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Asset Pricing



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

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