Investment under Uncertainty

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Investment under Uncertainty I ,

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Avinash K. Dixit and Robert S. Pindyck

Pri neeton U II hers it)' Press Princeton, New Jersey

Contents

x

4 5

Explaining Aggregate Investment Behavior Guide to the Literature

419

Preface

425

References

429

Symbol Glossary

...45

Author Index

......9

Subject Index

...55

THIS BOOK provides a systematic treatment of a new theoretical approach to capital investment decisions of firms, stressing the irreversibility of most investment decisions and the ongoing uncertainty of the economic environment in which those decisions arc made. This new approach recognizes the option value of waiting for better (but never complete) information. It exploits an analogy with the theory of options in financial markets, which permits a much richer dynamic framework than was possible with the traditional theory of investment. The new view of investment opportunities as options is the product of over a decade of research by many economists, and is still an active topic in journal articles. It has led to some dramatic departures from the orthodox theory. It has shown that the traditional "net present value" rule, which is taught to virtually every business school student and student of economics, can give very" wrong answers. The reason is that this rule ignores irreversibility and the option of delaying an investment. For the same reason, the new theory also contradicts the orthodox textbook view of production and supply going back to Marshall, according to which firms enter or expand when the price exceeds long-run average cost, and exit or contract when price falls below average variable cost. Policy prescriptions based on the traditional theory, for example, the use of interest rate cuts to stimulate investment and antitrust policies based on price-cost margins, are also called into question. In this book we have tried to present the new theory in a clear and systematic way, and to consolidate, synthesize, and in some places extend the various strands of this growing body of research. While there is a large and burgeoning literature of journal articles, including a survey article by XI

6

2

Introduction

The Option Approach

The net present value rule, however, is based on some implicit assumptions that are often overlooked. Most important, it assumes that either the investment is reversible, that is, it can somehow be undone and the expenditures recovered should market conditions turn out to be worse than anticipated, or, if the investment is irreversible, it is a now or never proposition, that is, if the firm does not undertake the investment now, it will not be able to in the future. Although some investments meet these conditions, most do not. Irreversibility and the possibility of delay are very important characteristics of most investments in reality. As a rapidly growing literature has shown, the ability to delay an irreversible investment expenditure can profoundly affect the decision to invest. It also undermines the simple net present value rule, and hence the theoretical foundation of standard neoclassical investment models. The reason is that a firm with an opportunity to invest is holding an "option" analogous to a financial call option-it has the right but not the obligation to buy an asset at some future time of its choosing. When a firm makes an irreversible investment expenditure, it exercises, or "kills." its option to invest. It gives up the possibility of waiting for new information to arrive that might affect the desirability or timing of the expenditure; it cannot disinvest should market conditions change adversely. This lost option value is an opportunity cost that must be included as part of the cost of the investment. As a result, the NPV rule "invest when the value of a unit of capital is at least as large as its purchase and installation cost" must be modified. The value of the unit must exceed the purchase and installation cost, by an amount equal to the value of keeping the investment option alive. Recent studies have shown that this opportunity cost of investing can be large, and investment rules that ignore it can be grossly in error. Also, this opportunity cost is highly sensitive to uncertainty over the future value of the project, so that changing economic conditions that affect the perceived riskiness of future cash flows can have a large impact on investment spending, larger than, say, a change in interest rates. This may help to explain why neoclassical investment theory has so far failed to provide good empirical models of investment behavior, and has led to overly optimistic forecasts of effectiveness of interest' rate and tax policies in stimulating investment. The option insight also helps explain why the actual investment behavior uf firms differs from the received wisdom taught III business schools. Firms invest in projects that are expected to yield a return in excess of a required,

7

A NCl\' ViclI' of Investment

or "hurdle.' rate. Observers of business practice lind that such hurdle rates arc typically three or four times the cost of capital.' In other words. firms do not invest until price rises substantially above long-run average cost. On the downside, firms stay in business for lengthy periods while absorbing operating losses. and price can fall substantially below average variable cost without inducing disinvestment or exit. This also seems to conflict with standard theory, but as we will sec. it can be explained once irreversibility and option value arc accounted for. Of course, one can always redefine NPY by subtracting from the conventional calculation the opportunity cost of exercising the option to invest. and then sav that the rule "invest if NPY is positive" holds once this correction has been made. However, to do so is to accept our criticism. To highlight the importance of option values, in this book we prefer to keep them separate from the conventional NPY. If others prefer to continue to use "positive NPY" terminology, that is fine as long as they arc careful to include all relevant option values in their definition of NPY. Readers who prefer that usage can readily translate our statements into that language, In this book we develop the basic theory of irreversible investment under uncertainty. emphasizing the option-like characteristics of investment opportunities. Weshow how optimal investment rules can be obtained from methods that have been developed for pricing options in financial markets. We also develop an equivalent approach based on the mathematical theory of optimal sequential decisions under uncertainty-dynamic programming. We illustrate the optimal investment decisions of firms in a variety of situations-new entry, determination of the initial scale of the firm and future costly changes of scale, choice between different forms of investment that offer different degrees of flexibility to meet future conditions, completion of successive stages of a complex multistage project, temporary shutdown and restart. permanent exit, and so forth. We also analyze how the actions of such firms are aggregated to determine the dynamic equilibrium of an industry. Tostress the analogy with options on financial assets. the opportunities to acquire real assets arc sometimes called "real options." Therefore this book could be titled "The Real Options Approach to Investment."

