John H. Cochrane

June 12, 2000

1

Acknowledgments

This book owes an enormous intellectual debt to Lars Hansen and Gene Fama. Most of the

ideas in the book developed from long discussions with each of them, and trying to make

sense of what each was saying in the language of the other. I am also grateful to all my col-

leagues in Finance and Economics at the University of Chicago, and to George Constantinides

especially, for many discussions about the ideas in this book. I thank George Constantinides,

Andrea Eisfeldt, Gene Fama, Wayne Ferson, Owen Lamont, Anthony Lynch, Dan Nelson,

Alberto Pozzolo, Michael Roberts, Juha Seppala, Mike Stutzer, Pietro Veronesi, an anony-

mous reviewer, and several generations of Ph.D. students at the University of Chicago for

many useful comments. I thank the NSF and the Graduate School of Business for research

support.

Additional material and both substantive and typographical corrections will be maintained

at

http://www-gsb.uchicago.edu/fac/john.cochrane/research/papers

Comments and suggestions are most welcome This book draft is copyright ° John H.

c

Cochrane 1997, 1998, 1999, 2000

John H. Cochrane

Graduate School of Business

University of Chicago

1101 E. 58th St.

Chicago IL 60637

773 702 3059

john.cochrane@gsb.uchicago.edu

June 12, 2000

2

Contents

Acknowledgments 2

Preface 8

Part I. Asset pricing theory 12

1 Consumption-based model and overview 13

1.1 Basic pricing equation 14

1.2 Marginal rate of substitution/stochastic discount factor 16

1.3 Prices, payoffs and notation 17

1.4 Classic issues in ¬nance 20

1.5 Discount factors in continuous time 33

1.6 Problems 38

2 Applying the basic model 41

2.1 Assumptions and applicability 41

2.2 General Equilibrium 43

2.3 Consumption-based model in practice 47

2.4 Alternative asset pricing models: Overview 49

2.5 Problems 51

3 Contingent Claims Markets 54

3.1 Contingent claims 54

3.2 Risk neutral probabilities 55

3.3 Investors again 57

3.4 Risk sharing 59

3.5 State diagram and price function 60

4 The discount factor 64

4.1 Law of one price and existence of a discount factor 64

4.2 No-Arbitrage and positive discount factors 69

3

4.3 An alternative formula, and x— in continuous time 74

4.4 Problems 76

5 Mean-variance frontier and beta representations 77

5.1 Expected return - Beta representations 77

5.2 Mean-variance frontier: Intuition and Lagrangian characterization 80

5.3 An orthogonal characterization of the mean-variance frontier 83

5.4 Spanning the mean-variance frontier 88

5.5 A compilation of properties of R— , Re— and x— 89

5.6 Mean-variance frontiers for m: the Hansen-Jagannathan bounds 92

5.7 Problems 97

6 Relation between discount factors, betas, and mean-variance frontiers 98

6.1 From discount factors to beta representations 98

6.2 From mean-variance frontier to a discount factor and beta representation 101

6.3 Factor models and discount factors 104

6.4 Discount factors and beta models to mean - variance frontier 108

6.5 Three riskfree rate analogues 109

6.6 Mean-variance special cases with no riskfree rate 115

6.7 Problems 118

7 Implications of existence and equivalence theorems 120

8 Conditioning information 128

8.1 Scaled payoffs 129

8.2 Suf¬ciency of adding scaled returns 131

8.3 Conditional and unconditional models 133

8.4 Scaled factors: a partial solution 140

8.5 Summary 141

8.6 Problems 142

9 Factor pricing models 143

9.1 Capital Asset Pricing Model (CAPM) 145

4

9.2 Intertemporal Capital Asset Pricing Model (ICAPM) 156

9.3 Comments on the CAPM and ICAPM 158

9.4 Arbitrage Pricing Theory (APT) 162

9.5 APT vs. ICAPM 171

9.6 Problems 172

Part II. Estimating and evaluating asset pricing models 174

10 GMM in explicit discount factor models 177

10.1 The Recipe 177

10.2 Interpreting the GMM procedure 180

10.3 Applying GMM 184

11 GMM: general formulas and applications 188

11.1 General GMM formulas 188

11.2 Testing moments 192

11.3 Standard errors of anything by delta method 193

11.4 Using GMM for regressions 194

11.5 Prespeci¬ed weighting matrices and moment conditions 196

11.6 Estimating on one group of moments, testing on another. 205

11.7 Estimating the spectral density matrix 205

11.8 Problems 212

12 Regression-based tests of linear factor models 214

12.1 Time-series regressions 214

12.2 Cross-sectional regressions 219

12.3 Fama-MacBeth Procedure 228

12.4 Problems 234

13 GMM for linear factor models in discount factor form 235

13.1 GMM on the pricing errors gives a cross-sectional regression 235

13.2 The case of excess returns 237

13.3 Horse Races 239

5

13.4 Testing for characteristics 240

13.5 Testing for priced factors: lambdas or b™s? 241

13.6 Problems 245

14 Maximum likelihood 247

14.1 Maximum likelihood 247

14.2 ML is GMM on the scores 249

14.3 When factors are returns, ML prescribes a time-series regression 251

14.4 When factors are not excess returns, ML prescribes a cross-sectional

regression 255

14.5 Problems 256

15 Time series, cross-section, and GMM/DF tests of linear factor models 258

15.1 Three approaches to the CAPM in size portfolios 259

15.2 Monte Carlo and Bootstrap 265

16 Which method? 271

Part III. Bonds and options 284

17 Option pricing 286

17.1 Background 286

17.2 Black-Scholes formula 293

17.3 Problems 299

18 Option pricing without perfect replication 300

18.1 On the edges of arbitrage 300

18.2 One-period good deal bounds 301

18.3 Multiple periods and continuous time 309

18.4 Extensions, other approaches, and bibliography 317

18.5 Problems 319

19 Term structure of interest rates 320

19.1 De¬nitions and notation 320

6

19.2 Yield curve and expectations hypothesis 325

19.3 Term structure models “ a discrete-time introduction 327

19.4 Continuous time term structure models 332

19.5 Three linear term structure models 337

19.6 Bibliography and comments 348

19.7 Problems 351

Part IV. Empirical survey 352

20 Expected returns in the time-series and cross-section 354

20.1 Time-series predictability 356

20.2 The Cross-section: CAPM and Multifactor Models 396

20.3 Summary and interpretation 409

20.4 Problems 413

21 Equity premium puzzle and consumption-based models 414

21.1 Equity premium puzzles 414

21.2 New models 423

21.3 Bibliography 437

21.4 Problems 440

22 References 442

Part V. Appendix 455

23 Continuous time 456

23.1 Brownian Motion 456

23.2 Diffusion model 457

23.3 Ito™s lemma 460

23.4 Problems 462

7

Preface

Asset pricing theory tries to understand the prices or values of claims to uncertain payments.

A low price implies a high rate of return, so one can also think of the theory as explaining

why some assets pay higher average returns than others.

