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The investorвЂ™s total risky portfolio is y 0 R. Hence, ОЈy gives the covariance of each return
with y0 R, and also with the investorвЂ™s overall portfolio y f Rf + y0 R. If all investors are
identical, then the market portfolio is the same as the individualвЂ™s portfolio so ОЈy also gives
the correlation of each return with Rm = yf Rf + y 0 R. (If investors differ in risk aversion О±,
the same thing goes through but with an aggregate risk aversion coefп¬Ѓcient.)
Thus, we have the CAPM. This version is especially interesting because it ties the market
price of risk to the risk aversion coefп¬Ѓcient. Applying (9.119) to the market return itself, we
have
E(Rm ) в€’ Rf
= О±.
Пѓ2 (Rm )

9.1.3 Quadratic value function, dynamic programming.

We can let investors live forever in the quadratic utility CAPM so long as we assume that
the environment is independent over time. Then the value function is quadratic, taking the
place of the quadratic second-period utility function. This case is a nice п¬Ѓrst introduction to
dynamic programming.

The two-period structure given above is unpalatable, since (most) investors do in fact live
longer than two periods. It is natural to try to make the same basic ideas work with less

148
SECTION 9.1 CAPITAL ASSET PRICING MODEL (CAPM)

restrictive and more palatable assumptions.
We can derive the CAPM in a multi-period context by replacing the second-period quadratic
utility function with a quadratic value function. However, the quadratic value function re-
quires the additional assumption that returns are i.i.d. (no вЂњshifts in the investment oppor-
tunity setвЂќ). This observation, due to Fama (1970), is also a nice introduction to dynamic
programming, which is a powerful way to handle multiperiod problems by expressing them
as two period problems. Finally, I think this derivation makes the CAPM more realistic, trans-
parent and intuitively compelling. Buying stocks amounts to taking bets over wealth; really
the fundamental assumption driving the CAPM is that marginal utility of wealth is linear in
wealth and does not depend on other state variables.
LetвЂ™s start in a simple ad-hoc manner by just writing down a вЂњutility functionвЂќ deп¬Ѓned
over this periodвЂ™s consumption and next periodвЂ™s wealth,

U = u(ct ) + ОІEt V (Wt+1 ).

This is a reasonable objective for an investor, and does not require us to make the very ar-
tiп¬Ѓcial assumption that he will die tomorrow. If an investor with this вЂњutility functionвЂќ can
buy an asset at price pt with payoff xt+1 , his п¬Ѓrst order condition (buy a little more, then
x contributes to wealth next period) is

pt u0 (ct ) = ОІEt [V 0 (Wt+1 )xt+1 ] .

Thus, the discount factor uses next periodвЂ™s marginal value of wealth in place of the more
familiar marginal utility of consumption

V 0 (Wt+1 )
mt+1 = ОІ
u0 (ct )

(The envelope condition states that, at the optimum, a penny saved has the same value as a
penny consumed u0 (ct ) = V 0 (Wt ). We could use this condition to express the denominator
in terms of wealth also.)
Now, suppose the value function were quadratic,
О·
V (Wt+1 ) = в€’ (Wt+1 в€’ W в€— )2 .
2

Then, we would have

Rt+1 (Wt в€’ ct ) в€’ W в€—
W
Wt+1 в€’ W в€—
mt+1 = в€’ОІО· = в€’ОІО·
u0 (ct ) u0 (ct )
В· ВёВ· Вё
ОІО·W в€— ОІО·(Wt в€’ ct ) W
= +в€’ Rt+1 ,
0 (c ) 0 (c )
ut ut

