Empirical Techniques in Finance

Springer Finance

Ramaprasad Bhar Shigeyuki Hamori

Empirical Techniques in Finance With 30 Figures and 30 Tables

123

Professor Ramaprasad Bhar School of Banking and Finance The University of New South Wales Sydney 2052 Australia E-mail: [email protected] Professor Shigeyuki Hamori Graduate School of Economics Kobe University Rokkodai, Nada-Ku, Kobe 657-8501 Japan E-mail: [email protected]

Mathematics Subject Classification (2000): 62-02, 62-07

Cataloging-in-Publication Data Library of Congress Control Number: 2005924539

ISBN 3-540-25123-5 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springeronline.com © Springer-Verlag Berlin Heidelberg 2005 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: design & production Production: Helmut Petri Printing: Strauss Offsetdruck SPIN 11401841

Printed on acid-free paper – 43/3153 – 5 4 3 2 1 0

Table of Contents

1 Introduction

1

2 Basic Probability Theory and Markov Chains 2.1 Random Variables 2.2 Function of Random Variable 2.3 Normal Random Variable 2.4 Lognormal Random Variable 2.5 Markov Chains 2.6 Passage Time 2.7 Examples and Exercises References 3 Estimation Techniques 3.1 Models, Parameters and Likelihood - An Overview 3.2 Maximum Likelihood Estimation and Covariance Matrix of Parameters 3.3 MLE Example - Classical Linear Regression 3.4 Dependent Observations 3.5 Prediction Error Decomposition 3.6 Serially Correlated Errors - Overview 3.7 Constrained Optimization and the Covariance Matrix 3.8 Examples and Exercises References

4 Non-Parametric Method of Estimation 4.1 Background 4.2 Non-Parametric Approach 4.3 Kernel Regression 4.4 Illustration 1 (EViews) 4.5 Optimal Bandwidth Selection 4.6 Illustration 2 (EViews) 4.7 Examples and Exercises References

19 19 20 22 23 24 25 27 28 29

X

Table of Contents

5 Unit Root, Cointegration and Related Issues 5.1 Stationary Process 5.2 Unit Root 5.3 Dickey-Fuller Test 5.4 Cointegration 5.5 Residual-based Cointegration Test 5.6 Unit Root in a Regression Model 5.7 Application to Stock Markets References 6 VAR Modeling

6.1 Stationary Process 6.2 Granger Causality 6.3 Cointegration and Error Correction 6.4 Johansen Test 6.5 LA-VAR 6.6 Application to Stock Prices References 7 Time Varying Volatility Models 7.1 Background 7.2 ARCH and GARCH Models 7.3 TGARCH and EGARCH Models 7.4 Causality-in-Variance Approach 7.5 Information Flow between Price Change and Trading Volume References

8 State-Space Models (I) 8.1 Background 8.2 Classical Regression 8.3 Important Time Series Processes 8.4 Recursive Least Squares 8.5 State-Space Representation 8.6 Examples and Exercises References

9 State-Space Models (11) 9.1 Likelihood Function Maximization 9.2 EM Algorithm

9.3 Time Varying Parameters and Changing Conditional Variance (EViews) 9.4 GARCH and Stochastic Variance Model for Exchange Rate (EViews) 9.5 Examples and Exercises References 10 Discrete Time Real Asset Valuation Model 10.1 Asset Price Basics 10.2 Mining Project Background 10.3 Example 1 10.4 Example 2 10.5 Example 3 10.6 Example 4 Appendix References

11 Discrete Time Model of Interest Rate I 1.1 Preliminaries of Short Rate Lattice 11.2 Forward Recursion for Lattice and Elementary Price 11.3 Matching the Current Term Structure 11.4 Immunization: Application of Short Rate Lattice 11.5 Valuing Callable Bond 11.6 Exercises References

12 Global Bubbles in Stock Markets and Linkages 12.1 Introduction 12.2 Speculative Bubbles 12.3 Review of Key Empirical Papers 12.4 New Contribution 12.5 Global Stock Market Integration 12.6 Dynamic Linear Models for Bubble Solutions 12.7 Dynamic Linear Models for No-Bubble Solutions 12.8 Subset VAR for Linkages between Markets 12.9 Results and Discussions 12.10 Summary References 13 Forward FX Market and the Risk Premium 13.1 Introduction

111 113 116 126

XI1

Table of Contents

Alternative Approach to Model Risk Premia The Proposed Model State-Space Framework 13.5 Brief Description of WolffICheung Model 13.6 Application of the Model and Data Description 13.7 Summary and Conclusions Appendix References 14 Equity Risk Premia from Derivative Prices 14.1 Introduction 14.2 The Theory behind the Modeling Framework 14.3 The Continuous Time State-Space Framework 14.4 Setting Up The Filtering Framework 14.5 The Data Set 14.6 Estimation Results 14.7 Summary and Conclusions References

Index About the Authors

1 Introduction

This book offers the opportunity to study and experience advanced empirical techniques in finance and in general financial economics. It is not only suitable for students with an interest in the field, it is also highly recommended for academic researchers as well as the researchers in the industry. The book focuses on the contemporary empirical techniques used in the analysis of financial markets and how these are implemented using actual market data. With an emphasis on implementation, this book helps focusing on strategies for rigorously combing finance theory and modeling technology to extend extant considerations in the literature. The main aim of this book is to equip the readers with an array of tools and techniques that will allow them to explore financial market problems with a fresh perspective. In this sense it is not another volume in econometrics. Of course, the traditional econometric methods are still valid and important; the contents of this book will bring in other related modeling topics that help more in-depth exploration of finance theory and putting it into practice. As seen in the derivatives analysis, modern finance theory requires a sophisticated understanding of stochastic processes. The actual data analyses also require new statistical tools that can address the unique aspects of financial data. To meet these new demands, this book explains diverse modeling approaches with an emphasis on the application in the field of finance. This book has been written for anyone with a general knowledge of the finance discipline and interest in its principles together with a good mathematical aptitude. For the presentation of the materials throughout the book, we therefore focused more on presenting a comprehensible discussion than on the rigors of mathematical derivations. We also made extensive use of actual data in an effort to promote the understanding of the topics. We have used standard software tools and packages to implement various algorithms. The readers with a computer programming orientation will enormously benefit from the available program codes. We have illustrated the implementation of various algorithms using contemporary data as appropriate and utilized either Excel spreadsheet (Microsoft Corporation), EViews (Quantitative Micro Software), or GAUSS

2

1 Introduction

(Aptech Systems Inc.) environments. These program codes and data would be made available through one of the author's website (www.bhar.id.au) with appropriate reference to the chapters in the book. We have implemented the routines using the software package versions currently in use so that most readers would be able to experiment with these almost immediately. We sincerely hope that the readers would utilize these software codes to enhance the capabilities and thus contribute to the empirical finance field in future.' Besides the first introductory chapter, the book comprises thirteen other chapters and the brief description these chapters follow. The chapter 2 reviews the basic probability and statistical techniques commonly used in quantitative finance. It also briefly covers the topic of Markov chains and the concept of first passage time. The chapter 3 is devoted to estimation techniques. Since most empirical models would consist of several unknown parameters as suggested by the underlying theory, the issue of inferring these parameters from available data is of paramount importance. Without these parameters the model is of little use in practice. In this chapter we mainly focus on maximum likelihood approach to model estimation. We discuss different ways to specify the likelihood function and these become useful for later chapters. We devote sufficient time to explain how to deal with placing restrictions on model parameters. As most commercially available optimization routines would automatically produce the covariance matrix of the parameters at the point of convergence, we include a careful analysis how to translate this to the constrained parameters that are of interest to the user. This translated covariance matrix is used to make inference about the statistical significance of the estimated parameters. In chapter 4 we cover the essential elements of non-parametric regression models and illustrate the principles with examples. Here we make use of the routines on Kernel density function available in the software package EViews. Chapters 5 and 6 then review the stationary and nonstationary time series models. The former chapter discusses the unit root, cointegration and related issues; the latter, multivariate time series models such as VAR, VECM and LA-VAR. Chapter 7 reviews time varying volatility models, such as ARCH, GARCH, T-GARCH and E-GARCH. Since these models have dominated the literature over the last several years we emphasize applications rather than theories. We extend this topic to include causality in

Datastream is a trademark of THOMSON FINANCIAL. EViews is a trademark of Quantitative Micro Software. Excel is a trademark of Microsoft Corporation. GAUSS is a trademark of Aptech Systems, Inc.

1 Introduction

3

variance and demonstrate its efficacy with an application to the commodity futures contracts. The chapter 8 is devoted to explaining the state-space models and its application to several time series data. We have attempted to demystifL the concepts underlying unobserved components in a dynamical system and how these could be inferred from an application of the filtering algorithm. The filtering algorithm and its various sophistication dominate the engineering literature. In this book we restrict ourselves to those problem settings most familiar to the researchers in finance and economics. Some of the examples in this chapter make use of the limited facility available in EViews to estimate such models. In chapter 9 we take up the issue of estimation of state-space models in greater detail by way of maximization of prediction error decomposition form of the likelihood function. This analysis gives the reader an insight into how various model parameters feed into the adaptive filtering algorithm and thus constitute the likelihood function. It becomes clear to the reader that for any reasonable practical system the likelihood function is highly non-linear in model parameters and thus optimization is a very complex problem. Although the standard numerical function optimization technique would work in most situations, there are cases where an alternative method based on Expectation Maximization is preferable. For the benefit of the readers we give a complete GAUSS program code to implement the EM algorithm. We sincerely hope that the readers would experiment with this code structure to enhance their understanding of this very important algorithm. In the next two chapters, 10 and 11, we move away from the state-space systems and take up the practical issues in modeling stochastic process in a discrete time framework. We will, however, take up more challenging modeling exercises using state-space models in the succeeding chapters. In chapter 10 we discuss the discrete time stochastic nature of real asset problem. This approach is suitable for resource based valuation exercises where there might be several embedded options available to the investor. After describing the basic issues in financial options valuation and the real asset options valuation, we describe the approach using a mining problem. We structure the development in this chapter and the next one following the excellent book by D. G. Luenberger on Investment Science (Luenberger DG (1997) Investment science. Oxford University Press, New York). We, however, add our interpretation of the issues as well as other illustrations. Besides, we include the relevant implementations in Excel spreadsheets. In chapter 1 1, we maintain the same discrete time theme and take up the issues with modeling interest rates and securities contingent on the term

4

1 Introduction

structure of interest rates. We explain the elegant algorithm of forward recursion in a fashion similar to that in D. G. Luenberger's Investment Science. There are several illustrations as well as the spreadsheets of the implementation. We hope that the readers would take full advantage of these spreadsheets and develop them further to suit their own research needs. In chapter 12 we highlight the recent advances in inferring the speculative component in aggregate equity prices. The topic area in this chapter should be very familiar to most researchers and students in finance. However, it may not be as obvious how to extend the standard present value models to infer unobserved speculative price component. The implementation of the models in this chapter relies upon the understanding of the contents of chapters 8 and 9. We not only infer the speculative components, we extend the analysis to investigate whether these components are related between different markets. The last two chapters, 13 and 14, deal with the important issue of risk premium in asset prices. Chapter 13 covers the foreign exchange prices and the chapter 14 deals with the equity market risk premium. Both these chapters require some understanding of the stochastic processes. Most students and researchers would be familiar with the topic of risk premium in a regression-based approach. That makes it a backward looking estimation process. We, however, exploit the rich theoretical structures in the derivatives market that connects the probability distribution in the risk-neutral world and the real world. Finally, this leads to a convenient mechanism, with minimum of assumptions, to uncover the market's belief about the likely risk premium in these markets. Since the methodology is based on the derivative securities, the inferred risk premium is necessarily forwardlooking. These two chapters completely rely on the unobserved component modeling framework introduced in the chapters 8 and 9. The associated GAUSS programs would be of immense benefit to all readers and would help them enhance their skills in this area as well.

2 Basic Probability Theory and Markov Chains

2.1 Random Variables Suppose x is a random variable that can take on any one of a finite number of specific values, e.g., x,,x ,,... x, . Also assume that there is a probability p, that represents the relative chance of an occurrencex, . In this case, pl satisfies

xm 1-1

pi = 1 and p, 2 0 for each i. Each p, can be thought of as the

relative frequency with which x, will occur when an experiment to observe x is carried out a large number of times. If the outcome variable can take on any real value in an interval, e.g., the temperature of a room, then a probability density function p(5) describes the probability. Since the random variable can take on a continuum of values, the probability density function has the following interpretation:

The probability distribution of the random variable x is the function F(5) defined as,

It therefore follows that F(-a) = 0 and F(a) = 1 . In the case of a continuum of values, if F is differentiable, then dF/d< = p(5) . Two random variables, x and y, are described by their joint probability density or joint probability distribution. The joint distribution is the function F defined as,

6

2 Basic Probability Theory and Markov Chains

The joint density is defined in terms of derivatives for random variables that take on a continuum of values. For discrete values, the joint density at a pair (x, ,y,) is p(x, ,y,) , which is equal to the probability of that pair occurring. This concept can be extended to n random variables. The distribution of any one of the random variables can be easily recovered from a joint distribution. Given the distribution F( 0 , we can change the spread by varying h if we define,

We can now define the weight function to be used in the weighted average as,

If h is very small, the averaging will be done with respect to a rather small neighborhood around each of the X, 's . If h is very large, the averaging will be over a large neighborhood of the X, 's . The degree of averaging amounts to adjusting the smoothing parameter, h, also known as bandwidth. Substituting equations (4.6) and (4.7) into equation (4.3) yields,

This is known as Nadaraya-Watson kernel estimator m, (x) of m(x) . Under a certain regularity condition on the shape of the kernel and the magnitude and behavior of the weights, we find that as the sample size grows, m,(x) + m(x) asymptotically. This convergence property holds for a large class of kernels. One of the most popular kernels is the Gaussian kernel defined by,

4.4 Illustration 1 (EViews)

35

In analyzing different examples with the EViews package, we would make use of this Gaussian kernel.

4.4 Illustration I (EViews) To illustrate the efficacy of kernel regression in capturing nonlinear relations, consider the smoothing technique for an artificially generated dataset using Monte Carlo simulation. Let {X,) denote a sequence of 500 observations that take on values between 0 and 2n: at evenly spaced increments, and let {Y,) be related to {X,) through the following nonlinear relation:

where {E,) is a sequence of IID pseudorandom standard normal variates. Using the simulated data pairs{X,,Y,) , we attempt to estimate the conditional expectation E[Y, I X,] = sin(X,) by kernel regression. We apply the Nadaraya-Watson estimator (4.8) with a Gaussian kernel to the data, and vary the bandwidth parameter h among 0.16,, 0.36,, 0.56, , where 6, is the sample standard deviation of {X,) . By varying h in units of standard deviation we are effectively normalizing the explanatory variable. The kernel estimator can be plotted for each variable. We notice from the plots that the kernel estimator is too choppy when the bandwidth is too small. It thus appears that for very low bandwidth, the information is too sparse to recover sin(X,) . While the kernel estimator succeeds in picking up the general nature of the function, it shows local variations due to noise. As the bandwidth climbs these variations can be smoothed out. At intermediate bandwidth, for example, the local noise is largely removed and the general appearance of the estimator is quite appealing. At still higher bandwidth, the noise is completely removed but the estimator fails to capture the genuine profile of the sine function. In the limit, the kernel estimator approaches the sample average of {Y,) and all the variability with respect to {Xi) is lost.

36

4 Non-Parametric Method of Estimation

This experiment may be carried out with other kernel functions (provided by EViews) as well. EViews also allows automatic selection of bandwidth. This brings us to the topic of optimal bandwidth selection.

4.5 Optimal Bandwidth Selection Choosing the proper bandwidth is critical in kernel regression. Among the several methods available for bandwidth selection, the most common is called cross-validation. This method is performed by choosing the bandwidth to minimize the weighted-average squared error of the kernel estimator. For a sample of T observations {X,,Yt):I: , let 1

m,,, (x,)= -Cwt,T(x,)yt T ttj

.

This is basically the kernel estimator based on the dataset with the jth observation deleted, evaluated at the j& value X, . The cross validation function CV(h) is defined as,

where 6(Xt) is a non-negative weight function required to reduce the effect of edges of the boundary (for additional information, see Hardle (1990)). The function CV(h) is called cross-validation since it validates the success of the kernel estimator in fitting {Y,) across T subsamples (Xt ,Yt , each with one observation omitted. The optimal bandwidth is the bandwidth that minimizes this function.

4.6 Illustration 2 (EViews) This example will acquaint you with the use of EViews for applying both non-parametric and parametric estimations procedures in the modeling of the short-term interest rate. Academic researchers and practitioners have

4.6 Illustration 2 (EViews)

37

been investigating ways to model the behavior of the short-term interest rate for many years. The short-term interest rate plays very important roles in financial economics. To cite just one use, its function is crucial in the pricing of fixed income securities and interest rate contingent claims. The following example deals with a method to model the mean and variance of changes in the short-term interest rate. The behavior of the short-term interest rate is generally represented by,

where r, is the interest rate, dWt is the standard Brownian motion, M(.) is the conditional mean function ofdr, , and V(.) is the conditional variance function of dr, . In estimating the model in equation (4.13), a non-parametric method does not need to specify the functions M(.) and V(-) . As part of the exercise, a standard parametric form may also be estimated for equation (4.13). The volatility estimated by both methods may then be compared. The standard non-parametric regression model is,

where, Y, is the dependent variable, X, is the independent variable, and v, is the IID with mean zero and finite variance. The aim is to obtain a non-parametric estimate of f(-) using the Nadaraya-Watson estimator. The conditional mean and variance of the interest rate changes can be defined as,

38

4 Non-Parametric Method of Estimation 2

Y,, = Art, YZt= (Ar,) . Estimates of the conditional means, EIYlt 1 X, = x] and E[YZt( Xt = x] are obtained from the following non-

where, X,

= rt,

parametric regressions:

The means of equations (4.17) and (4.18) can be estimated using the estimator in equation (4.8). This provides the estimates of ~ ( r , and ) ?(rt) in the equations (4.15) and (4.16), respectively. In this process, we use a Gaussian kernel and the optimal bandwidth suggested by Silverman (available in EViews). The variance estimate obtained from this non-parametric method may be compared with that from a parametric specification. The popular parametric specification for the short-term interest rate is the following GARCH-M (GARCH-in-Mean) model,

We encourage you to study the two different variance estimates-the non-parametric one obtained from equation (4.16) and the parametric one obtained from equation (4.20). Both of these methods have been applied in practice for different applications. The dataset for this exercise is described in the next section.

4.7 Examples and Exercises Exercise 4.1 : The empirical models for equity return in the CAPM (capital asset pricing model) framework most commonly adopted by researchers is given by,

References

39

The left-hand side represents the excess return from the asset and the right-hand side is a linear function of the excess return from the market. The aim of this exercise is to explore, using nonparametric kernel regression, whether such a linear relation holds for a given dataset. The dataset consists of weekly Japanese market data spanning January 1990 to December 2000. It contains data on excess return in the banking sector as well as excess return in the total market. You may also examine the above relationship using the GARCH(1,l) structure for the residual term.

References Hafner CM (1998) Nonlinear time series analysis with applications to foreign exchange rate volatility. Physica-Verlag, Berlin Hardle W (1990) Applied nonparametric regression. Cambridge University Press, Cambridge Hart JD (1997) Nonparametric smoothing and lack-of-fit tests. Springer, Berlin Niizeki MK (1998) Empirical tests of short-term interest rate models: a nonparametric approach. Applied Financial Economics 8: 347-352 Pagan A, Ullah A (1999) Nonparametric econometrics. Cambridge University Press, Cambridge Scott DW (1992) Multivariate density estimation. John Wiley & Sons, New York Silverman BW (1986) Density estimation for statistics and data analysis. Chapman and Hall, New York

5 Unit Root, Cointegration and Related Issues

5.1 Stationary Process A stochastic process {y,) is weakly stationary or covariance stationary if it satisfies the following conditions. 1. E[y, ] is a constant; 2. V[y, ] is a constant; 3. Cov[y, ,y,,] is a function of s, but not oft, where s = +1,+2,-.-. In other words, a stochastic process (y,) is said to be weakly stationary (or covariance stationary) if its mean, variance and autocovariances are unaffected by changes of time. Note that the covariance between observations in the series is a function only of how far apart the observations are in time. Since the covariance between y, and y,, is a function of s, we can define the autocovarince function:

Equation (5.1) shows that for s=O, y(0) is equivalent to the variance of y, . Further, the autocorrelation function (ACF) or correlogram between y, and y,, is obtained by dividing y(s) by the variance y(0) as follows:

Enders (1995) and Hamilton (1994) are good reference to understand time series models. The explanation of chapters 5 and 6 relies on them.