To value an asset, we have to account for the delay and for the risk of its payments. The

effects of time are not too dif¬cult to work out. However, corrections for risk are much

more important determinants of an many assets™ values. For example, over the last 50 years

U.S. stocks have given a real return of about 9% on average. Of this, only about 1% is due

to interest rates; the remaining 8% is a premium earned for holding risk. Uncertainty, or

corrections for risk make asset pricing interesting and challenging.

Asset pricing theory shares the positive vs. normative tension present in the rest of eco-

nomics. Does it describe the way the world does work or the way the world should work?

We observe the prices or returns of many assets. We can use the theory positively, to try to

understand why prices or returns are what they are. If the world does not obey a model™s pre-

dictions, we can decide that the model needs improvement. However, we can also decide that

the world is wrong, that some assets are “mis-priced” and present trading opportunities for

the shrewd investor. This latter use of asset pricing theory accounts for much of its popular-

ity and practical application. Also, and perhaps most importantly, the prices of many assets

or claims to uncertain cash ¬‚ows are not observed, such as potential public or private invest-

ment projects, new ¬nancial securities, buyout prospects, and complex derivatives. We can

apply the theory to establish what the prices of these claims should be as well; the answers

are important guides to public and private decisions.

Asset pricing theory all stems from one simple concept, derived in the ¬rst page of the

¬rst Chapter of this book: price equals expected discounted payoff. The rest is elaboration,

special cases, and a closet full of tricks that make the central equation useful for one or

another application.

There are two polar approaches to this elaboration. I will call them absolute pricing and

relative pricing. In absolute pricing, we price each asset by reference to its exposure to fun-

damental sources of macroeconomic risk. The consumption-based and general equilibrium

models described below are the purest examples of this approach. The absolute approach is

most common in academic settings, in which we use asset pricing theory positively to give

an economic explanation for why prices are what they are, or in order to predict how prices

might change if policy or economic structure changed.

In relative pricing, we ask a less ambitious question. We ask what we can learn about an

asset™s value given the prices of some other assets. We do not ask where the price of the other

set of assets came from, and we use as little information about fundamental risk factors as

possible. Black-Scholes option pricing is the classic example of this approach. While limited

in scope, this approach offers precision in many applications.

8

Asset pricing problems are solved by judiciously choosing how much absolute and how

much relative pricing one will do, depending on the assets in question and the purpose of the

calculation. Almost no problems are solved by the pure extremes. For example, the CAPM

and its successor factor models are paradigms of the absolute approach. Yet in applications,

they price assets “relative” to the market or other risk factors, without answering what deter-

mines the market or factor risk premia and betas. The latter are treated as free parameters.

On the other end of the spectrum, most practical ¬nancial engineering questions involve as-

sumptions beyond pure lack of arbitrage, assumptions about equilibrium “market prices of

risk.”

The central and un¬nished task of absolute asset pricing is to understand and measure the

sources of aggregate or macroeconomic risk that drive asset prices. Of course, this is also the

central question of macroeconomics, and this is a particularly exciting time for researchers

who want to answer these fundamental questions in macroeconomics and ¬nance. A lot of

empirical work has documented tantalizing stylized facts and links between macroeconomics

and ¬nance. For example, expected returns vary across time and across assets in ways that

are linked to macroeconomic variables, or variables that also forecast macroeconomic events;

a wide class of models suggests that a “recession” or “¬nancial distress” factor lies behind

many asset prices. Yet theory lags behind; we do not yet have a well-described model that

explains these interesting correlations.

In turn, I think that what we are learning about ¬nance must feed back on macroeco-

nomics. To take a simple example, we have learned that the risk premium on stocks “ the

expected stock return less interest rates “ is much larger than the interest rate, and varies a

good deal more than interest rates. This means that attempts to line investment up with inter-

est rates are pretty hopeless “ most variation in the cost of capital comes from the varying risk

premium. Similarly, we have learned that some measure of risk aversion must be quite high,

or people would all borrow like crazy to buy stocks. Most macroeconomics pursues small

deviations about perfect foresight equilibria, but the large equity premium means that volatil-

ity is a ¬rst-order effect, not a second-order effect. Standard macroeconomic models predict

that people really don™t care much about business cycles (Lucas 1987). Asset prices are be-

ginning to reveal that they do “ that they forego substantial return premia to avoid assets that

fall in recessions. This fact ought to tell us something about recessions!

This book advocates a discount factor / generalized method of moments view of asset

pricing theory and associated empirical procedures. I summarize asset pricing by two equa-

tions:

pt = E(mt+1 xt+1 )

mt+1 = f(data, parameters).

where pt = asset price, xt+1 = asset payoff, mt+1 = stochastic discount factor.

9

The major advantage of the discount factor / moment condition approach are its simplicity

and universality. Where once there were three apparently different theories for stocks, bonds,

and options, now we see each as just special cases of the same theory. The common language

also allows us to use insights from each ¬eld of application in other ¬elds.

This approach also allows us to conveniently separate the step of specifying economic

assumptions of the model (second equation) from the step of deciding which kind of empiri-

cal representation to pursue or understand. For a given model “ choice of f (·) “ we will see

how the ¬rst equation can lead to predictions stated in terms of returns, price-dividend ra-

tios, expected return-beta representations, moment conditions, continuous vs. discrete time

implications and so forth. The ability to translate between such representations is also very

helpful in digesting the results of empirical work, which uses a number of apparently distinct

but fundamentally connected representations.

Thinking in terms of discount factors often turns out to be much simpler than thinking in

terms of portfolios. For example, it is easier to insist that there is a positive discount factor

than to check that every possible portfolio that dominates every other portfolio has a larger

price, and the long arguments over the APT stated in terms of portfolios are easy to digest

when stated in terms of discount factors.

The discount factor approach is also associated with a state-space geometry in place of

the usual mean-variance geometry, and this book emphasizes the state-space intuition behind

many classic results.

For these reasons, the discount factor language and the associated state-space geometry

is common in academic research and high-tech practice. It is not yet common in textbooks,

and that is the niche that this book tries to ¬ll.

I also diverge from the usual order of presentation. Most books are structured follow-

ing the history of thought: portfolio theory, mean-variance frontiers, spanning theorems,

CAPM, ICAPM, APT, option pricing, and ¬nally consumption-based model. Contingent

claims are an esoteric extension of option-pricing theory. I go the other way around: con-

tingent claims and the consumption-based model are the basic and simplest models around;

the others are specializations. Just because they were discovered in the opposite order is no

reason to present them that way.

I also try to unify the treatment of empirical methods. A wide variety of methods are pop-

ular, including time-series and cross-sectional regressions, and methods based on generalized

method of moments (GMM) and maximum likelihood. However, in the end all of these ap-

parently different approaches do the same thing: they pick free parameters of the model to

make it ¬t best, which usually means to minimize pricing errors; and they evaluate the model

by examining how big those pricing errors are.

As with the theory, I do not attempt an encyclopedic compilation of empirical procedures.

The literature on econometric methods contains lots of methods and special cases (likelihood

ratio analogues of common Wald tests; cases with and without riskfree assets and when

factors do and don™t span the mean variance frontier, etc.) that are seldom used in practice. I

10

try to focus on the basic ideas and on methods that are actually used in practice.