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CHAPTER 9 FACTOR PRICING MODELS

or, once again,

mt+1 = at + bt RW ,
t+1

the CAPM!
LetвЂ™s be clear about the assumptions and what they do.
1) The value function only depends on wealth. If other variables entered the value func-
tion, then в€‚V /в€‚W would depend on those other variables, and so would m. This assumption
bought us the п¬Ѓrst objective of any derivation: the identity of the factors. The ICAPM, be-
low, allows other variables in the value function, and obtains more factors. (Actually, other
variables could enter so long as they donвЂ™t affect the marginal value of wealth. The weather
is an example: You like me might be happier on sunny days, but you do not value additional
wealth more on sunny than on rainy days. Hence, covariance with weather does not affect
how you value stocks.)
2) The value function is quadratic. We wanted the marginal value function V 0 (W ) be
linear, to buy us the second objective, showing m is linear in the factor. Quadratic utility and
value functions deliver a globally linear marginal value function V 0 (W ). By the usual Taylor
series logic, linearity of V 0 (W ) is probably not a bad assumption for small perturbations, and
not a good one for large perturbations.

Why is the value function quadratic?
You might think we are done. But economists are unhappy about a utility function that
has wealth in it. Few of us are like DisneyвЂ™s Uncle Scrooge, who got pure enjoyment out
of a daily swim in the coins in his vault. Wealth is valuable because it gives us access to
more consumption. Utility functions should always be written over consumption. One of the
few real rules in economics that keep our theories from being vacuous is that ad-hoc вЂњutility
functionsвЂќ over other objects like wealth (or means and variances of portfolio returns, or
вЂњstatusвЂќ or вЂњpolitical powerвЂќ) should be defended as arising from a more fundamental desire
for consumption.
More practically, being careful about the derivation makes clear that the superп¬Ѓcially
plausible assumption that the value function is only a function of wealth derives from the
much less plausible, in fact certainly false, assumption that interest rates are constant, the
distribution of returns is i.i.d., and that the investor has no risky labor income. So, let us see
what it takes to defend the quadratic value function in terms of some utility function.
Suppose investors last forever, and have the standard sort of utility function
в€ћ
1 Xj
U = в€’ Et ОІ u(ct+j ).
2 j=0

Again, investors start with wealth W0 which earns a random return RW and they have no
other source of income. In addition, suppose that interest rates are constant, and stock returns

150
SECTION 9.1 CAPITAL ASSET PRICING MODEL (CAPM)

are i.i.d. over time.
Deп¬Ѓne the value function as the maximized value of the utility function in this environ-
ment. Thus, deп¬Ѓne V (W ) as7
в€ћ
X
ОІ j u(ct+j ) (9.120)
V (Wt ) в‰Ў max{ct ,ct+1 ,ct+2 ...О±t ,О±t+1 ,...} Et
j=0

= RW (Wt в€’ ct ); RW = О±0 Rt ; О±0 1 = 1
s.t. Wt+1 t+1 t t t
ВЈ В¤0
(I used vector notation to simplify the statement of the portfolio problem; R в‰Ў R1 R2 ... RN ,
etc.) The value function is the total level of utility the investor can achieve, given how much
wealth he has, and any other variables constraining him. This is where the assumptions of
no labor income, a constant interest rate and i.i.d. returns come in. Without these assump-
tions, the value function as deп¬Ѓned above might depend on these other characteristics of the
investorвЂ™s environment. For example, if there were some variable, say, вЂњD/PвЂќ that indicated
returns would be high or low for a while, then the investor would be happier, and have a
high value, when D/P is high, for a given level of wealth. Thus, we would have to write
V (Wt , D/Pt )
Value functions allow you to express an inп¬Ѓnite period problem as a two period problem.
Break up the maximization into the п¬Ѓrst period and all the remaining periods, as follows
пЈ± пЈ№пЈј
пЈ®
пЈІ пЈЅ
в€ћ
Xj
V (Wt ) = max{ct ,О±t } u(ct ) + ОІEt пЈ° ОІ u(ct+1+j )пЈ»
max Et+1 s. t. ..
пЈі пЈѕ
{ct+1 ,ct+2 ..,О±t+1 ,О±t+2 ....}
j=0

or

(121)
V (Wt ) = max{ct ,О±t } {u(ct ) + ОІEt V (Wt+1 )} s.t. ...