42

5 Unit Root, Cointegration and Related Issues

The property of stationary stochastic process is represented by autocorrelation function. Example 5.1 : white noise process The simplest stationary stochastic processes is called the white noise process, u, . This process has the following properties:

.(.)=

('

s=o, 0 for s=+1,+2,... .

A sequence u, is a white noise process if each value in the sequence has a mean of zero, a constant variance and no autocorrelation. Example 5.2: MA(1) process The first order moving average process, i.e., the MA(1) process, is written as follows:

where u, is a white noise process. Thus,

5.1 Stationary Process

002 Cov[~t,~t-,I=~(s)= 0

C (

43

for s = +l, for s=+2,f3,..-,

1 0

for

s=O

for

s=+l

0

for s =i2,+3,--a.

Generally speaking, the MA(q) process is written as follows:

Note that the finite moving average processes are stationary regardless of the parameter values. Example 5.3: AR(1) process The first order autoregressive process, i.e., the AR(1) process, is written as :

where ut is a white noise process. If/ $ I< 1, we characterize the AR(1) process as stationary. Thus,

44

5 Unit Root, Cointegration and Related Issues

P(S>=

1 for s = 0, 4"or s = f l , f 2 , . . ..

As is clear from equation (5.17), p(s) + 0 as I s I+ co . A more general AR(p) process would be written as follows:

The AR(p) process is stationary if the roots of the characteristic equation,

have modulus greater than 1 or lie outside the unit circle. The AR(1) process is the simplest case and its characteristic equation is,

with a single root of 114. This lies outside the unit circle if

I 4 I< 1.

5.2 Unit Root There are important differences between stationary and nonstationary time series. Shocks to a stationary time series are temporary, while a nonstationary series has permanent components. Consider the following AR (I) process:

5.2 Unit Root

45

where u, is a white noise process. If I@I (E[x])'.

70

7 Time Varying Volatility Models

The GARCH model developed by Bollerslev (1986) is an extension of the ARCH model. The ARCH(p) process specifies the conditional variance solely as a linear function of past sample variances, whereas the GARCH(p,q) process allows lagged conditional variances to enter as well. This corresponds to some sort of adaptive learning mechanism. The variance dynamics is thus specified as follows:

The conditional variance at time t depends on three factors: a constant (o), past news about volatility taken as the squared error from the past (the ARCH term, i.e., Celaia:i ), and past forecast variance (the GARCH term, i.e.,

xq r=l

~,o:, ). The (p,q) in GARCH(p,q) refers to p ARCH terms

and q GARCH terms. The condition 0 2 0 , a, 2 0 , Pi 2 0 guarantees the non-negativity of ~ariance.~ This specification is logical since variance at time t is predicted by forming a weighted average of the forecast from the past and either a long-term average or constant variance. Example 7.1 : GARCH(1,l) Model Let us consider the simple GARCH(1,l) model as follow^:^

As clearly seen from equation (7.8), the GARCH (1,l) model includes one ARCH term and one GARCH (o:,) term. If equation (7.8) is lagged by one period and substituted for the lagged variance on the righthand side, an expression with two lagged squared errors and a two-period

Nelson and Cao (1992) show that inequality constraints less severe than those commonly imposed are sufficient to keep the conditional variance non-negative. In the GARCH(2,l) case, for example, w > 0 , al 2 0 ,

PI 2 0 , and

P,al+a22 0 are sufficient to ensure

or > 0 , such that a>may be negative.

The parameter subscripts are not necessary for the GARCH(1,1), TGARCH(l,l), and EGARCH(1,l) models and are suppressed for the remainder of this section.

7.3 TGARCH and EGARCH Models

71

lagged variance is obtained. By successively substituting for the lagged conditional variance, the following expression is found:

A sample variance would give each of the past squares an equal weight rather than a declining weight. The GARCH variance is thus like a sample variance, but one that emphasizes the most recent observations. Since o: is the variance forecasted one period ahead based on past information, we call it the conditional variance or volatility. The unpredictable in squared returns is given by

an equation which, by definition, is unpredictable based on the past. Substituting equation (7.10) into equation (7.8) yields the following alternative expression:

We can immediately see that the squared errors ( E: ) follow an ARMA(1,l) process. The autoregressive root is the sum of a and p , a value that governs the persistence of volatility shocks.

7.3 TGARCH and EGARCH Models In the models discussed so far, the variance dynamics are treated as symmetric, wherein a positive shock and negative shock of the same magnitude will have the same effect on the present volatility. Christie (1982) and Schwert (1989), however, pointed out that downward movements in the market are often followed by higher volatility than upward movements of the same magnitude. Glosten, Jagannathan and Runkle (1993), Zakoian (1994) and Nelson (1991) explicitly treat this asymmetry in variance in their extensions of the GARCH model.

72

7 Time Varying Volatility Models

Glosten, Jagannathan and Runkle (1993) and Zakoian (1994) proposed the threshold GARCH model (TGARCH model) to specify the asymmetry of volatility. The TGARCH(p,q) model is specified as follows:

where the dummy variable D,, is equal to 0 for a positive shock (E,, > 0 ) and 1 for a negative shock (E,, < 0). Provided that yi > 0 , the TGARCH model generates higher values for ot given E,, < 0 , than for a positive shock of equal magnitude. As with the ARCH and GARCH models, the parameters of the conditional variance are subject to non-negativity constraints. Example 7.2: TGARCH(1,l) Model As a special case, the TGARCH(1,l) model is given as:

In this case, equation (7.13) becomes

for a positive shock ( E,, > 0 ), and

for a negative shock (E,, < 0). Thus, the presence of a leverage effect can be tested by the hypothesis that y = 0 , where the impact is asymmetric if y+O. An alternative way of describing the asymmetry in variance is through the use of the EGARCH (exponential GARCH) model proposed by Nelson (199 1). The EGARCH(p,q) model is given by

7.3 TGARCH and EGARCH Models

73

where z, = E, /ot . Note that the left-hand side of equation (7.16) is the log of the conditional variance. The log form of the EGARCH(p,q) model ensures the non-negativity of the conditional variance without the need to constrain the coefficients of the model. The asymmetric effect of positive and negative shocks is represented by inclusion of the term z,, . If yi > 0 , volatility tends to rise (fall) when the lagged standardized shock, z ~=- ~~ ~ - ,~is/positive o ~ - (negative). ~ The persistence of shocks to the conditional variance is given by

x"i 1=1

. Since negative coefficients are not

precluded, the EGARCH models allows for the possibility of cyclical behavior in volatility. Example 7.3: EGARCH(1,l) Model As a special case, the EGARCH(1,l) model is given as follows:

Equation (7.17) becomes

for a positive shock (z, > 0), and

for a negative shock (z, < 0). Thus, the presence of a leverage effect can be tested by the hypothesis that yi = 0 , where the impact is asymmetric if y, # 0 . Furthermore, the sum of a and governs the persistence of volatility shocks in the GARCH (1,l) model, whereas only parameter P governs the persistence of volatility shocks in the EGARCH(1,l) model.

74

7 Time Varying Volatility Models

7.4 Causality-in-Variance Approach Cheung and Ng (1996) developed a testing procedure for causality-inmean and causality-in-variance. This test is based on the residual crosscorrelation function (CCF) and is robust to distributional assumptions. Their procedure to test for causality-in-mean and causality-in-variance consists of two steps. The first step involves the estimation of univariate timeseries models that allow for time variation in both conditional means and conditional variances. The second step constructs the residuals standardized by conditional variances and the squared residuals standardized by conditional variances. The CCF of standardized residuals is used to test the null hypothesis of no causality-in-mean, while the CCF of squaredstandardized residuals is used to test the null hypothesis of no causality-invariance. In the vein of Cheung and Ng (1996) and Hong (200 I), let us summarize the two-step procedure of testing causality. Suppose that there are two stationary time-series, X, and Yt . When I,, , I,, and I, are three information sets defined by I,,, = (X, ,X,,,..-), I,,, = (Y,, Y,-,,-..) and I, = (X,, X,, ,. ..Y,, Y,-,,-.-), Y is said to cause X in mean if

Similarly, X is said to cause Y in mean if

Feedback in mean occurs if Y causes X in mean and X causes Y in mean. On the other hand, Y is said to cause X in variance if

where px , is the mean of X, conditioned on 11,,,. Similarly, X is said to cause Y in variance if

7.4 Causality-in-Variance Approach

75

where py , is the mean of Yt conditioned on I,,,, . Feedback in variance occurs if X causes Y in variance and Y causes X in variance. The causality-in-variance has its own interest since it is directly related to volatility spillover across different assets or markets. As the concept defined in equations (7.20) through (7.23) is too general to test empirically, additional structure is required to make the general causality concept applicable in practice. Suppose X, and Yt can be written as:

where E, and 6, are two independent white noise processes with zero mean and unit variance. For the causality-in-mean test, we have the standardized innovation as follows:

Since both

E,

and

6, are unobservable, we have to use their estimates, 6,

and t t , to test the hypothesis of no causality-in-mean. Next, the sample cross-correlation coefficient at lag k, FEc(k), is computed from the consistent estimates of the conditional mean and variance of X, and Y, . This gives us

76

7 Time Varying Volatility Models

where c, 0 with positive probability for some t. The fact that there are risk neutral probabilities in the binomial model can be used to show that this model does not allow arbitrage opportunity. The binomial model is also said to be complete as any contingent claim can be replicated by a self-financing trading strategy.

10.2 Mining Project Background Options associated with investment opportunities are not always financial options. A factory manager may have the option to hire more employees, buying new equipment etc. Similarly, drilling for oil in a piece of land or offshore may be viewed as a series of operational options. These are referred to as real options. The option valuation framework has great scope since it can value virtually any contingent decision. In product development, for example, different design choices lead to different follow-on opportunities. In large irreversible investments the framework can be used to evaluate modifications to construction schedule or the trade-offs between options to delay, abandon, expand or accelerate against additional value they create. Both real and financial options valuation can be less precise in practice than in theory because certain asset and market features can affect law of one price from holding. The option valuation framework provides a clear image of the magnitude of the imprecision in valuation. The option valuation method essentially relies on constructing a tracking portfolio, which is dynamically updated as the value of the underlying asset changes. This states that the option and the tracking portfolio are affected by the same source of uncertainty. Two real asset features cause tracking errors: the costs of tracking and the quality of tracking. When it is costly to change the portfolio composition frequently it may result in the portfolio to wander away from the value of the option. For real options the tracking portfolio may include specific features or commodities that make dynamic tracking difficult. For example, real options have private risk that is not contained in traded securities. The risk of failing to develop a new technology is a private risk carried by the high-tech firm.The risk of not finding large amount of oil in a particular prospect is a private risk borne by the oil firm. For more information regarding real options see excellent books by, Amran and Kulatilaka (1 999) and Trigeorgis (1997).

130

10 Discrete Time Real Asset Valuation Model

Real options can usually be analyzed by the same methods used to analyze financial options. We need to set out an appropriate representation of uncertainty of the underlying asset e.g. using a binomial lattice and work backward to find the value of the option. We will analyze the issues in this approach through some examples. The examples here are taken from Luenberger (1998). We explain the intuitions behind these examples as well as provide the spreadsheets to implement the models.

10.3 Example I A straightforward gold mine has a great deal of remaining gold deposits and you are part of team that is considering leasing the mine from its owners for a period of 10 years. Gold can be extracted from this mine at a rate of up to 10,000 ounces per year at a cost of $200 per ounce. This cost is the total of mining and refining and does not include leasing cost. The current market price of gold is $400 per ounce. The interest rate is 10% per annum. Assuming that the price of gold, the operating cost and the interest rate remains constant over the 10 year period, what is the value of the lease? This is very straightforward and it is clear that the mine should be operated at full capacity to maximize profit. The annual profit is, therefore,

If we further assume that this cash flow occurs at the end of each year, the present value is,

This is the value of the lease. There is, however, some inherent contradiction in analyzing the problem in this simple way. These would become clearer as we proceed to more complex and realistic situations.

10.4 Example 2

131

10.4 Example 2 We extend the gold mine of example 1 to the case where the gold price fluctuates randomly. We still maintain the assumption the term structure of interest rate is flat. We also follow the convention that the price obtained from gold mined during the year is the price that prevailed at the beginning of that year and the cash flow occurs at the end of the year. We represent the random nature of gold price by a binomial lattice. Each year the price can increase by a factor of 1.2 with a probability of 0.75 and can decrease by a factor of 0.9 with a probability of 0.25. All other relevant parameters of the problem remain same as described in example 1. We wish to find out the value of the 10-year lease. We find the value of the lease by the methods developed for option pricing. The trick is to note that the value of the mining lease can be regarded as a financial instrument whose value fluctuates with the price of gold. In fact, the value of the lease at any point can only be dependent on the price of gold and the interest rate (assumed constant). The value of the gold mine lease is a derivative instrument dependent on the price of gold. In the attached spreadsheet labeled 'Example-2-3 (Table 10.1) the top panel describes the movement of the gold price based on the up and down factor assumed for this problem. The value of the lease can be entered node by node based on the gold price lattice. This is done in the second panel. The lease values are easily determined on the last nodes where these are all zeros since the mine must be returned to the owner at that time. Consider a node in year 9 where the lease has one more year to go. The lease value must be equal to the profit that can be made from the gold mined in that year. Remember that our assumption is that the gold can be sold at the price that prevailed at the beginning of the year and the cash flow occurs at the end of the year. Focusing on the top node in year 9 on the second panel, the value of the lease is,

Following the same reasoning all the nodes in year 9 can be filled in with lease values.

132

10 Discrete Time Real Asset Valuation Model

Table 10.1 Example -2-3 0

1

2

3

4

5

6

7

8

9

10

Panel 1: Gold Price Lattice 400.00

480.00 360.00

576.00 432.00 324.00

691.20 518.40 388.80 291.60

829.44 622.08 466.56 349.92 262.44

995.33 746.50 559.87 419.90 314.93 236.20

1194.39 895.80 671.85 503.88 377.91 283.44 212.58

37.74 26.41 17.91 11.54 6.76 3.19

37.15 26.28 18.12 12.01 7.42 3.98 1.46

1433.27 1 7 1 9 1074.95 1289.95 806.22 967.46 604.66 725.59 453.50 544.20 340.12 408.15 255.09 306.11 191.32 229.58 172.19

. 9 3 1 1547.93 1160.95 870.71 653.03 489.78 367.33 275.50 206.62 154.97

2476.69 1857.52 1393.14 1044.86 783.64 587.73 440.80 330.60 247.95 185.96 139.47

27.81+7( 19.99 12.25 14.13 8.74 9.73 6.10 6.43 4.12 3.95 2.63 2.10 1.52 0.71 0.69 0.04 0.06 0.00

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

Panel 2: Lease Values ($ Million) 24.22

27.90 18.04

31.3520.85 12.97

34.37 23.34 15.07 8.87

36.63 25.30 16.80 10.42 5.64

34.15 24.37 17.03 11.53 7.41 4.31 1.99 0.44

Panel 3: Lease Values Assuming Enhancements in Place ($ Million) 27.21

31.99 19.68

36.53 23.39 13.55

40.53 26.74 16.41 8.68

43.62 29.45 18.82 10.85 4.95

45.28 31.12 20.50 12.53 6.56 2.33

44.86 31.26 21.07 13.43 7.69 3.41 0.84

41.45 29.22 20.05 13.17 8.01 4.15 1.31 0.15

33.90 24.12 16.79 11.29 7.17 4.07 1.75 0.25 0.00

20.73 14.86 10.47 7.17 4.69 2.84 1.45 0.40 0.00 0.00

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

40.86 27.26 18.12 12.01 7.42 3.98 1.46

37.45 25.22 17.03 11.53 7.41 4.31 1.99 0.44

29.90 20.12 14.13 9.73 6.43 3.95 2.10 0.71 0.04

16.94 12.25 8.74 6.10 4.12 2.63 1.52 0.69 0.06 0.00

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

Panel 4: Lease with Option for Enhancement 24.80

28.79 18.14

32.75 21.01 12.97

36.53 23.60 15.07 8.87

39.62 25.73 16.80 10.42 5.64

41.28 27.12 17.91 11.54 6.76 3.19

See main text for explanation of entries.

The lease value at any node before year 9 would be equal to the profit that can be made from the gold mined in that year plus the expected value discounted from the two succeeding nodes.

10.5 Example 3

133

For example again, we focus on the top node in year 8 and the computation is detailed below:

In the above expression p is the risk-neutral probability of gold price to go up. This is easily seen to be,

The lease value can, therefore, be calculated by this backward recursion. We should also note that at those nodes where the price of gold is less than the cost of extraction (i.e. $200) we do not mine that year. In this way, the value of the lease today is $24.2m. (Based upon the level of accuracy you carry through the lattice you may get slightly different results). You probably are able to see the similarity of pricing financial options in the binomial lattice and this example. In example 1 we maintain the gold price constant and it is clearly not realistic. The assumption of flat term structure of interest rate, however, is commonly employed in problems of this kind. You should also note that if gold price were known to be constant then it would act as a risk-free asset with zero rate of return. In that case it would be incompatible with assumption of risk-free rate of 10%. Indeed, for the lattice of gold price must be constructed such that, u > (1 + r) > d , where u is the up factor, d is the down factor and r is the risk-free rate of interest. We can now move on to the next level of complexity, where more realistic operational scenario is analyzed.

10.5 Example 3 The gold mine we are analyzing already contains several real options, e.g. the yearly options to carry out the operations. In fact, the value of the lease can be expressed as the sum of these individual options. More interesting, however, is to consider the following situation.

134

10 Discrete Time Real Asset Valuation Model

Assume that there is possibility of enhancing the production rate of the above mine by making some structural changes and buying and installing new machines. This enhancement would cost $4 million, but would raise the mine capability to produce 12,500 ounces of gold per year and raise the operating cost to $240 per ounce. This enhancement alternative is an option since it need not be carried out and it is available to you over the whole term of the lease. We also assume that the enhancement can be undertaken in any year (beginning) and once in place it applies to all future years. We further assume that at the termination of the lease, the enhancement becomes the property of the original owner of the mine. We focus on the panels 3 and 4 in the spreadsheet labeled 'Example-2-3 to analyze this enhancement option. We proceed as follows. We first calculate the value of the lease assuming that the enhancement is already in place. This is shown in panel 3. This is same as the panel 2 except for the annual production is 12,500 ounces of gold and the operating cost of $240 per ounce. This shows the value of the lease is $27.2 million. Remember that this figure does not include the cost of enhancement, which is $4 million. If the enhancement were to be implemented at the beginning the lease value would be $23.2 million, which is slightly less than the $24.2 million valuation we got before. This indicates that it is not optimal to carry out the enhancement at the beginning. We have the option to carry out the enhancement at any time later and we proceed as follows to value that option. We construct the panel 4 in the spreadsheet labeled 'Example23 and we use the original parameters i.e. production rate of 10,000 ounces per year and production cost of $200 per ounce. But, at each node, in addition to the computation explained before, we also compare the value at a node with the value of the corresponding node in panel 3. If the node value in panel 3 is $4 million more than that computed for panel 4 we take that value. This ensures that the benefit of enhancement is taken into account correctly. The figures in bold in panel 4 indicate where we find it is advantageous to implement the enhancement. The overall value of the lease turns out to be $24.8 million - this is a slight improvement (since $4 million cost of enhancement has already been factored in) over the original value of $24.2 million. Finally, we illustrate additional operational complexity that is often encountered in real life situations.