The accent in this book is on understanding statements of theory, and working with that

theory to applications, rather than rigorous or general proofs. Also, I skip very lightly over

many parts of asset pricing theory that have faded from current applications, although they

occupied large amounts of the attention in the past. Some examples are portfolio separation

theorems, properties of various distributions, or asymptotic APT. While portfolio theory is

still interesting and useful, it is no longer a cornerstone of pricing. Rather than use portfolio

theory to ¬nd a demand curve for assets, which intersected with a supply curve gives prices,

we now go to prices directly. One can then ¬nd optimal portfolios, but it is a side issue for

the asset pricing question.

My presentation is consciously informal. I like to see an idea in its simplest form and

learn to use it before going back and understanding all the foundations of the ideas. I have or-

ganized the book for similarly minded readers. If you are hungry for more formal de¬nitions

and background, keep going, they usually show up later on in the chapter.

Again, my organizing principle is that everything can be traced back to specializations of

the basic pricing equation p = E(mx). Therefore, after reading the ¬rst chapter, one can

pretty much skip around and read topics in as much depth or order as one likes. Each major

subject always starts back at the same pricing equation.

The target audience for this book is economics and ¬nance Ph.D. students, advanced MBA

students or professionals with similar background. I hope the book will also be useful to

fellow researchers and ¬nance professionals, by clarifying, relating and simplifying the set of

tools we have all learned in a hodgepodge manner. I presume some exposure to undergraduate

economics and statistics. A reader should have seen a utility function, a random variable, a

standard error and a time series, should have some basic linear algebra and calculus and

should have solved a maximum problem by setting derivatives to zero. The hurdles in asset

pricing are really conceptual rather than mathematical.

11

PART I

Asset pricing theory

12

Chapter 1. Consumption-based model

and overview

I start by thinking of an investor who thinks about how much to save and consume, and

what portfolio of assets to hold. The most basic pricing equation comes from the ¬rst-order

conditions to that problem, and say that price should be the expected discounted payoff, using

the investor™s marginal utility to discount the payoff. The marginal utility loss of consuming

a little less today and investing the result should equal the marginal utility gain of selling the

investment at some point in the future and eating the proceeds. If the price does not satisfy

this relation, the investor should buy more of the asset.

From this simple idea, I can discuss the classic issues in ¬nance. The interest rate is

related to the average future marginal utility, and hence to the expected path of consumption.

High real interest rates should be associated with an expectation of growing consumption. In

a time of high real interest rates, it makes sense to save, buy bonds, and then consume more

tomorrow.

Most importantly, risk corrections to asset prices should be driven by the covariance of

asset payoffs with consumption or marginal utility. For a given expected payoff of an asset,

an asset that does badly in states like a recession, in which the investor feels poor and is

consuming little, is less desirable than an asset that does badly in states of nature like a boom

when the investor feels wealthy and is consuming a great deal. The former assets will sell for

lower prices; their prices will re¬‚ect a discount for their riskiness, and this riskiness depends

on a co-variance. This is the fundamental point of the whole book.

Of course, the fundamental measure of how you feel is marginal utility; given that assets

must pay off well in some states and poorly in others, you want assets that pay off poorly in

states of low marginal utility, when an extra dollar doesn™t really seem all that important, and

you™d rather that they pay off well in states of high marginal utility, when you™re hungry and

really anxious to have an extra dollar. Most of the book is about how to go from marginal

utility to observable indicators. Consumption is low when marginal utility is high, of course,

so consumption may be a useful indicator. Consumption is also low and marginal utility is

high when the investor™s other assets have done poorly; thus we may expect that prices are

13

CHAPTER 1 CONSUMPTION-BASED MODEL AND OVERVIEW

low for assets that covary positively with a large index such as the market portfolio. This

is the Capital Asset Pricing Model. The rest of the book comes down to useful indicators

for marginal utility, things against which to compute a covariance in order to predict the

risk-adjustment for prices.

1.1 Basic pricing equation

An investor™s ¬rst order conditions give the basic consumption-based model,

·0 ¸

u (ct+1 )

pt = Et β 0 xt+1 .

u (ct )

Our basic objective is to ¬gure out the value of any stream of uncertain cash ¬‚ows. I start

with an apparently simple case, which turns out to capture very general situations.

Let us ¬nd the value at time t of a payoff xt+1 . For example, if one buys a stock today,

the payoff next period is the stock price plus dividend, xt+1 = pt+1 +dt+1 . xt+1 is a random

variable: an investor does not know exactly how much he will get from his investment, but he

can assess the probability of various possible outcomes. Don™t confuse the payoff xt+1 with

the pro¬t or return; xt+1 is the value of the investment at time t + 1, without subtracting or

dividing by the cost of the investment.

We ¬nd the value of this payoff by asking what it is worth to a typical investor. To do this,

we need a convenient mathematical formalism to capture what an investor wants. We model

investors by a utility function de¬ned over current and future values of consumption,

U(ct , ct+1 ) = u(ct ) + βEt [u(ct+1 )] ,

where ct denotes consumption at date t. We will often use a convenient power utility form,

1 1’γ

u(ct ) = c .

1’γ t

The limit as γ ’ 1 is

u(c) = ln(c).

The utility function captures the fundamental desire for more consumption, rather than

posit a desire for intermediate objectives such as means and variance of portfolio returns.

Consumption ct+1 is also random; the investor does not know his wealth tomorrow, and

hence how much he will decide to consume. The period utility function u(·) is increasing,

re¬‚ecting a desire for more consumption, and concave, re¬‚ecting the declining marginal value

of additional consumption. The last bite is never as satisfying as the ¬rst.

14

SECTION 1.1 BASIC PRICING EQUATION

This formalism captures investors™ impatience and their aversion to risk, so we can quan-

titatively correct for the risk and delay of cash ¬‚ows. Discounting the future by β captures

impatience, and β is called the subjective discount factor. The curvature of the utility func-

tion also generates aversion to risk and to intertemporal substitution: The consumer prefers a

consumption stream that is steady over time and across states of nature.

Now, assume that the investor can freely buy or sell as much of the payoff xt+1 as he

wishes, at a price pt . How much will he buy or sell? To ¬nd the answer, denote by e the

original consumption level (if the investor bought none of the asset), and denote by ξ the

amount of the asset he chooses to buy. Then, his problem is,

max u(ct ) + Et βu(ct+1 ) s.t.

{ξ}

ct = et ’ pt ξ

ct+1 = et+1 + xt+1 ξ

Substituting the constraints into the objective, and setting the derivative with respect to ξ

equal to zero, we obtain the ¬rst-order condition for an optimal consumption and portfolio

choice,

pt u0 (ct ) = Et [βu0 (ct+1 )xt+1 ] (1)

or,

·0 ¸

u (ct+1 )

(2)

pt = Et β 0 xt+1 .

u (ct )

The investor buys more or less of the asset until this ¬rst order condition holds.

Equation (1.1) expresses the standard marginal condition for an optimum: pt u0 (ct ) is

the loss in utility if the investor buys another unit of the asset; Et [βu0 (ct+1 )xt+1 ] is the

increase in (discounted, expected) utility he obtains from the extra payoff at t+1. The investor

continues to buy or sell the asset until the marginal loss equals the marginal gain.