Thus, we have defended the existence of a value function. Writing down a two period
вЂњutility functionвЂќ over this periodвЂ™s consumption and next periodвЂ™s wealth is not as crazy as
it might seem.
The value function is also an attractive view of how people actually make decisions. You
donвЂ™t think вЂњIf I buy a sandwich today, I wonвЂ™t be able to go out to dinner one night 20 years
from nowвЂќ вЂ“ trading off goods directly as expressed by the utility function. You think вЂњI canвЂ™t
afford a new carвЂќ meaning that the decline in the value of wealth is not worth the increase
in the marginal utility of consumption. Thus, the maximization in (9.121) describes your
psychological approach to utility maximization.

There is also a transversality condition or a lower limit on wealth in the budget constraints. This keeps the
7

consumer from consuming a bit more and rolling over more and more debt, and it means we can write the budget
constraint in present value form.

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CHAPTER 9 FACTOR PRICING MODELS

The remaining question is, can the value function be quadratic? What utility function
assumption leads to a quadratic value function? Here is the fun fact: A quadratic utility
function leads to a quadratic value function in this environment. This is not a law of nature;
it is not true that for any u(c), V (W ) has the same functional form. But it is true here and
a few other special cases. The вЂњin this environmentвЂќ clause is not innocuous. The value
function вЂ“ the achieved level of expected utility вЂ“ is a result of the utility function and the
constraints.
How could we show this fact? One way would be to try to calculate the value function
by brute force from its deп¬Ѓnition, equation (9.120). This approach is not fun, and it does
not exploit the beauty of dynamic programming, which is the reduction of an inп¬Ѓnite period
problem to a two period problem.
Instead solve (9.121) as a functional equation. Guess that the value function V (Wt+1 )
is quadratic, with some unknown parameters. Then use the recursive deп¬Ѓnition of V (Wt ) in
(9.121), and solve a two period problemвЂ“п¬Ѓnd the optimal consumption choice, plug it into
(9.121) and calculate the value function V (Wt ). If the guess was right, you obtain a quadratic
function for V (Wt ), and determine any free parameters.
LetвЂ™s do it. Specify
1
u(ct ) = в€’ (ct в€’ cв€— )2 .
2
Guess
Оі
V (Wt+1 ) = в€’ (Wt+1 в€’ W в€— )2
2
with Оі and W в€— parameters to be determined later. Then the problem (9.121) is (I donвЂ™t write
the portfolio choice О± part for simplicity; it doesnвЂ™t change anything)
В· Вё
1 Оі
в€—2 в€—2
s. t. Wt+1 = RW (Wt в€’ ct ).
V (Wt ) = max в€’ (ct в€’ c ) в€’ ОІ E(Wt+1 в€’ W ) t+1
2 2
{ct }

(Et is now E since I assumed i.i.d.) Substituting the constraint into the objective,
В· Вё
Оі ВЈW В¤2
1
V (Wt ) = max в€’ (ct в€’ cв€— )2 в€’ ОІ E Rt+1 (Wt в€’ ct ) в€’ W в€— (122)
.
2 2
{ct }

The п¬Ѓrst order condition with respect to ct , using c to denote the optimal value, is
Л†
ct в€’ cв€— = ОІОіE RW (Wt в€’ ct ) в€’ W в€— Rt+1
Л† Л†
t+1

Solving for ct ,
Л†
В©ВЈ W 2 В¤ВЄ
ct = cв€— + ОІОіE Rt+1 Wt в€’ ct Rt+1 в€’ W в€— Rt+1
Л† W2 W
Л†

152
SECTION 9.1 CAPITAL ASSET PRICING MODEL (CAPM)

ВЈ В¤
ct 1 + ОІОіE(RW 2 ) = cв€— + ОІОіE(Rt+1 )Wt в€’ ОІОіW в€— E(RW )
W2
Л† t+1 t+1

cв€— в€’ ОІОіE(RW )W в€— + ОІОіE(Rt+1 )Wt
W2
t+1
(123)
ct =
Л†
1 + ОІОіE(RW 2 )
t+1