10.6 Example 4

135

10.6 Example 4 In this case we analyze the problem of mining lease when the cost of extraction depends on the amount of gold remaining. If you lease the mine, you must decide how much to mine each period, taking into account that mining in one period affects future mining costs. The practical relevance of such a scenario can be looked at this way. If the mine has been worked heavily in the past and it is reaching depletion then it becomes increasingly difficult to extract rich ore. Hence the cost of extraction depends on the ore body remaining. Referring to the spreadsheet labeled 'Example4 (Table 10.2) the top panel gives the fluctuation of gold price over the ten-year period as for the previous example. We assume that that the cost of extraction is given by the function,

where x is the amount of gold remaining at the beginning of the year and z is the amount of gold extracted in ounces. Initially we assume that there are x, = 50,000 ounces of gold in the mine. We continue with assumption of 10% flat term structure of interest rates, and the profit from mining is determined by the price of gold at the beginning of the year. We assume further that all cash flows occur at the beginning of the year. As before, the preliminary analysis suggests that the value of the lease is zero at the final time and we enter zero at all the nodes for year 10. At any node in end of year 9 we must decide the optimal amount to mine during the tenth year. Accordingly, we need to solve,

where g is the price of gold at that particular node. We find the maximum by differentiating (see appendix for these steps) this with respect to z, and equating to zero. This gives,

136

10 Discrete Time Real Asset Valuation Model

Table 10.2 Example4

Panel 1: Gold Price Lattice 400.00

480.00 360.00

576.00 432.00 324.00

691.20 518.40 388.80 291.60

829.44 622.08 466.56 349.92 262.44

995.33 746.50 559.87 419.90 314.93 236.20

1194.39 895.80 671.85 503.88 377.91 283.44 212.58

1433.27 1074.95 806.22 604.66 453.50 340.12 255.09 191.32

865.4 579.9 391.7 260.7 170.0 108.5

1062.1 695.9 464.3 305.8 197.4 124.6 77.1

1316.8 833.0 543.7 351.5 222.9 138.3 84.2 50.4

1719.93 2063.91 2476.69 1289.95 1547.93 1857.52 967.46 1160.95 1393.14 725.59 870.71 1044.86 544.20 653.03 783.64 408.15 489.78 587.73 440.80 306.11 367.33 229.58 275.50 330.60 172.19 206.62 247.95 154.97 185.96 139.47

Panel 2: Factor of Proportionality 326.2

395.9 273.9

480.4 331.3 226.8

583.3 400.1 273.3 183.5

709.3 482.2 328.0 219.6 144.1

1658.7 996.0 622.4 387.6 237.4 142.9 84.6 49.5 28.7

2129.9 1198.0 673.9 379.1 213.2 119.9 67.5 37.9 21.3 12.0

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

See main text for explanation of the entries.

Therefore,

This shows that the value of the lease is proportional to x, and the proportionality constant, K, is given by,

10.6 Example 4

137

Thus,

We can now set up a lattice of K values with nodes corresponding to various gold prices. We have already shown that K,, = 0 for all the nodes in year 10. We can easily enter the entries for year 9 following the above relation by picking up the appropriate value of g from gold price lattice in the upper panel. The next important point to realize is that for nodes in year 8 we can write the following:

where, K, = p x K9 + (1 - p)K$, and K9 is the value on the node directly to the right (i.e. up state) and Ki is the value on the node just below that (i.e. down state). The discount factor given by d is (for our example) 111.10. This leads to,

and

where

138

10 Discrete Time Real Asset Valuation Model

We repeat this process for each node in year 8 and then for all other nodes till year 0. The lattice shows that KO=326.2 and this gives the lease value at the beginning, Vo = Koxo= 326.2 x 50,000

= $16.2 million.

(10.22)

This particular approach has been based on the particular cost function assumed in the process. It may not generally be true that for all cost functions such a convenient proportionality constant could be defined. In that case some form dynamic optimization algorithm needs to be adopted.

Appendix Node at t=9:

gx, 9'

( X ~ ) = ~ G -

500g2x: 10002X9

Appendix

139

Thus,

Node at t=8:

where d is the discount factor based on the interest rate assumed constant, and K, is the expected value of the factor K at t = 9 with respect to the two succeeding nodes. Thus,

Note that the factor K only depends upon the gold price in the respective nodes and not on the amount of gold remaining in the deposit or extracted. dV8 -dz,

-g---

1oooz,

x8

dK, =O.

Substituting z, from the above equation in equation (A10.6) leads to,

which simplifies to,

140

10 Discrete Time Real Asset Valuation Model

References Amran M, Kulatilaka N (1999) Real options, managing strategic Investment in an uncertain world. Harvard Business School Press, Boston Luenberger DG (1997) Investment science. Oxford University Press, New York Trigeorgis L (1997) Rea options, managerial flexibility and strategy in resource allocation. The MIT Press, Cambridge

11 Discrete Time Model of Interest Rate

11.I Preliminaries of Short Rate Lattice This chapter draws on the book by Luenberger (1998), and enhances its interpretation as well as illustrates its application to several examples. Binomial lattice provides a framework for constructing interest rate models. The basic time span is selected to be of interest to the analyst e.g. a week or a month etc. We then assign a short rate to each node. Short rate is just one period forward rate. This rate is applicable over the next period. To obtain the full probabilistic behaviour of the process each node may be assigned probabilities. In pricing securities dependent on interest rate "real" probabilities are not important. We, therefore, assign risk-neutral probabilities. In this approach risk-neutral probabilities are assigned rather than derived from the replication argument (as in equity option). It is, therefore, convenient to assign these probabilities as 0.50. To be able feel comfortable with the development in this area we need to have a good understanding of the symbols normally used and what it really means. In this respect the following Figure 11.1 introduces readers to the environment appropriately. From a given node (t, i) two successor nodes that are reachable; (t+l, i) and (t+l, i+l). Let Vt,ibe the value of a security at (t, i) and D,; be the cash inflow at (t, i). Then, the lattice rule suggests,

where rt,i > 0 is the short rate at node (t, i).

142

11 Discrete Time Model of Interest Rate

Fig. 11.1 Short rate lattice and naming conventions

The spot rate implied by a short rate lattice can be extracted as follows. Poo(2)is the price of a bond that matures in period 2 and pays $1.

1

PO,(2) = -[0.5x Plo(2) + 0.5 x PI,(2)l. 1+ roo

Thus, the 2 period spot rate, s2, is obtained from

1

This process can be adapted to infer spot rates for other maturities. Besides, the process can be applied with respect to any other node (t, i). Thus, the short rate lattice helps generate a family of spot rate curves. This describes the evolution of the term structure. The term structure obtained from the short rate lattice is arbitrage-free. To understand this, consider the one period risk-neutral pricing formula,

1 1 . l Preliminaries of Short Rate Lattice

143

For the security to represent arbitrage we must have, P,,, 1 0 , V,,, , 2 0, Vt+,,i+l2 0 with one of these inequalities being a strict one. This is clearly impossible since all values are positive by construction. Hence, no arbitrage is possible over one period. This argument can be extended to any other periods. Hence, the short rate lattice provides an environment for modeling interest rate movements. Example: Consider the following 6 period short rate lattice. To construct this we used an up factor, u=1.3, and down factor d=0.9. Risk-neutral probabilities are 0.5.

The four period spot rates can be obtained as follows. First, assign the final cash flow from the discount bond at period 4, then compute discounted present value at the prior nodes using the short rates applicable to that node.

The computation involved is explained below.

Similarly, all other prices are obtained. The bond that pays $1 at t=4 is priced today 0.7334. The four period spot rate is,

11 Discrete Time Model of Interest Rate

144

Similarly, the other spot rates can be obtained. With respect to the above short rate lattice the spot rates are 0.0700, 0.0734, 0.0769, 0.0806 and 0.0844. Now consider the price of a call option on the 4 period bond with maturity at t=2 and strike price 84.00. We reproduce the bond price lattice up to t=2 for $100 face value.

The payoff from the call option at t=2, give that the strike price is 84.00,

coo

CII clo

0.00 0.81 5.09

We want find Coo. Referring to the short rate lattice,

Thus, the value of the call option today is 1.4703.

11.2 Forward Recursion for Lattice and Elementary Price

145

The discussion above relies upon the short rate lattice. The most critical task is to obtain this lattice such that it matches the observed term structure. This may be achieved by following the methodologies suggested by Ho and Lee (1986) or Black, Derman and Toy (1990). In the next section we illustrate how this may be achieved. Again, before embarking on the actual process, we need to understand how to speed up the algorithm of spot rate computation given a short rate lattice, without traversing the tree several times.

11.2 Forward Recursion for Lattice and Elementary Price We have seen how to use lattice for valuing interest rate sensitive securities using a backward recursion approach. To construct the term structure of interest rates from the short rate lattice many passes have to be made through the lattice. There is another approach - forward recursion -, which is more efficient in term of computation time. A single pass through the lattice will determine the whole term structure. It, however, depends upon the concept of elementary prices. With reference to the short rate lattice, the elementary price, Po(k, s) , refers to the price of a security at time 0 that pays $1 at the node (k, s) and zero everywhere else. Here k refers to the time axis and s refers to the state at that time. Fig. 11.2 shows the naming conventions. Po(1,l) refers to the price of a security at time 0, which pays $1 at that node and zero everywhere else. These are called elementary prices since they refer to a security with $1 payoff at only one node. Given a short rate lattice we can construct the elementary prices for each node in one forward pass. Once the elementary prices are known, constructing the term structure is straightforward. Assume that all elementary prices have been computed up to and including time k starting at 0. There are (k+l) states (s) at that time period and these are labelled 0 through k. Consider now the time period (k+l) and any state in the range 1 through k at time ( k i l ) i.e. excluding the two end states. For a node (k+l,s) there are two predecessor nodes at time k and these are: (k, s) and (k, s-1).

146

11 Discrete Time Model of Interest Rate

Fig. 11.2 Elementary prices

Fig. 11.3 Forward recursion

In Fig. 11.3, d,,, , and d,,,-, are the discount factors determined by the corresponding short rate lattice. If r represents short rate then d,,, = 1/(I+ r,,,) , similarly for the other one. Recall that we have assumed Po(k, s) and Po(k, s - 1) have already been determined. Corresponding to an elementary security that pays $1 at node (k+l,s), the time zero values at the two predecessor nodes are 0.5xdk,,-, x Po(k,s - 1) and 0.5 x d,,, x Po(k, s) respectively. Therefore, the time zero value of the elementary security corresponding to the node (k+l,s) is the sum of these two components. For the two end nodes at time (k+l) since

11.2 Forward Recursion for Lattice and Elementary Price

147

there is only one predecessor node the values will consist of only one component. This is summarized below:

Once all the elementary prices are computed, price of any security can be easily found. For, example the time zero price of a discount bond that

Po(n, s) .

pays $1 at time n, is s=o

Fig. 1 1.4 presents another intuitive explanation of the efficacy of the elementary prices. Other aspect of the forward recursion algorithm is also highlighted. The elementary prices at t=2 are labelled. The discount factor applicable at two different nodes and branches are also identified.

Fig. 11.4 Elementary prices and discount bond

148

11 Discrete Time Model of Interest Rate

From the elementary prices identified in the diagram, we can write the price of two-period zero coupon bond as,

11.3 Matching the Current Term Structure The previous discussion relied upon having the short rate lattice. We will now explore how to build the short rate lattice from the current observed term structure of interest rates. The procedure should also account for the fluctuation or volatility in the interest rates. There are number of methods available. We outline the method based on Ho-Lee (1986) modelling framework. The short rate is given by, r,, =a, + b,s , where a, and b, are parameters to be determined. It is clear that in Ho-Lee framework these parameters are time varying. At any given time period the differences in the short rate between states is controlled by b, . In fact it can be shown that this parameter represents the volatility of rate movement and a, represents the aggregate drift from period 0 through k. In the basic application we will assume that this volatility is constant. The problem, therefore, is to choose the parameter a, such that the short rate lattice produces the spot rate structure that is observed at time zero covering periods 0,1,2 ...n . The process is described in the context of a eight period term structure in the spreadsheet snap shot presented below. Notes: The relation between the Ho-Lee model parameters and the spot rate is an indirect one. The solution has to be based upon numerical methods of optimising some suitable criterion. It should become apparent that the method could be extended to account for the situation when the volatility parameter is not constant and in fact there is a term structure of volatility. The figures seen in the print out of the spreadsheet is essentially the at the point of convergence of the optimisation algorithm. In Excel, we have used Solver function to minimise the sum of

11.4 Immunization: Application of Short Rate Lattice

149

squared error differences of the model given spot rate and the given spot rates. Of course, any other meaningful optimisation criterion could be used depending on the application, The best way to get the appropriate feel of the model implementation, the reader has to experiment with it in Excel or any other suitable software environment. Period Spot rate

0 0.0767 0.0766

ak bk

0.0001 Short rate

State 7 6 5 4 3 2 1 0

0.0766

State Elementay price

p(0) Model spot rate Squared error Sum squared error

1.0000

0.9289 0.0766

0.8530 0.0827

0.7767 0.0879

0.6993 0.0935

0.6305 0.0966

0.5564 0.1026

0.4990 0.1044

0.4376 0.1088

1.51E-08 1.04E-09 4.66E-08 1.898-07 7.47E-07 l.lOE-06 6.15E-07 1.12E-07 2.838-06

11.4 Immunization: Application of Short Rate Lattice One of the key risk management in bond portfolio is to immunize it fiom anticipated changes in interest rats. Some of the methods normally used assume that the term structure changes in a parallel fashion. In other words, interest rate uncertainty is not properly addressed in such an approach. The short rate lattice can be used for immunising a bond portfolio while accounting for such uncertainties. The aim in this section is to explain how such a strategy might be implemented.

150

11 Discrete Time Model of Interest Rate

Consider a case where a series of cash obligations have to be met at specific times in future. Compute the initial value of the obligation stream using the lattice i.e. the present value of the obligation. We then need a bond portfolio with the same present value. After the first period the obligation stream can take one of two possible values depending on the two successor nodes. The bond portfolio will also have two possible values at the two successor nodes. If the bond portfolio values match the obligation values, the portfolio is immunised for one period. Therefore, for one period immunisation we need to match the present values at three places - the initial node and the two successor nodes. Due to the arbitrage-free property of the lattice, this matching can be achieved in a straightforward way using different bonds. For example, using two different bonds we can construct a portfolio having the same values (as that of the obligation stream) at each of the two successor nodes. The no-arbitrage property then makes sure that the initial value of the portfolio will also match the obligation stream. After one period, the portfolio can be rebalanced to obtain immunisation for the following period. By continuing this rebalancing for each period, complete immunisation for all periods can be achieved. Consider the immunisation problem below. We have a $1 million liability in 5 years time and we need to invest in two bonds so that the portfolio is immunised for one period. We will use the short rate lattice discussed before as our starting point and the bonds are: Bond 1: 5 year, 10% coupon Bond 2: 6 year, 6% coupon We assume that bonds pay annual coupon and the short rate lattice use year as period.

11.4 Immunization: Application of Short Rate Lattice

15 1

The attached spreadsheet describes the prices of these bonds with respect to the short rate lattice and the present value of the liability is also computed similarly. To construct the immunisation, we assume that x is the number of bond 1 and y is the number of bond 2 required. The following two equations can be solved to find x and y.

The first equation implies that the present value of the liability at t=O is met by the value of the bond portfolio. The second equation states that the same situation holds at t=l and state 1. There is no need to replicate the same for t=l and state 0 explicitly, since the arbitrage free nature of the lattice will ensure that happens. The solution can be easily found to be: x = 522.15 and y = 70 13.72. Bond 1. 5 year, 10%

10.00

10.00

10.00

10.00

110.00

Bond 2: 6 year 6%

Obligation: $lm 5 year

Just to sum up, the present values for the bonds and the liabilities are computed using the short rate tree developed before. The driving equation is (1 1.2). The solution for x and y shown above are computed using any of

152

11 Discrete Time Model of Interest Rate

the pairs of nodes as outlined above. The values displayed above are only two decimal places but the Excel internally would carry much more floating point precision. If you just use ordinary calculator and use the two decimal place values displayed above you would not get the x and y values given above. This is also due to the fact that the present values of the two bonds and that of the liability are order of magnitude different.

11.5 Valuing Callable Bond We will now illustrate how to use a short rate lattice to value a callable bond, and in turn find the value of the call feature, given a short rate tree. Consider a non-callable bond with coupon rate 6% per period and face value of $100. We are interested in finding its price today. This bond provides cash flow $6 at t=l, t=2 and $106 at t=3. As shown in the spreadsheet below the straight bond value is $102.90. Short Rate

Straight bond price

102.90

105.77 109.30

Callable bond price

104.51 106.70 108.27

106 106 106 106

11.6 Exercises

153

Assume the bond has one period worth of call protection i.e. the issuer may not exercise the call provision prior to one period from now i.e. the issue date. The issuer may call the bond back at the end of period 1 or 2 at face value plus one period coupon (i.e. $106). The issuer would want to call this bond whenever this reduces the value that the bondholders' claim would otherwise have on the firm. In other words, this action would increase the value of the shareholders' claim on the firm. From the previous tree we can identify the nodes at which the bond would be called as those at which the bond's value exceed the current call price. That is whenever the bond's value exceeds $106. Since the bondholders would be aware of the issuer's optimal call strategy they would never pay more than $106 at the nodes where call is imminent. To value the callable bond we simply replace the bond value as $106 at those identified nodes (shown in bold). These nodes are identified as bold italic characters. After replacing the bond value as $106 we recomputed the bond's price today using the same interest tree. This turns out to be $10 1.17. Therefore, comparing this value with the non-callable bond value the value of the issuer's call provision is ($102.90-$101.17) = $1.73. Once we have the short rate tree we can use it to value any type of complex bond, as long as, all of the uncertainty is confined to future interest rate movements captured by the tree.

11.6 Exercises Exercise 11.1 : Valuing a digital option as described here. Given the following short rate tree your task is to find the value of the following bet: you win $10 if one period interest rate is greater than 10% four periods from now.

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References Black FE, Derman E, Toy W (1990) A one-factor model of interest rates and its application to Treaswy bond options. Financial Analysts Journal, JanuaryFebruary, 46: 33-39 Ho TSY, Lee SB (1986) Term structure movements and pricing interest rate contingent claims. Journal of Finance, 41 : 1011-1029 Luenberger DG (1997) Investment science. Oxford University Press, New York

12 Global Bubbles in Stock Markets and Linkages

12.1 Introduction During the past 30 years, theoretical models and empirical testing in asset pricing have been motivated by market efficiency. This theory claims that the price of an asset today reflects correctly the future random payoffs of this asset conditioned on today's information and appropriately discounted by a stochastic factor. The opportunity to make a riskless profit using arbitrage strategies ensures that markets are efficient. Campbell (2000) articulates this paradigm in detail. Many economists accept market efficiency as the well-established paradigm of financial economics but also acknowledge that asset prices are too volatile. For example, NASDAQ, from its peak on March 10,2000 when it stood at 5048.62 to its low of 1638.80 on April 4, 2001, it declined by 67.5%. Is this significant decline due to substantial revisions of the expected payoffs andlor changes in the discount factor? This chapter considers the speculative bubble approach as an alternative to the present value model of market efficiency. Section 12.2 reviews the general ideas of rational bubbles and section 12.3 identifies several key statistical tests proposed by economists. In section 12.4 we offer a modification of the existing tests and apply it to the stock markets of the US, Japan, England and Germany. We find evidence of rational bubbles and then proceed to examine whether these bubbles travel across mature economies. Section 12.5 presents the relevant literature that supports the hypothesis of global integration. In sections 12.6, 12.7 and 12.8 we elaborate in detail our methodological procedures to test for bubbles and linkages of such bubbles between mature stock markets. Our main findings and the conclusions are given in the last two sections.