'Summers (19H7. p. 31111) found hurdle rates ranging from X to 311 percent. with a median of 15 and a mean of 17 percent. The cost of riskless capital was much lower; allowing for the deductibility of interest expenses, the nominal interest rate was -! percent, and the real rate close tn '7lF"rn

c...,.,. '..Ien nprt(U17~1t.: •• t ~ll (Il)QO

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hi' The hllnllt' rate nnnroortate for investments with

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not by enough 't:,

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the numbers used by

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Chapter

3

Stochastic Processes and Ito's Lemma

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THIS CHAPTER and the next provide the mathematical tools-stochastic calculus, dynamic programming, and contingent claims analysis-that will be used throughout the rest of this book. With these tools, we can study investment decisions using a continuous-time approach, which is both intuitively appealing and quite powerful. In addition, the concepts and techniques that we introduce here arc becoming widely used in a number of areas of economics and finance, and so arc worth learning even apart from their application to investment problems. This chapter begins with a discussion of stochastic processes. We will begin with simple discrete-time processes, and then turn to the Wiener process (or Brownian motion), an important continuous-time process that is a fundamental building block for many of the models that we will develop in this book. We will explain the meaning and properties of the Wiener process, and show how it can be derived as the continuous limit of a discrete-time random walk. We will then see how the Wiener process can be generalized to a broad class of continuous-time stochastic processes, called Ito processes. Ito processes can be used to represent the dynamics of the value of a project, output prices, input costs, and other variables that evolve stochastically over time and that affect the decision to invest. As we will see, these processes do not have a time derivative in the conventional sense, and as a result, cannot always be manipulated using the ordinary rules of calculus. To work with these processes, we must make usc

59

60

Mathematical Background

of Ito's Lemma. This lemma, sometimes called the Fundamental Theorem of stochastic calculus, is an important result that will allow us to differentiate and integrate functions of stochastic processes. We will provide a heuristic derivation of Ito's Lemma and then, through a variety of examples. show how it can be used to perform simple operations on functions of Wiener processes. We will also show how it can be used to derive and solve stochastic differential equations. Next, we will introduce jump processes-processes that make infrequent but discrete jumps, rather than fluctuate continuously-and show how they can be analyzed using a version of Ito's Lemma. Finally. in the Appendix to this chapter we introduce the Kolmogorov equations, which describe the dynamics of the probability density function for a stochastic process, and show how they can be applied.

1 Stochastic Processes A stochastic process is a variable that evolves over time in a way that is at least in part random. The temperature in downtown Boston is an example; its variation through time is partly deterministic (rising during the day and falling at night, and rising towards summer and falling towards winter), and partly random and unpredictable.' The price of IBM stock is another example; it fluctuates randomly, but over the long haul has had a positive expected rate of growth that compensated investors for risk in holding the stock. Somewhat more formally, a stochastic process is defined by a probability law for the evolution XI of a variable X over time t. Thus. for given times t, < t: < t3, etc., we are given, or can calculate, the probability that the corresponding values X" X2, X3, etc .. lie in some specified range. for example