This is a linear function of Wt . Writing (9.122) in terms of the optimal value of c, we get
Оі ВЈW В¤2
1
V (Wt ) = в€’ (Л†t в€’ cв€— )2 в€’ ОІ E Rt+1 (Wt в€’ ct ) в€’ W в€— (124)
c Л†
2 2
This is a quadratic function of Wt and c. A quadratic function of a linear function is a
Л†
quadratic function, so the value function is a quadratic function of Wt . If you want to spend
a pleasant few hours doing algebra, plug (9.123) into (9.124), check that the result really is
quadratic in Wt , and determine the coefп¬Ѓcients Оі, W в€— in terms of fundamental parameters
ОІ, cв€— , E(RW ), E(RW 2 ) (or Пѓ2 (RW )). The expressions for Оі, W в€— do not give much insight,
so I donвЂ™t do the algebra here.

9.1.4 Log utility

Log utility rather than quadratic utility also implies a CAPM. Log utility implies that
consumption is proportional to wealth, allowing us to substitute the wealth return for con-
sumption data.

The point of the CAPM is to avoid the use of consumption data, and so to use wealth
or the rate of return on wealth instead. Log utility is another special case that allows this
substitution. Log utility is much more plausible than quadratic utility.
Suppose that the investor has log utility

u(c) = ln(c).

Deп¬Ѓne the wealth portfolio as a claim to all future consumption. Then, with log utility, the
price of the wealth portfolio is proportional to consumption itself.
в€ћ в€ћ
X X
u0 (ct+j ) ct ОІ
ОІj 0 ОІj
pW = Et ct+j = Et ct+j = ct
t
u (ct ) ct+j 1в€’ОІ
j=1 j=1

The return on the wealth portfolio is proportional to consumption growth,
ОІ
pW + ct+1 1в€’ОІ + 1 ct+1 1 u0 (ct )
1 ct+1
= t+1 W
W
Rt+1 = = = .
ОІ ОІ u0 (ct+1 )
ct ОІ ct
pt 1в€’ОІ

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CHAPTER 9 FACTOR PRICING MODELS

Thus, the log utility discount factor equals the inverse of the wealth portfolio return,
1
(125)
mt+1 = .
RW
t+1

Equation (9.125) could be used by itself: it attains the goal of replacing consumption data
by some other variable. (Brown and Gibbons 1982 test a CAPM in this form.) Note that log
utility is the only assumption so far. We do not assume constant interest rates, i.i.d. returns
or the absence of labor income.
Log utility has a special property that вЂњincome effects offset substitution effects,вЂќ or in
an asset pricing context that вЂњdiscount rate effects offset cashп¬‚ow effects.вЂќ News of higher
consumption = dividend should make the claim to consumption more valuable. However,
through u0 (c) it also raises the discount rate, lowering the value of the claim to consumption.
For log utility, these two effects exactly offset.

9.1.5 Linearizing any model: Taylor approximations and normal distributions.

Any nonlinear model m = f(z) can be turned into a linear model m = a + bz in discrete
time by assuming normal returns.

It is traditional in the CAPM literature to try to derive a linear relation between m and
the wealth portfolio return. We could always do this by a Taylor approximation,

mt+1 в€ј at + bt RW .
= t+1

We can make this approximation exact in a special case, that the factors and all asset returns
are normally distributed. (We can also take the continuous time limit, which is really the
same thing. However, this discrete-time trick is common and useful.) First, I quote without
proof the central mathematical trick as a lemma

Lemma 1 (SteinвЂ™s lemma) If f, R are bivariate normal, g(f ) is differentiable and E |
g0 (f) |< в€ћ, then

cov [g(f), R] = E[g 0 (f )] cov(f, R). (126)

Now we can use the lemma to state the theorem.

Theorem 2 If m = g(f ), if f and a set of the payoffs priced by m are normally distributed
returns, and if |E[g0 (f )]| < в€ћ, then there is a linear model m = a + bf that prices the
normally distributed returns.