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12.2 Speculative Bubbles Economists have long conjectured that movements in stock prices may involve speculative bubbles, as trading is often said to generate over-priced markets. Some economists, however, believe that stock price fluctuations reflect changes in the values of the underlying market fundamentals. The standard definition of fundamental value is the summed discounted value of all future cash flows. The difference, if any, between the market value of the security and its fundamental value is termed a speculative bubble. Yet conhsion persists about what factors generate bubbles. Fads and irrationality have always figured prominently, and the hypothesis that these factors are important has gained some empirical support from the literature on asset price volatility. Another bubble-producing factor is the structure of information in the market. In a partial-equilibrium setting, Allen and Gorton (1988) showed that rational bubbles could exist with a finite number of agents who had asymmetric information. The existence of bubbles is inherently an empirical issue that has not been settled yet. A number of studies such as Blanchard and Watson (1982) and West (1988) have argued that dividend and stock price data are not consistent with the "market fundamentals" hypothesis, in which prices are given by the present discounted values of expected dividends. These results have often been construed as evidence for the existence of bubbles or fads. According to Shiller (1981), and LeRoy and Porter (1981) the variability of stock price movements is too large to be explained by the present value of future earnings. Over the past century US stock prices are five to thirteen times more volatile than can be justified by new information about future dividends. Campbell and Shiller (1988 a, b) and West (1987, 1988) remove the assumption of constant discount rate. But a variable discount rate provides only marginal support in explaining stock price volatility. They reject the null hypothesis of no bubbles. See also Rappoport and White (1993, 1994). A major problem with such arguments is that evidence for bubbles can be reinterpreted in terms of market fundamentals that are unobserved by the econometricians (Flood and Garber 1980; Hamilton and Whiteman 1985; Hamilton 1986). Diba and Grossman (1984, 1988a, 1988b) have recommended the alternative strategy of testing for rational bubbles by investigating the stationarity properties of asset prices and observable fundamentals. In essence, the argument for equities is that if stock prices are not more explosive than dividends, then it can be concluded that rational bubbles are

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not present, since they would generate an explosive component in stock prices. Using unit-root tests, autocorrelation patterns, and cointegration tests to implement this procedure, several authors reach the conclusion that stock prices do not contain explosive rational bubbles (see Dezhbakhsh and Demirgue-Kunt (1990)). Evans (1991) criticizes tests for bubbles based on an investigation of the stationarity properties of stock prices and dividends. By Monte-Carlo simulations he demonstrates that an important class of rational bubbles cannot be detected by these tests even though the bubbles are explosive. Froot and Obstfeld (1991), introduce the concept of intrinsic bubble, which they define as exclusively dependent on market fundamentals and not on extraneous events. Assuming that a stock price should go to zero as dividends go to zero, they derive a bubble solution composed of stable and unstable components. This bubble is unstable and implies an explosive price-dividend ratio. They find significant evidence of such a bubble and demonstrate that incorporating an intrinsic bubble into the simple presentvalue model helps account for the long-run variability of the US stock data. Furthermore, as their bubble is a deterministic function of dividends, once the bubble gets started, it will never burst as long as dividends remain positive. In practice, tests for intrinsic bubbles are very easily implemented only when dividends are assumed to follow a very simple process, for example, a geometric random walk. When a more general dynamic specification, such as ARIMA (p, 1, q) process is introduced for dividends, the test procedure for intrinsic bubbles becomes virtually intractable. Using a stochastic dividend-growth model, Ikeda and Shibata (1992) specify the stock price as a function of dividends (i.e. market fundamentals) as well as of time. The resulting bubble solution will bridge a gap between time-driven bubbles (Flood and Garber 1980) and bubbles exclusively depending on fundamentals (Froot and Obstfeld 1991). Depending on a parameter that decides relative degrees of fundamental dependency and time dependency, the bubble solution obtained exhibits various dynamic properties, which cannot be derived by combining linearly the two special solutions. Wu (1997) examines a stochastic bubble, able to burst and restart continuously. The specification is parsimonious and allows easy estimation. The model fits the data reasonably well, especially during several bull and bear markets in this century. Such rational stochastic bubbles can explain much of the deviation of US stock prices from the simple present-value model. Miller and Weller (1990), and Buiter and Pesenti (1990) examine the effects of fundamental-dependents bubbles on exchange rate dynamics, us-

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12 Global Bubbles in Stock Markets and Linkages

ing log linear models in which the consideration of free-disposal or pricesensitivity constraint is not required. A rational speculative bubble is nonnegative by definition: it represents what an investor might be willing to pay to buy a stock forever stripped of its dividends. As soon as either dividends or discount rates depend on the presence or absence of a bubble, however, the fundamental is affected by the presence of a bubble. For instance, the existence of a bubble may lead to an increase in interest rates, which so depresses the fundamental that the sum of the positive bubble and the bubbly fundamental falls short of the non-bubbly fundamental. Hence, a positive rational bubble may in fact decrease the overall price of a stock, contrary to what is commonly believed. Weil(1990), for example, provides empirical tests of this hypothesis. Most of the references above address issues of rational bubbles either theoretically or in the context of a mature economy. To complete this rapid bibliographical review we need to mention two additional trends in this literature. First several studies have tested for the existence of bubbles in emerging markets. For example, Richards (1996) claims that emerging markets have not consistently been subject to fads or bubbles. Chan et al. (1998) test for rational bubbles in Asian stock markets, and Chen (1999) specializes his search for bubbles in the Hong Kong market. Sarno and Taylor (1999) find evidence of bubbles in all East Asian economies. Significant increases in cross-markets linkages after a shock have now become a topic of important research under the term "contagion". Several important papers collected in Claessens and Forbes (2001) discusses both methodological issues and case studies of contagion. Beyond the existence or not of bubbles, economists have also studied in detail the implications of a stock market bubble to the economy at large. Biswanger (1999) offers a comprehensive review of these issues and Chirink0 and Schaller (1996) argue that bubbles existed in the US stock market but real investment decisions were based on fundamentals.

12.3 Review of Key Empirical Papers To motivate our methodological contribution we review several influential papers.

12.3.1 Flood and Garber (1980) We wish to begin with the classic Flood and Garber (1980). The authors test the hypothesis that price-level bubbles did not exist in a particular his-

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159

torical period. The existence of a price-level bubble places such extraordinary restrictions on the data that such bubbles are not an interesting research problem during normal times. Since hyperinflations generated series of data extraordinary enough to admit the existence of a price-level bubble, the German episode is an appropriate and interesting period to search for bubbles. The authors build a theoretical model of hyperinflation in which they allow price-level bubbles. Than, they translate the theoretical model into data restrictions and use these restrictions to test the hypothesis that price-level bubbles were not partly responsible for Germany's massive inflation during the early 1920s. Cagan (1956) used the following monetary model in his study of seven hyperinflations:

The quantities m and p are the natural logarithms of money and price at time t. The anticipated rate of inflation between t and t+l is n and E is a stochastic disturbance term. The rational-expectations assumption requires:

where n, = p,+, - p, is the mathematical expectations operator, and I is the information set available for use at time t. The solution of the equation (12.1) is:

where y = ( a - 1 ) / a >1, p,,, = m,+,+I-m,+,, w,+~ = E,+~+,-E,+, and A is an arbitrary constant. For this model, market fundamental is defined as

price level bubbles are then captured by the term -aAoyt .

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12 Global Bubbles in Stock Markets and Linkages

Rational-expectations models normally contain the assumption A=O, which prevents bubbles. Notice that ifA + 0 , and then price will change with t even if market fundamentals are constant. The definition of a pricelevel bubble as a situation in which A + 0 is appropriate for two reasons. First, A is an arbitrary and self-fulfilling element in expectations. Second, ifA ;t 0 , then agents expect prices to change through time at an everaccelerating rate, even if market fundamentals do not change. Since economics usually consider price bubbles to be episodes of explosive price movement, which are unexplained by the normal determinants of market price, A st 0 will produce a price-level bubble. Finally, the results of the empirical analysis support the hypothesis of no price-level bubbles. 12.3.2 West (1987)

The test compares two sets of estimates of the parameters needed to calculate the expected present discounted value (PDV) of a given stock's dividend stream, with expectations conditional on current and all past dividends. In a constant discount rate model the two sets are obtained as follows. One set may be obtained simply by regressing the stock price on a suitable set of lagged dividends. The other set may be obtained indirectly from a pair of equations. One of the pair is an arbitrage equation yielding the discount rate, and the other is the ARIMA equation of the dividend process. The Hansen and Sargent (1980) formulas, familiar from rational expectations tests of cross-equation restrictions, may be applied to this pair of equations' coefficients to obtain a second set of estimates of the expected PDV parameters. Under the null hypothesis that the stock price is set in accord with a standard efficient markets model, the regression coefficients in all equations may be estimated consistently. When the two sets of estimates of the expected PDV parameters are compared, then, they should be the same, apart from sampling error. But this equality will not hold under the alternative hypothesis that the stock price equals the sum of two components: the price implied by the efficient markets model and a speculative bubble. A stock price is determined by the arbitrage condition:

12.3 Review of Key Empirical Papers

161

where p, is the real stock price in period t, b the constant ex ante real discount rate, OO). The log-linear approximation of (12.1 1) can be written as follows:

Where q is the required log gross return rate, Y is the average ratio of the stock price to the sum of the stock price and the dividend, k is -ln(Y)-(1Y)ln(l/Y-1), pt is ln(Pt), and dt is In(Dt). The general solution to (12.12) is given by:

where b, satisfies the following homogeneous difference equation:

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12 Global Bubbles in Stock Markets and Linkages

In equation (12.12), the no-bubble solution p, is exclusively determined by dividends, while b can be driven by events extraneous to the market and is referred to as a rational speculative bubble. After defining the stock price equation, the parametric bubble process and the dividend process in a state-space form, the bubble is treated as an unobserved state vector, which can be estimated by the Kalman filtering technique. The author finds statistically significant estimate of the innovation variance for the bubble process. During the 1960s bull market the bubble accounts for between 40% and 50% of the actual stock prices. Negative bubbles are found during the 1919-1921 bear market, in which the bubble explains between 20% and 30% of the decline in stock prices.

The same model has been used also to estimate the unobserved bubbly component of the exchange rate and test whether it is significantly different from zero. Using the monetary model of exchange rate determination, the solution for the exchange rate is the sum of two components. The first component, called the fundamental solution, is a function of the observed market fundamental variables. The second component is an unobserved process, which satisfies the monetary model and is called the stochastic bubble. The monetary model, the market fundamental process and the bubble process are expressed in the state-space form, with the bubble being treated as a state variable. The Kalman filter can than be used to estimate the state variable. The author finds no significant estimate of a bubble component was found at any point in the period 1974-1988. Similar results were obtained for the sub-sample, 1981 through 1985, in which the results US dollar appreciated most drastically and a bubble might more likely have occurred.

12.4 New Contribution The purpose of our study is to search empirically for bubbles in national stock markets using state-of-the-art methodology such as Wu (1995, 1997) with emphasis on the U.S., Japan, Germany and the United Kingdom. We focus on the post-war period in these four countries as opposed to Wu (1997), which concentrates on only the U.S. annual data series dating back to 1871. All data are monthly returns of the S&P 500, Nikkei 225, Dax-30 and FT- 100 indexes ranging from January 1951 to December 1998, that is,

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165

576 observations. All data are converted to real values using the corresponding CPI measures and Global Financial Data provided the data. In order to establish the soundness of our methodology we attempted to reproduce the results from Wu (1997) using annual U.S. data (also obtained from Global Financial Data) covering the period 1871 - 1998. Although we employ the unobserved component modeling approach similar to Wu (1997), our implementations of the state-space form (or the Dynamic Linear Model, DLM, (chapter 8) is quite different from that of Wu. We treat both the dividend process and the bubble process as part of the unobserved components i.e. the state vector. The state equations also include their own system error, which are assumed uncorrelated. The measurement vector in this case contains the price and the realized dividend without any measurement errors. The advantage of this way modeling is that the comparison with the no bubble solution becomes much more straightforward. Wu (1997) had to resort to alternative way (GMM) of estimating the no bubble solution and the model adequacy tests are not performed there. Besides, the precise moment conditions used in the GMM estimation are not reported there. On the other hand, in our approach we are able to subject both the bubble and the no bubble solutions to a battery of diagnostics test applicable to state-space systems. In the following subsections we describe in detail the mathematical structures of our models and the estimation strategies. Once bubbles are confirmed empirically, we proceed to test linkages between the four markets in terms of both the fundamental price and the bubble price series. In this context we adopt a sub-set VAR methodology (Lutkepohl 1993 p.179). The approach builds into it the causal relations between the series and this gives us the opportunity to analyze the potential global contagion among these national equity markets through the speculative component of the prices. The potential existence of global linkages among equity markets will further decrease the expected benefits of a global diversification. The review of the literature associated with global integration and diversification is briefly presented next.

12.5 Global Stock Market Integration During the past thirty years, world stock markets have become more integrated, primarily because of financial deregulation and advances in computer technology. Financial researchers have examined various aspects of the evolution of this particular aspect of world integration. For example, the early studies by Grubel (1968), Levy and Sarnat (1970), Grubel and

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12 Global Bubbles in Stock Markets and Linkages

Fadner (1971), Agmon (1972, 1973), Ripley (1973), Solnik (1974), and Lessard (1973, 1974, 1976) have investigated the benefits from international portfolio diversification. While some studies, such as Solnik (1976), were exclusively theoretical in extending the capital asset pricing model to a world economy, others such as Levy and Sarnat (1970) used both theory and empirical testing to confirm the existence of financial benefits from international diversification. Similar benefits were also confirmed by Grubel (1968), Grubel and Fadner (197 I), Ripley (1973), Lessard (1973, 1974, 1976), Agmon (1972, 1973), Makridakis and Wheelwright (1974), and others, who studied the relations among equity markets in various countries. Specifically, Agmon (1972, 1973) investigated the relationships among the equity markets of the U.S., United Kingdom, Germany and Japan, while Lessard (1973) considered a group of Latin American countries. By 1976, eight years after the pioneering work of Grubel (1968), enough knowledge had been accumulated on this subject to induce Panton, Lessing and Joy (1976) to offer taxonomy. It seems reasonable to argue that although these studies had used different methodologies and diverse data from a variety of countries, their main conclusions confirmed that correlations among national stock market returns were low and that national speculative markets were largely responding to domestic economic fundamentals. Theoretical developments on continuous time stochastic processes and arbitrage theory were quickly incorporated into international finance. Stulz (1981) developed a continuous time model of international asset pricing while Solnik (1983) extended arbitrage theory to an international setting. Adler and Dumas (1983) integrated international portfolio choice and corporate finance. Empirical research also continued to flow such as Hilliard (1979), Moldonado and Saunders (198 I), Christofi and Philippatos (1987), Philippatos, Christofi and Christofi (1983) and also Grauer and Hakansson (1987), Schollhammer and Sand (1987), Wheatley (1988), Eun and Shim (1989), von Furstenberg and Jeon (1989), Becker, Finnerty and Gupta (1990), Fisher and Palasvirta (1990), French and Poterba (I 99 1) and Harvey (1991). These numerous studies employ various recent methodologies and larger databases than the earlier studies to test for interdependencies between the time series of national stock market returns. The underlying issue remains the empirical assessment of how much integration exists among national stock markets. In contrast to earlier results, and despite some reservations, several of these new studies find high and statistically significant

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167

level of interdependence between national markets supporting the hypothesis that global stock markets are becoming more integrated. In comparing the results of the earlier studies with those of the more recent ones, one could deduce that greater global integration implies fewer benefits from international portfolio diversification. If this is true, how can one explain the ever-increasing flow of big sums of money invested in international markets? To put differently, while Tesar and Werner (1992) confirm the home bias in the globalization of stock markets, why are increasing amounts of funds invested in non-home equity markets? For instance, currently about 10% of all trading in U.S. equities take place outside of the United States. The June 14, 1993 issue of Barron's reported that US investors have tripled their ownership of foreign equities over the past five years from $63 billion to over $200 billion in 1993. The analysis of the October 19, 1987 stock market crash may offer some insight in answering this question. Roll (1988, 1989), King and Wadhwani (1990), Hamao, Musulis and Ng (1990) and Malliaris and Urrutia (1992) confirm that almost all stock markets fell together during the October 1987 crash despite the existing differences of the national economies while no significant interrelationships seem to exist for periods prior and post the crash. Malliaris and Urrutia (1997) also confirm the simultaneous fall of national stock market returns because of the Iraqi invasion of Kuwait in July 1990. This evidence supports the hypothesis that certain global events, such as the crash of October 1987 or the Invasion of Kuwait in July, 1990, tend to move world equity markets in the same direction, thus reducing the effectiveness of international diversification. On the other hand, in the absence of global events, national markets are dominated by domestic fundamentals, and international investing increases the benefits of diversification. Exceptions exist, as in the case of regional markets, such as the European stock markets reported in Malliaris and Urrutia (1996). Longin and Solnik (2001) distinguish between bear and bull markets in international equity markets and find that correlation increases in bear markets, but not in bull markets.

12.6 Dynamic Linear Models for Bubble Solutions Our starting point in this approach is the equations (12.13) and (12.14) described earlier. As our preliminary investigations reveal that both the log real price and log real dividend series are non-stationary, we choose to work with the first differenced series. Thus, the equation (12.13) becomes,

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12 Global Bubbles in Stock Markets and Linkages

Apt = Apf + Ab,,

x m

where, Ap: = (1 - y ~ ) W'E,[d,,, ] - (1 - W)

(12.15)

x m

yiEt-,[dt-,+,I . Assuming the

following parametric representation of equation (12.14),

In order to express the fundamental component of the price, hpf ,in term of the dividend process, we fit an appropriate AR model of sufficient order so that the information criterion AIC is minimized. We find that for the Japanese data a AR(1) model is sufficient whereas for the other three countries we need AR(3) models. The infinite sums in the expression for ~~f may be expressed in terms of the parameters of the dividend process once we note the following conditions: The differenced log real dividend series is stationary, therefore the infinite sum converges, Any finite order AR process can be expressed in companion form (VAR of order 1) by using extended state variables i.e. suitable lags of the original variables, (Campbell, Lo and MacKinlay 1997 p.280), Using demeaned variables the VAR(1) process can be easily used for multiperiod ahead forecast (Campbell, Lo and MacKinlay 1997 p.280). Assuming the demeaned log real dividend process has the following AR(3) representation, Ad, =$,Adt-, +$,Adt-, +$,Adt-, + c g , g - ~ ( ~ 7 . : ) , the companion form may be written as,

(12.18)

12.6 Dynamic Linear Models for Bubble Solutions

X,

=

ax,-,+ E,,

169

(12.20)

where the definitions of X,, 0 , and E, are obvious from comparison of equations (12.19) and (12.20). Following Campbell, Lo and MacKinlay (1997, p. 280), Apf may be expressed as, (with I being the identity matrix of the same dimension as @ ) Ap:

= Ad,

+ v @(I - v@)-'AX,.

(12.21)

We can now express equation (12.15) in terms of fundamental component and the bubble component, ~ p =,~ d+,e f v O(I - v@)-'AX, + Ab,,

(12.22)

where e' = [1 0 01. The equation (12.22) represents the measurement equation of the DLM and we need to suitably define the state equation for the model. An examination of the equation (12.17) and (12.19) suggests that the following state equation adequately represent the dynamics of the dividend and the bubble process:

12 Global Bubbles in Stock Markets and Linkages

170

-

1 0 0

O I 0

0

0

O O O O 0 0 0 1 0 0 1 0 0 -

-0

0

0

4 ) 1 $ 2 4 ) 3 O

-

=

Adt-3 bt - bt-1 -

-

-

-

v

0

1

O O 0 0

-

-

A

t

Eg

0 Adt-2 Adt-3 + 0 0 Adt-4 0 bt-l 0 -0 0 - - bt-2 -

0 -, 0 0 0 '

(12.23a)

E,

0-

We are in a position now to define the measurement equation of the DLM in term of the state vector in equation (12.23a). This is achieved by examining equation (12.22) and defining a row vector, M = elv@ (I - yr@)-' = [m,, m2,m3 as follows:

1,

Equation (12.24) determines the measurement equation of the DLM without any measurement error. In other words, the evolution of the state

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171

vector in equation (12.23a) results in the measurement of the measurement vector through equation (1 2.24). Equations (12.23a) and (12.24) represent the DLM for the bubble solution when the dividend process is described by the AR(3) system in equation (12.19). In our sample this is the case for Germany, U.K. and the U.S.A data. Since the data for Japan required only a AR(1) process for the dividend in equation (12.19), the DLM, in this case, may be written directly as:

Similarly, the measurement equation for the DLM of the bubble solution for the Japanese data becomes,

[+,I

where, M r e'WQ,(l- yD)-l = [ml], since er=[l], Q, = . We have now set up the DLM for the bubble solution for the data for Germany, U.K., and the U.S.A. given by the equations (12.23a) and (12.24). For the Japanese data, on the other hand, these are given by the equations (12.25a) and (12.26). The parameters of the models embedded in these equations and the filtered and the smoothed estimates of the bubble series are to be estimated from the observed price and the dividend series.

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The details of the estimation procedure are described in chapter 9. In the next section we proceed to set up the DLMs for the no-bubble solutions.

12.7 Dynamic Linear Models for No-Bubble Solutions In order to compare the performance of the bubble solution discussed in the previous sub-section we develop the DLM for a no-bubble solution. We maintain the same framework so that comparison is more meaningful. This is opposed to the approach taken in Wu (1997) where the no-bubble solution was estimated in the GMM framework. We also note that the model should account for the correlations in the variance of the stock return series. This is done by incorporating the GARCH(1,l) effect in the price equation (12.15) without the bubble component. In this context we adopt the methodology of Harvey, Ruiz and Sentana (1992) and follow Kim and Nelson (1999, page 144) to suitably augment the state vector of the DLM. For Germany, U.K. and the U.S.A date set the state equation (12.23a) becomes,

and a,-,is the information set at time t-1. The corresponding measurement equation becomes,

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173

For the Japanese data with an AR(1) dividend process, the no-bubble DLM may be written following the approach above. The state equation (12.25a) becomes,

The corresponding measurement (12.26) becomes,

In the no-bubble solutions the parameters to be estimated are those of the dividend process and the GARCH(1,l) coefficients. The procedure for this is same as that for the bubble solutions and is described in detail in appendix A. The next sub-section takes up the issues in modeling the linkages between the markets in the subset VAR framework.

