RT2D,DET2D(VTD), XT2D]. By continuing to work backward, it is shown that GtZVt. QED. Lemma 2 and its proof closely follow the Theorem 1 of Chen and Yeh (2002) and their method of proof. However, there are some important differences here. Let us first reproduce their Theorem 1 below. Theorem 1 of Chen and Yeh (2002): ‘‘An American option is bounded from above by the risk neutral expectation of its maturity payoff if this expectation is greater than the intrinsic value at all times.’’

This theorem is quite general and only assumes that the risk-neutral measure exists and the discount factor is strictly between 0 and 1 over all sample paths. In our terminology, the generalized claim G is an American claim in Chen and Yeh’s Theorem 1. The risk-neutral expectation of the maturity payoff of G is in fact the price of a pure European option that is a European option with futures-style margining. If the value of a pure European option always exceeds its intrinsic value, then the early exercise feature does not add any value and as such the pure American option value equals the pure European option value. Since a pure American option is clearly more valuable than a conventional American option, the pure option value serves as an upper bound.9 For spot options, however, the pure European option value may fall below the intrinsic value. This may occur for call options if there is a high positive leakage on the underlying asset. For put options, just a low enough asset price may cause the pure European option value to fall below its intrinsic value. Therefore, the pure American option value exceeds the pure European option value (expected maturity payoff).10 Although the pure American option value exceeds the conventional American option value, the

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pure European option value or the expected maturity payoff is no longer an upper bound for the conventional American option value over the entire range of asset prices. In other words, if the claim G is an American call option with high leakage or a standard American spot put option, then the expected maturity payoff or the pure European value of G is not an upper bound for G.11 There are three critical ways our Lemma 2 improves upon Chen and Yeh’s Theorem 1. First, Chen and Yeh’s theorem concerns bounding the American option value by its own expected maturity payoff or pure European value. Lemma 2 here, on the other hand, uses a generalized claim G, possibly different, from the target American option V that we wish to bind from above. Thus our Lemma 2 enlarges the set of bounding claims compared to Chen and Yeh.12 Second, Lemma 2 is general enough to allow generalized bounding claims that are European as long as the intrinsic value condition of Lemma 2 is met. This flexibility is due to the fact that G is possibly different from V, although both are contingent claims on the same spot asset and have same maturity. The simplest examples of Gt are St to bind the standard American call option and K to bind the standard American put option. Since European claims are less valuable than pure European claims (Expected Maturity Payoff), our Lemma 2 opens up a set of tighter upper bounds. Third, the American or early exercise feature is most valuable for the standard American put options. However, Chen and Yeh’s Theorem 1 does not directly apply to standard American put options. Our Lemma 2, on the other hand, applies to all American put options including the standard ones. It is, of course, possible to combine Chen and Yeh’s Theorem 1 and our Lemma 2 to suggest the requirement for the expected maturity payoff or pure European value Et(XG,T) of a generalized American claim to be an upper bound for the standard American spot option value Vt. Lemma 3. A standard American spot option’s value Vt is bounded from above by Et(XG,T), the risk neutral expectation of the maturity payoff (or the pure European option value) of a generalized American option, if the generalized American option G satisfies the conditions: (a) Et(XG,T) is never less than its own intrinsic value XG,t, and (b) Gt never falls below the intrinsic value Xt of V. Proof: If condition (a) above is met, then by Chen and Yeh’s Theorem 1, Et(XG,T) is an upper bound for the American option value Gt of the generalized claim. If condition (b) is satisfied, then by Lemma 2, Gt bounds Vt


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from above. Therefore, combining (a) and (b), Et(XG,T) is an upper bound for Vt. QED. The following corollary gives the sufficient condition for the condition (b) of Lemma 3 to hold. Corollary 1. A sufficient condition for the generalized claim’s American option value Gt to stay above the intrinsic value Xt of the standard American option V is that G’s intrinsic value XG,t never falls below V’s intrinsic value Xt. Proof: If XG,t never falls below Xt, then the claim G dominates the claim V in terms of payoff under all circumstances. Hence, to prevent arbitrage, the price Gt must be at least as high as the price Vt . But by the intrinsic value boundary condition, VtZXt. Therefore, Gt ZXt follows. QED. It is important to note two things. First, Corollary 1 provides a condition that relates just the intrinsic values of the bound and the American option. To our knowledge, such a relationship is novel. Second, Lemma 3 and Corollary 1 apply to generalized American claims. According to Lemma 2, this American aspect in and of itself is not necessary to bound Vt. A good example of a European generalized claim that satisfies Lemma 2 and thus bounds the standard American option is Margrabe’s (1978) AKE option. We shall shortly discuss such claims.

3. NEW GENERALIZED CLAIMS AS UPPER BOUNDS We now turn to the important task of structuring a generalized claim G such that Gt satisfies the requirements for bounding the American spot option value Vt. We propose a general structure, not specific to any stochastic process for the underlying asset. Although it is not essential, we assume for convenience a constant interest rate r and a constant leakage rate d.13 The net risk-neutral drift of the asset is thus assumed to be a constant y ¼ (rd)>0.14 Lemma 4. Suppose G is a European put option with the following maturity payoff function: XG,T ¼ [er(Tt)eTK–ed (Tt)ZTST]+, where eT and ZT are positive random variables with Et(eT) ¼ 1, Et(ZT) ¼ 1, Variancet(eT) ¼ se2, Variancet(ZT) ¼ sZ2, and Covariancet (eT,ST) ¼ Covariancet (ZT,ST) ¼ Covariancet(eT,ZT) ¼ 0, for all t under the risk-neutral measure. Then, the generalized European put option’s value Gt is an upper bound for the standard American put option’s value Vt.

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Proof: G t ¼ erðTt-Þ E t ½erðTtÞ T K edðTtÞ ZT S T þ erðT -tÞ ½KE t ðerðTtÞ T Þ E t ðedðTtÞ ZT ST Þþ ðrdÞðTtÞ

¼ ½K e ¼ ½K S t þ

ðby convexityÞ

þ

E t ðST Þ ðby assumptionÞ ðgiven the drift of the assetÞ

Since GtZ[KSt]+ ¼ Xt, for all t, then by Lemma 2, GtZVt. QED. The European put option G generalizes the conventional put option to a situation where the buyer at time t has the right to sell, at time T, a random number ed(Tt) ZT of asset units for a random total price of er(Tt)eTK. The upper bound Gt, on the other hand, is easy to compute once the risk-neutral distribution for the terminal asset price is specified. The AKE bound is a special case of the generalized European put option here, with deterministic eT ¼ ZT ¼ 1. With constant interest and leakage rates, the ASPE bound for the American put option is: G CY ;t ¼ E t ðK eðrdÞðTtÞ STÞþ ; this bounding put option is a pure European option with eT ¼ ZT ¼ er(Tt-) Either compounding the strike price (without slicing the optioned amount of the asset), or slicing the optioned amount of asset (leaving the strike unchanged), essentially enhances moneyness of the bounding claim relative to V.15 We shall revisit this interesting insight later in the paper. The economic intuition behind the AKE and ASPE bounds and the generalized European claims here is that by enhancing the moneyness of the option, these bounds are effectively factoring in the present value of expected net interest earnings in the exercise region.16 The value of these bounds never falls below the intrinsic value of the standard American option and as such the implicit boundary is the strike price K of the standard American option. The strike price K is of course higher than the actual time varying exercise boundary of the standard American option. Consequently, the upper bounds reflect a higher present value of expected net earnings in the extended exercise region and end up bounding the standard American option’s value. The potential role that we have in mind for the randomization of the maturity payoff is to make the lock-in value (notional early exercise value) of the generalized European option uncertain although maintaining the same expected value. More work is needed to fully explore the implications of the randomization scheme. Note that, in AKE and ASPE and in Lemma 4, a different generalized European option bounds the American option value at different points of the latter’s life. This can be realized from the presence of the time to