154
SECTION 9.1 CAPITAL ASSET PRICING MODEL (CAPM)

Proof: First, the deп¬Ѓnition of covariance means that the pricing equation can be
rewritten as a restriction between mean returns and the covariance of returns with
m:

(127)
1 = E(mR) в‡” 1 = E(m)E(R) + cov(m, R).

Now, given m = g(f), f and R jointly normal, apply SteinвЂ™s lemma (9.126) and
(9.127),

1 = E[g(f)]E(R) + E[g0 (f)]cov(f, R)

1 = E[g(f)]E(R) + cov(E[g 0 (f)]f, R)

Exploiting the в‡ђ part of (9.127), we know that an m with mean E(g(f )) and that
depends on f via E(g 0 (f ))f will price assets,

m = E[g(f )] + E[g0 (f )][f в€’ E(f )].

ВҐ

Using this trick, and recalling that we have not assumed i.i.d. so all these moments are
conditional, the log utility CAPM implies the linear model
"Вµ В¶2 #
Вµ В¶
ВЈW В¤
1 1
Rt+1 в€’ Et (RW ) (128)
mt+1 = Et в€’ Et t+1
W RW
Rt+1 t+1

if RW and all asset returns to be priced are normally distributed. From here it is a short
t+1
step to an expected return-beta representation using the wealth portfolio return as the factor.
In the same way, we can trade the quadratic utility function for normal distributions in the
dynamic programming derivation of the CAPM. Starting from
ВЈ В¤
V 0 RW (Wt в€’ ct )
V 0 (Wt+1 ) t+1
mt+1 = ОІ =ОІ
0 (c ) u0 (ct )
ut

we can derive an expression that links m linearly to RW by assuming normality.
t+1
Using the same trick, the consumption-based model can be written in linear fashion, i.e.
expected returns can be expressed as a linear function of betas on consumption growth rather
than betas on consumption growth raised to a power. However, for large risk aversion co-
efп¬Ѓcients (more than about 10 in postwar consumption data) or other transformations, the
inaccuracies due to the normal or lognormal approximation can be very signiп¬Ѓcant in dis-
crete data.
The normal distribution assumption seems rather restrictive, and it is. However, the most
popular class of continuous-time models specify instantaneously normal distributions even
for things like options that have very non-normal distributions for discrete time intervals.

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CHAPTER 9 FACTOR PRICING MODELS

Therefore, one can think of the SteinвЂ™s lemma tricks as a way to get to continuous time
approximations without doing it in continuous time. I demonstrate the explicit continuous
time approach with the ICAPM, in the next section.

9.1.6 Portfolio intuition

The classic derivation of the CAPM contains some useful intuition. The classic derivation
starts with a mean-variance objective for portfolio wealth, max Eu(W ). Beta drives average
returns because beta measures how much adding a bit of the asset to a diversiп¬Ѓed portfolio
increases the volatility of the portfolio.
The central insight that started it all is that investors care about portfolio returns, not about
the behavior of speciп¬Ѓc assets. Once the characteristics of portfolios replaced demand curves
for individual stocks, modern п¬Ѓnance was born.

9.2 Intertemporal Capital Asset Pricing Model (ICAPM)

Any вЂњstate variableвЂќ zt can be a factor. The ICAPM is a linear factor model with wealth
and state variables that forecast changes in the distribution of future returns or income.