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12 Global Bubbles in Stock Markets and Linkages

12.8 Subset VAR for Linkages between Markets Linkages among international stock markets have been extensively investigated and a good review of the literature can be found in McCarthy and Najand (1995). These authors adopt the state-space methodology to infer the linkage relationships between the stock markets in Canada, Germany, Japan, U.K and the U.S.A. The authors claim that this approach not only determines the causal relationship (in the Granger sense) but it delivers the result with minimum number of parameters necessary. They report that the U.S. market exerts the most influence on other markets. Since these authors use daily data there is some overlap in the market trading time and they attempt to take care of that in the interpretation of their results. The main finding is consistent with similar findings by other researchers e.g. Eun and Shim (1989), who examine nine stock markets in the North America and Europe over period 1980-1985 in a VAR framework. In this chapter we adopt a similar approach but at two different levels. The methodology developed in this chapter allows us to decompose the stock prices in their fundamental and the bubble components. We, therefore, analyze the linkage relationship both through the fundamental as well as through the speculative component. This helps us understand whether the market linkages are through the fundamental or through the speculative components of the price series. Also, since we are dealing with monthly data, the time overlap problem between markets is largely non-existent. The econometric procedure we adopt is referred to as the subset VAR. Use of standard VAR approach to study causal relations between variables is frequently employed. A typical VAR model involves a large number of coefficients to be estimated and thus estimation uncertainty remains. Some of the coefficients may in fact be zero. When we impose zero constraints on the coefficients in full VAR estimation problem what results is the subset VAR. But, since most often no a priori knowledge is available that will guide us to constrain certain coefficients, we base the modeling strategy on information based model selection criterion e.g. AIC (Akaike Information Criterion) and HQ (Hannan-Quinn). Actual mathematical definitions and the details of this approach can be found in Lutkepohl(1993, chapter 5). In the paragraphs below we very briefly summarize the procedure. We first obtain the order of the VAR process for the four variables using the information criterion mentioned above. The top-down strategy starts from this full VAR model and the coefficients are deleted one at a time (from the highest lag term) from the four equations separately. Each time a coefficient is deleted the model is estimated using least-square algorithm and the information criterion is compared with the previous minimum one.

12.9 Results and Discussions

175

If the current value of the criterion is greater than the previous minimum value, the coefficient is maintained otherwise it is deleted. The process is repeated for each of the four equations in the system. Once all the zero restrictions are determined the final set of equations are estimated again, which gives the most parsimonious model. We also check for the adequacy of this model by examining the multivariate version of the portmanteau test for whiteness of the residuals (Lutkepohl 1993, p. 188). Once the subset VAR model is estimated there is no further need for testing causal relations and/or linkages between the variables. The causality testing is built into the model development process. We will, therefore, examine linkages between the four markets in our study using this subset VAR model. As mentioned earlier we will explore linkages between these markets in two stages. In the first stage, the fundamental price series are all found to be stationary, and hence in this case the modeling is done using the levels of the variables. We find evidence of one unit root in the speculative components of the price series for all the four markets. As we suspect existence of a cointegrating relation between these speculative components, we explore this using Johansen's cointegration test and find evidence of one cointegrating vector. It is, therefore, natural to estimate a vector error correction model, which is essentially a restricted VAR model with the cointegrating relation designed into it. As suggested in Lutkepohl (1993, p. 378) we examine the causal relation between these variables in the same way as for a stable system. In other words, we explore the linkages as for the fundamental price component but in this case we use first differenced form and use the lagged values of the cointegrating vector as well.

12.9 Results and Discussions First we discuss the estimations results of the dynamic linear model with the bubble solution for the annual U.S. data series. In Table 12.1 we find all the parameter estimates are statistically significant. The significance of the parameter, IS, implies highly variable bubble component of the price through out the period 1871 to 1998.. The parameters describing the real dividend process are very close to the univariate estimation (not included) results of the dividend series. Besides, the discount parameter, , is close to its sample value. In Table 12.2 we present the estimates of the no-bubble solution with a GARCH (1,l) error structure for the price equation for the same U.S. annual data. Here also, most of the parameters are statistically significant.

176

12 Global Bubbles in Stock Markets and Linkages

The significance of the GARCH parameter, PI, implies persistence in the residual volatility. This model is used to compare the results of the bubble solution. We would like to stress the fact we implemented the GARCH (1,l) model also in the state-space framework so that the comparison with the bubble solution would be more realistic. This is, however, not the case with Wu (1997), which uses the GMM methodology. This approach also allows us to check the performance of both the models by analyzing the residual diagnostics. We present these test results in Table 12.3. The portmanteau tests support the whiteness of the residuals and the ARCH tests indicate no remaining heteroscedasticity in the residuals. Besides, the Kolmogorov-Smirnov tests support the normality of the residuals. These three tests overwhelmingly support the modeling approach adopted here and, therefore, the conclusions drawn are statistically meaningful. In addition to the three tests just outlined above, we also include two additional tests particularly designed for recursive residuals produced by the dynamic linear systems developed in this study. The modified von Neumann ratio tests against serial correlations in the residuals where as the recursive t-test is used to check for correct model specification. As the entries Table 12.3 suggest both the models, i.e. the dynamic linear models of the bubble and the no-bubble solution, both perform extremely well in respect of these two tests. There is overwhelming support for the adequacy of the models in describing the price process. In view of the battery of tests discussed in the preceding two paragraphs we can now proceed to analyze the other observations. As discussed in Wu (1997) the rational stochastic bubble can alternate between positive and negative values. It is argued there that the securities may be overvalued when the participants are bullish and these may be undervalued when the participants are bearish. The Figure 12.1 shows negative bubble in the very early part of the sample as well as during the early 1920s. It is obvious though that the stochastic bubbles account for a substantial percentage of the stock price in the sample. It is also interesting to note that in spite of the drop in the bubble percentage during the oil shock of the 1970 s and the stock market crash of 1987 there has been an upward trend of the bubble percentage through out the later part of the sample period considered. Next, we compare the performance of the bubble and the no-bubble solutions by examining the in sample frtting the stock prices. In Table 12.4 we display the criteria used and these are defined as,

12.9 Results and Discussions

177

and

where 6,is the fitted price and T is the number of observations. The entries in Table 12.4 clearly demonstrate the superiority of the bubble solution to capture the price process over the sample period. We next proceed to analyze the monthly data, covering the post war period, for the four countries, Germany, Japan, U.K. and the U.S.A. In Table 12.5 we present the estimation results of the bubble solutions and it is clear that most of the parameters are statistically significant. The discount parameter, y , as before is close to the respective sample values while the significant o, for all the four countries imply highly variable bubble components. Needless to say that the estimated parameters of the dividend processes are close to their respective univariate estimation (not reported here) results. As evident from the Table 12.6, the significant ARCH and the GARCH parameters indicate appropriateness of the error specification for the log price difference series. There is substantial persistence in the variance process. We now move to analyze the residual diagnostics in order to ascertain the appropriateness of the model for the monthly data series for all the four countries. As with the annual data (for the U.S.) we find evidence of whiteness on residuals from the portmanteau test and the lack of ARCH effect in the residuals from ARCH test results. The U.S. data also supports the normality of the residuals. More importantly though the tests for model adequacy are captured by the von Neumann ratio and the recursive t-test. As pointed out in Harvey (1990, page 157) von Neurnann test provides the most appropriate basis for a general test of misspecification with recursive residuals. In this context the dynamic linear models for the bubble and the no-bubble solutions both perform extremely well. Figure 12.2 plots the bubble price ratio for the sample period and the substantial variation of the bubble component is visible for all the countries. Except for the U.S. there is evidence of negative bubble for the other three countries in the initial part of the sample period. All countries were affected by the oil price shock of the 1970s but by different extent and the most severe appear to be in the U.K. The fall in the bubble percentage during the October 1987 stock market crash is evident for all the countries. It is also interesting to note that there is a general upward trend for the bub-

178

12 Global Bubbles in Stock Markets and Linkages

ble price ratio toward the later part of sample period for Germany, U.K. and the U.S.A. but not for Japan. This provides the visual evidence of the collapsing and self-starting nature of the rational stochastic bubble we have attempted to capture in this study. In order to quantify the performance improvement of the bubble solution compared to the no-bubble case with GARCH (1,l) errors we present the in sample fitting statistics, RMSE and MAE, in Table 12.8. The entries in Table 12.8 prove beyond doubt that the bubble solution does a credible job in terms of both metrics. For example, the bubble solution reduces the metric RMSE to 7% and the metric MAE to 52% of the no-bubble solution respectively, for the U.S. monthly data. We indicated earlier the importance and the extent of investigation into the study of market linkages by various researchers. In this chapter we are able to focus on this aspect in two different levels. The study of rational stochastic bubble through the dynamic linear models enables us to separate the price series into a fundamental and the bubble component. It is, therefore, natural to examine whether the market linkages exist via both these components. McCarthy and Najand (1995) demonstrated the influence of the U.S. market on several other OECD countries using daily data which might have unintended consequences of trading time overlap in these markets. Using monthly data over a period of 48 years we are in a better position to analyze the market interrelationships. VAR methodology is often employed to study causal relationships. If some variables are not Granger-causal for the others then zero coefficients are obtained. Besides, the information in the data may not be sufficient to provide precise estimates of the coefficients. In this context the top-down strategy of the subset VAR approach described in the earlier section is most suitable. For the fundamental price series we adopt this approach in the levels of the variables since these are all found to be stationary. Using the Hannan-Quinn criterion we start our VAR model with a lag of one and follow the subset analysis process described before. This gives us the model presented in Table 12.9. As with McCarthy and Najand (1995) we find strong evidence of the U.S. dominance on all the other three countries, but no reverse causality. This is a particularly important finding in the sense that this causality exists in the fundamental components of the prices. Intuitively, this evidence suggests that the US economy, as represented by the stock market data, acts as the engine of global growth. For Germany and Japan the causality from the U.S. are significant at 5% level whereas for the U.K it is significant at 1% level only. The overall significance of this modeling approach is also established by testing the multivariate version of the portmanteau test to detect whiteness of the residuals.

12.9 Results and Discussions

179

We also apply the top-down strategy for the subset VAR approach to the bubble components as well to examine the causality between the four markets. Since the bubble components are found to be non-stationary (results for the unit root tests not included) we model this using the first difference of the log prices. With the non-stationary bubble price series it is natural to expect some long-term equilibrium relationship between these variables. We detected one cointegrating vector using Johansen's procedure and this has been described Table 12.10. We follow the same process (as for the fundamental prices) to obtain the subset VAR model, including the cointegrating vector that describes the causal relationship between these markets. We find (from Table 12.10) that the causality also exists fiom the U.S. to the other three markets and the these linkages are significant at 5% level for Germany and Japan and only at 1% level for the U.K. Similar to the fundamental prices there is no reverse causality in the bubble price components as well. It is also observed that the strength of this causality from the U.S. to Japan is slightly stronger for the bubble price process, 0.1915 as opposed to 0.1878 for the fundamental prices. It is also noted fiom Table 12.10 that the coefficients of the error correction term i.e. 'Coint (-1)' are statistically significant. This implies that the modeled variables i.e. the changes in log prices adjust to departures from the equilibrium relationship. The magnitude of the coefficient 'Coint (-1)' for the Japanese log price difference is much higher than the others, capturing, first the upward and later, the downward trend in the Japanese market. Although, the existence of an error correction model implies some form of forecasting ability (see e.g. Ghosh (1993)), we do not pursue this in this chapter. Finally, we note the multivariate portmanteau test for whiteness of residuals in Table 12.10. This again supports the model adequacy and hence the inferences drawn are statistically meaningful. Table 12.1 Parameter estimates of bubble solution, USA yearly data

Estimates reported here are obtained from maximizing the innovation form of the likelihood function. Numerical optimization in GAUSS is used without any parameter restriction. The standard errors (reported below the parameters in parentheses) are obtained from the Hessian matrix at the point of convergence. These estimates are robust to different starting values including different specification of the prior covariance matrix. Significance at 5% level is indicated by *

180

12 Global Bubbles in Stock Markets and Linkages

Table 12.2 No-bubble GARCH(1,l) solution, USA yearly data

(0.132) .... c 0-8 6 L Q?.085) @,008) tO.OO6J (0.127) (0.264) Notes for Table 12.1 apply here. GARCH(1,l) error for state-space system implemented following Harvey, Ruiz, Sentana (1992).

Table 12.3 Diagnostics and model adequacy tests, USA yearly data MNR Rec. T Port. ARCH KS "--" Bubble 0.035 0.385 0.138 0.53 1 0.952 No Bubble 0.033 0.519 0.119 0.422 0.931 Entries are p-values for the respective statistics except for the KS statistic. These diagnostics-are computed from the recursive residual of the measurement equation, which corresponds to the real dividend process. The null hypothesis in portmanteau test is that the residuals are serially uncorrelated. The ARCH test checks for no serial correlations in the squared residual up to lag 26. Both these test are applicable to recursive residuals as explained in Wells (1996, page 27). MNR is the modified Von Neumann ratio test using recursive residual for model adequacy (see Harvey (1990, chapter 5). Similarly, if the model is correctly specified then Recursive T has a Student's t-distribution (see Harvey (1990, page 157). KS statistic represents the Kolmogorov-Smirnov test statistic for normality. 95% and 99% significance levels in this test are 0.121 and 0.145 respectively. When KS statistic is less than 0.121 or 0.145 the null hypothesis of normality cannot be rejected at the indicated level of significance. ----*--.--p--*---p-.mm-m-o"

Table 12.4 Models compared, USA yearly data Bubble

RMSE 0.25

MAE 0.34

RMSE and MAE stand for 'root mean squared error' and 'mean absolute error' respectively. These are computed from the differences between the actual log prices and the fitted log prices from the corresponding estimated model. Additional details are in the text.

12.9 Results and Discussions

181

Table 12.5 Parameter estimates of bubble solution, monthly data

Germany

Japan

U.K.

U.S.A. ---

(0.001 0) $1

-0.0009 (0.0400)

$2

0.0611" (0.0210)

4'3

0.0947* (0.0271)

(S6

0.0475"

Notes of Table 12.1 apply here. Table 12.6 Parameter estimates no-bubble solution, monthly data

Jxan Germany .-.-au.u.-.-.-. 0.8526* 0.5437* (0.0391) (0.0372)

U.K. 0.2830* (0.0380)

U.S.A. 0.3 189* (0.0344)

$1

0.0047 (0.0407)

-0.5331* (0.041 1)

-0.7213* (0.0413)

$2

0.063 1 (0.0409)

-0.34253 (0.0440)

-0.3271* (0.0484)

43

0.0848* (0.0415)

-0.1 148* (0.0399)

-0.0901* (0.0400)

66

0.0475* (0.0014) 0.0001* (5.14E-05) 0.1108* (0.0299) 0.8633*

0.0407* (0.0012) 0.0004* (0.0001) 0.2307* (0.0541) 0.6107*

0.0288* (0.0008) 0.0001* (4.62E-05) 0.0657* (0.0274) 0.8365*

"

""

W

a, '4 P1

Notes of Table 12.2 apply here.

-0.0906* (0.0407)

0.051 l* (0.0015) 0 0, 0.0988* (0.0232) 0.8869*

182

12 Global Bubbles in Stock Markets and Linkages

Table 12.7 Diagnostics and model adequacy tests, monthly data

-Bubble Germany Japan U.K. U.S.A.

Port.

ARCH

KS-

MNR

Rec. T

0.253 0.061 0.366 0.377

0.158 0.206 0.199 0.327

0.176 0.093 0.136 0.048

0.586 0.379 0.467 0.425

0.903 0.972 0.93 1 0.894

m -

No Bubble 0.254 0.195 0.175 0.466 0.806 Germany Japan 0.017 0.194 0.089 0.186 0.771 U.K. 0.307 0.179 0.139 0.571 0.907 U.S.A. 0.353 0.283 0.047 0.418 0.846 Notes of Table 12.3 apply. Critical values for KS statistic are 0.057 and 0.068, respectively.

Table 12.8 Models compared, monthly data

--

Bubble Germany Japan U.K. U.S.A.

No Bubble Garch(1,l) Germany Japan U.K. Notes for Table 12.4 apply here.

RMSE ---------0.796 1.730 0.247 0.117

2.945 4.394 0.719

------

MAE----"-- --0.795 1.730 0.366 0.895

12.9 Results and Discussions

183

Table 12.9 Subset VAR results: linkages in fundamental prices GR(- 1)

JP(- 1)

UK(- 1)

US(-1)

Constant

P

Details of the methodology for determining the subset VAR relations are given in the text. This has been done in the level variables since the fundamental price series are stationary. The numbers in parentheses are t-statistics for the corresponding coefficient. Significance at 5% and 10% level are indicated by * and ** respectively. The p-value for the multivariate portmanteau statistic for residual white noise is 0.017. This is described in Lutkepohl (1993) page 188. This indicates that the model adequately represents the relationship documented here.

Table 12.10 Subset VAR results: linkages in bubble prices AGR(-1) AGR AJP AUK AUS

AJP(-1)

AUK(-1)

AUS(1)

Coint(-1)

Const.

0.007* (2.47) 0.017~ (4.76)

0.096* (1.99)

0.194~ (3.91) 0.195* (3.20) 0.106"" (1.73)

0.003 (1.74) 0.005* (2.09) 0.002 (0.74) 0.004*

0.129~ (2.94) -0.144~ (-2.67)

0.001*

The bubble prices are found non-stationary and Johansen's procedure identified existence of one cointegrating vector. The lagged value of this cointegrating vector (COINT) has been used in estimating the subset VAR relations for the linkages between the markets. The details of the unit root and the cointegration tests are not reported here but can be obtained from the authors. The estimated cointegrating vector (normalized on GR) including TREND and constant terms is given below. The numbers in parentheses are t-statistics for the corresponding coefficient. Significance at 5% and 10% level are indicated by * and ** respectively. GR(-1)-1.5826JP(-1)+2.7303UK(-1)-3.2545US(-1)+0.0054T~ND+2.3772 The p-value for the multivariate portmanteau statistic for residual white noise is 0.068. This is described in Lutkepohl (1993) page 188. This indicates that the model adequately represents the relationship documented here.

184

12 Global Bubbles in Stock Markets and Linkages

BubblePrice Ratio ('33)

Fig. 12.1 Plot using smoothed estimates from bubble, yearly US data

BubblePrice Ratio (%)

Jan- Jan- Jan- Jan- Jan- Jan- Jan- Jan- Jan- Jan- Jan- Jan- Jan- Jan- Jan- Jan78 81 84 87 90 93 96 69 72 75 51 54 57 60 63 66

Fig. 12.2a Plot using smoothed estimates from bubble, monthly German data

12.9 Results and Discussions

BubblePrice Ratio (YO)

Jan- Jan- Jan- Jan- Jan- Jan- Jan- Jan- Jan- Jan- Jan- Jan- Jan- Jan- Jan- Jan51 54 57 60 63 66 69 72 75 78 81 84 87 90 93 96

Fig. 12.2b Plot using smoothed estimates from bubble, monthly Japanese data

BubblePrice Ratio ( O h )

Jan- Jan- Jan- Jan- Jan- Jan- Jan- Jan- Jan- Jan- Jan- Jan- Jan- Jan- Jan- Jan51 54 57 60 63 66 69 72 75 78 81 84 87 90 93 96

Fig. 1 2 . 2 ~Plot using smoothed estimates fkom bubble, monthly UK data

185

12 Global Bubbles in Stock Markets and Linkages

186

BubblePrice Ratio ( O h ) 40

-----

30

-

20

-

--

I

i -10

0

I

1

Jan- Jan- Jan- Jan- Jan- Jan- Jan- Jan- Jan- Jan- Jan- Jan- Jan- Jan- Jan- Jan51 54 72 75 78 81 57 60 63 66 69 84 87 90 93 96

Fig. 12.2d Plot using smoothed estimates f?om bubble, monthly US data

12.10 Summary Economists have long conjectured that movements in stock prices may involve speculative bubbles because trading often generates over-priced markets. A speculative bubble is usually defined as the difference between the market value of a security and its fundamental value. Although there are several important theoretical issues surrounding the topic of asset bubbles, the existence of bubbles is inherently an empirical issue that has not been settled yet. This chapter reviews several important tests and offers a new methodology that improves upon the existing ones. In addition, the new methodology is applied to the four mature markets of the US, Japan, England and Germany to test whether a bubble was present during the period of January 1951 to December 1998. Once we find evidence of bubbles in these four mature stock markets, we next ask the question whether these bubbles are interrelated. We avoid using the technical term of contagion because it has a very specific meaning. Several authors use contagion to mean a significant increase in crossmarket linkages, usually after a major shock. For example, when the Thai economy experienced a major devaluation of its currency during the summer of 1997, the spreading of the crisis across several Asian countries has

References

187

been viewed as a contagion. Unlike the short-term cross-market linkages that emerge as a result of a major, often regional economic shock, we are here interested in long-run linkages. Bubbles often take long time, that is several years to inflate and one is interested in knowing if such processes travel from one mature economy to another. The bursting of a bubble, as in the case of the Thai market with its impact on the Asian stock markets, can be viewed as a contagion. However, our methodology captures long-term characteristics describing the markets studied over the entire sample period. Our statistical tests of the long-term linkages between the four mature stock markets provide evidence that US bubbles cause bubbles in the other three markets but we find no evidence for reverse causality.