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maturity (Tt) in the maturity payoff. In contrast, the time to maturity does not appear in the maturity payoff function for standard European and American options. The bounding G at time t corresponds to a European option to sell ed (Tt)ZT units of asset for a total price of er(Tt)eTK at time T. But the bounding G at time t+j corresponds to a European option to sell ed (Ttj) ZT units of asset for a total price of er(Ttj)eTK at time T. In other words, to bound future values of the standard American put option, the bounding option G would call for selling rights on fewer units of the asset at a lower total price at maturity. As maturity approaches, the bounding G’s optioned number of asset units approach the still unknown random number ZT and the total exercise price tends to K times the still unknown random number eT. Thus, once the random numbers realize at maturity, the payoff of Lemma 4’s G may not bound V’s payoff. This terminal weakness of Lemma 4’s European G arises because it never has a meaningful intrinsic or exercise value of its own as the proposed randomization leaves the number of optioned asset units and the total exercise price undetermined until at maturity. Under stochastic interest and leakage rates, a similar situation arises with the AKE and ASPE bounds. While the expected strike price at maturity (AKE) and the expected number of optioned asset units (ASPE) are always known as they are in Lemma 4 here, the exact numbers are known only at maturity. Interestingly, perhaps not unexpectedly, the G that bounds Vt, continues to bound the future values of V until exactly at maturity. Let us denote the value of t-specific bounding claim G’s value at time t+j as Gt,t+j. Corollary 2. Suppose G is a European put option with the following maturity payoff function: XG,T ¼ [er(T-t)eTK–ed(Tt)ZTST]+, where eT and ZT are positive random variables with Et(eT) ¼ 1, Et(ZT) ¼ 1, Variancet(eT) ¼ se2, Variancet(ZT) ¼ sZ2, and Covariancet(eT,ST) ¼ Covariancet(ZT,ST) ¼ Covariancet(eT,ZT) ¼ 0, for all t under the risk-neutral measure. Then, the generalized European put option’s value Gt,t+j is an upper bound for the standard American put option’s value Vt+j for 0rjoT. Proof: G t;tþj ¼ erðTtjÞ E tþj ½erðTtÞ T K edðTtÞ ZT S T þ erðTtjÞ ½KE tþj ðerðTtÞ T Þ E tþj ðedðTtÞ ZT ST Þþ rj

¼ ½Ke e

ðrdÞðTtÞ

rj

dj þ

¼ ½Ke S tþj e ½K St

þ

rj þ

E tþj ðS T Þe

ðby assumptionÞ

ðgiven the drift to the assetÞ

ðif r4d as assumedÞ

ðby convexityÞ

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Since Gt,t+jZ[K–St+j]+ ¼ Xt+j, for 0rjoT, then by Lemma 2, Gt,t+jZVt+j. QED. Our specifications so far for the bounding claim G have been of European type. One key reason why European type bounding claims may be preferred is because European claims are valued less than their American counterparts and as such are likely to provide tighter upper bounds for the standard American option value. Further, if G is of American type but its (intrinsic) value may exceed its expected maturity payoff in violation of Lemma 3, then it does not help the cause of skipping the computation of an American claim. Consider, for example, the ASPE bound. While its expected maturity payoff or pure option value, G CY ;t ¼ E t ðK eðrdÞðTtÞ S T Þþ ; never falls below [K–St]+, there is no guarantee that GCY,t will not below its own intrinsic value ðK eðrdÞðTtÞ S t Þþ ; if it were of American type. This is because it’s intrinsic value ðK eðrdÞðTtÞ S t Þþ also exceeds [K–St]+ as long as the asset has a positive risk-neutral drift (r>d). Consider a numerical example to see this. Suppose the Black–Scholes setup applies with current time t ¼ 0, maturity time T ¼ 3, current stock price St ¼ $80, strike price K ¼ $100, constant risk-free rate r ¼ 10%, dividend yield or leakage rate d ¼ 0%, and the constant volatility rate s ¼ 30%. Using these values, the maturity payoff function of Chen and Yeh’s G is XG,T ¼ Max(0,1000.7408 ST) . The (risk-neutral) expected maturity payoff of G at time t, namely the pure European option value of G is Et[XG,T] ¼ $30.08. It is above the $20 intrinsic value of the standard American option and indeed it is an upper bound for the standard American option value of Vt ¼ $21.27. 17 However, Et[XG,T] is below its own intrinsic value, XG,t ¼ Max(0,1000.7408*80) ¼ 40.73. Therefore, if the G is of American type, then it is potentially useful as an upper bound for V only if G satisfies Lemma 3, since in that case the upper bound, namely the expected maturity payoff of G, should be easy to calculate. We now propose generalized American claims, the expected maturity payoff of which can be used as upper bounds. In this context, we abstract from randomizing the terminal payoff. As mentioned earlier, the early exercise value and premium of an American option becomes difficult to interpret, if at all possible, when dealing with a randomized payoff function. Further, for simplicity, the generalization involves only the number of optioned asset units. The following lemma presents a class of bounding American claims under these circumstances. Lemma 5. Suppose, at time t, the intrinsic value of the generalized American put option G is XG,t ¼ Max(0, K–ltSt), where 0oltr1 is a


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deterministic monotonically decreasing function of time (i.e., @lt/@to0) and lToltSt/Et(ST). Then, the expected maturity payoff, Et(XG,T), is an upper bound for the conventional American spot put option value, Vt. Proof: Let t* be the optimal stopping time for the conventional spot put option V. Then, Z t ru du ðK St Þþ V t ¼ E t exp t Z t þ ru du fðK lt S t Þ þ S t ðlt 1Þg ¼ E t exp t Z t þ ru du ðK lt S t Þ E t exp t

GA t In the first inequality, we have used the property that for real numbers a and b, (a++b+)Z(a+b)+, and that 0olt*r1. The second inequality follows from the fact that t* is the optimal stopping time for V and not for G. So far, we have shown that the generalized claim’s American option value is an upper bound for the conventional American put option. Now, the generalized claim G’s pure European option value or expected maturity payoff is Et(KlTST)+. By the convexity of payoff, we have: E t ½K lT S T þ ½K lT E t ðS T Þþ ¼ ½K lt S t flT E t ðST Þ=lt S t gþ ½K lt S t þ The last inequality follows from the restriction 0olToltSt/Et(ST) or 0olTEt(ST)/ltSto1. Thus the pure European option value or expected maturity payoff of G never dips below its intrinsic value. This means the pure American value of G equals its expected maturity payoff. Since the pure American value of G is greater than the American value of G (GtA), its expected maturity payoff is an upper bound for GtA. As we have shown that GtA is an upper bound for Vt, it follows that the generalized claim G’s pure European option value or expected maturity payoff is indeed an upper bound for the American spot option value Vt. QED. Note that in the above proof, we did not use any specific assumption about the stochastic processes of the underlying asset price and its volatility. Nor did we make any assumption about the stochastic process for the riskfree rate or the leakage rate. Thus, Lemma 5 applies to arbitrary risk-neutral stochastic processes.

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4. IMPLICATIONS OF UPPER BOUNDS Among the various implications of the upper bound requirements and specifications that we have discussed so far, we focus here on two areas. First, we discuss the implications for relative pricing of options. Second, relevance in the context of empirical option pricing is explored.

4.1. Relative Pricing of Options Options of different strikes and maturities on an asset are traded as separate securities but they share the same underlying stochastic process. This forces many restrictions on rational (arbitrage-free) pricing of options relative to the underlying asset and relative to each other. Most primitive of these are the upper and lower bounds on individual option prices reflecting option pricing relative to the asset. Put-Call Parity (or bounds) imposes pricing discipline on call and put options of the same strike relative to the asset. Pricing conditions concerning options alone are numerous. Some examples are lower vs. higher strike, shorter vs. longer maturity, combinations of call and put options, etc. Ideally, upper bounds for options should retain such relative pricing discipline. While a full investigation is beyond the scope of this paper, we examine below two arbitrage conditions that relate to put options of different strikes to get a sense of whether the relative pricing conditions of option prices carry over to their upper bounds. For this purpose, we assume a constant interest rate and set the leakage rate to zero, and we use the nonrandomized version of the generalized European option as an upper bound with the number of optioned assets set to one: Gt ¼ er(Tt)Et[er(Tt)K–ST]+. 4.1.1. Lower vs. Higher Strike American Put Options Suppose we have two American put options V1 and V2 with strike prices K1>K2, both maturing at time T. One arbitrage condition between the prices of these two options is that V1>V2. The following corollary shows that this basic pricing discipline is carried over to their generalized European upper bounds. Corollary 3. Let G1t and G2t be the generalized European option upper bounds for the T-maturity standard American put option prices V1t and V2t with strike prices K1>K2: G1t ¼ er(Tt)Et[er(Tt)K1–ST]+, G2t ¼ er(Tt)Et[er(Tt)K2–ST]+. Then, G1t ZG2t.