Equation (1.2) is the central asset-pricing formula. Given the payoff xt+1 and given the

investor™s consumption choice ct , ct+1 , it tells you what market price pt to expect. Its eco-

nomic content is simply the ¬rst order conditions for optimal consumption and portfolio for-

mation. Most of the theory of asset pricing just consists of specializations and manipulations

of this formula.

Notice that we have stopped short of a complete solution to the model, i.e. an expression

with exogenous items on the right hand side. We relate one endogenous variable, price,

to two other endogenous variables, consumption and payoffs. One can continue to solve

this model and derive the optimal consumption choice ct , ct+1 in terms of the givens of the

model. In the model I have sketched so far, those givens are the income sequence et , et+1 and

a speci¬cation of the full set of assets that the investor may buy and sell. We will in fact study

15

CHAPTER 1 CONSUMPTION-BASED MODEL AND OVERVIEW

such fuller solutions below. However, for many purposes one can stop short of specifying

(possibly wrongly) all this extra structure, and obtain very useful predictions about asset

prices from (1.2), even though consumption is an endogenous variable.

1.2 Marginal rate of substitution/stochastic discount factor

We break up the basic consumption-based pricing equation into

p = E(mx)

u0 (ct+1 )

m=β 0

u (ct )

where mt+1 is the stochastic discount factor.

A convenient way to break up the basic pricing equation (1.2) is to de¬ne the stochastic

discount factor mt+1

u0 (ct+1 )

(3)

mt+1 ≡β 0

u (ct )

Then, the basic pricing formula (1.2) can simply be expressed as

(4)

pt = Et (mt+1 xt+1 ).

When it isn™t necessary to be explicit about time subscripts or the difference between

conditional and unconditional expectation, I™ll suppress the subscripts and just write p =

E(mx). The price always comes at t, the payoff at t + 1, and the expectation is conditional

on time t information.

The term stochastic discount factor refers to the way m generalizes standard discount

factor ideas. If there is no uncertainty, we can express prices via the standard present value

formula

1

(5)

pt = xt+1

Rf

where Rf is the gross risk-free rate. 1/Rf is the discount factor. Since gross interest rates

are typically greater than one, the payoff xt+1 sells “at a discount.” Riskier assets have

lower prices than equivalent risk-free assets, so they are often valued by using risk-adjusted

16

SECTION 1.3 PRICES, PAYOFFS AND NOTATION

discount factors,

1

pi = Et (xi ).

t t+1

Ri

Here, I have added the i superscript to emphasize that each risky asset i must be discounted

by an asset-speci¬c risk-adjusted discount factor 1/Ri .

In this context, equation (1.4) is obviously a generalization, and it says something deep:

one can incorporate all risk-corrections by de¬ning a single stochastic discount factor “ the

same one for each asset “ and putting it inside the expectation. mt+1 is stochastic or random

because it is not known with certainty at time t. As we will see, the correlation between the

random components of m and xi generate asset-speci¬c risk corrections.

mt+1 is also often called the marginal rate of substitution after (1.3). In that equation,

mt+1 is the rate at which the investor is willing to substitute consumption at time t + 1 for

consumption at time t. mt+1 is sometimes also called the pricing kernel. If you know what a

kernel is and express the expectation as an integral, you can see where the name comes from.

It is sometimes called a change of measure or a state-price density for reasons that we will

see below.

For the moment, introducing the discount factor m and breaking the basic pricing equa-

tion (1.2) into (1.3) and (1.4) is just a notational convenience. As we will see, however, it

represents a much deeper and more useful separation. For example, notice that p = E(mx)

would still be valid if we changed the utility function, but we would have a different func-

tion connecting m to data. As we will see, all asset pricing models amount to alternative

models connecting the stochastic discount factor to data, while p = E(mx) is a convenient

accounting identity with almost no content. At the same time, we will study lots of alter-

native expressions of p = E(mx), and we can summarize many empirical approaches to

p = E(mx). By separating our models into these two components, we don™t have to redo all

that elaboration for each asset pricing model.

1.3 Prices, payoffs and notation

The price pt gives rights to a payoff xt+1 . In practice, this notation covers a variety of

cases, including the following:

17

CHAPTER 1 CONSUMPTION-BASED MODEL AND OVERVIEW

Price pt Payoff xt+1

Stock pt pt+1 + dt+1

Return 1 Rt+1

³ ´

pt+1 dt+1

pt

Price-dividend ratio dt+1 + 1

dt dt

Excess return e a b

0 Rt+1 = Rt+1 ’ Rt+1

Managed portfolio zt zt Rt+1

Moment condition E(pt zt ) xt+1 zt

One-period bond pt 1

Risk free rate Rf

1

Option C max(ST ’ K, 0)

The price pt and payoff xt+1 seem like a very restrictive kind of security. In fact, this

notation is quite general and allows us easily to accommodate many different asset pricing

questions. In particular, we can cover stocks, bonds and options and make clear that there is

one theory for all asset pricing.

For stocks, the one period payoff is of course the next price plus dividend, xt+1 = pt+1 +

dt+1 . We frequently divide the payoff xt+1 by the price pt to obtain a gross return

xt+1

Rt+1 ≡

pt

We can think of a return as a payoff with price one. If you pay one dollar today, the return is

how many dollars or units of consumption you get tomorrow. Thus, returns obey

1 = E(mR)

which is by far the most important special case of the basic formula p = E(mx). I use capital

letters to denote gross returns R, which have a numerical value like 1.05. I use lowercase

letters to denote net returns r = R ’ 1 or log (continuously compounded) returns ln(R), both

of which have numerical values like 0.05. One may also quote percent returns 100 — r.

Returns are often used in empirical work because they are typically stationary over time.

(Stationary in the statistical sense; they don™t have trends and you can meaningfully take an

average. “Stationary” does not mean constant.) However, thinking in terms of returns takes

us away from the central task of ¬nding asset prices. Dividing by dividends and creating a

payoff

µ ¶

pt+1 dt+1

xt+1 = 1 +

dt+1 dt

corresponding to a price pt /dt is a way to look at prices but still to examine stationary vari-

ables.

Not everything can be reduced to a return. If you borrow a dollar at the interest rate Rf

and invest it in an asset with return R, you pay no money out-of-pocket today, and get the

18

SECTION 1.4 CLASSIC ISSUES IN FINANCE

payoff R ’ Rf . This is a payoff with a zero price, so you obviously can™t divide payoff by

price to get a return. Zero price does not imply zero payoff. It is a bet in which the chance of

losing exactly balances its chance of winning, so that it is not worth paying extra to take the

bet. It is common to study equity strategies in which one short sells one stock or portfolio and

invests the proceeds in another stock or portfolio, generating an excess return. I denote any

such difference between returns as an excess return, Re . It is also called a zero-cost portfolio

or a self-¬nancing portfolio.

In fact, much asset pricing focuses on excess returns. Our economic understanding of

interest rate variation turns out to have little to do with our understanding of risk premia, so

it is convenient to separate the two exercises by looking at interest rates and excess returns

separately.