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Sarno L. Taylor M (1999) Moral hazard, asset price bubbles, capital flows and the East Asian crisis: the First test. Journal of International Money and Finance, 18: 637-57 Schollhammer H, Sand OC (1987) Lead-lag relationships among national equity markets: an empirical investigation. In: Khoury SJ, Ghosh A (eds) Recent Developments in International Banking and Finance, Lexington Books, Lexington Shiller RJ (1978) Rational expectations and the dynamic structure of macroeconomic models. Journal of Monetary Economics, 4: 1-44 Shiller RJ (1981) Do stock price move too much to be justified by subsequent changes in dividends? American Economic Review, 71: 421-436 Shumway RH, Stoffer DS (2000) Time series analysis and its applications, Springer, New York Solnik BH (1974) Why not diversify internationally rather than domestically? Financial Analyst Journal, 30: 91-135 Solnik BH (1976) An equilibrium model of the international capital market. Journal of Economic Theory, 8: 500-524 Solnik BH (1983) International arbitrage pricing theory. Journal of Finance, 38: 449-457 Stulz R (1981) A model of international asset pricing. Journal of Financial Economics, 9: 383-406 Tesar L, Werner I (1992) Home bias and the globalization of securities markets. NBER Working paper No. 4218. Tirole J (1982) On the possibility of speculation under rational expectations. Econornetrica, 50: 1163-1181 Tirole J (1985) Asset bubbles and overlapping generations. Econornetrica, 53: 1499-1528 Von Fustenberg GM, Jeon BN (1989) International stock price movements: links and messages. Brookings Papers on Economic Activity, 1:125-167 Weil P (1990) On the possibility of price decreasing bubbles. Econornetrica, 58: 1467-1474 Wells C (1996) The Kalman filter in finance, Kluwer Academic Publishers, Amsterdam West KD (1987) A specification test for speculative bubbles. Quarterly Journal of Economics, 102: 553-580 West KD (1988) Bubbles, fads, and stock price volatility tests: a partial evaluation. NBER Working Paper No. 2574 Wheatley S (1988) Some tests of international equity integration. Journal of Financial Economics, 2 1: 177-212 Wu Y (1995) Are there rational bubbles in foreign exchange markets? Journal of International Money and Finance, 14: 27-46 Wu Y (1997) Rational bubbles in the stock market: accounting for the U.S. stockprice volatility. Economic Inquiry, 35: 309-3 19

13 Forward FX Market and the Risk Premium

13.1 Introduction Several regression based studies attempted to explain the ability (or otherwise) of the forward exchange rates in predicting the future realized spot exchange rates. Although with the improvements in econometric theory the nature of the tests employed have changed, but the basic approach has remained essentially within the regression framework. For example, Wu and Zhang (1997) employ a non-parametric test and not only reject the unbiasedness hypothesis but also conclude that the forward premium either contains no information or wrong information about the future currency depreciation. On the other hand, Bakshi and Naka (1997) derive an error correction model under the assumption that the spot and the forward rates are cointegrated and conclude using the generalized method of moments that the unbiasedness hypothesis cannot be rejected. Phillips and McFarland (1997) develop a robust test and reject the unbiasedness hypothesis but conclude that the forward rate has an important role as a predictor of the fbture spot rate. The failure of the unbiasedness hypothesis has been attributed to the existence of a foreign exchange risk premium. This has led to a great deal of research on the modeling of the risk premia in the forward exchange rate market. However, models of risk premia have been unsuccessful in explaining the magnitude of the failure of unbiasedness (Engel 1996, page 124). We define the term rp, = f, - E,[s,+,] as the foreign exchange risk premium. Under risk-neutrality the market participants would behave in such a way that f , , equals E, [st+,] and the expected profit from forward market speculation would be zero. Stulz (1994) discusses a model of foreign exchange risk premium based on optimizing behavior of international investors. However, alongside such theoretical developments pure time series studies of rp, have also assumed a renewed importance. These are useful in describing the behavior of f,., - E,[s,+,] . Models of the foreign ex-

,

194

13 Forward FX Market and the Risk Premium

change risks premium that assume rational expectations should be able to explain the observed time series properties. Examples of such studies include Backus et a1 (1993) and Bekaert (1994). Modeling of the time varying risk premia has been inadequately addressed in the literature since there is little theory to guide us in this respect. Wolff (1987) and Cheung (1993) have modeled the risk premia as an unobserved component and estimated it using the Kalman filter. In their signal extraction approach they empirically determine the temporal behavior of the risk premium using only data of forward exchange rate and the spot exchange rate. Although the signal extraction approach avoids specifying any particular form of the risk premia, it offers little insight into the risk premia and other economic variables. In fact Cheung (1993) attempts to link the estimated risk premia with other macro economic variables in the intertemporal asset-pricing model of Lucas (1982). However, the results are not very encouraging and the estimated regression models have very low R-squares. Both Wolff (1987) and Cheung (1993) analyze the quantity (f,,, -st+,) to determine the time series characteristic of the unobserved risk premia. This in turn determines the dynamics of the unobserved component, the risk premia. For the different currencies they examine the dynamics of the risk premia can be captured by a low order ARMA process. Wolff (2000) further extends the number of currencies studied in the same framework. In those papers, therefore, the observed difference between the forward exchange rate at time t for the period t+k and the subsequently realized spot rate at time t+k is the main driver for the structure of the risk premia. This is assumed to be composed of the unobserved risk premia and the unexpected depreciation of the exchange rate. In this chapter we also adopt the unobserved component model approach and its estimation by Kalman filter. However, we attempt to model the market price of risk (and hence the risk premia) by utilizing the noarbitrage relation between the spot and the forward markets and by assuming a certain dynamic process for the market price of risk. This allows us to obtain the state-space system for the spot and the forward exchange rates and the market price of risk. The filtered estimates of the market price of risk and the other parameters of its dynamic process allow us to compute the risk premia. This approach essentially differs from those of Wolff (1987) and Cheung (1993) in that it models the spot and forward dynamics as well as the market price of risk. We obtain similar characterization of risk premia to Wolff and Cheung, which we interpret as confirmation of this methodology of using the noarbitrage relation under the historical measure. The advantage of the meth-

13.2 Alternative Approach to Model Risk Premia

195

odology is that it is extendible to other derivative markets such as FX options, which are a rich source of untapped information about markets' view of risk premia.

13.2 Alternative Approach to Model Risk Premia It is apparent from the preceding discussion that the risk premium plays an important role in explaining the divergence between the forward exchange rate and subsequently realised spot exchange rate. Most previous studies have employed some form of regression based approaches and the conclusions are based on asymptotic inferences. To avoid data correlation problems with overlapped sample the researchers normally align the data for the realised spot exchange rate with the time period spanned by the forward exchange rate. For example, when the one-month forward rate is used the data frequency for the spot exchange rate should also be of one month. This has the unwanted side effect of reducing effective sample size even if one starts with a large data set. This procedure may, therefore, lead to loss of information in the intervening period. Furthermore, the use of just the forward exchange rate and the subsequently realised spot exchange rate, as in Wolff (1987) and Cheung (1993), does not make use of the no-arbitrage relation that exists between the spot asset and the derivative instrument written on that asset. The derivative market being essentially forward looking it impounds a great deal of market information. As explained in Hull (1997, chapter 13) it is the market price of risk that connects the spot asset and all the derivative assets written on that spot asset. In this chapter we make use of this interconnection in a framework that starts with the usual assumptions of the Black-Scholes option-pricing model and let the spot exchange rate follow a geometric diffusion process. The standard arbitrage argument is then applied to relate the forward exchange rate (a derivative instrument) to the spot exchange rate through the contract period, and the related interest rates in the two countries. By applying Ito's lemma we are able to express the dynamics of the forward price as another stochastic differential equation. Following the argument in Hull (1997, chapter 13) the investors in the forward contracts may be assumed risk-neutral provided they are compensated by additional return from holding the derivative asset. This additional return is related to the market price of risk and the volatility of the underlying spot asset. Since this market price of risk is not observable, we adopt

196

13 Forward FX Market and the Risk Premium

the unobserved component modelling approach after specifying a suitable stochastic process for the market price of risk. Since we are not pricing the forward contracts as such in this chapter we incorporate the market price of risk and treat this as an unobserved state variable in the system dynamics under the historical measure. This is where the main innovation of this chapter enters. Once we express the dynamics of the market price of risk we can treat the observations of the forward exchange rates and the spot exchange rates in the historical measure as opposed to an equivalent risk neutral measure. This leads to a partially observed system involving three variables, the spot exchange rate, the forward exchange rate and the market price of risk. This system can be put into a state-space form and then suitably discretised for estimation by the Kalman filter. The advantage of this approach is that we get the filtered estimates of the market price of risk, which form the basis for estimation of the risk premia. Since we are modelling the dynamics of the three variables simultaneously through the discretisation period, there is no longer a need to align the spot exchange rates with the forward exchange rate period. We, therefore, have the advantage of benefiting from the utilisation of the information generated through the discretisation period. This is normally not possible in regression-based approaches. However, Dunis and Keller (1995) suggest a panel approach that avoids such a loss of data in the regression-based approach

13.3 The Proposed Model Let the spot exchange rate follow the one-dimensional geometric diffusion process, dS = pSdt + o , ~ (t) d ,~

(13.1)

where p is the expected return from the spot asset, o, is the volatility of this return, both measured per unit of time and dW is the increment of a Wiener process under the so-called historical (statistical) probability measure Q , r is the domestic risk-free interest rate and r, as the counterpart in the foreign currency. Since r, can be interpreted as a continuous dividend yield, the instantaneous return to an investor holding foreign exchange is @+I-,). Thus the relationship between the excess return demanded and the market price of risk (A)may be written

13.3 The Proposed Model

197

Thus, under the historical measure Q equation (13.1) can be rewritten dS = (r - rf + ho)Sdt + o,SdW(t) , under Q .

(13.3)

Alternatively under the risk neutral measure Q the last equation becomes

where, ~ ( t=)W(t) + Ih(u)du . 0

We recall that under Q , the process ~ ( t is) not a standard Wiener process since ~ [ d ~ ( = t )hdt ] # 0 in general. However, Girsanov's theorem allows us to obtain the equivalent measure Q under which ~ ( t does ) become a standard Wiener process. The measures Q and Q are related via the Radon-Nikodym derivative. Using standard arguments for pricing derivative securities (see for example, Hull (1997), chapter 13), the forward price at time t for a contract maturing at T(> t) , is

But from equation (l3.4), by Ito's lemma,

so that under Q , the quantity S(t)e-"-'f" is a martingale and it follows immediately that

6,(s,

) = ~,e('-'~

,1.e. '

198

13 Forward FX Market and the Risk Premium

If the maturity date of the contract is a constant period, x, ahead then (13.6) may be written as

Then from (13.3), (13.4) and (13.7) and by a trivial application of Ito's lemma we obtain the stochastic differential equation for F under Q and Q . Thus, under Q

whilst under Q ,

with, F(0, x) = ~,e"-~~'" . We now assume that under historical measure Q the market price of risk, 1 , follows the mean reverting stochastic process

where % is the long-term average of the market price risk, K defines the speed of mean reversion. Here, we assume that the same noise process drives both the spot exchange rate and the market price of risk. It would of course also be possible to consider a second independent Wiener process driving the stochastic differential equation for h . However, we leave investigation of this issue for future research. It should be pointed out here that when discretised the stochastic differential equation (13.10) would become a low order ARMA type process of the kind reported in Wolff (1987) and Cheung (1993). The parameters in

13.3 The Proposed Model

199

equation (13.10) may be estimated from the data using the Kalman filter as pointed out earlier. Considering we have one forward price, f(t, x) , then we have a system of 3 stochastic differential equations. These are (under the measure Q )

where, S(0) = So, i(0) = Lo , f (0, x) = ~ ~ e ". - ~ ~ ' ~ It should be noted that the information contained in equations (13.1 la) (13.1 1c) is also contained in the pricing relationships,

To estimate the parameters in the filtering framework, however, we choose to work with the equation (1 3.1 1c). From equation (13.3), we can write the spot price at time t + x as, using s(t) = In S(t) ,as

From equation (I 3.13) we can write the expected value of s(t + x) as

The calculations outlined in appendix allow us to then write,

200

13 Forward FX Market and the Risk Premium

The above equation may also be expressed (via use of equation (13.7)) as,

Let 7~ (t, X) represent the risk premium (under Q ) for the x period ahead spot rate, then from equation (13.16)

We pointed out in the introduction that previous studies attributed the difference between the forward rate and the subsequently realised spot exchange rate to a risk premium and the unexpected depreciation of the exchange rate. Equation (13.17) gives an explicit expression for the risk premium, characterising how the market price of risk enters the expectation formation and thus influence the risk premium. The integral terms involving the Wiener increments in equation (13.13) should be related to the noise terms identified in Wolff (1987) and Cheung (1993). We would now like to compare the time variation of risk premia for one-month forward rates obtained from equation (13.17) for different exchange rates. This will require estimates of the parameters describing the stochastic process for A given by equation (13.10). In the next section, we describe the state-space formulation of the system and estimation of these parameters as well as the filtered and smoothed estimates of h(t) . We would also like to compare this with the risk premia obtained from the approach outlined in Wolff (1987) and Cheung (1993). It should be pointed out that our method can be easily applied to multiple forward exchange rates and thereby help us examine the term structure of forward risk premia present in quoted forward exchange rates. Besides, our method is not reliant upon data synchronisation with respect to matching the forward rate period as opposed to the methods in Wolff (1987) and Cheung (1993). In

13.4 State-Space Framework

20 1

the following section we also briefly describe the method of WolffICheung so as to facilitate comparison with the approach presented in this chapter.

13.4 State-Space Framework The dynamics of the spot exchange rate, forward exchange rate and the market price of risk are described in equations (13.1 1a) through (13.1 lc). The main consideration in state-space formulation is the separation of the noise driving the system dynamics and the observational noise. The measurements in practical systems are not necessarily the variables driving the system but some transformation of these masked by the measurement noise. Also in most cases all the driving variables are not directly observable, thus leading to partially observed systems. Similarly, in our model the market price of risk is not observable and hence we are dealing with a partially observed system of three variables. We assume that the contribution to the observation noise in our system is from different sources eg the time of measurement (i.e. at the beginning of the trading day or the end of the trading day, etc.), or the spread in the quoted forward exchange rates. We also assume that this measurement noise is independent of the noise sources driving the system dynamics i.e. W . For the purposes of implementation and estimation we need to discretise the continuous time dynamics given by the equations (13.1 1 a) through (13.1 lc). Although a number of different approaches are available (see Kloeden and Platen (1992)) we choose to work with the Euler-Maruyama scheme. As can be seen from the equations (13.1 1a) and (13.1 1c), the diffusion terms are dependent on the state variables themselves and are thus stochastic in nature. We can avoid dealing with the stochastic diffusion system by a simple transformation of variable and application of Ito's lemma. We use the natural logarithm of the spot and forward exchange rates and this transforms the system with constant diffusion terms. The transformed stochastic differential equations with s = ln(S), and f = ln(F) are,

202

13 Forward FX Market and the Risk Premium

After discretisation of the equations (13.1 lb), (13.18) and (13.19) we obtain for the time interval between k and k+ 1:

a)

where AW(t) = W(t) - W(t - At) - N(O, . The equations (13.20) - (13.22) describe the dynamics of the partially observed system and in the state-space framework they are generally referred to as the state transition equations. In a multivariate situation it is convenient to express these in matrix notation and following Harvey (1990) this turns out as follows:

where

13.4 State-Space Framework

203

and q, is a (2x1) vector of noise sources that are serially uncorrelated, with expected values zero and the covariance matrix,

The observations in our system are related to the state variables in an obvious way as

where

The variance of the measurement errors is represented by h for both the observed variables. We are also assuming in this set up that the noise sources in the state and the measurement equations are independent of each other. The estimation process for the state-space system is adaptive in nature and thus requires specification of the initial state vector. As suggested in Harvey, Ruiz and Shepherd (1994) the first observations can be used to initialise it if non-stationarity is suspected. Another application of state-space formulation by the authors in the context of non-Markovian term structure model can be found in Bhar and Chiarella (1997). The full mathematical details of the Kalman filter algorithm are discussed in chapter 8 and 9.

204

13 Forward FX Market and the Risk Premium

--

13.5 Brief Description of WolffICheung Model The main idea in modelling the forward exchange rate risk premia in their model is the assumption that the forecast error resulting from the forward rate as the predictor of futures spot exchange rate consists of a premium component and a white noise error. In this context,

Here E~ is an uncorrelated zero-mean sequence and Pt is the unobserved risk premium. In terms of state-space system representation this is the measurement equation. Both Wolff and Cheung determine the dynamic of the risk premium by studying the time series properties (in the Box-Jenkins sense) of the quantity (Ft,t+,-St+,). As suggested in Wolff (2000) for most currencies either an ARMA(1,l) or an MA(1) representation is adequate. The corresponding equations for Pt for AR(l), ARMA(1,l) and MA(]), respectively, are given by,

The state equation matrices (with reference to equation (13.23)) are given below for each of these time series representations (assuming v2 is the variance of the innovation 9 ) . The estimation process for these models is same as described earlier in the context of our model. For the AR(1) representation:

For the ARMA(1,l) representation:

13.6 Application of the Model and Data Description

205

For the MA(1) representation the matrices are similar to that in case of ARMA(1,l) with restriction that 4 = 0 .

13.6 Application of the Model and Data Description As part of our empirical investigation we apply the methodology developed here to five different exchange rates, all against U.S. dollars. These are Australian dollar (AUD), German marks (DEM), French frank (FRF), British pound (GBP) and the Japanese yen (JPY). The data set covers the period January 1986 to December 1998 with 156 observations in each series. We use only the one-month forward exchange rates so that the results can be compared directly with those from the implementation of WolffICheung methodology. It should be pointed out that our modelling approach does not require that the observations should be properly aligned with the maturity of the forward rates. The exchange rate data reflects the daily 4PM London quotation obtained from Datastream and the interest rate data are the daily closing one-month Euro currency deposit rates. To start the adaptive algorithm of the Kalman filter we initialise the state vector with the first observations. The algorithm also requires specifying the prior covariance matrix for the state vector. In the absence of any specific knowledge about the prior distribution we use the diffuse prior specification following Harvey (1990, p. 121). See also chapter 8 and 9 for additional information in this respect. The parameter estimates are obtained by maximizing the log likelihood function given by the equation (8.35) in Chapter 8. The numerical optimization algorithm called 'Newton' in GAUSS is used for this purpose without any parameter constraints. The results of the estimation procedure are shown in Table 13.1. The t-statistics reported in that table are computed from the standard error obtained from the heteroscedasticity consistent covariance matrix of the parameters at the point of convergence. All the parameters of the model except the long-term average market price of risk are statistically significant for each of the currencies. The estimated parameter o, compares favourably with the sample estimates obtained from the spot exchange rate series (not reported separately). How the model fits the data is best analysed by examining the residual from the

206

13 Forward FX Market and the Risk Premium

estimation process. These are reported in Table 13.2. One of the main requirements is that the residual be serially uncorrelated both in its level and its squared form. The portmanteau test and the ARCH test support this requirement for all the currencies examined. As the Kalman filter generated residuals are recursive in nature two other tests are carried out to judge the model adequacy. These are modified Von Neumann ratio and the recursive t-tests (Harvey 1990, page 157). Both these tests support our modelling approach. Finally, the conditional normality assumption made in the modelling is also supported by the Kolmogorov-Smirnov test statistics reported in Table 13.2. Since we are interested in comparing and contrasting the modelling approach developed here with that reported earlier by WolffICheung we present the estimation results of their models for the same set of currencies in Table 13.3. As can be seen most parameters are statistically significant. We subject the model residuals to the same set of tests as in Table 13.2 (although in their original papers, WolffICheung do not report these diagnostics). The results reported in Table 13.4 support all the model adequacy tests. For the model developed in this chapter the risk premia contained in the one-month forward exchange rate can be computed easily from equation (13.17) with the help of the estimated parameters from Table 13.1 and the filtered (or smoothed) estimates of the market price of risk. Since WolffICheung method does not provide this risk premia directly we dot not analyse this aspect any further. Next, we compare the one-month ahead prediction of the spot exchange rate with the realised exchange rate and thus generate the mean absolute prediction error and the root mean squared prediction error for each of the exchange rate series. In the context of Kalman filter this is really ex ante prediction error since the prediction of the measurement variable for time k+l is made utilising information up to and including time k. This is true for WolffICheung model as well because the way we have implemented it. The comparative results are shown in Table 13.5 for our model, WolffICheung model as well as a martingale process. Overall conclusion from examining the Table 13.5 is that both our model and WolffICheung model perform better than the martingale process. There is, however, not much difference in forecasting performance between our model and WolffICheung model. It should, however, be remembered that our model could be implemented for data set of any observed frequency whereas WolffICheung approach is limited to data set where the spot exchange rate frequency aligns with the forward rate data used.