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Proof: It suffices to show that Et[er(Tt)K2ST]+rEt[er(Tt)K1ST]+ E t ½erðTtÞ K 2 S T þ ¼ E t ½erðTtÞ fK 1 þ ðK 2 K 1 Þg S T þ ¼ E t ½ferðTtÞ K 1 ST g þ erðTtÞ ðK 2 K 1 Þþ E t ½erðTtÞ K 1 ST þ þ erðTtÞ ðK 2 K 1 Þþ ¼ E t ½erðTtÞ K 1 ST þ :QED. In the first inequality, we have used the property that for real numbers a and b, (a+b)+r(a++b+). The last equality follows from K1>K2. Further, G1t and G2t are themselves tradable European options. Therefore, in an arbitrage-free market, G1trG2t cannot prevail. To see this, suppose G1trG2t. Then, sell one G2 option and buy one G1 option. The net proceed now is (G2tG1t)Z0. At maturity time T: if SToK2 , the payoff is +(K1–K2), if K2oSTrK1, the payoff is +(K1–ST), and if K1oST, the payoff is 0. Thus the arbitrage strategy leads to nonnegative proceeds now, nonnegative payoff at maturity, and nonzero probability of positive payoff at maturity. Therefore, in the absence of arbitrage, G1tZG2t should prevail. 4.1.2. Put Option (Money) Spreads Suppose we have two American put options V1 and V2 with strike prices K1>K2, both maturing at time T. An important arbitrage condition on the prices of these two options is that, in the absence of arbitrage, the long bear spread cannot be worth more than the difference in strikes, i.e., (V1V2)o(K1K2). The following corollary shows that this pricing discipline is carried over to their generalized European upper bounds. Corollary 4. Let G1t and G2t be the generalized European option upper bounds for the T-maturity standard American put option prices V1t and V2t with strike prices K1>K2: G1t ¼ er(Tt)Et[er(Tt)K1–ST]+, G2t ¼ er(Tt)Et[er(Tt)K2–ST]+. Then, (G1t–G2t)r(K1–K2). Proof: G 1t ¼ erðTtÞ E t ½erðTtÞ K 1 S T þ ¼ erðTtÞ E t ½erðTtÞ ðK 1 K 2 Þ þ ðerðTtÞ K 2 S T Þþ ðK 1 K 2 Þ þ erðTtÞ E t ½erðTtÞ K 2 S T þ ¼ ðK 1 K 2 Þ þ G2t ) ðG1t G2t Þ ðK 1 K 2 Þ:QED

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It is perhaps premature to say that all arbitrage conditions involving the American options would carryover to the generalized European upper bounds. However, in light of the fact that these upper bounds here are European options and based on Corollaries 3 and 4 above, we are optimistic that the rational option pricing bounds would apply to the upper bounds. The importance of this carryover property for empirical option pricing will be discussed shortly. 4.1.3. Trading Early Exercise Options A standard American option is a package of a standard European option and an early exercise option. In practice, we observe trading of American options but not the early exercise options separately. Based on Margrabe (1978) and our analysis in this paper, it looks like one can trade the early exercise options indirectly using the European options alone. To see this, we first present an implication of the generalized European claims in this regard. Corollary 5. Let Gt be the generalized European option upper bound for the T-maturity standard American put option value Vt with strike prices K: G t ¼ erðTtÞ E t ½erðTtÞ K S T þ Then, (a) there exists a standard European option of strike K*, with KoK*oer(Tt)K , such that its value n*t equals the American option value Vt, and (b) the early exercise feature of the American option is valued at EEPK ¼ n*tnt, where vt ¼ er(Tt)Et[K–ST]+ is the value of a standard European option of strike K. Part (a) of the statement above follows directly from the fact that the European option value is a monotonic increasing function of the strike price, while part (b) simply reflects the two components of an American option value. A long put spread strategy involves a long position in the higher strike put option and a short position in the lower strike put option. A striking interpretation of Corollary 5 is that all long put (money) European spreads essentially represent ratio positions in the early exercise option associated with the lower strike. While determination of the strike K* is equivalent to calculating the American option value Vt and thus provides no computational relief, the economic insight is that it is not necessary to trade American options in order to trade the early exercise option. One practical difficulty in using the spread in lieu of the American option itself is that the investor needs to dynamically adjust the strike K*, or equivalently the ratio in the spread.


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4.2. Relevance for Empirical Option Pricing A clear strength of the arbitrage-based theoretical models of option pricing is that by definition they incorporate arbitrage-free relative pricing of the asset and all derivatives including the options. These relative pricing restrictions are obviously of greater importance in the context of American options because of their early exercise feature. However, the recent parametric theoretical models are already quite complex to implement in the context of European options. As such, in empirical studies of American options using parametric models, the relative pricing bounds do not receive much attention either. The European claims that we have proposed as upper bounds for American options should be helpful in empirical testing of parametric American option pricing models. For example, one can estimate the parameters of the asset price process from the observed asset returns, form the risk neutral terminal distribution given the theoretical valuation model, and then compute the value of the bounding generalized European claims using the risk-neutral distribution. To the extent the risk-neutral return dynamics is properly captured by the theoretical model, the estimated upper bounds should all be above the observed American option prices. If this leads to a failure of the theoretical model, then the much more complex task of estimating the theoretical American option prices may not be worthwhile. Given the limited nature of success of the various parametric theoretical extensions of the Black–Scholes model in explaining the patterns of observed option prices, a number of researchers have explored nonparametric alternatives. These nonparametric methods attempt to extract an empirical option valuation model from the actual option prices themselves. Semiparametric versions arise when guidance from some theoretical model(s) is used to improve dimensionality of the estimation problem. For example, AitSahalia and Lo (1998) estimate empirical pricing function for European options on the S&P 500 Index and the implied state price density using nonparametric and semiparametric methods. Broadie, Detemple, Ghysels, and Torres (2000), on the other hand, study the properties of nonparametric empirical American option pricing function and exercise boundary for the S&P 100 Index options. It seems that nonparametric studies such as the above do not quite consider whether the estimated pricing functions obey the various arbitrage bounds including the upper and lower bounds for option prices.18 Imposing or testing for these bounds is even more important for nonparametric models, especially in the context of American options, as there is no built-in

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arbitrage-free structure of prices here. It is hoped that the upper bounds can be helpful in controlling the quality of nonparametric option models. For example, suppose the researcher estimates an empirical European option pricing function for the S&P 500 (SPX) and an empirical American option pricing function for the S&P 100 (OEX) using kernels on several predictors including moneyness and volatility. Based on our results, adjusting for slight changes in volatility and leakage and for the index level, the option price predicted for a K-Strike T-Maturity OEX put option should be lower than the predicted price for a T-maturity SPX put option with strike Ku ¼ Ker(Tt) . If not, the empirical pricing functions are such that would permit arbitrage across the SPX and OEX contracts. The CBOE has of late introduced European option contracts (XEO) on the S&P 100 Index. As the volume and open interest of the XEO options grow, the arbitrage (upper) bounds tests will likely become easier in future. Later in this paper we examine sample quotes for these options.

5. A QUASI-BOUND A potential weakness of upper bounds that do not rely on approximating the early exercise boundary is that the bounds may be quite wide. In the context of the generalized European claims in a Black–Scholes setup, we now propose a claim that holds as an upper bound except for a range of moneyness not commonly traded on organized exchanges. Corollary 6. Suppose Q is a European put option with the following maturity payoff function: XQ,T ¼ [K–lRTST]+, where lT ¼ [St–K{1– EtRt,Tt}]/Et[Rt,TtST], and Rt,Tt ¼ exp( Ttrudu) is the discount factor between t and T. Then, the European put option Q’s value Qt ¼ Et[Rt,Tt(K–lTST)+] is an upper bound for the standard American put option’s value Vt and Q is meaningfully defined as a put option for (St/K)>(1–EtRt,Tt). Proof: Qt ¼ E t ½Rt;Tt ðK lT S T Þþ ½KE t Rt;Tt lT E t fRt;Tt STgþ ¼ ½K St þ ðusing the given value of lT Þ By Lemma 2, then QtZVt and Q is meaningfully defined as a put option as long as lT>0, that is (St/K)>(1–EtRt,Tt). QED. The European claim Q is a quasi-bound since it is not meaningfully defined as a put option when the American put option is too deep-in-the-money, that