Stochastic Processes and Ito's Lemma ,',

The temperature in Boston and the price of IBM stock are processes that differ in an important respect. The temperature in Boston is a stationary process. This means, roughly, that the statistical properties of this variable are constant over long periods of time.' For example, although the expected temperature tomorrow may depend in part on today's temperature. the expectation and variance of the temperature on January I of next year is largely independent of today's temperature, and is equal to the expectation and variance of the temperature on January 1 two years from now. three years from now, etc. The price of IBM stock, on the other hand, is a nonstationary process. The expected value of this price can grow without bound. and. as we will soon see, the variance of price T years from now increases with T. The temperature in Boston and the price of IBM stock are both continuous-time stochastic processes, in the sense that the time index t is a continuous variable. (Even though we might only measure the temperature or stock price at particular points in time. these variables vary continuously through time.) Although we will work mostly with continuous-time processes in this book, it is easiest to begin with some examples of discrete-time processes, that is, variables whose values can change only at discrete points in time. Similarly, the set of all logically conceivable values for XI (often called the states) can be continuous or discrete. Our definition above is general enough to allow all these possibilities. One of the simplest examples of a stochastic process is the discrete-time discrete-slate random walk. Here, XI is a random variable that begins at a known value Xo, and at times t = 1.2,3, ..., takes a jump of size I either up or down. each with probability ]: Since the jumps arc independent of each other, we can describe the dynamics of XI with the following equation: X,

where When time t, arrives and we observe the actual value XI, we can condition the probability of future events on this information."

'One might argue that the randomness is a reflection of the limitations of meteorology. and that in principle it could be eliminated if we could build sufficiently complete and accurate meteorological models. Perhaps. but from an operational point of view. next week's temperature is indeed a random variable. 2 In this book we will not attempt any detailed or rigorous treatment of stochastic processes, offering instead the minimal explanations and intuitions that suffice for our applications. For detailed and general treatments. see Cox and Miller (1\165). Feller (I \171). and Karlin and Taylor ( (975).

61

EI

= XI_I + EI•

( I)

is a random variable with probability distribution probrs,

=

I)

=

probrs,

=

-1)

=

!

(t =

I. 2.... ).

We call XI a discrete-state process because it can only take on discrete values. For example, set Xo = O. Then for odd values of t, possible values -I, I, t), and for even values of t, possible values of of x, are (-t XI are (-t, , -2. 0, 2 t). The probability distribution for XI is found

'This ignores the very long-run possibilities of global warming or cooling.

Mathematical Background

from the binomial distribution. For I steps, the probability that there are downward jumps and I - II upward jumps is

Therefore, the probability that

XI

probi.r,. --

will take on the value I - 7_11) --

(I) II

7-1 .

I -

211 at time

I

II

is

Stochastic Processes and Ito \ Lcnuna

for XI+I depends only on X" and not additionally on what happened before time I. For example. in the ease of the simple random walk given hI' equation (1), if XI = 6. then X'+I can equalS or 7. each with prohabilitv ~. The values of XI_I, XI-c. etc .. are irrelevant once we know XI' The Marko~ property is important because it can greatly simplify the analysis of a stochastic process, We will see this shortly as we turn to continuous-time processes.

(2)

2

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We will use this probability distribution in the next section when we derive the Wiener process as the continuous limit of the discrete-time random walk. At this point, however, note that the range of possible values that XI can take on increases with I, as does the variance of X,, Hence XI is a nonstutionary process. Because the probability of an upward or downward jump is ~, at time t = D the expected value of XI is zero for all t . (Likewise, at time t, the expected value of XT for T > t is XI') One way to generalize this process is by changing the probabilities for an upward or downward jump. Let p be the probability of an upward jump and £/ = (I - p) the probability of a downward jump, with P > II. Now we have a random walk with driji; at time t = D, the expected value of XI for t > Dis greater than zero, and is increasing with t . Another way to generalize this process is to let the size of the jump at each time t be a continuous random variable. For example, we might let the size of each jump be normally distributed with mean zero and standard deviation a . Then, we refer to X, as a discrete-lime continuous-state stochastic process. Another example of a discrete-time continuous-state stochastic process is the.first-order autoregressive process, abbreviated as AR( I). It is given by the equation (3)

where 8 and p are constants, with -I < p < 1, and ~I is a normally distributed random variable with zero mean. This process is stationary, and XI has the long-run expected value 8jn - p), irrespective of its current value. [This long-run expected value is found by setting XI = XI_I = x in equation (3) and solving for x.] The AR(l) process is also referred to as a mean-reverting process, because x, tends to revert back to this long-run expected value. We will examine a continuous-time version of this process later in this chapter. Both the random walk (with discrete or continuous states, and with drift or without) and the AR( I) process satisfy the Markov property, and are therefore called Markov processes. This property is that the probability distribution