We also want to think about the managed portfolios, in which one invests more or less

in an asset according to some signal. The “price” of such a strategy is the amount invested

at time t, say zt , and the payoff is zt Rt+1 . For example a market timing strategy might put

a weight in stocks proportional to the price-dividend ratio, investing less when prices are

higher. We could represent such a strategy as a payoff using zt = a ’ b(pt /dt ).

When we think about conditioning information below, we will think of objects like zt as

instruments. Then we take an unconditional expectation of pt zt = Et (mt+1 xt+1 )zt , yielding

E(pt zt ) = E(mt+1 xt+1 zt ). We can think of this operation as creating a “security” with

payoff xt+1 zt+1 , and “price” E(pt zt ) represented with unconditional expectations.

A one period bond is of course a claim to a unit payoff. Bonds, options, investment

projects are all examples in which it is often more useful to think of prices and payoffs rather

than returns.

Prices and returns can be real (denominated in goods) or nominal (denominated in dol-

lars); p = E(mx) can refer to either case. The only difference is whether we use a real or

nominal discount factor. If prices, returns and payoffs are nominal, we should use a nomi-

nal discount factor. For example, if p and x denote nominal values, then we can create real

prices and payoffs to write

·µ 0 ¶ ¸

pt u (ct+1 ) xt+1

= Et β0

Πt u (ct ) Πt+1

where Π denotes the price level (cpi). Obviously, this is the same as de¬ning a nominal

discount factor by

·µ 0 ¶ ¸

u (ct+1 ) Πt

pt = Et β 0 xt+1

u (ct ) Πt+1

To accommodate all these cases, I will simply use the notation price pt and payoff xt+1 .

These symbols can denote 0, 1, or zt and Rt , rt+1 , or zt Rt+1 respectively, according to the

e

case. Lots of other de¬nitions of p and x are useful as well.

19

CHAPTER 1 CONSUMPTION-BASED MODEL AND OVERVIEW

1.4 Classic issues in ¬nance

I use simple manipulations of the basic pricing equation to introduce classic issues in ¬-

nance: the economics of interest rates, risk adjustments, systematic vs. idiosyncratic risk, ex-

pected return-beta representations, the mean-variance frontier, the slope of the mean-variance

frontier, time-varying expected returns, and present value relations.

A few simple rearrangements and manipulations of the basic pricing equation p = E(mx)

give a lot of intuition and introduce some classic issues in ¬nance, including determinants of

the interest rate, risk corrections, idiosyncratic vs. systematic risk, beta pricing models, and

mean variance frontiers.

1.4.1 Risk free rate

Rf = 1/E(m).

With lognormal consumption growth,

γ2 2

f

rt = δ + γEt ∆ ln ct+1 ’ σt (∆ ln ct+1 )

2

Real interest rates are high when people are impatient (δ), when expected consumption

growth is high (intertemporal substitution), or when risk is low (precautionary saving). A

more curved utility function (γ) or a lower elasticity of intertemporal substitution (1/γ) means

that interest rates are more sensitive to changes in expected consumption growth.

The risk free rate is given by

Rf = 1/E(m). (6)

The risk free rate is known ahead of time, so p = E(mx) becomes 1 = E(mRf ) =

E(m)Rf .

If a risk free security is not traded, we can de¬ne Rf = 1/E(m) as the “shadow” risk-free

rate. (In some models it is called the “zero-beta” rate.) If one introduced a risk free security

with return Rf = 1/E(m), investors would be just indifferent to buying or selling it. I use

Rf to simplify formulas below with this understanding.

To think about the economics behind real interest rates in a simple setup, use power utility

20

SECTION 1.4 CLASSIC ISSUES IN FINANCE

u0 (c) = c’γ . Start by turning off uncertainty, in which case

µ ¶γ

1 ct+1

Rf = .

β ct

We can see three effects right away:

1. Real interest rates are high when people are impatient, when β is low. If everyone wants

to consume now, it takes a high interest rate to convince them to save.

2. Real interest rates are high when consumption growth is high. In times of high interest

rates, it pays investors to consume less now, invest more, and consume more in the

future. Thus, high interest rates lower the level of consumption today, while raising its

growth rate from today to tomorrow.

3. Real interest rates are more sensitive to consumption growth if the power parameter γ is

large. If utility is highly curved, the investor cares more about maintaining a consumption

pro¬le that is smooth over time, and is less willing to rearrange consumption over time

in response to interest rate incentives. Thus it takes a larger interest rate change to induce

him to a given consumption growth.

To understand how interest rates behave when there is some uncertainty, I specify that

consumption growth is lognormally distributed. In this case, the real riskfree rate equation

becomes

γ2 2

f

(7)

rt = δ + γEt ∆ ln ct+1 ’ σ (∆ ln ct+1 )

2t

f

where I have de¬ned the log riskfree rate rt and subjective discount rate δ by

f f

rt = ln Rt ; β = e’δ ,

and ∆ denotes the ¬rst difference operator,

∆ ln ct+1 = ln ct+1 ’ ln ct .

To derive expression (1.7) for the riskfree rate, start with

"µ ¶’γ #

ct+1

f

Rt = 1/Et β .

ct

Using the fact that normal z means

2

1

E (ez ) = eE(z)+ 2 σ (z)

21

CHAPTER 1 CONSUMPTION-BASED MODEL AND OVERVIEW

(you can check this by writing out the integral that de¬nes the expectation), we have

· ¸’1

2

’δ ’γEt (∆ ln ct+1 )+ γ σ2 (∆ ln ct+1 )

f

Rt = e e .

t

2

Then take logarithms. The combination of lognormal distributions and power utility is one of

the basic tricks to getting analytical solutions in this kind of model. Section 1.5 shows how

to get the same result in continuous time.

Looking at (1.7), we see the same results as we had with the deterministic case. Real in-

terest rates are high when impatience δ is high and when consumption growth is high; higher

γ makes interest rates more sensitive to consumption growth. The new σ2 term captures pre-

cautionary savings. When consumption is more volatile, people with this utility function are

more worried about the low consumption states than they are pleased by the high consump-

tion states. Therefore, people want to save more, driving down interest rates.

We can also read the same terms backwards: consumption growth is high when real

interest rates are high, since people save more now and spend it in the future, and consumption

is less sensitive to interest rates as the desire for a smooth consumption stream, captured by

γ, rises. . Section 2.2 below takes up the question of which way we should read this equation

“ as consumption determining interest rates, or as interest rates determining consumption.

For the power utility function, the curvature parameter γ simultaneously controls in-

tertemporal substitution “ aversion to a consumption stream that varies over time, risk aver-

sion “ aversion to a consumption stream that varies across states of nature, and precautionary

savings, which turns out to depend on the third derivative of the utility function. This link is

particular to the power utility function. We will study utility functions below that loosen the

links between these three quantities.

1.4.2 Risk corrections

E(x)

p= + cov(m, x)

Rf

¡ ¢

E(Ri ) ’ Rf = ’Rf cov m, Ri .

Payoffs that are positively correlated with consumption growth have lower prices, to com-

pensate investors for risk. Expected returns are proportional to the covariance of returns with

discount factors.