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is the compound interest value on K is too high. For longer maturity options, Q reaches this threshold level sooner than for shorter maturity options. However, this shortcoming is not practically that important since below the threshold, Qt can be set to the American option’s intrinsic value. The reason Q tightens the AKE and ASPE bounds is because Q adjusts the number of optioned units (lT) of the underlying asset depending on the moneyness of the option. While the AKE and ASPE adjustments are fixed for a time to maturity, lT decreases (increases) with the moneyness of the put option (asset price). For at-the-money put options (St ¼ K), the adjustment factor lT of Q is equal to the adjustment factor of ASPE, and for in-the-money (out-of-the-money) put options lT is lower (higher). To have a general feeling about the bounds, let us now present some numerical results for the BlackScholes setup: constant interest rate of 10%, zero leakage rate, time to maturity of 0.25 years, and constant volatility of 30%. The strike price is set to 100 and the asset price is varied from 80 to 120. The American put option price is calculated using a 100-step Binomial Tree. Fig. 1 plots five series against the measure of moneyness American Put Value, Bounds and Approximation, T - t = 0.25 25.00 v 20.00

V G

15.00

Value

Q v(G)

10.00 5.00

-30.00

-20.00

-10.00

0.00 0.00

10.00

20.00

30.00

K-S

Fig. 1. Black–Scholes Setup: Put Option Values, Bounds, and Approximation. Note: The parameter values used for this chart are: interest rate r ¼ 10%, leakage rate d ¼ 0%, volatility r ¼ 30%, time to maturity Tt ¼ 0.25 year, and K ¼ 100. The legends are as follows: v ¼ European value, V ¼ American value, G ¼ Generalized European option value, Q ¼ Quasi-Bound of this paper, and v(G) ¼ discounted value of G. The American option value is calculated using a 100-step Binomial Tree.

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KS: v (European option value), V (American option value), G (the ASPE bound), Q (the Quasi-Bound value), and v(G) (discounted value of the ASPE bound). Although v(G) is not an upper bound, we have included v(G) to see how well this approximates the American option value V. Several observations can be made from Fig. 1. First, as expected, G indeed bounds V from above and so does Q given the parameter values. Second, the curvature of all the bounds and the approximation (G, Q, and v(G)) are very similar to that of the American option. This is rather encouraging as the hedge ratios based on the bounds and the approximations are expected to be good estimates for the true hedge ratio. Third, the bounds and the approximation track the American option value very closely for in-the-money put options. This is also encouraging since in practice in-the-money observed option prices are believed to be notoriously unreliable. The bounds and the approximation here can thus provide good valuation guidance for these options. Fourth, as expected, the quasi-bound (Q) provides a tighter bound than the ASPE bound. Fifth, the discounted value v(G) of the ASPE bound provides a nice approximation although its theoretical relationship to V is unclear. The results in Fig. 1 are for short maturity options. Fig. 2 presents the results for time to maturity of 1.0 year, other parameters remaining the same American Put Value, Bounds and Approximation, T - t = 1.0 25.00 v V

20.00

Value

G Q

15.00

v(G) 10.00 5.00

-30.00

-20.00

-10.00

0.00 0.00 K-S

10.00

20.00

30.00



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as in Fig. 1. As expected the bounds and the approximation widen relative to the American option value with a substantially longer maturity as they do not consider the true exercise boundary and overestimate the expected interest value. However, both G and Q still track the curvature well. As the option goes deep in-the-money, the American option’s hedge ratio approaches –1.0 faster than the bounds and the approximation. Once again this is due to the fact that the intrinsic value of the bounds here always stays above the intrinsic value of the American option by design. It is also to be noted that for deep-in-the-money option, the Quasi-Bound Q hits its threshold level with the longer time to maturity and the v(G) approximation deteriorates as well. Next we consider the joint effect of a lower volatility (15%) and a lower interest rate (5%) in Figs. 3 and 4. Unlike the European option component, the early exercise component of the American option tends to go up with a lower volatility. Lowering the interest rate of course reduces the value of the American put option. The 50% reduction in both the volatility and the interest rate, however, reduced the American option value in the current experiment. As expected, the bounds and the approximation seem to track

American Put Value, Bounds and Approximation, T - t = 0.25

Value

25.00 v V G Q v (G)

20.00 15.00 10.00 5.00

-30.00

-20.00

-10.00

0.00 0.00 K-S

10.00

20.00

30.00


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MO CHAUDHURY American Put Value, Bounds and Approximation, T - t = 1.0 25.00 v 20.00

V Value

G 15.00

Q v(G)

10.00 5.00

-30.00

-20.00

-10.00

0.00 0.00 K-S

10.00

20.00

30.00


the American option value better with a lower volatility–lower interest rate combination, especially for the short maturity options. Overall, it appears that the European nature of the bounds and the approximation help retain the essential convexity-of-payoff related properties of American option values. However, one weakness that needs further attention is that the American option value is more convex than the bounds and the approximation and it approaches the intrinsic value faster as the option goes deeper in-the-money.

6. SAMPLE S&P 100 OPTION QUOTES The purpose of this section is to see, on a very preliminary basis, if the upper bound properties hold in practice. To our knowledge, only the S&P 100 Index has both American (OEX) and European (XEO) option contracts available. This should greatly facilitate empirical study of American options, their bounds, and the nature of early exercise premium (EEP). However, the


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European (XEO) contracts are relatively new and their volume is currently far less than that of the well-known American (OEX) contracts.19 Meantime, the bid–ask quotes of the XEO and OEX contracts can still provide useful insights into the pricing of American options vis-a`-vis their European counterparts. In particular, it will be interesting to see if the European option based bounds proposed in this paper apply to the American options in practice. In Panel A of Table 1, we report a sample of CBOE option quotes for the XEO and OEX June, 2002 contracts. The Bid and Ask quotes are 15-minute delayed quotes retrieved from the CBOE web site at about 1:56 PM on March 14, 2002; the S&P 100 Index was hovering about the 585.00 level around that time (largely unchanged from its level around 1:41 PM). Given that the XEO market is not as liquid as the OEX market, the last sale prices of the OEX and XEO contracts may not match. Also, the last sale prices of the XEO and OEX contracts of various strikes may not be comparable. But the Bid and Ask quotes are updated much more frequently and as such are more representative of the respective option values. In line with empirical option pricing tradition, we take the mid-point of the Bid–Ask spread as an estimate of the fair price of the option. To see if the observed American option price is bounded by the price of a compounded strike European option, we estimate the compounded strike, Ku ¼ Ker(Tt), using a risk-free rate of 5% and the 0.3671 year time to maturity of the options. Luckily, the compounded strike is fairly close to the next available strike of the sample options and as such the observed quote of the XEO option closest to the compounded strike Ku can be used as a proxy for the upper bound of the K-strike OEX option. It seems that the observed American option (mid) quotes are indeed bounded by the corresponding compounded strike European option counterparts. For example, the K ¼ 570 OEX contract’s mid-quote $12.75 is less than the K ¼ 580 (Ku ¼ 581) XEO contract’s mid-quote $15.95. Similarly, the K ¼ 580 OEX contract’s mid-quote $16.35 is less than the K ¼ 590 (Ku ¼ 591) XEO contract’s mid-quote $20.35. Panel A data also provides an opportunity to see if the observed American option spread is bounded by the compounded strike European option spread. The long spread reported against a strike, say 570 (Ku ¼ 581), represents the net cost of buying the option of that strike (570, Ku ¼ 581) at the Ask quote and selling the immediately lower strike (560, Ku ¼ 570) option at the Bid quote. The long XEO spread (long 580, short 570) cost reported against 580 is then used as a proxy for bounding the long OEX spread (long 570, short 560) cost reported against 570. Indeed, the observed bounding

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Table 1.

March 14, 2002 CBOE Option Quotes for the S&P 100 Options.