The ICAPM generates linear discount factor models

mt+1 = a + b0 ft+1

in which the factors are вЂњstate variablesвЂќ for the investorвЂ™s consumption-portfolio decision.
The вЂњstate variablesвЂќ are the variables that determine how well the investor can do in
his maximization. Current wealth is obviously a state variable. Additional state variables
describe the conditional distribution of income and asset returns the agent will face in the
future or вЂњshifts in the investment opportunity set.вЂќ In multiple good or international models,
relative price changes are also state variables.
Optimal consumption is a function of the state variables, ct = g(zt ). We can use this fact
once again to substitute out consumption, and write
u0 [g(zt+1 )]
mt+1 =ОІ 0 .
u [g(zt )]
From here, it is a simple linearization to deduce that the state variables zt+1 will be factors.
Alternatively, the value function depends on the state variables

V (Wt+1 , zt+1 ),

156
SECTION 9.2 INTERTEMPORAL CAPITAL ASSET PRICING MODEL (ICAPM)

so we can write
VW (Wt+1 , zt+1 )
mt+1 = ОІ
VW (Wt , zt )
(The marginal value of a dollar must be the same in any use, so I made the denominator pretty
by writing u0 (ct ) = VW (Wt , zt ). This fact is known as the envelope condition.)
This completes the п¬Ѓrst step, naming the proxies. To obtain a linear relation, we can take
a Taylor approximation, assume normality and use SteinвЂ™s lemma, or, most conveniently,
move to continuous time (which is really just a more convenient way of making the normal
approximation.) We saw above that we can write the basic pricing equation in continuous
time as
Вµ В¶
dp dО› dp
f
E в€’ r dt = в€’E .
p О›p
(for simplicity of the formulas, IвЂ™m folding any dividends into the price process). The dis-
count factor is marginal utility, which is the same as the marginal value of wealth,
du0 (ct )
dО›t dVW (Wt , zt )
=0 =
О›t u (ct ) VW
Our objective is to express the model in terms of factors z rather than marginal utility or
value, and ItoвЂ™s lemma makes this easy
dVW W VW W dW VW z 1
dz + (second derivative terms)
= +
VW VW W VW 2
(We donвЂ™t have to grind out the second derivative terms if we are going to take rf dt =
Et (dО›/О›) , though this approach removes a potentially interesting and testable implication
of the model). The elasticity of marginal value with respect to wealth is often called the
coefп¬Ѓcient of relative risk aversion,
W VW W
rra в‰Ў в€’ .
VW
Substituting, we obtain the ICAPM, which relates expected returns to the covariance of re-
turns with wealth, and also with the other state variables,
Вµ В¶ Вµ В¶
dp dW dp VW z dp
f
E в€’ r dt = rra E в€’ E dz .
p Wp VW p

From here, it is fairly straightforward to express the ICAPM in terms of betas rather than
covariances, or as a linear discount factor model. Most empirical work occurs in discrete
time; we often simply approximate the continuous time result as

E(R) в€’ Rf в‰€ rra cov(R, в€†W ) + О»z cov(R, в€†z).

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CHAPTER 9 FACTOR PRICING MODELS

One often substitutes covariance with the wealth portfolio for covariance with wealth, and
one uses factor-mimicking portfolios for the other factors dz as well. The factor-mimicking
portfolios are interesting for portfolio advice as well, as they give the purest way of hedging
against or proп¬Ѓting from state variable risk exposure.
This short derivation does not do justice to the beauty of MertonвЂ™s portfolio theory and
ICAPM. What remains is to actually state the consumerвЂ™s problem and prove that the value
function depends on W and z, the state variables for future investment opportunities, and that
the optimal portfolio holds the market and hedge portfolios for the investment opportunity
variables.

9.3 Comments on the CAPM and ICAPM

Conditional vs. unconditional models.
Do they price options?
Why bother linearizing?
The wealth portfolio.
Ex-post returns.
The implicit consumption-based model.
What are the ICAPM state variables?
CAPM and ICAPM as general equilibrium models

Is the CAPM conditional or unconditional?
Is the CAPM a conditional or an unconditional factor model? I.e., are the parameters a
and b in m = a в€’ bRW constants, or do they change at each time period, as conditioning in-
formation changes? We saw above that a conditional CAPM does not imply an unconditional
CAPM, so additional steps must be taken to say anything about observed average returns.
The two period quadratic utility based derivation results in a conditional CAPM, since the
parameters at and bt depend on consumption which changes over time. Also we know that a
and b must vary over time if the conditional moments of RW , Rf vary over time. This two-
period investor chooses a portfolio on the conditional mean variance frontier, which is not on
the unconditional frontier. The multiperiod quadratic utility CAPM only holds if returns are
i.i.d. so it only holds if there is no difference between conditional and unconditional models.
The log utility CAPM expressed with the inverse market return is a beautiful model, since
it holds both conditionally and unconditionally. There are no free parameters that can change