Using the de¬nition of covariance cov(m, x) = E(mx) ’ E(m)E(x), we can write

22

SECTION 1.4 CLASSIC ISSUES IN FINANCE

equation (1.2) as

(8)

p = E(m)E(x) + cov(m, x).

Substituting the riskfree rate equation (1.6), we obtain

E(x)

(9)

p= + cov(m, x)

Rf

The ¬rst term in (1.9) is the standard discounted present value formula. This is the asset™s

price in a risk-neutral world “ if consumption is constant or if utility is linear. The second

term is a risk adjustment. An asset whose payoff covaries positively with the discount factor

has its price raised and vice-versa.

To understand the risk adjustment, substitute back for m in terms of consumption, to

obtain

E(x) cov [βu0 (ct+1 ), xt+1 ]

(10)

p= +

Rf u0 (ct )

Marginal utility u0 (c) declines as c rises. Thus, an asset™s price is lowered if its payoff co-

varies positively with consumption. Conversely, an asset™s price is raised if it covaries nega-

tively with consumption.

Why? Investors do not like uncertainty about consumption. If you buy an asset whose

payoff covaries positively with consumption, one that pays off well when you are already

feeling wealthy, and pays off badly when you are already feeling poor, that asset will make

your consumption stream more volatile. You will require a low price to induce you to buy

such an asset. If you buy an asset whose payoff covaries negatively with consumption, it

helps to smooth consumption and so is more valuable than its expected payoff might indicate.

Insurance is an extreme example. Insurance pays off exactly when wealth and consumption

would otherwise be low“you get a check when your house burns down. For this reason, you

are happy to hold insurance, even though you expect to lose money”even though the price

of insurance is greater than its expected payoff discounted at the risk free rate.

To emphasize why the covariance of a payoff with the discount factor rather than its

variance determines its riskiness, keep in mind that the investor cares about the volatility of

consumption. He does not care about the volatility of his individual assets or of his portfolio,

if he can keep a steady consumption. Consider what happens to the volatility of consumption

if the investor buys a little more ξ of payoff x:

σ2 (c) becomes σ 2 (c + ξx) = σ 2 (c) + 2ξcov(c, x) + ξ 2 σ2 (x)

For small (marginal) portfolio changes, the covariance between consumption and payoff de-

termines the effect of adding a bit more of each payoff on the volatility of consumption.

We use returns so often that it is worth restating the same intuition for the special case that

23

CHAPTER 1 CONSUMPTION-BASED MODEL AND OVERVIEW

the price is one and the payoff is a return. Start with the basic pricing equation for returns,

1 = E(mRi ).

I denote the return Ri to emphasize that the point of the theory is to distinguish the behavior

of one asset Ri from another Rj .

The asset pricing model says that, although expected returns can vary across time and

assets, expected discounted returns should always be the same, 1.

Applying the covariance decomposition,

1 = E(m)E(Ri ) + cov(m, Ri ) (11)

and, using Rf = 1/E(m),

E(Ri ) ’ Rf = ’Rf cov(m, Ri ) (12)

or

cov[u0 (ct+1 ), Ri ]

t+1

E(Ri ) ’ Rf = ’ (13)

.

0 (c

E[u t+1 )]

All assets have an expected return equal to the risk-free rate, plus a risk adjustment. Assets

whose returns covary positively with consumption make consumption more volatile, and so

must promise higher expected returns to induce investors to hold them. Conversely, assets

that covary negatively with consumption, such as insurance, can offer expected rates of return

that are lower than the risk-free rate, or even negative (net) expected returns.

Much of ¬nance focuses on expected returns. We think of expected returns increasing

or decreasing to clear markets; we offer intuition that “riskier” securities must offer higher

expected returns to get investors to hold them, rather than saying “riskier” securities trade for

lower prices so that investors will hold them. Of course, a low initial price for a given payoff

corresponds to a high expected return, so this is no more than a different language for the

same phenomenon.

1.4.3 Idiosyncratic risk does not affect prices

Only the component of a payoff perfectly correlated with the discount factor generates an

extra return. Idiosyncratic risk, uncorrelated with the discount factor, generates no premium.

You might think that an asset with a high payoff variance is “risky” and thus should have

a large risk correction. However, if the payoff is uncorrelated with the discount factor m, the

asset receives no risk-correction to its price, and pays an expected return equal to the risk-free

24

SECTION 1.4 CLASSIC ISSUES IN FINANCE

rate! In equations, if

cov(m, x) = 0

then

E(x)

p= .

Rf

This prediction holds even if the payoff x is highly volatile and investors are highly risk

averse. The reason is simple: if you buy a little bit more of such an asset, it has no ¬rst-order

effect on the variance of your consumption stream.

More generally, one gets no compensation or risk adjustment for holding idiosyncratic

risk. Only systematic risk generates a risk correction. To give meaning to these words, we can

decompose any payoff x into a part correlated with the discount factor and an idiosyncratic

part uncorrelated with the discount factor by running a regression,

x = proj(x|m) + µ.

Then, the price of the residual or idiosyncratic risk µ is zero, and the price of x is the same

as the price of its projection on m. The projection of x on m is of course that part of x

which is perfectly correlated with m. The idiosyncratic component of any payoff is that part

uncorrelated with m. Thus only the systematic part of a payoff accounts for its price.

Projection means linear regression without a constant,

E(mx)

proj(x|m) = m.

E(m2 )

You can verify that regression residuals are orthogonal to right hand variables E(mµ) = 0

from this de¬nition. E(mµ) = 0 of course means that the price of µ is zero.

µ ¶ µ ¶

E(mx) 2 E(mx)

p (proj(x|m)) = p m =E m = E(mx) = p(x).

E(m2 ) E(m2 )

The words “systematic” and “idiosyncratic” are de¬ned differently in different contexts,

which can lead to some confusion. In this decomposition, the residuals µ can be correlated

with each other, though they are not correlated with the discount factor. The APT starts with

a factor-analytic decomposition of the covariance of payoffs, and the word “idiosyncratic”

there is reserved for the component of payoffs uncorrelated with all of the other payoffs.

1.4.4 Expected return-beta representation

25

CHAPTER 1 CONSUMPTION-BASED MODEL AND OVERVIEW

We can write p = E(mx) as

E(Ri ) = Rf + β i,m »m

We can express the expected return equation (1.12), for a return Ri , as

µ ¶µ ¶

cov(Ri , m) var(m)

E(Ri ) = Rf + (14)

’

var(m) E(m)

or

E(Ri ) = Rf + β i,m »m (15)

where β im is the regression coef¬cient of the return Ri on m. This is a beta pricing model.

It says that expected returns on assets i = 1, 2, ...N should be proportional to their betas in a

regression of returns on the discount factor. Notice that the coef¬cient »m is the same for all

assets i,while the β i,m varies from asset to asset. The »m is often interpreted as the price of

risk and the β as the quantity of risk in each asset.

Obviously, there is nothing deep about saying that expected returns are proportional to

betas rather than to covariances. There is a long historical tradition and some minor conve-

nience in favor of betas. The betas refer to the projection of R on m that we studied above,

so you see again a sense in which only the systematic component of risk matters.