Panel A CBOE 15-Minutes Delayed June-Maturity Put Option Quotes for OEX and XEO Trading Date: March 14, 2002 Quotes Retreived from CBOE Site at about 1:56 PM S&P 100 Index Around 585.00 During 1:40 and 2:00 PM

r

0.0500

Tt

0.3671

exp[r(Tt)]

1.0185

XEO Bid

XEO Ask

XEO Mid

XEO Ask– Bid

OEX Bid

OEX Ask

OEX Mid

OEX Ask– Bid

K* exp[r(Tt)]

550 560 570 580 590 600 610 620

7.00 9.20 11.60 15.20 19.60 24.60 31.00 38.20

7.70 9.90 13.10 16.70 21.10 26.80 33.20 40.40

7.35 9.55 12.35 15.95 20.35 25.70 32.10 39.30

0.70 0.70 1.50 1.50 1.50 2.20 2.20 2.20

7.30 9.50 12.00 15.60 20.00 25.30 31.70 39.10

8.00 10.20 13.50 17.10 22.20 27.50 33.90 41.30

7.65 9.85 12.75 16.35 21.10 26.40 32.80 40.20

0.70 0.70 1.50 1.50 2.20 2.20 2.20 2.20

560 570 581 591 601 611 621

Long XEO Spread

3.90 5.10 5.90 7.20 8.60

Long OEX Spread

2.90 4.00 5.10 6.60 7.50

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K

K

EEP Mid ¼ OEX–Mid Less XEO–Mid

EEP Ask ¼ OEX–Ask Less XEO–Bid

EEP Bid ¼ OEX–Bid Less XEO–Ask

EEP Ask–Bid

550 560 570 580 590 600 610 620

0.30 0.30 0.40 0.40 0.75 0.70 0.70 0.90

1.00 1.00 1.90 1.90 2.60 2.90 2.90 3.10

0.40 0.40 1.10 1.10 1.10 1.50 1.50 1.30

1.40 1.40 3.00 3.00 3.70 4.40 4.40 4.40

Panel A of this table contains the Bid and Ask quotes for the S&P 100 European options (XEO) and American options (OEX) maturing in June, 2002. These 15-min delayed quotes were retrieved from the CBOE web site at about 1:56 PM; the S&P 100 Index was about 585.00 around that time (largely unchanged from its level around 1:41 PM). The mid-point of the Bid–Ask spread is an estimate of the fair value of the option. The compounded strike Kexp[r(Tt)] is estimated using a risk-free rate of 5% and the 0.3671 year time to maturity of the June options (it seems to be fairly close to the next available strike). The long spread reported against a strike (say 570) represents the net cost of buying the option of that strike (570) at the Ask quote and selling the immediately lower strike (560) option at the Bid quote. Panel B of this table first estimates the early exercise premium (EEP) as the difference between the mid quotes of the OEX and XEO options. The EEP Ask quote is then estimated as the net cost of buying the OEX option at the Ask quote and selling the XEO option of same strike at the Bid quote. The EEP Bid quote is estimated as the net proceeds of selling the OEX option at the Bid quote and buying the XEO option of same strike at the Ask quote.


Panel B

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XEO spread cost of $5.10 is greater than the OEX spread cost of $4.00. This is also the case for spreads involving other strikes. In Panel B of Table 1, we estimate the EEP of a given strike as the difference between the mid-quotes of the OEX and XEO contracts. The magnitude of the EEP is not large. However, the behavior of the estimated EEP is largely in line with theory. The estimated EEP seems to increase with the strike price and the increment appears larger for deeper in-the-money options. Given the existence of both European (XEO) and American (OEX) options, one can trade the EE option synthetically. The cost of buying a synthetic EE option (EEP Ask) is estimated as the OEX Ask net of the XEO Bid. Similarly, the proceeds of selling a synthetic EE option (EEP Bid) are estimated as the OEX Bid net of the XEO Ask. From the sample information in Panel B of Table 1, buying a synthetic EE option seems quite expensive while shorting a synthetic EE option is not feasible at all. This may in part explain why the XEO contracts are not as popular as the OEX contracts although the European options are cheaper and should have attracted more speculators, hedgers, and portfolio insurers. The primary reason for the synthetic EE option anomaly is that the (dollar) Bid–Ask spreads for the index option (both XEO and OEX) contracts are too wide relative to the size of the EEP. For example, in Panel B of Table 1, the EEP is estimated at about $0.40 for K ¼ 570 but the Bid–Ask spread is $1.50 for both XEO and OEX contracts. This poses a challenging measurement problem for empirical options researchers. Further, a policy question also arises as to whether the CBOE should act to sufficiently reduce the spreads in both types of contracts so that investors are not limited to only long positions in (synthetic) EE options. One way to make this possible is to replace the OEX contracts with EE contracts. For example, an EE contract can be designed to pay the buyer the excess of the intrinsic value over the XEO midquote in case the EE option is exercised. This suggestion is in line with the fact that in the presence of transaction costs, synthetic replication may not closely track the value of directly tradable derivatives. Further, the European component of the OEX contract is clearly redundant given the XEO contract.

7. SUMMARY AND CONCLUSIONS This paper has provided a fuller characterization of the analytical upper bounds for American options by establishing properties that are required of the bounds. A key property is that if a claim’s value never falls below the intrinsic value of the American option, then the claim is an upper bound for


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the American option value. While the literature primarily relies upon bounds for the exercise boundary, we have shown that a class of generalized European options can be made to satisfy the key property and hence serve as upper bounds. This class contains the analytical bounds of Margrabe (1978) and Chen and Yeh (2002). An important benefit of having generalized European options as upper bounds is that they are in closed form and are easy to implement since a direct treatment of the early exercise boundary is avoided. They are also intuitively tractable. When the valuation situation involves multiple state variables, the class of European upper bounds suggested in this paper can significantly ease the burden of computation and still serve as useful benchmarks.20 This characteristic is also quite beneficial in practice where only a general valuation range is desired. The upper bounds seem to have many desirable properties and interesting implications. For example, we have shown that the across-strike arbitrage conditions on option prices seem to carry over into the bounds and that one can trade early exercise options using merely European option spreads and never trading the American options. We believe both parametric and nonparametric empirical option pricing models can improve the quality of estimation using the bounds results of this paper. So far empirical attempts to incorporate various arbitrage bounds have been lacking, especially in nonparametric models. In an attempt to tighten the European-type bound, we proposed a quasibound that holds as an upper bound for most practical circumstances. We also suggest an approximation based on the bound of Chen and Yeh (2002). Our limited numerical results in the traditional Black–Scholes setup are encouraging. The bounds and the approximation of this paper seem to track the American option value and its curvature rather well for short maturity options and in-the-money options. Hence the bounds here should be useful in estimating American option hedge ratios and as valuation benchmarks or proxies for in-the-money options for which observed option prices are believed to be notoriously unreliable. A caveat is that as the maturity gets longer and the volatility and interest rate increase, the bounds and the approximation widen relative to the American option value. Another potential weakness that needs further attention is that the European-type bounds and approximation do not change fast enough as the American option goes deep in-the-money. This is, of course, a tradeoff that arises from not explicitly considering the early exercise boundary of the American option. In this paper we did not undertake any empirical study of the upper bounds. However, as a first attempt, we did examine sample (March 14, 2002) option quotes for the European (XEO) and the American (OEX)

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options on the S&P 100 Index. These quotes appear well behaved with respect to the upper bound properties. But a synthetic long position in the early exercise option seems quite expensive and a synthetic short position in the early exercise option is not feasible. This is because the bid–ask spreads are too wide relative to the magnitude of the EEP, and is likely one of the factors why the XEO contracts are not as popular as the OEX contracts. Redesigning the OEX contracts purely as early exercise options would eliminate the redundancy of the European option (XEO) and allow investors to take long as well as short positions in the early exercise option directly.

NOTES 1. Analytic solutions that require numerical evaluation of the early exercise boundary are available in Kim (1990), Jamshidian (1992), Geske and Johnson (1984), Jacka (1991), Carr, Jarrow, and Myneni (1992), Bunch and Johnson (2000), and Broadie, Detemple, Ghysels, and Torres (2000). Analytic approximations based on approximation of the early exercise boundary include Johnson (1983), Omberg (1987), Huang, Subrahmanyam, and Yu (1996), Ju (1998), Broadie and Detemple (1996), and Bunch and Johnson (2000). Broadie and Detemple (2004) provide a recent survey of American option valuation. 2. For example, in nonparametric estimation of American option pricing function, upper bounds should be useful in controlling the quality of estimation. 3. Based on the upper bound of Chaudhury and Wei (1994), Chaudhury (1995) developed several Black-Scholes type closed form analytic approximations for American futures options that are quite accurate and provide better approximation than the quadratic approximation of MacMillan (1986) and Barone-Adesi and Whaley (1987) for most actively traded futures options. 4. Applications of analytical upper bound for American futures option prices (Chaudhury & Wei, 1994) include Bates (2000), Melick and Thomas (1997), Leahy and Thomas (1996), Soderlind and Svensson (1997), Flamouris and Giamouridis, (2002), Galeti and Melick (2002), and Beber and Brandt (2003). The bounds of this paper will be helpful in extracting information from American spot options data. 5. References on option bounds based on equilibrium pricing kernel can found in Huang (2004). 6. Closed form bounds require closed form terminal distribution of the asset price. 7. For example, in a two-factor random volatility model, the upper bound of Chen and Yeh (2002) is more than 3700 times faster than the American finite difference algorithm. 8. For example, Chen and Yeh (2002) have given several examples involving stochastic interest rates, leakage, and volatility. The bounds of this paper also apply to such cases. Both in Chen and Yeh and in this paper, the only requirements are that: (a) the risk neutral measure exists, (b) the values of the stochastic discount factor are less than one for all sample paths, and (c) the instantaneous expected net growth process is strictly positive (to make the American spot put option problem interesting).