The Wiener Process

A Wiener process-also called a Brownian 1l1Olioll-is a continuous-lime stochastic process with three important properties..) First. it is a Markov process. As explained above, this means that the probability distribution for all future values of the process depends only on its current value. and is unaffected by past values of the process or by any other current information. As a result. the current value of the process is all one needs to make a hest forecast of its future value. Second. the Wiener process has independent iIlCH'IIIt'lIls. This means that the probability distribution for the change in the process over any time interval is independent of any other (nonoverlapping) time interval. Third. changes in the process over any finite interval of time arc uonnallv distributed, with a variance that increases linearly with the time interval. . The Markov property is particularly important. Again. it implies that onlv current information is useful for forecasting the future path of the process. Stock prices are often modelled as Markov processes. on the grounds that public information is quickly incorporated in the current price of the stock. so that the past pattern of prices has no forecasting value. (This is called the weak form of market efficiency. If it did not hold. investors could in principle "heal the market" through technical analysis. that is, by using the past pattern of ~rices to forecast the luturc.) The fact that a Wiener process has independent Increments means that we can think of it as a continuous-time version of a random walk. a point that we will return to below. The three conditions discussed above-the Markov property. independent increments, and changes that are normally distributed-may seem quite restrictive, and might suggest that there arc very few real-world variables

'In IX27. the botanist Robert Brown first observed and described the motion of small particles suspended in a liquid. resulting from the apparent successive and random impactx of neighboring particles; hence the term Brownian motion. In 1l)(L'i, Albert Einstein prop"s",1 'I mathematical theory of Brownian motion, which was further developed and maul: more riuorou-, by Norbert Wiener in IlJ23. '

Mathematical Background

64

that can be realistically modelled with Wiener processes. For example, while it probably seems reasonable that stock prices satisfy the Markov property and have independent increments, it is not reasonable to assume that price changes are normally distributed; after all, we know that the price of a stock can never fall below zero. It is more reasonable to assume that changes in stock prices are lognorma/ly distributed, that is, that changes in the logarithm of the price are normally distributed.' But this just means modelling the logarithm of price as a Wiener process, rather than the price itself. As we will see, through the use of suitable transformations, the Wiener process can be used as a building block to model an extremely broad range of variables that vary continuously (or almost continuously) and stochastically through time. It is useful to restate the properties of a Wiener process somewhat more formally. If z(t) is a Wiener process, then any change in z, t:.z, corresponding to a time interval /:!"t, satisfies the following conditions:

1. The relationship between

/:!,,::

and S: is given by /:!"z

== ErJiJ,

where Er is a normally distributed random variable with a mean of zero and a standard deviation of 1. 2. The random variable Er is serially uncorrelated, that is, E[EtE.>] == 0 for t i- s. Thus the values of /:!"Z for any two different intervals of time are independent. [Thus z(t) follows a Markov process with independent increments.]

Stochastic Processes and Ito's Lemma

We will make considerable use of this property later. Also. note that the Wiener process is nonstationary. Over the long run its variance will go to infinity. By letting /:!"t become infinitesimally small, we can represent the increment of a Wiener process, d z, in continuous time as dz

Let us examine what these two conditions imply for the change in z over some finite interval of time T. We can break this interval up into n units of length /:!"t each, with n == T / /:!"t. Then the change in z over this interval is given by n

z(s

+ T)

- z(s) ==

L

Ej

v'IS:i.

(4)

;=1

The E;'S are independent of each other. Therefore we can apply the Central Limit Theorem to their sum, and say that the change z(s+ T) -z(s) is normally distributed with mean zero and variance n /:!"t == T. ihis last point, which follows from the fact that /:!"z depends on ~ and not on /:!"t, is particularly important; the variance ofthe change in a Wiener process grows linearly with the time horizon.

'We always use natural logarithms. that is. those with base e.

== ErJJi.

(5)

has zero mean and unit standard deviation, E(dz) == O. and V[d.:] == == dt. Note, however, that a Wiener process has no time ~erivative ,. in a conventional sense; /:!"z//:!"t == Et (/:!"t)-1/2. which becomes infi itc as L\/ approaches zero. \ At times we may want to work with two or more Wiener processes, and we will be interested in their covariances, Suppose that ::1 (t) and ':2 (t) are Wiener \ \ processes. Then we can write E(d': l d:: 2) == PI2 dt , where PI2 is the coefficient of correlation between the two processes. Because a Wiener process has a variance and standard deviation per unit of time equal to I (E[ (d.:)21/dt == I), PI2 is also the covariance per unit of time for the two processes." Since

Er

E[(dz)2]

2.A

Brownian Motion with Drift

We mentioned earlier that the Wiener process can easily be generalized into more complex processes. The simplest generalization of equation (5) is the Brownian motion with drift: dx ==

I.