’γ

With m = β (ct+1 /ct ) , we can take a Taylor approximation of equation (1.14) to

express betas in terms of a more concrete variable, consumption growth, rather than marginal

utility. The result, which I derive more explicitly and conveniently in the continuous time

limit below, is

E(Ri ) = Rf + β i,∆c »∆c (1.16)

»∆c = γvar(∆c).

Expected returns should increase linearly with their betas on consumption growth itself. In

addition, though it is treated as a free parameter in many applications, the factor risk premium

»∆c is determined by risk aversion and the volatility of consumption. The more risk averse

people are, or the riskier their environment, the larger an expected return premium one must

pay to get investors to hold risky (high beta) assets.

1.4.5 Mean-variance frontier

All asset returns lie inside a mean-variance frontier. Assets on the frontier are perfectly

correlated with each other and the discount factor. Returns on the frontier can be generated

as portfolios of any two frontier returns. We can construct a discount factor from any frontier

26

SECTION 1.4 CLASSIC ISSUES IN FINANCE

return (except Rf ), and an expected return-beta representation holds using any frontier return

(except Rf ) as the factor.

Asset pricing theory has focused a lot on the means and variances of asset returns. Inter-

estingly, the set of means and variances of returns is limited. All assets priced by the discount

factor m must obey

¯ ¯

¯E(Ri ) ’ Rf ¯ ¤ σ(m) σ(Ri ). (17)

E(m)

To derive (1.17) write for a given asset return Ri

1 = E(mRi ) = E(m)E(Ri ) + ρm,Ri σ(Ri )σ(m)

and hence

σ(m)

E(Ri ) = Rf ’ ρm,Ri σ(Ri ). (18)

E(m)

Correlation coef¬cients can™t be greater than one in magnitude, leading to (1.17).

This simple calculation has many interesting and classic implications.

1. Means and variances of asset returns must lie in the wedge-shaped region illustrated

in Figure 1. The boundary of the mean-variance region in which assets can lie is called the

mean-variance frontier. It answers a naturally interesting question, “how much mean return

can you get for a given level of variance?”

2. All returns on the frontier are perfectly correlated with the discount factor: the frontier

¯ ¯

is generated by ¯ρm,Ri ¯ = 1. Returns on the upper part of the frontier are perfectly negatively

correlated with the discount factor and hence positively correlated with consumption. They

are “maximally risky” and thus get the highest expected returns. Returns on the lower part of

the frontier are perfectly positively correlated with the discount factor and hence negatively

correlated with consumption. They thus provide the best insurance against consumption

¬‚uctuations.

3. All frontier returns are also perfectly correlated with each other, since they are all

perfectly correlated with the discount factor. This fact implies that we can span or synthesize

any frontier return from two such returns. For example if you pick any single frontier return

Rm then all frontier returns Rmv must be expressible as

¡ ¢

Rmv = Rf + a Rm ’ Rf

for some number a.

4. Since each point on the mean-variance frontier is perfectly correlated with the discount

27

CHAPTER 1 CONSUMPTION-BASED MODEL AND OVERVIEW

E(R)

Mean-variance frontier

Slope σ(m)/E(m)

Idiosyncratic risk Ri

Rf

Some asset returns

σ(R)

Figure 1. Mean-variance frontier. The mean and standard deviation of all assets priced by

a discount factor m must line in the wedge-shaped region

factor, we must be able to pick constants a, b, d, e such that

m = a + bRmv

Rmv = d + em.

Thus, any mean-variance ef¬cient return carries all pricing information. Given a mean-

variance ef¬cient return and the risk free rate, we can ¬nd a discount factor that prices all

assets and vice versa.

5. Given a discount factor, we can also construct a single-beta representation, so expected

returns can be described in a single - beta representation using any mean-variance ef¬cient

return (except the riskfree rate),

£ ¤

E(Ri ) = Rf + β i,mv E(Rmv ) ’ Rf .

The essence of the β pricing model is that, even though the means and standard deviations

of returns ¬ll out the space inside the mean-variance frontier, a graph of mean returns versus

betas should yield a straight line. Since the beta model applies to every return including

Rmv itself, and Rmv has a beta of one on itself, we can identify the factor risk premium as

» = E(Rmv ’ Rf ).

The last two points suggest an intimate relationship between discount factors, beta models

and mean-variance frontiers. I explore this relation in detail in Chapter 6. A problem at the

end of this chapter guides you through the algebra to demonstrate points 4 and 5 explicitly.

28

SECTION 1.4 CLASSIC ISSUES IN FINANCE

6. We can plot the decomposition of a return into a “priced” or “systematic” component

and a “residual,” or “idiosyncratic” component as shown in Figure 1. The priced part is

perfectly correlated with the discount factor, and hence perfectly correlated with any frontier

asset. The residual or idiosyncratic part generates no expected return, so it lies ¬‚at as shown

in the ¬gure, and it is uncorrelated with the discount factor or any frontier asset..

1.4.6 Slope of the mean-standard deviation frontier and equity premium puzzle

The Sharpe ratio is limited by the volatility of the discount factor. The maximal risk-return

tradeoff is steeper if there is more risk or more risk aversion

¯ ¯

¯ E(R) ’ Rf ¯

¯ ¤ σ(m) ≈ γσ(∆ ln c)

¯

¯ σ(R) ¯ E(m)

This formula captures the equity premium puzzle, which suggests that either people are very

risk averse, or the stock returns of the last 50 years were good luck which will not continue.

The ratio of mean excess return to standard deviation

E(Ri ) ’ Rf

= Sharpe ratio

σ(Ri )

is known as the Sharpe ratio. It is a more interesting characterization of any security than

the mean return alone. If you borrow and put more money into a security, you can increase

the mean return of your position, but you do not increase the Sharpe ratio, since the standard

deviation increases at the same rate as the mean. The slope of the mean-standard deviation

frontier is the largest available Sharpe ratio, and thus is naturally interesting. It answers “how

much more mean return can I get by shouldering a bit more volatility in my portfolio?”

Let Rmv denote the return of a portfolio on the frontier. From equation (1.17), the slope

of the frontier is

¯ ¯

¯ E(Rmv ) ’ Rf ¯

¯ = σ(m) = σ(m)Rf .

¯

¯ σ(Rmv ) ¯ E(m)

Thus, the slope of the frontier is governed by the volatility of the discount factor.

For an economic interpretation, again consider the power utility function, u0 (c) = c’γ ,

¯ ¯

¯ E(Rmv ) ’ Rf ¯ ’γ

¯ = σ h t+1 /ct ) ]i .

[(c

¯ (19)

¯ σ(Rmv ) ¯

E (ct+1 /ct )’γ

The standard deviation is large if consumption is volatile or if γ is large. We can state this

29

CHAPTER 1 CONSUMPTION-BASED MODEL AND OVERVIEW

approximation again using the lognormal assumption. If consumption growth is lognormal,

¯ ¯

¯ E(Rmv ) ’ Rf ¯ p 2 2

¯ ¯ (20)

γ σ (∆ ln ct+1 ) ’ 1 ≈ γσ(∆ ln c).