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9. This is the argument used by Chaudhury and Wei (1994) and Chaudhury (1995) for American futures options. For these options, the pure European value always stays above the intrinsic value (Lieu, 1990; Chen & Scott, 1992). 10. Unless mentioned otherwise, all expectations in this paper are expectations under the risk-neutral or equivalent martingale measure. 11. Chen and Yeh (2002, p. 119 and FootNote 4, p. 120) recognize these limitations of their Theorem 1. 12. Chen and Yeh (2002, p. 118) mention that an upper bound for an American option always stays above both the continuation and the exercise value of the American option. Of course, this is definitional of an upper bound. 13. See Chen and Yeh (2001) for the treatment of stochastic interest rates. For interested readers, the author of this paper can provide the proof that the results here are unaffected by stochastic interest rates and leakage. 14. Examples of further drift adjustment include Bakshi, Cao and Chen (1997) and Bates (2000) for jumps in asset price in a stochastic volatility framework. 15. For r>d, maturity payoff of Chung and Chang’s (2005) bound is equivalent to equal adjustments to the strike price and the number of units of the optioned asset; in that case, their bound translates to adjusting the number of standard or conventional European options on the asset. For rod, their bound is like European option on exp(dT) units of stocks for a total strike of K exp(rT), that is an implied strike of K exp{(rd)T}oK per unit of stock; in this case, it is like an adjustment of the strike price alone. In either case, Chung and Chung’s bounds work because they satisfy Corollary 1 of this paper. Thus, Chung and Chang’s bounds can be considered special cases of the generalized European claim G in Lemma 4 here. In addition, in this paper, the bounding claim G can be American too. Of course, Chung and Chang do not consider possible stochastic adjustments as in Lemma 4 here. 16. Merton (1973), pp. 154–155, first showed that, for an American call warrant, if the rate of increase in the strike price is less than the interest rate, then a premature exercise is not optimal. Accordingly, the American warrant value will equal the European warrant value. However, he did not use this result to establish upper bounds for call or put options. Also, Merton did not consider adjustments in the number of optioned asset units for this purpose. 17. The value of the standard American option is calculated using a 100-step Binomial tree. 18. Arbitrage condition violations are reported by Ackert and Tian (2000) for the S&P 500 European option contracts, and by Capelle-Blancard and Chaudhury (2001) for the CAC 40 European option contracts. 19. The CBOE launched the OEX contract on March 11, 1983 and the XEO contract on July 23, 2001. Both contracts are cash-settled with a multiple of 100. Since its inception, more than a billion contracts of OEX have been traded. By the close of trading on March 14, 2002, a total of 59,315 OEX traded, of which 27,908 (31,407) are call (put) option contracts. In comparison, a total of 10,776 XEO contracts traded on that day, of which 6,897 (3,879) are call (put) option contracts. 20. That this line of research is promising is demonstrated by the recent work of Chung and Chang (2005). They have extended the theoretical results of Chen and Yeh (2002) and this paper in deriving upper bounds for American options on multiple assets.

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MO CHAUDHURY

ACKNOWLEDGMENT Thanks are due to San-Lin Chung, Jerome Detemple, David Hsieh, Lars Norden and seminar participants of the 2004 Asian Finance Association in Taipei for helpful comments, and to Ren-Raw Chen for continued dialogue on American option valuation over many years. The author is responsible for any remaining errors.

REFERENCES Ackert, L. F., & Tian, Y. S. (2000). Evidence on the efficiency of index options markets. Federal Reserve Bank of Atlanta Economic Review, (First Quarter), 40–52. Ait-Sahalia, Y., & Lo, A. W. (1998). Nonparametric estimation of state-price densities implicit in financial asset prices. Journal of Finance, 53(2), 499–547. Andersen, L., & Broadie, M. (2004). A primal-dual simulation algorithm for pricing multidimensional American options. Management Science, 50(9), 1222–1234. Bakshi, G., Cao, C., & Chen, Z. (1997). Empirical performance of alternative option pricing models. Journal of Finance, 52(5), 2003–2049. Barone-Adesi, G., & Whaley, R. (1987). Efficient analytic approximation of American option values. Journal of Finance, 42(2), 301–320. Bates, D. (2000). Post-’87 crash fears in the S&P 500 futures option market. Journal of Econometrics, 94, 145–180. Beber, A., & Brandt, M. W. (2003). The effect of macroeconomic news on beliefs and preferences: Evidence from the options market. NBER Working Paper 9914. Broadie, M., & Detemple, J. (1996). American option valuation: New bounds, approximations, and a comparison of existing methods. Review of Financial Studies, 9(4), 1211–1250. Broadie, M., Detemple, J., Ghysels, E., & Torres, O. (2000). American options with stochastic dividends and volatility: A nonparametric investigation. Journal of Econometrics, 94, 53–92. Broadie, M., & Detemple, J. (2004). Option pricing: Valuation models and applications. Management Science, 50(9), 1145–1177. Bunch, D. S., & Johnson, H. (2000). The American put option and its critical stock price. Journal of Finance, 55(5), 2333–2356. Capelle-Blancard, G., & Chaudhury, M. (2001). Efficiency tests of the French index (CAC 40) options market. McGill Finance Research Centre Working Paper. Carr, P., Jarrow, R., & Myneni, R. (1992). Alternative characterizations of American put options. Mathematical Finance, 2(2), 87–106. Chaudhury, M., & Wei, J. (1994). Upper bounds for American futures options: A note. Journal of Futures Markets, 14(1), 111–116. Chaudhury, M. (1995). Some easy-to-implement methods of calculating American futures option prices. Journal of Futures Markets, 15(3), 303–344. Chen, R.-R., & Scott, L. (1992). Pricing interest rate futures options with futures-style margining. Journal of Futures Markets, 13, 15–22. Chen, R.-R., & Yeh, S.-K. (2002). Analytical upper bounds for American option prices. Journal of Financial and Quantitative Analysis, 37(1), 117–135.


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Chung, S.-L., & Chang, H.-C. (2005). Generalized analytical upper bounds for American option prices. Working Paper, National Taiwan University. Flamouris, D., & Giamouridis, D. (2002). Estimating implied PDFs from American options on futures: A new semiparametric approach. Journal of Futures Markets, 22(1), 1–30. Galeti, G., & Melick, W. (2002). Central bank intervention and market expectations. BIS Papers No. 10, Monetary and Economic Department, Bank of International Settlements. Geske, R., & Johnson, H. (1984). The American put option valued analytically. Journal of Finance, 39, 1511–1524. Grundy, B. (1991). Option prices and the underlying asset’s return distribution. Journal of Finance, 46(3), 1045–1069. Huang, J. (2004). Option pricing bounds and the elasticity of the pricing kernel. Review of Derivatives Research, 7, 25–51. Huang, J.-z., Subrahmanyam, M. G., & Yu, G. G. (1996). Pricing and hedging American options: A recursive integration method. Review of Financial Studies, 9, 277–300. Jacka, S. D. (1991). Optimal stopping and the American put. Mathematical Finance, 1(2), 1–14. Jamshidian, F. (1992). An analysis of American option. Review of Futures Markets, 11(1), 72–82. Johnson, H. (1983). An analytic approximation for the American put price. Journal of Financial and Quantitative Analysis, 18(1), 141–148. Ju, N. (1998). Pricing an American option by approximating its early exercise boundary as a multipiece exponential function. Review of Financial Studies, 11(3), 627–646. Kim, I. J. (1990). The analytic valuation of American options. Review of Financial Studies, 3(4), 547–572. Leahy, M. L., & Thomas, C. P. (1996). The sovereignty option: The Quebec referendum and market views on the Canadian dollar. International Finance Discussion Paper No. 555, Board of Governors of the Federal Reserve System. Lieu, D. (1990). Option pricing with futures-style margining. Journal of Futures Markets, 10, 327–338. Lo, A. (1987). Semiparametric upper bounds for option prices and expected payoffs. Journal of Financial Economics, 19, 373–388. MacMillan, L. (1986). Analytical approximation for the American put option. Advances in Futures and Options Research, 1, 119–139. Margrabe, W. (1978). The value of an option to exchange one asset for another. Journal of Finance, 33(1), 177–186. Melick, W. R., & Thomas, C. P. (1997). Recovering an asset’s implied PDF from option prices: An application to crude oil during the Gulf crisis. Journal of Financial and Quantitative Analysis, 32(1), 91–115. Merton, R. (1973). Theory of rational option pricing. Bell Journal of Economics and Management Science, 4, 141–183. Omberg, E. (1987). The valuation of American puts with exponential exercise policies. Advances in Futures and Options Research, 2, 117–142. Perrakis, S., & Ryan, P. J. (1984). Option pricing bounds in discrete time. Journal of Finance, 39, 519–525. Rogers, L. C. G. (2002). Monte Carlo valuation of American options. Mathematical Finance, 12, 271–286. Soderlind, P., & Svensson, L. E. O. (1997). New techniques to extract market expectations from financial instruments. Journal of Monetary Economics, 40(2), 373–429.