65

0:

dt + (f dz,

(6)

where dz is the increment of a Wiener process as defined above. In equation (6), ex is called the drift parameter, and a the variance parameter. Note that over any time interval Ar, the change in x, denoted by /:!"x, is normally distributed, and has expected value E(/:!"x) == ex t:./ and variance V(/:!"x) == (f2/:!"t. Figure 3.1 shows three sample paths of equation (6), with trend ex == 0.2 per year, and standard deviation a == 1.0 per year. Although the graph is shown in annual terms (over the time period 1950 to 2000), each sample path was generated by taking a time interval, /:!"/,of one month, and then calculating a trajectory for x (L) using the equation X, ==Xt-I

+O.01667+0.2887E r •

"Recall that if X and Yare random variables. their coefficient of correlation is Cov( XYl/(I1.r 11,.). In this case 11.r = l1y = I.

(7)

fl.\1'

i

Mathematical Background

\

92

\

stochastically, and (54) is a natural representation.l ' Although x(t) will fluctuate stochastically, it is of interest to study the probability density for x in

1

steady-state equilibrium. In steady-state equilibrium,

I, but as McDonald and Siegel demonstrated, this is incorrect. Because future values of V are unknown, there is an opportunity cost to investing today. Hence the optimal investment rule is to invest when V is at least as large as a critical value V* that exceeds I. As we will see, for reasonable parameter values, this critical value may be two or three times as large as I. Hence the simple NPY rule is not just wrong; it is often very wrong. After describing the basic model in more detail, we will show how the optimal investment rule (that is, the critical value V*) can be found by dynamic programming. An issue that arises, however, is the choice of discount rate. If capital markets are "complete" (in a sense that will be made clear), the investment problem can be viewed as a problem in option pricing, and solved using the techniques of contingent claims analysis. We will re-solve the optimal investment problem in this way, and then examine the characteristics of the firm's option to invest and its dependence on key parameters. Finally, we will extend the model by considering alternative stochastic processes for the value of the project, V. In particular, we will find and characterize the optimal investment rules that apply when V follows a mean-reverting process, and then when it follows a mixed Brownian motion/Poisson jump process.

;

.'I

1 The Basic Model Our starting point is a model first developed by McDonald and Siegel (1986). They considered the following problem: At what point is it optimal to pay a sunk cost I in return for a project whose value is V, given that V evolves according to the following geometric Brownian motion: dV =

IX

V dt +'J" V dz,

(1 )

where dz is the increment of a Wiener process. Equation (1) implies that the current value of the project is known, but future values are lognormally distributed with a variance that grows linearly with the time horizon; the exact formulas are in Section 3(a) of Chapter 3. Thus although information arrives over time (the firm observes V changing), the future value of the project is always uncertain. Equation (1) is clearly an abstraction from most real projects. For example, suppose the project is a widget factory with some capacity. If variable

Investment Opportunities and Investment Timing

137

costs are positive and managers have the option to shut down the factory temporarily when the price of output is below variable cost, and/or the option to abandon the project completely, V will not follow a geometric Brownian motion even if the price of widgets does. (We will develop models in which the output price follows a geometric Brownian motion and the project can be temporarily shut down and/or abandoned in Chapters 6 and 7.) If variable cost is positive and managers do not have the option to shut down (perhaps because of regulatory constraints), V can become negative, which is again in conflict with the assumption of lognormality. In addition, one might believe that a competitive product market will prevent the price from wandering too far from long-run industry-wide marginal cost. or that stochastic changes in price are likely to be infrequent but large. so that /1" should follow a meanreverting or jump process. For the time being we ignore these possibilities in order to provide the simplest introduction to the basic ideas and techniques. We allow exogenously specified mean reversion in Section 5(a) of this chapter, and consider industry equilibrium in Chapters 8 and 9. Note that the firm's investment opportunity is equivalent to a perpetual call option-the right but not the obligation to buy a share of stock at a prespecificd price. Therefore the decision to invest is equivalent to deciding when to exercise such an option. Thus. the investment decision can be viewed as a problem of option valuation (as we saw in the simple examples presented in Chapter 2).1 Alternatively, it can be viewed as a problem in dynamic programming. We will derive the optimal investment rule in two ways, first using dynamic programming, and then using option pricing (contingent claims) methods. This will allow us to compare these two approaches and the assumptions that each requires. We will then examine the characteristics of the solution. In what follows. we will denote the value of the investment opportunity (that is, the value of the option to invest) by F( n. We want a rule that maximizes this value. Since the payoff from investing at time t is VI - I, we want to maximize its expected present value:

(2) where E denotes the expectation, T is the (unknown) future time that the investment is made, p is a discount rate, and the maximization is subject to

'The investment opportunity is analogous to a perpetual call option on a dividend-paying stock. (The payout stream from the completed project is equivalent to the dividend on the stock.) A solution to this option valuation and exercise problem was first found by Samuelson (I