¯ σ(Rmv ) ¯ = e

(A problem at the end of the chapter guides you though the algebra of the ¬rst equality.

The relation is exact in continuous time, and thus the approximation is easiest to derive by

reference to the continuous time result; see section 1.5.)

Reading the equation, the slope of the mean-standard deviation frontier is higher if the

economy is riskier “ if consumption is more volatile “ or if investors are more risk averse.

Both situations naturally make investors more reluctant to take on the extra risk of holding

risky assets. This expression is also the slope of the expected return beta line of the consump-

tion beta model, (1.16). (Or, conversely, in an economy with a high Sharpe ratio, low risk

aversion investors should take on so much risk that their consumption becomes volatile.)

In postwar US data, the slope of the mean-standard deviation frontier, or of expected

return-beta lines is much higher than reasonable risk aversion and consumption volatility

estimates suggest. This is the “equity premium puzzle.” Over the last 50 years in the U.S.,

real stock returns have averaged 9% with a standard deviation of about 16%, while the real

return on treasury bills has been about 1%. Thus, the historical annual market Sharpe ratio

has been about 0.5. Aggregate consumption growth has been about 1%. Thus, we can only

reconcile these facts with (1.20) if investors have a risk aversion coef¬cient of 50!

Obvious ways of generalizing the calculation just make matters worse. Equation (1.20)

relates consumption growth to the mean-variance frontier of all contingent claims. The mar-

ket indices with 0.5 Sharpe ratios are if anything inside that frontier, so recognizing market

incompleteness will only make matters worse. Aggregate consumption has about 0.2 cor-

relation with the market return, while the equality (1.20) takes the worst possible case that

consumption growth and asset returns are perfectly correlated. If you add this fact, you need

risk aversion of 250 to explain the market Sharpe ratio in the face of 1% consumption volatil-

ity! Individuals have riskier consumption streams than aggregate, but as their risk goes up

their correlation with any aggregate must decrease proportionally, so to ¬rst order recogniz-

ing individual risk will not help either.

Clearly, either 1) people are a lot more risk averse than we might have thought 2) the stock

returns of the last 50 years were largely good luck rather than an equilibrium compensation

for risk, or 3) something is deeply wrong with the model, including the utility function and

use of aggregate consumption data. This “equity premium puzzle” has attracted the attention

of a lot of research in ¬nance, especially on the last item. I return to the equity premium in

more detail in Chapter 21.

1.4.7 Random walks and time-varying expected returns

30

SECTION 1.4 CLASSIC ISSUES IN FINANCE

If investors are risk neutral, returns are unpredictable, and prices follow martingales. In

general, prices scaled by marginal utility are martingales, and returns can be predictable if

investors are risk averse and if the conditional second moments of returns and discount factors

vary over time. This is more plausible at long horizons.

So far, we have concentrated on the behavior of prices or expected returns across assets.

We should also consider the behavior of the price or return of a given asset over time. Going

back to the basic ¬rst order condition,

pt u0 (ct ) = Et [βu0 (ct+1 )(pt+1 + dt+1 )]. (21)

If investors are risk neutral, i.e. if u(c) is linear or there is no variation in consumption,

if the security pays no dividends between t and t + 1, and for short time horizons where β is

close to one, this equation reduces to

pt = Et (pt+1 ).

Equivalently, prices follow a time-series process of the form

pt+1 = pt + µt+1 .

If the variance σ2 (µt+1 ) is constant, prices follow a random walk. More generally, prices

t

follow a martingale. Intuitively, if the price today is a lot lower than investor™s expectations

of the price tomorrow, then people will try to buy the security. But this action will drive

up the price of the security until the price today does equal the expected price tomorrow.

Another way of saying the same thing is that returns should not be predictable; dividing by

pt , expected returns Et (pt+1 /pt ) = 1 should be constant; returns should be like coin ¬‚ips.

The more general equation (1.21) says that prices should follow a martingale after adjust-

ing for dividends and scaling by marginal utility. Since martingales have useful mathematical

properties, and since risk-neutrality is such a simple economic environment, many asset pric-

ing results are easily derived by scaling prices and dividends by marginal utility ¬rst, and

then using “risk-neutral” formulas and economic arguments.

Since consumption and risk aversion don™t change much day to day, we might expect

the random walk view to hold pretty well on a day-to-day basis. This idea contradicts the

still popular notion that there are “systems” or “technical analysis” by which one can predict

where stock prices are going on any given day. It has been remarkably successful. Despite

decades of dredging the data, and the popularity of television and radio reports that purport

to explain where markets are going, trading rules that reliably survive transactions costs and

do not implicitly expose the investor to risk have not yet been reliably demonstrated.

However, more recently, evidence has accumulated that long-horizon excess returns are

quite predictable, and to some this indicates that the whole enterprise of economic explana-

tion of asset returns is ¬‚awed. To think about this issue, write our basic equation for expected

31

CHAPTER 1 CONSUMPTION-BASED MODEL AND OVERVIEW

returns as

covt (mt+1 , Rt+1 )

f

(1.22)

Et (Rt+1 ) ’ Rt =’

Et (mt+1 )

σt (mt+1 )

= σt (Rt+1 )ρt (mt+1 , Rt+1 )

Et (mt+1 )

≈ γ t σ t (∆ct+1 )σt (Rt+1 )ρt (mt+1 , Rt+1 ).

I include the t subscripts to emphasize that the relation applies to conditional moments.

Sometimes, the conditional mean or other moment of a random variable is different from its

unconditional moment. Conditional on tonight™s weather forecast, you can better predict rain

tomorrow than just knowing the average rain for that date. In the special case that random

variables are i.i.d. (independent and identically distributed), like coin ¬‚ips, the conditional

and unconditional moments are the same, but that is a special case and not likely to be true of

asset prices, returns, and macroeconomic variables. In the theory so far, we have thought of

an investor, today, forming expectations of payoffs, consumption, and other variables tomor-

row. Thus, the moments are really all conditional, and if we want to be precise we should

include some notation to express this fact. I use subscripts Et (xt+1 ) to denote conditional

expectation; the notation E(xt+1 |It ) where It is the information set at time t is more precise

but a little more cumbersome.

Examining equation (1.22), we see that returns can be somewhat predictable. First, if

the conditional variance of returns changes over time, we might expect the conditional mean

return to vary as well “ the return can just move in and out a line of constant Sharpe ratio.

This explanation does not seem to help much in the data; variables that forecast means do

not seem to forecast variances and vice versa. Unless we want to probe the conditional

correlation, predictable excess returns have to be explained by changing risk “ σt (∆ct+1 )

“ or changing risk aversion γ. It is not plausible that risk or risk aversion change at daily

frequencies, but fortunately returns are not predictable at daily frequencies. It is much more

plausible that risk and risk aversion change over the business cycle, and this is exactly the

horizon at which we see predictable excess returns. Models that make this connection precise

are a very active area of current research.

1.4.8 Present value statement

∞

X

pt = Et mt,t+j dt+j .

j=0

It is convenient to use only the two period valuation, thinking of a price pt and a payoff