A SPREAD-BASED MODEL FOR THE VALUATION OF CREDIT DERIVATIVES WITH CORRELATED DEFAULTS AND COUNTER-PARTY RISKS Chuang-Chang Chang and Yu Jih-Chieh ABSTRACT We set out, in this paper, to extend the Das and Sundaram (2000) model as a means of simultaneously considering correlated default risk structure and counter-party risk. The multinomial model established by Kamrad and Ritchken (1991) is subsequently modified in order to facilitate the development of a computational algorithm for valuing two types of active credit derivatives, credit-spread options and default baskets. From our numerical examples, we find that along with the correlated default risk, the existence of counter-party risk results in a substantially lower valuation of credit derivatives. In addition, we find that different settings of the term structure of interest rate volatility also have a significant impact on the value of credit derivatives.


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194

CHUANG-CHANG CHANG AND YU JIH-CHIEH

1. INTRODUCTION Within the overall derivatives markets in the early 1990s, the market for credit derivatives was virtually non-existent; however, over recent years, credit derivatives have received much attention. By 2003, the notional outstanding value of the derivatives market was estimated at slightly over US$3 trillion, and according to Lehman Brothers Inc. market analysts’ reports, it is expected to exceed US$7 trillion by 2006. Such rapid growth in the credit derivatives market has ultimately led to the creation of a new set of financial instruments, which has been hailed as a major new risk management tool for the management of credit exposure. By making large and important risks tradable, these new financial instruments form an important step toward market completion and efficient risk allocation. Nowadays, several types of credit derivatives are available in the market; however, credit derivative payoffs will depend, first and foremost, on the default event itself, or the credit quality of a certain issuer. This credit quality can be measured by the credit rating of the issuer, or by the yield spread of the issuer’s bonds over the yields of comparable default-free bonds. From such a perspective, these derivatives can be roughly categorized into ‘default-based credit derivatives’ (for examples default swaps and total return swaps) or ‘spread-based credit derivatives’ (for examples creditspread options and credit-spread swaps) (see Schonbucher, 2000). In addition to standard credit derivative products, such as credit default swaps (CDS) or total return swaps based upon a single underlying credit risk, many new products are now being associated with credit risk portfolios. A typical example nowadays is a product with payment contingent upon the time and identity of the ‘first to default’ or ‘second to default’ in a given credit risk basket. The types of credit derivatives can be basically divided as the one underlying with single name (for examples default swaps and credit-spread options) or with multiple name (for examples firstto-default basket and CDOs). The key to the valuing of credit derivatives written on a credit portfolio is the ability to effectively deal with the default correlation, an issue which, if we do not assume the independence of default between the reference asset and the default swap seller, even arises in the valuation of a simple credit default swap with one underlying reference asset; i.e., the so-called ‘counterparty risk’. More specifically, credit derivatives such as default swaps and default baskets both involve more than one correlated default process; indeed, default swaps involve two default processes, counter-party default and issuer default, while default baskets involve multiple default processes,

A Spread-Based Model for the Valuation of Credit Derivatives

195

including counter-party default and defaults of multiple issuers. Hence, all of the above considerations are essential elements in the process of accurately pricing credit derivatives. Default can be modeled in various ways, with one very popular way being through the use of Poisson distribution; correlation is not, however, clearly defined between two Poisson processes. Another popular method is the use of credit spread as a default measure, since the co-movements of each firm’s credit spreads, under credit-spread-based models, can directly measure the default correlations. However, thus far, there has been very little development in terms of the consideration of correlated defaults in the credit risk modeling literature (see Chen & Sopranzetti, 2003). Indeed, as was pointed out by Kothari (2002): While the modeling of single-name credit derivatives is relatively understood, and there is available data for the estimation and calibration of theses models, this is not the case for multi-name credit derivatives, and generalizing single-name credit derivatives models to the multivariate case is not that simple. Usually straightforward generalizations are not applicable to multi-name credit derivatives; therefore new models have to be developed.

This paper therefore sets out to extend the Das and Sundaram (2000) model in an effort to develop a general ‘spread-based’ reduced-form model which can be used to price different types of credit derivatives involving several default correlations. Based upon our model, we demonstrate that the Das and Sundaram model is a special case. The basic idea behind our model is to describe the evolution of the ‘correlated’ forward rate and multiple forward spreads by means of a risk-neutral lattice. At the same time, we derive the default probabilities and recovery rates, at each node on this lattice, which are consistent with the credit spreads at that node. We begin by briefly presenting some of the features of our approach. First of all, by taking existing spreads as a model input, we are able to derive the evolution of spreads directly, as opposed to deriving them from the implications of default probabilities and recovery rates; this also guarantees that our model is consistent with any observed term structure of credit spreads. Second, as compared with the ‘intensity-based’ reduced-form model, our approach facilitates the pricing of credit derivatives whose payoffs depend directly on the spread. Third, our model incorporates not only the correlative market and credit risk, but also the interdependent default risk structure and counter-party risk. Fourth, the parameters can be deduced by using readily available market data. Finally, as opposed to providing an exogenous recovery rate setting, a much more rational rate is presented in our model.

196


The remainder of this paper is organized as follows. Section 2 undertakes a review of some of the relevant literature on credit-risk modeling. This is followed, in Section 3, by the introduction of our approach to the modeling of the correlated spreads and defaults, and the collation of all the necessary information required to construct the lattice. Some numerical examples of price credit risk derivatives are provided in Section 4, along with the presentation and discussion of the pricing results. Section 5 provides the conclusions drawn from this study.

2. LITERATURE REVIEW There are two well-known approaches to the modeling of credit risk, the first of which is the so-called ‘structure-form approach’ proposed by Merton (1974), which views equity shares and debts as derivatives on the firm’s assets. Since limited liability provides shareholders with the option of abandoning the firm, and putting it in the hands of bondholders, these bondholders will then have a short position in this put option. Conversely, one can regard equity as a call option on the value of the firm, with the strike being equal to the notional amount of outstanding debt. The typical method here posits a process for the evolution of firm value, specifying the conditions leading to bankruptcy, as well as the payoffs to various parties in the event of bankruptcy. The value of the debt is then derived as a consequence. In practice, however, the structure-form approach does have several important weaknesses. The first of these is that many of the firm’s assets are typically not traded; therefore, the firm’s value process is fundamentally unobservable, which clearly makes implementation difficult. Second, under this approach, in valuing a particular tranche of corporate debt, one also has to simultaneously value all debt further up the ladder, thus increasing computational complexity.1 As opposed to modeling firm value, the ‘reduced-form approach’ directly models the default process of risky debt, while also making use of observable market data to derive the model parameters; this is an approach which has gained in popularity over recent years. Representative models of this type have been developed by Jarrow and Turnbull (1995), Duffie and Singleton (1999) and Madan and Unal (2000). Jarrow and Turnbull considered the simplest case, where the default was driven by a Poisson process with constant intensity and a known payoff at default. Other models in this group have extended the key concepts deriving the time of default as