13.6 Application of the Model and Data Description

207

Table 13.1 Parameter estimates for the model based on market price of risk

--

"

AUD

DEM

FRF GBP JPY

0.0272 (2.16) 0.0797 (9.48) 0.0528 (3.45) 0.0609 (5.12) 0.0960

8.0884

0.7226

--

21.2137

Numbers in parentheses are t-statistics computed from standard errors obtained using the het&oscedasticity consistent cova&nce matrix at the point of convergence.

Table 13.2 Residual diagnostics and model (In Table 13.1) adequacy tests AUD DEM FRF GBP JPY

Port. 0.226 0.080 0.482 0.091 0.286

ARCH 0.702 0.474 0.494 0.342 0.608

MNR 0.832 0.996 0.871 0.897 0.600

Rec. T 0.597 0.887 0.917 0.857 0.956

-

KS 0.042 0.055 0.082 0.068 0.064

]on; diagnostics are computed from the recursive residual of the measurement equation, which corresponds to the spot index process. The null hypothesis in portrnanteau test is that the residuals are serially uncorrelated. The ARCH test checks for no serial correlations in the squared residual up to lag 26. Both these test are applicable to recursive residuals as explained in Wells (1996, page 27). MNR is the modified Von Neumann ratio test using recursive residual for model adequacy (see Harvey (1990, chapter 5). Similarly, if the model is correctly specified then Recursive T has a Student's t-distribution (see Harvey (1990, page 157). KS statistic represents the Kolmogorov-Smirnov test statistic for normality. 95% significance level in this test is 0.109. When KS statistic is less than 0.109 the null hypothesis of normality cannot be rejected.

208

13 Forward FX Market and the Risk Premium

Table 13.3 Parameter estimates for Wolff (1987) and Cheung (1993) models

-

p -

AUD

4)

0.9439 (23.18) -0.7150 (-6.57) -0.7183 (-7.40)

--

8

v

E

0.0261 0.0000 (17.01) (0.000) DEM 0.9206 0.0142 0.0277 (5.12) (1.64) (4.28) FRF 0.9189 0.0077 0.0288 (9.70) (5.56) (15.31) GBP 0.6318 0.0269 0.0159 (5.90) (6.14) (2.25) JPY 0.9311 0.0368 0.0054 (3.91) (10.27) (0.57) Numbers in parentheses are t-statistics computed from standard errors obtained using the heteroscedasticity consistent covariance matrix at the point of convergence.

---

Table 13.4 Residual diagnostics and model (In Table 13.3) adequacy tests Port. ARCH 0.130 0.769 AUD DEM 0.428 0.938 FRF 0.591 0.937 GBP 0.270 0.420 JPY 0.458 0.551 Notes of Table 13.2 apply here. 7

-

P

MNR 0.297 0.604 0.379 0.486 0.539 -

Rec. T 0.925 0.482 0.275 0.287 0.942

KS 0.035 0.055 0.059 0.083 0.063

?

13.7 Summary and Conclusions Table 13.5 One step ahead forecast error for spot exchange rate

----

MAE -----"-------

--"-----"PP

209

-

MSE

AUD Market price of risk CheungIWolff Martingale process

0.0205 0.0206 0.0279

0.0007 0.0007 0.0013

Market price of risk CheungNolff Martingale process

0.0258 0.0254 0.0451

0.001 1 0.0010 0.0035

Market price of risk Cheung/Wolff Martingale process

0.0248 0.0241 0.1446

0.0010 0.0009 0.0344

Market price of risk CheungIWolff Martingale process

0.0235 0.0250 0.0143

0.0010 0.001 1 0.0004

Market price of risk CheungNolff

0.0286 0.0290

0.0014 0.0014

DEM

FRF

GBP

JPY

'MAE' and 'MSE' represent mean absolute error and mean squared error respectively. These are computed from the one step ahead forecast error obtained during Kalman filter recursion. These forecast errors are used to develop the prediction error form of the likelihood function. CheungNolff model refers to our somewhat modified implementation of their approach.

13.7 Summary and Conclusions In this chapter we have presented a new approach to analyse the risk premium in forward exchange rates. This involves exploiting the relationship that links the spot exchange rate and the forward exchange rate through the market price of risk. By directly modelling the market price of risk as a mean reverting process we are able to show how the market price of risk enters into expectation formation for a future spot exchange rate. This methodology allows us to quantify the risk premium associated with a particular forward exchange rate in terms of the parameters of the process describing the market price of risk. We also demonstrate how these parameters can be estimated in a state-space framework by application of the Kalman filter. This procedure, in turn, generates the filtered and the smoothed estimates for the unobserved market price of risk.

210

13 Forward FX Market and the Risk Premium

We apply the procedure developed in this chapter to AUD, DEM, FRF, GBP and JPY all against USD and use one-month forward exchange rates. Various model diagnostics support the modelling approach. We also compare our results with the models proposed by Wolff (1987) and Cheung (1993). We also point out that our approach can be applied to any frequency of data whereas the model of WolffICheung would not work unless the forward rate maturity matches the observation frequency.

Appendix: Calculation of Et

[ j' f l h(r)dr

By an application of Ito's lemma the stochastic differential equation for h (equation 13.10) can be expressed as

Integrating A 13.1 from t to t (> t)

from which

Now integrating (A13.3) from t to t+x,

References

2 11

The first two integrals in the foregoing equation are readily evaluated. However, in order to proceed, the third integral needs to be expressed as a standard stochastic integral, having the dW(u) term in the outer integration. This is achieved by an application of Fubini's theorem (see Kloeden and Platen (1992)) which essentially allows us to interchange the order of integration in the obvious way. Thus,

Thus,

References Backus D, Gregory A, Telmer C (1993) Accounting for forward rates in markets for foreign currency. Journal of Finance, 48: 1887-1908 Baillie R, Bollerslev T (1990) A multivariate generalized ARCH approach to modelling risk premia in forward exchange markets. Journal of International Money and Finance, 9: 309-324 Bakshi GS, Naka A (1997) Unbiasedness of the forward exchange rates. The Financial Review, 32: 145-162 Bekaert G (1994) Exchange rate volatility and deviation from unbiasedness in a cash-in-advance model. Journal of International Economics, 36: 29-52 Bhar R, Chiarella C (2000) Analysis of time varying exchange rate risk premia. In: Dunis CL (ed) Advances in Quantitative Asset Management, Kluwer, Dordrecht, pp. 255 - 273 Bhar R, Chiarella C (1997) Interest rate futures: estimation of volatility parameters in an arbitrage-free framework. Applied Mathematical Finance, 4: 1-19 Boudoukh J, Richardson M, Smith T (1993) Is the ex ante risk premium always positive? Journal of Financial Economics, 34: 387-408

2 12

13 Forward FX Market and the Risk Premium

Canova, F. (1991), An empirical analysis of ex ante profits from forward speculation in foreign exchange markets. Review of Economics and Statistics, 73, 489-496. Canova F, Ito T (1991) The time series properties of the risk premium in the yenldollar exchange market. Journal of Applied Econometrics, 6: 125-142 Canova F, Marrinan J (1993) Profits, risk and uncertainty in foreign exchange markets. Journal of Monetary Economics, 32: 259-286 Cheung Y (1993) Exchange rate risk premiums. Journal of International Money and Finance, 12: 182-194 Cochrane JH (1999) New facts in finance. NBER Working Paper No. 7169 Dumas B (1993) Partial- vs. general-equilibrium models of the international capital market. NBER Working Paper No. 4446 Dunis C, Keller A (1995) Efficiency tests with overlapping data: an application to the currency option market. European Journal of Finance, 1: 345-66 Engel C (1996) The forward discount anomaly and the risk premium: a survey of recent evidence. Journal of Empirical Finance, 3: 123-192 Harvey AC (1990) Forecasting structural time series models and the Kalman filter. Cambridge University Press, Cambridge Harvey AC, Ruiz E, Shephard N (1994) Multivariate stochastic variance model. Review of Economic Studies, 6 1: 247-264 Hull JC (1997) Options, futures, and other derivatives. 31d edn. Prentice Hall International Inc Jazwinski AH (1970), Stochastic processes and filtering theory. Academic Press, New York Kloeden PE, Platen E (1992) Numerical solution of stochastic differential equations. Springer-Verlag, Berlin Lucas RE (1982) Interest rates and currency prices in a two country world. Journal of Monetary Economics, 10: 335-360 Nijman TE, Palm FC, Wolff CCP (1993) Premia in forward exchange rate as unobserved components. Journal of Business and Economic Statistics, 11: 361365 Ostdiek B (1998) The world ex ante risk premium: an empirical investigation. Journal of International Money and Finance, 17: 967-999 Phillips PCB, Mcfarland JW (1997) Forward exchange market unbiasedness: the case of Australian dollar since 1984. Journal of International Money and Finance, 16: 885-907 Ross SA, Westerfield RW, Jordan BJ (1998) Fundamentals of corporate finance. Irwin-McGraw-Hill Stulz R (1994) International portfolio choice and asset pricing an integrative survey. NBER Working Paper No. 4645 Wolff CCP (1987) Forward foreign exchange rates, expected spot rates, and premia: a signal-extraction approach. Journal of Finance, 42: 395-406 Wolff CCP (2000) Measuring the forward exchange risk premium: multi-country evidence from unobserved component models. Journal of International Financial markets, Institutions and Money, 10: 1-8

References

2 13

Wu Y, Zhang H (1997) Forward premiums as unbiased predictors of future currency depreciation: a non-parametric analysis. Journal of International Money and Finance, 16: 609-623

14 Equity Risk Premia from Derivative Prices

14.1 Introduction This chapter focuses on a topical and important area of finance theory and practice, namely the analysis of the equity market risk premium. In particular the chapter suggests a new approach to the estimation of the equity market risk premium by making use of the theoretical relationship that links it to the prices of traded derivatives and their underlying assets. The volume of trading in equity derivatives, particularly on broad indices, is enormous and it seems reasonable that prices of the underlying and the derivative should impound in them the market's view on the risk premium associated with the underlying. To our knowledge no attempt has been made to get at the risk premium from this perspective. The approach we adopt also has the advantage of quite naturally leading to a dynamic specification of the equity risk premium. This aspect of our framework is pertinent in the context that a great deal of recent research has pointed to the significance of time varying risk premia. Consider for instance research on the predictability of asset returns and capital market integration. If markets are completely integrated then assets with the same risk should have the same expected return irrespective of the particular market. Bekaert and Harvey (1995) use a time varying weight to capture differing price of variance risk across countries. Ferson and Harvey (1991) and Evans (1994) showed that although changes in covariance of returns induce changes in betas, most of the predictable movements in returns could be attributed to time changes in risk premia. Some authors have investigated the time variation of both the systematic and specific risks of portfolios in a number of equity markets using suitable dynamic specifications (such as GARCH-M type models) for return volatility e.g. Giannopoulos (1995). According to the equilibrium capital asset pricing model (CAPM), expected return from a risky asset is directly related to the market risk premium through its covariance with the market return (i.e. its beta). Although

216

14 Equity Risk Premia from Derivative Prices

in CAPM beta is assumed to be time invariant, many studies (e.g. Bos and Newbold (1984), Bollerslev, Engle and Wooldridge (1988), Chan, Karolyi and Stulz (1992)) have confirmed instability of betas over time. These authors also show that betas of financial assets can be better described by some type of stochastic model and hence explores the conditional CAPM. It is in this context that the modeling of risk-premia across time is important, particularly from the point of view of domestic f h d managers looking to diversify their portfolios internationally. The fact that risk is time varying has significant implications for portfolio managers. This is because many risk management strategies are based on the assumption of a static measure of risk, which does not offer satisfactory guide to its possible future evolution. The modeling of the dynamic behaviour of risk premia is a difficult exercise since it is not directly observable in the financial market. It can only be inferred from the prices of other related observable financial variables. Evans (1994) points out a number of information sources that can be used to measure risk-premia. These are, for example, lagged realized return on a one-month Treasury bill, the spread between the yield on one- and sixmonth Treasury bills, the spread between dividend-price ratio on the S&P500 and one month Treasury bill. However, one encounters some significant econometric problems such as multicollinearity when attempting to estimate risk-premia from these variables. Besides, the dynamic behaviour of risk-premia is still not well captured by such regression-based techniques. In this chapter we propose to model the dynamic behaviour of riskpremia using the stochastic differential equations for underlying price processes that arise from an application of the arbitrage arguments used to price derivatives on the underlying, such as index futures and options on such futures contracts. This stochastic differential system is considered under the so-called historical (or real world) probability measure rather than the risk neutral probability measure required for derivative security pricing. The link between these two probability measures is the risk-premium. The price process can thus be expressed in a dynamic form involving observable prices of the derivative securities and their underlying assets and the unobservable risk-premium. A mean reverting process for the dynamics of the risk premium is considered. This system of prices and riskpremium can be treated as a partially observed stochastic dynamic system. In order to cater for the time variation of volatility we use the option implied volatility in the dynamic equations for the index and its derivatives. This quantity is in a sense treated as a signal that impounds the market's forward looking view on the equity risk premium. The resulting system of

14.2 The Theory behind the Modeling Framework

217

stochastic differential equations can then be cast into a state-space form from which the risk-premia can be estimated using Kalman filtering methodology. We apply this approach to estimate the market risk premium at monthly frequency in the Australian and US markets over the period January 1995 to December 1999. The plan of the chapter is as follows. Section 14.2 lays out the theoretical framework linking the index, the futures on the index and the index futures option. The stochastic differential equations driving these quantities are expressed under both the risk-neutral measure and the historical measure. The role of the equity risk premium linking these two measures is then made explicit. In section 14.3 a stochastic differential equation modeling the dynamics of the market price of equity risk is specified. The dynamics of the entire system of index, index futures, index futures option and market price of equity risk is then laid out and interpreted in the language of state-space filtering, as in chapter 8. Section 14.4 describes the Kalman filtering set-up and how the equity risk premium is estimated following the ideas in chapter 9. Section 14.5 describes the data set used for empirical implementation. Section 14.6 gives the estimation results and various interpretations. Section 14.7 summarizes and concludes and makes suggestions for future research.

14.2 The Theory behind the Modeling Framework We use S to denote the index value, F a futures contract on the index and C an option on the futures. We assume that S follows the standard lognormal diffusion process,

where Z is a Wiener process under the historical probability measure P, p is the expected instantaneous index return and o its volatility. The spotlfutures price relationship is,

where q is the continuous dividend yield on the index and T is the maturity date of the index futures. Applying Ito's lemma to (14.2)' we derive the stochastic differential equation (SDE) for F, viz.

218

14 Equity Risk Premia from Derivative Prices

dF=p,Fdt+o, F d Z ,

(14.3)

where

Application of the standard Black-Scholes hedging argument to a portfolio containing the call option and a position in the futures yields the stochastic differential equations for S, F and C, namely dS = (r-q) S dt + o S d 2 ,

(14.5a)

Here, 2 is a Wiener process under the risk-neutral measure and is related to the Wiener process Z under the historical measure P according to,

where i ( t ) is the instantaneous market price of risk of the index. This latter quantity can be interpreted from the expected excess return relation' p - (r-q) = Lo,

(14.7)

We recall that the expected excess return relation equation (14.7) arises from expressing the condition of no riskless arbitrage between the index option and the pc-r-p~-runderlying as ----I. '=c

'=F

14.2 The Theory behind the Modeling Framework

219

as the amount investors require instantaneously to be compensated for a unit increase in the volatility of the index. In this study we interpret ho as the risk premium of the market, as it measures the compensation that an investor would require above the cost-of-carry ( = r - q for the index) to hold the market portfolio. The option return volatility o, is given by,

and the partial derivative is the option delta with respect to the futures price. Equation (14.5) is converted into the traditional Black's (1976) futures call option pricing formula via the observation that Ce-* is a martingale under the risk-neutral probability measure and is given by,

where,

In the expression (14. lo), T is the maturity of the option contract and is typically a few days before the futures delivery date2. Our purpose is to use market values of S, F and C to extract information about the market price of risk, 3L. Thus, we use equation (14.6) to convert the dynamic system (14.5) into a diffusion process under the historical measure P, namely, (14.1 la)

In this study we treat the option maturity and futures maturity as contemporaneous

220

14 Equity Risk Premia from Derivative Prices

Where o, can be calculated from Black's model as,

Equations (14.1 1) describe the dynamic evolution of the value of the index, its futures price and the price of a call option on the futures under the historical probability measure and assuming that there are no arbitrage opportunities between these assets. The volatility o and the market price of risk h are the only unobservable quantities. In the next section we describe how filtering techniques may be used to infer these quantities from the market prices.

14.3 The Continuous Time State-Space Framework A fundamental question is how should the time variation of A be modelled? Here we have little theory to guide us, though we could appeal to a dynamic general equilibrium framework. However this in turn requires many assumptions such as specification of utility function and process(es) for underlying factor(s). For our empirical application we prefer to simply assume h follows the mean reverting diffusion process,

Here, h is the long-run value of h , K is the speed of reversion and o, is the standard deviation of changes in h . We assume that the procedure for h is driven by the same Wiener process that drives the index. The motivation for this assumption is the further assumption that the market price of risk is some function of S and t. An application of Ito's Lemma would then imply that the dynamics for h are driven by the Wiener process Z(t).

14.3 The Continuous Time State-Space Framework

221

The specification (14.13) has a certain intuitive appeal. Through the mean reverting drift it captures the observation that ex-post empirical estimates of h appear to be mean reverting. The only open issue with the specification (14.13) is whether we should specify a more elaborate volatility structure rather than just assuming o, is constant. Here we prefer to let the data speak; if the specification (14.13) does not provide a good fit then it would seem appropriate to consider more elaborate volatility structures (and indeed also for the drift). Thus we end up considering a four dimensional stochastic dynamic system for S, F, C and h which we write in full here: dS = (r-q

+ Lo) Sdt + o SdZ ,

(14.14a)

It will be computationally convenient to express the system (14.14) in terms of logarithms of the quantities S, F and C. Thus our system becomes

222

14 Equity Risk Premia from Derivative Prices

where we set s = ln(S), f = ln(F) and c = In (C). In filtering language equation (14.15) is in state-space form and we are dealing with a partially observed system since the prices s, f and c are observed but the market price of risk, h, is not. In setting up the filtering framework in the next section it is most convenient to view h as the unobserved state vector (here a scalar) and changes in s, f and c as observations dependent on the evolution of the state. We know from a great deal of empirical work that the assumption of a constant o is not valid. Perhaps the most theoretically satisfactory way to cope with the non-constancy of o would be to develop a stochastic volatility model. However we then would not have a simple option pricing model such as (14.9), furthermore this would introduce a further market price of risk- namely that for volatility, into our framework. Thus as a practical solution to handling the non-constancy of o we shall use implied volatility calculated from market prices using Black's model. Given a set of observations f and c we can use equation (14.9) to infer the implied volatility &(f,c,t). Here we use a notation that emphasizes the hnctional dependence of & on f, c and t. This dependence becomes important when we set up the filtering algorithm in the next section. The corresponding option price volatility o, would be calculated from equation (14.12), bearing in mind that the quantity d, in equation (14. lo), also is now viewed as a function of &(f,c, t) . Thus we write

6,(f, C,t) = &(f,C,t)e(f-c)e-r(T-t)~(d, (f, &(f,c, t), t)) We can view the system (14.15) as a state-space system with (s, f, c, h ) being the state vector. This is a partially observed system in that we have observations of s, f and c but not of h . It is worth making the point that by using the implied volatility we are using a forward-looking measure of volatility as this quantity can be regarded as a signal that impounds the market's most up-to-date view about risk in the underlying index.