197

being directly modeled as the time of the first jump in a Poisson process with random intensity (i.e., a Cox process). Other studies, such as Das and Tufano (1996), Jarrow, Lando, and Turnbull (1997) and Lando (1998) have used a credit-rating based approach wherein default is depicted through a gradual change in ratings driven by a Markov transition matrix. Das (1998) and Schonbucher (1999, 2000) subsequently proposed a credit-spread-based model,2 with their approach being set in a simple discrete time Heath, Jarrow, and Morton (1992) framework (hereafter referred to as the HJM model or HJM framework) which allows for defaultable securities. Credit spread, under this framework, follows a stochastic process, and is the sort of setting which is more suited to valuing spread-based credit derivatives. Das and Sundaram (2000) constructed a ‘defaultable’ discrete-time termstructure model, as proposed by Heath et al. (1992), allowing for the valuation of a credit derivatives model. Their approach was based upon an expansion of the HJM term-structure model to allow for defaultable debt; however, as opposed to following a procedure in which the behavior of spreads was implied from assumptions concerning the default process, they worked directly with the evolution of spreads. They also used a logistic regression model to estimate the default probability from market data, using the default-free interest rate and credit spread as explanatory variables. Wilson (1997) provided strong support for a specification of this type. Being the first to model default rates as functions of macroeconomic variables, he found that a logit regression fitted the default rates for many 2 countries, with the R values being in the range of 80–90%. The Das and Sundaram (2000) approach differed from that of Wilson, in that Das and Sundaram employed only those variables that were available on the riskneutral lattice. Combining a ‘recovery of the market value’ (RMV) condition, they were able to implement all of the default information on a lattice to undertake the pricing of a variety of credit derivatives. The recursive algorithm programming of the Das and Sundaram model enabled it to easily handle the path-dependence and early-exercise features. The model also accommodated the consideration that market and credit risk were correlated; that is, a correlation existed between the risk-free forward rate process and one forward spread process. A similar concept can be found in other studies, such as Duffee (1999), which assumed risk-neutral intensity to be lt ¼ a þ lt þ b1 ðs1t s1t Þ þ b2 ðs2t s2t Þ

(1)

198


where S1t and S2t are the default-free factors inferred from treasury yields through the short rate model rt ¼ ar+S1t+S2t ; s1t and s2t are the respective sample means; and lt* is the firm-specific factor. Despite the wealth of literature on credit risk modeling, there has been surprisingly little development, in terms of the consideration of default correlations, under a reduced-form framework. Here we review some of the more famous studies within the literature. Duffie and Singleton (1999) modeled the correlation between default processes by combining (summing or subtracting) independent Poisson processes; however, the limitation of this approach was that it was necessary to pre-specify the sign of the correlation. Furthermore, when there are many issuers, the correlation structure under this model will be quite limited. In order to explain the clustering defaults around an economic recession, Jarrow and Yu (2001) suggested that the credit risk induced by the interdependence structure between firms could be taken into consideration by generalizing the intensity-based models, thus allowing a firm to be exposed not only to common risk factors, but also to some firm-specific default risk. They then set up a ‘primary–secondary framework’ to describe the default intensities dependent upon the default of the counter-party. However, given the complexity of the analysis, they confined their discussion to a situation where the default intensity followed a simple point process, and therefore only priced the ‘idealized’ default swaps under the simplified assumption that the recovery payment was made at the maturity of the credit default swaps. Chen and Sopranzetti (2003) presented a simple model for default correlation, structuring defaults non-parametrically by the use of simple Bernoulli events and conditional default probabilities for any given time period. Using conditional default probabilities, they were able to describe the dependency of two default events (as opposed to specifying the correlation). However, the method was not so objective in terms of deciding the degree of interdependency, and when extending their model to multiple assets, the calculations of the probabilities became multi-dimensional.

3. THE MODEL In this study we extend the model of Das and Sundaram (2000), while incorporating the concept proposed by Boyle, Evnine, and Gibbs (1989) and Kamrad and Ritchken (1991) to deal with the joint probability of several stochastic processes; our main aim is to take account of the correlated


199

default risk structure. We adopt the concept of Das and Sundaram (2000), which extended the HJM term-structure model to describe the evolution of default-free forward rate and forward credit spread (as reviewed in the preceding section), while also generalizing the model to contain more than one credit-spread process. The main notations for the construction of the model are summarized in Table 1. First of all, we consider that our model is built up in an economy on a finite time interval [0, T*]. A time period is represented of length h; thus, an arbitrary time-point t will have the form kh for some integer k with 0rtrTrT*–h. It is assumed that at all times t, there will be a full range of default-free zero-coupon bond trades, as well as a full range of risky

Table 1.

The Model Notation.

Default-Free Term Structure of Interest Rates F(t,T) a(t,T) s(t,T) P(t,T) r(t)

Default-free instantaneous forward rate Drift of the default-free forward rate F(t,T) Volatility of the default-free forward rate F(t,T) Default-free zero coupon bond price Default-free instantaneous short rate, r(t) ¼ F(t,t)

Defaultable Term Structure of Interest Rates i

ji(t,T) Si(t,T) bi(t,T) Zi(t,T) Bi(t,T)

i ¼ 1, 2, y , n our model allows more than one defaultable counterpart, subscript i in credit derivatives can refer to a firm i, a bond i, or a risky name i Defaultable instantaneous forward rate, where i ¼ 1, 2, y , n Defaultable instantaneous forward rate spread by definition: Si(t,T) ¼ ji(t,T)F(t,T) Drift of forward spread Si(t,T) Volatility of forward spread Si(t,T) Defaultable zero coupon bond price

Credit Risk Model Di

li(t) liP(t) fI(t) xi(t) H

( Default event; where Di ¼

0;

default event does not occur

1;

default event occurs

Default probability in the risk-neutral world Default probability in the real world Recovery rate, when default events occur Premium for bearing credit risk Time interval

200


zero-coupon bond trades. We also assume that markets are free from arbitrage; thus, an equivalent Martingale measure Q exists for this economy. The default-free continuous-time forward rate process is denoted as dF c ðt; T Þ ¼ ac ðt; T Þ dt þ sc ðt; T Þ dW

(2)

while its discrete-time counterpart is denoted as F ðt þ h; T Þ ¼ F ðt; T Þ þ aðt; T Þ h þ sðt; T Þ X

pffiffiffi h

(3)

At the same time, there also exist multiple forward credit-spread processes; we let these spreads adopt the following continuous-time process: dSci ðt; T Þ ¼ bci ðt; T Þ dt þ Zci ðt; T Þ dW i

(4)

where dW and dWi are normally distributed random variables with zero mean and variance dt,3 and a discrete-time process of pffiffiffi (5) S i ðt þ h; T Þ ¼ S i ðt; T Þ þ bi ðt; T Þ h þ Zi ðt; T Þ X i h where the terms X and Xi are both taken to be binomial variables, each of which takes on the value of 71 with probability 1/2. We place no restrictions on any of the correlations between dFc (t,T) and all of the dSc (t,T). Taking n ¼ 2 as an example, the relationships existing between dFc (t,T), dSc1 (t,T) and dSc2 (t,T) are as illustrated in Fig. 1, and we assume that the correlations between dF (t,T), dS1(t,T) and dS2(t,T) are the same as those in the continuous time setting. We let n ¼ 2 be the simplest description of an interdependent default structure. Terms S1 and S2 can represent the credit spreads of either a risky underlying bond and a defaultable derivatives seller, or two risky underlying bonds. The former is the so-called ‘counter-party risk’. Under a framework in which correlations are taken into consideration, our task is to find a set of appropriately determined probabilities that can

ρ FS

dF c (t,T )

1

c

dS 1 (t,T )

ρ FS

2

c

ρS S

dS 2 (t,T )

1 2

Fig. 1.

The Correlation between the Forward Rate and Forward Spreads.


201

suitably describe the joint distribution of the forward rate and forward spreads (we will deal with this problem later). When illustrating an implementation of the model in the latter context, we will focus on the discretetime settings; however, referring to Das and Sundaram (2000), by taking limits as the time interval h-0, the continuous time expressions (2) and (4) can be approximated by the discrete time expressions (3) and (5). We can also define P (t,T) as 8 9 < T=h1 = X Pðt; T Þ ¼ exp F ðt; khÞ h (6) : k¼t=h ; and Bi (t,T) as

8

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