14.4 Setting Up The Filtering Framework

223

14.4 Setting Up The Filtering Framework The ideal framework to deal with estimation of partially observed dynamical systems is the Kalman filter. See for example, Jaminski (1970) and Lipster and Shiryaev (2000) as general references, and Harvey (1989) and Wells (1996) for economic and financial applications. Financial implementations of the Kalman filter are usually carried out in a discrete time setting as data are observed discretely. To this end we discretise the system (14.15) using the Euler-Maruyama discretisation, which has as one advantage that it retains the linear (conditionally) Gaussian feature of the continuous time counterpart. Considering first equation (14.15d) for the (unobserved) state variable X ( = A), after time discretisation its evolution from time period k (t = kAt) to k+l is given by

where,

and, the disturbance term ~k - N ((),I) is serially uncorrelated. In filtering terminology the equation (14.17) is known as the state transition equation. The observation equation in this system consists of changes in log of the spot index, index futures and the call option prices (obtained by discretising equations (14.15a- 14.15~).In matrix notation these are,

224

14 Equity Risk Premia from Derivative Prices

Here, Hk is the matrix

h

and we use ok and o,,, to denote the values of 6 and o, respectively at time kAt . In addition to the system noise€, we have assumed in (14.19) the existence of an observation noise term Qkqk, where q, - N(O,I) is serially uncorrelated and independent of the E, . The ( 3 x 3 ) diagonal matrix Q, has elements whose values would depend on features (such as bid-ask spread) of the market for each of the assets in the observation vector. Equation (14.19) can be written more compactly as, Yk

= dk

+ DkXk + H A + Qkrlk

7

(14.21)

where we use Yk to indicate the observation vector over the interval k to k+l, and its elements consist of the log price changes in s, f and c. In order to express the observation equation (14.21) in standard form we define the combined noise term Vk = Hk~k+ Qkrlk

9

so that vk - N (0, V, ) where

With these notations the observation equation (14.21) may then be written

14.4 Setting Up The Filtering Framework

225

The state transition equation (14.17) together with observation equation (14.24) constitute a state-space representation to which the Kalman filter as outlined in Jazwinski (1970) and Harvey (1989) may be applied. The relevant issues are also given in chapter 8 and 9. It needs to be noted that we are dealing with the case in which there is correlation between the system noise and observation noise3since

With the system now in state-space form, the recursive Kalman filter algorithm can be applied to compute the optimal estimator of the state at time k, based on the information available at time k. This information set consists of the observations of Y up to and including at time k. We also note that the basic assumption of Kalman filtering viz. that the distribution of the evolution of the state vector is conditionally normal is satisfied in our case since the Wiener increments are normal and the implied volatilities o, and o,,,(that affect the coefficients in the observation equation) depend on Y up to time (k-1). Therefore, the state variable is completely specified by the first two moments. It is these quantities that the Kalman filter computes as it proceeds from one time step to the next. Here we merely summarize these updating equations, full details of which are available in Jazwinski (1970), Lipster and Shiryaev (2000), Harvey (1989) and Wells (1996). Given the values of X, and P, , the optimal one step a head predictor of X,,, is given by (for k=O, 1, , ,N- 1)

while the covariance matrix (here a scalar) of the predictor is given by,

See Jazwinski (1970), section 7.3, pp 209-210.

226

14 Equity Risk Premia from Derivative Prices

Pk+,lk = TPkIkT + R R'.

(14.27)

The equations (14.26) and (14.27) are known as the prediction equations. Once the next new observation becomes available, the estimator of Xk+,in equation (14.26) can be updated as,

and

where

In order to clarifj the notation we note that Xk 1 k, Xk+lI k, Xk+lI k+l, Pk+lI k, Pk+l1 k+l, ak, T and R are scalars, dk, Dk and vk are 3-dimensional column vectors, Ckis a 3-dimensional row vector and, Fk, Vk are 3 x 3 matrices. The set of equations (14.26)-(14.30) essentially describes the Kalman filter and these are specified in terms of the initial values Xo and Var(Xo)= Po. Once these initial values are given, the Kalman filter produces the optimal estimator of the state vector, as each new observation becomes available. It should be noted that the equations (14.28) and (14.29) assume that the inverse of the matrix Fk+,exists. It may, however, be replaced, if needed, by a pseudo inverse. The updating equations step forward through the N observations. For insample estimation, as we are doing here, it is possible to improve the estimates of the state vector based upon the whole sample information. This is referred to as Kalman smoother and it uses as the initial conditions the last observation, N, and steps backwards through the observations at each step adjusting the mean and covariance matrix so as to better fit the observed data. The estimated mean and the associated covariance matrix at the N"

14.4 Setting Up The Filtering Framework

227

observation are XNIN, PNI,respectively. The following set of equations describes the smoother algorithm, for k = N, N-1, 2. Although the smoothing procedure has been introduced in chapter 8, for the sake of completion it is just summarized below.

where

Clearly to implement the smoothing algorithm the quantities Xklk,Pklk generated during the forward filter pass must be stored. The quantity within the second parentheses on the R.H.S. in equation (14.28) is known as the prediction error. For the conditional Gaussian model studied here, it can be used to form the likelihood function viz.,

-

where m is the number of elements in the state vector (in this study equal to 1). To estimate the parameter vector 0 ( K , % , G ~ )the likelihood function (14.32) can be maximized using a suitable numerical optimization procedure. This will yield the consistent and asymptotically efficient estimator 6 (Lo 1988).

228

14 Equity Risk Premia from Derivative Prices -

-

-

14.5 The Data Set The estimation methodology is applied to monthly data from the Australian and US markets for the period January 1995 to December 1999. For the Australian market, we use the market index (All Ordinaries Index), index futures, and call options on the index futures for all the four delivery months (March, June, September and December). For the US market we use S&P 500, index futures and call options on the index futures for all the four delivery months (March, June, September and December). The data were taken for the first trading day of each month. To avoid possible thin trading problems we construct a time series that uses only the last three months of a particular futures contract before switching to the next. For the Australian market, we collected all futures and futures options market data, including the implied volatility from the Sydney Futures Exchange and all the spot market data from DataStream. The 13-weeks Treasury note approximates the data for the risk-free interest rate and the information on dividend yield is provided by the Australian Stock Exchange. For the US market futures and futures options market data, including the implied volatility, were collected from the Futures Industry Institute and the US T-bill3 month rate was taken from DataStream.

14.6 Estimation Results The estimation results are set out in Tables 14.1 and 14.2. Table 14.1 gives the results for the estimation of the coefficients K , h and o,for both the Australian and US markets. The numbers in parentheses below the parameters represent t-ratios and * indicates significance at the 5% level. The t-statistic focuses on the significance of parameter estimates. We have also applied a range of other tests that focus on goodness-of-fit of the model itself, in particular residual diagnostics and model adequacy. The relevant tests are the portmanteau test, ARCH test, KS (KolmogorovSmirnov) test, the MNR (modified von Neuman ratio) test and the recursive t-test. The results of these are displayed in Table 14.2. Entries are pvalues for the respective statistics except for the KS statistic. These diagnostics are computed from the recursive residual of the measurement equation, which corresponds to the spot index process. The null hypothesis in the portmanteau test is that the residuals are serially uncorrelated and this hypothesis is clearly accepted. The ARCH test checks for no serial correlations in the squared residual up to lag 26 and the results in Table 14.2 indicate there are very little ARCH effects in the residuals. Both these test are

14.6 Estimation Results

229

applicable to recursive residuals as explained in Wells (1996, page 27). MNR is the modified Von Neumann ratio test using recursive residuals for model adequacy (Harvey 1990, chapter 5) and the results confirm model adequacy. Similarly, we conclude correct model specification on the basis of the recursive T since if the model is correctly specified then the recursive T has a Student's t-distribution (Harvey 1990, page 157). The KS (Kolmogorov-Smirnov) statistic represents the test statistic for normality. The 95% and 99% significance levels in this test are 0.088 and 0.105 respectively (when the KS statistic is less than 0.088 or 0.105 the null hypothesis of normality cannot be rejected at the indicated level of significance) and so the results provide support for the normality assumption underpinning the Kalman filter approach. Overall the set of tests in Table 14.2 indicate a good fit for the model in both markets. Table 14.1 Estimated parameters of market price of risk K

Australia (AOI) USA (S&P)

17.41* (1.90) 9.81*

3L

0,

1.1541* (0.3526) 2.0218*

0.0468* (0.0075) 0.0242*

Data set spans monthly (beginning) observations from January 1995 to December 1999. The numbers in parentheses below the parameters represent standard errors. Significance at 5% level is indicated by * and at 1% level is indicated by **. Table 14.2 Residual diagnostics and model adequacy tests "-ARCH -KS Test MNR Portmanteau Australia 0.785 0.603 0.099 0.465 USA 0.493 0.748 0.133 0.956 Entries are p-values for the respective statistics except for the KS statistic. These diagnostics are computed from the recursive residual of the measurement equation, which corresponds to the spot index process. The null hypothesis in portmanteau test is that the residuals are serially uncorrelated. The ARCH test checks for no serial correlations in the squared residual up to lag 26. Both these test are applicable to recursive residuals as explained in Wells (1996, page 27). MNR is the modified Von Neumann ratio test using recursive residual for model adequacy (Harvey 1990, chapter 5). KS statistic represents the Kolmogorov-Smirnov test statistic for normality. 95% and 99% significance levels in this test are 0.179 and 0.214 respectively. When KS statistic is less than 0.179 or 0.214 the null hypothesis of normality cannot be rejected at the indicated level of significance. ,

"e

230

14 Equity Risk Premia from Derivative Prices

Mar- Jul- Nov- Mar- Jul- Nov- Mar- Jul- Nov- Mar- Jul- Nov- Mar- Jul- Nov99 99 95 96 96 96 97 97 97 98 98 98 99 95 95

1 -Exnost --- Smoothed ....-Smoothed-2SD

Smoothed+2SDI

Fig. 14.1 Inferred risk premium for S&P (USA)

Mar- Jul- NOV- Mar- Jul- NOV- Mar- Jul- NOV- Mar- Jul- NOV- Mar- Jul- NOV98 99 99 99 96 97 97 97 98 98 95 95 95 96 96

/ -~ x ~ o-s- -t Smoothed . - - - Smoothed-2SD Fig. 14.2 Inferred risk premium for A01 (Australia)

Smoothed+2SD

/

14.6 Estimation Results

23 1

Table 14.3 Filtered mean and standard deviations (S.D.) for S&P risk premium Month

E

m

Model

Model-2S.D.

Model+2S.D.

232

14 Equity Risk Premia from Derivative Prices

Table 14.3 Continued.

----

Month

Average S.D. Corr.

Model

0.2071

Model-2S.D. ----------

Model+2S.D.

0.2744 0.1541 0.5488

Corr indicates correlation between model risk premium and the ex-post risk pre-

14.6 Estimation Results

233

Table 14.4 Filtered mean and standard deviations (S.D.) for A01 risk premium Month Ex-= -- Model Model-2S.D. Model+2S.D.

--

234

14 Equity Risk Premia from Derivative Prices

Table 14.4 Continued.

-

Month

Average S.D. Corr.

Ex-post

0.0943

Model

Model-2S.D.

Model+2S.D.

0.1412 0.2053 0.7391

Corr indicates correlation between model risk premium and the ex-post risk premium.

14.7 Summary and Conclusions

235

The result of the procedure of stepping forward and then stepping backward through the filter updating equations yield us estimates of the conditional mean X,,, and variance P,, of the distribution of the market price of risk A. These are turned into estimates of the conditional mean and variance of the equity risk premium at each k by appropriately scaling with the implied volatility 6,. In Fig. 14.1 (for the S&P 500) and Fig. 14.2 (for the SFE) we plot the estimates of the conditional mean of the equity risk premium together with a two standard deviations band. The actual computed vales are given in Tables 14.3 and 14.4. For comparison purposes we have also calculated the ex-post equity risk premium. This has been calculated simply by subtracting from monthly returns, the proxy for the risk free interest rate. For the S&P 500 the ex-post estimates remain within the two standard deviations band about 60% of the time, furthermore most movements out of the band are in the downward direction. So compared to the estimates of the equity risk premium implied by index futures options prices the ex-post estimates tend to be underestimates. The two standard deviation band of the SFE is wider than that for the S&P 500, and the ex-post estimates remain within the band about 84 % of the time. This could indicate a greater degree of uncertainty about the equity risk premium in the smaller Australian market.

14.7 Summary and Conclusions In this chapter we have expressed the no-riskless arbitrage relationship between the value of the stock market index, the prices of futures on the index and the prices of options on the futures as a system of stochastic differential equations under the historical probability measure, rather than the risk neutral measure used for derivative pricing. As a consequence the stochastic differential equation system involves the market price of risk for the stochastic factor driving the index. This market price of risk is an unobserved quantity and we posit for its dynamics a simple mean-reverting process. We view the resulting stochastic dynamic system in the statespace framework with the changes in index value, futures prices and option prices as the observed components and the market price of risk as the unobserved component. In order to cater for time varying (and possibly stochastic) volatility we replace the volatility of the index by the implied volatility calculated by use of Black's model. We use Kalman filtering methodology to estimate the parameters of this system and use these to es-

236

14 Equity Risk Premia from Derivative Prices

timate the time varying conditional normal distribution of the equity risk premium implied by futures options prices. The method has been applied to daily data on the Australian All Ordinaries index and options on the SPI futures and the S&P 500 and index futures options for the period 1995-1999. Estimations were performed at monthly frequency. As well as applying the usual t-test to determine significance of the parameter estimates a range of tests were conducted to determine the adequacy of the model. It was found that parameter estimates are significant and the model fit is quite good based on a range of goodness-of-fit tests. The estimates of the conditional mean and standard deviation of the distribution of the equity risk premium seem reasonable, when compared with point estimates computed simply from ex-post returns. For the S&P 500 the filtered estimates yield a much tighter band than the expost estimates. Overall we conclude that the approach of using filtering methodology to infer risk premia from derivative prices is a viable one and is worthy of further research effort. One advantage as we have discussed in section 3 is that it gives a forward-looking measure of the risk premium. Also it gives a time varying distribution of the equity risk premium as opposed to the point estimates of the ex-post calculation. A number of avenues for future research suggest themselves. First, a careful comparison of the equity risk premium computed by the methods of this chapter with that calculated using the traditional method based on ex-post returns should be carried out. Second, the technique could be extended to options on heavily traded stocks and risk premia for individual stocks could be calculated. These could be used to determine the beta for the stock implied by the option prices. These in turn could be used as the basis of portfolio strategies and the results could be compared with use of the beta calculated by traditional regression based methods.

References Bekaert G, Harvey CR (1995) Time-varying world market integration. Journal of Finance, 50: 403-444 Black F (1976) The pricing of commodity contracts. Journal of Financial Economics, 3: 167-179 Bollerslev T, Engle RF, Wooldridge JM (1988) A capital asset pricing model with time varying covariances. Journal of Political Economy, 96: 116-131 Bos T, Newbold P (1984) An empirical investigation of the possibility of stochastic systematic risk in the market model. Journal of Business, 57: 35-41

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Chan KC, Karolyi GA, Stulz RM (1992) Global financial markets and the risk premium on U.S. equity. Journal of Financial Economics, 32: 137-167 Evans MD (1994) Expected returns, time varying risk, and risk premia. Journal of Finance, 49: 655-679 Ferson WE, Harvey CR (1991) The variation of economic risk premiums. Journal of Political Economy, 99: 385-415 Giannopoulos K (1995) Estimating the time varying component of international stock market risk. European Journal of Finance, 1: 129-164 Harvey AC (1989) Forecasting structural time series models and the Kalman filter. Cambridge University Press, Cambridge, New York Jazwinski AH (1970) Stochastic processes and filtering theory. Academic Press, New York, London Lipster RS, Shiryaev AN (2000) Statistics of random processes 11. Springer Ver1% Lo AW (1988) Maximum likelihood estimation of generalized Ito processes with discretely sampled data. Econometric Theory, 4: 23 1-247 Wells C (1996) The Kalman filter in finance. Kluwer Academic Publishers, Boston

Index

ACF(autocorre1ation function), 41 ADF(augmented Dickey-fuller) test, 47-48 AIC(Akaike information criterion), 48,56 AR(autoregressive),43, 86 AR(l), 43 AR(P), 44,86 ARMA(autoregressivemoving average), 86 ARMA(I>,q),87 ARCH(autoregressiveconditional heteroskedasticity), 26, 68-7 1,111 asset price basics, 127-129 callable bond, 152 causality-in-meantest, 75-76 causality-in-variancetest, 76-77 CCF(cross correlation function), 74 classical regression, 83-86 coincident indicator, 99- 101 cointegration, 49-50, 59-61 constrained optimization, 27-28 continuous time state space framework, 220-222 correlogram, 4 1 countable stochastic process, 10 decomposition of earnings, 121- 125 dependent observations, 23-24 DLM(dynamic linear model), 83 Dickey-Fuller test, 46-49

discrete time model of interest rate, 141 discrete time real asset valuation model, 127 dynamic linear models for bubble solutions, 167-172 dynamic linear models for no-bubble solutions, 172-173 EGARCH(exponentia1GARCH), 7 173 EGARCH(1, l), 73 EM(expectation maximization) algorithm, 108-111, 116-118 equity risk premia from derivative prices, 2 15 error correction, 59 evolution of commodity prices, 125126 forward FX market and the risk premium, 193 forward recursion for lattice and elementary price, 145 Friedman's plucking model of business fluctuations, 118- 121 function of random variable, 7-8 GARCH(genera1ized ARCH), 26-27, 68-71, 113-116 GARCH(1, I), 26-27,70-71, 113-116

240

Index

GARCH-M (GARCH-in-Mean), 38 Granger causality, 57-59 global bubbles in stock markets, 155 global stock market integration, 165167

I@), 46 I(1), 46 information flow between price change and trading volume, 77-8 1 immunization, 149 Johansen test, 61-62 kernel regression, 33-35 kurtosis, 69 lognormal random variable, 9- 10 LA-VAR(1ag-augmented VAR), 6264

MA(moving average), 42 MA(l), 42 MA(q), 43987 Markov chains, 10-14 matching the current term structure, 148-149 mining project, 129 MLE(maximum likelihood estimation), 21-22,22-23 maximum eigenvalue test, 62 Nadaraya-Watson (kernel) estimator, 34-35,37 non-parametric approach, 32-33

normal random variable, 8-9 optimal bandwidth selection, 36 passage time, 14-15 prediction error decomposition, 24-25 random variables, 5-6 recursive least squares, 89-9 1 residual-based cointegration test, 50-5 1

SBIC(Schwarz Bayesian Information Criterion), 48, 57 short rate lattice, 141-145, 149 signal extraction, 95-99 speculative bubbles, 156-158 spurious regression, 5 1 SSM (state-space model), 83, 105 state-space framework, 20 1-203 state-space representation, 9 1-94 state-space representation of a VARMA(2,l) model, 94-95 stationary process, 41-44, 55-57 stochastic regression, 102 stochastic variance, 113-116 term structure of interest rate, 148 TGARCH(thresho1d GARCH), 7 1-73 TGARCH(1, l), 72 trace test, 6 1 unit root, 44-46 unit root in a regression model, 5 1-54

Index

valuing callable bond, 152-153 VAR(vector autoregression), 55, 86 VARMA(vector autoregressive moving average), 86 VECM (vector error correction model), 61 volatility, 68 white noise process, 42 WolffICheung model, 204-205

241

About the Authors

Dr. Ramaprasad Bhar is an Associate Professor in the School of Banking and Finance at The University of New South Wales in Australia.

Dr. Shigeyuki Hamori is a Professor in the Graduate School of Economics at Kobe University in Japan.