158
SECTION 9.3 COMMENTS ON THE CAPM AND ICAPM

with conditioning information:
Вµ В¶ Вµ В¶
1 1
1 = Et Rt+1 в‡”1=E Rt+1 .
RW RW
t+1 t+1

In fact there are no free parameters at all! Furthermore, the model makes no distributional as-
sumptions, so it can apply to any asset, including options. Finally it requires no speciп¬Ѓcation
of the investment opportunity set, or (macro language) no speciп¬Ѓcation of technology.
Linearizing the log utility CAPM comes at enormous price. The expectations in the lin-
earized log utility CAPM (9.128) are conditional. Thus, the apparent simpliп¬Ѓcation of linear-
ity destroys the nice unconditional feature of the log utility CAPM.
Should the CAPM price options?
As I have derived them, the quadratic utility CAPM and the nonlinear log utility CAPM
should apply to all payoffs: stocks, bonds, options, contingent claims, etc. However, if we as-
sume normal return distributions to obtain a linear CAPM from log utility, we can no longer
hope to price options, since option returns are non-normally distributed (thatвЂ™s the point of
options!) Even the normal distribution for regular returns is a questionable assumption. You
may hear the statement вЂњthe CAPM is not designed to price derivative securitiesвЂќ; the state-
ment refers to the log utility plus normal-distribution derivation of the linear CAPM.
Why linearize?
Why bother linearizing a model? Why take the log utility model m = 1/RW which
should price any asset, and turn it into mt+1 = at + bt Rt+1 that loses the clean conditioning-
W

down property and cannot price non-normally distributed payoffs? These tricks were de-
veloped before the p = E(mx) expression of asset pricing models, when (linear) expected
return-beta models were the only thing around. You need a linear model of m to get an ex-
pected return - beta model. More importantly, the tricks were developed when it was hard to
estimate nonlinear models. ItвЂ™s clear how to estimate a ОІ and a О» by regressions, but estimat-
ing nonlinear models used to be a big headache. Now, GMM has made it easy to estimate and
evaluate nonlinear models. Thus, in my opinion, linearization is mostly intellectual baggage.
The desire for linear representations and this normality trick is one of the central reasons
why many asset pricing models are written in continuous time. In most continuous time
models, everything is locally normal. Unfortunately for empiricists, this approach adds time-
aggregation and another layer of unobservable conditioning information into the predictions
of the model. For this reason, most empirical work is still based on discrete-time models.
However, the local normal distributions in continuous time, even for option returns, is a good
reminder that normal approximations probably arenвЂ™t that bad, so long as the time interval is
kept reasonably short.
What about the wealth portfolio?
The log utility derivation makes clear just how expansive is the concept of the wealth
portfolio. To own a (share of) the consumption stream, you have to own not only all stocks,

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CHAPTER 9 FACTOR PRICING MODELS

but all bonds, real estate, privately held capital, publicly held capital (roads, parks, etc.), and
human capital вЂ“ a nice word for вЂњpeople.вЂќ Clearly, the CAPM is a poor defense of common
proxies such as the value-weighted NYSE portfolio. And keep in mind that since it is easy to
п¬Ѓnd ex-post mean-variance efп¬Ѓcient portfolios of any subset of assets (like stocks) out there,
taking the theory seriously is our only guard against п¬Ѓshing.
Implicit consumption-based models
Many users of alternative models clearly are motivated by a belief that the consumption-
based model doesnвЂ™t work, no matter how well measured consumption might be. This view is
not totally unreasonable; as above, perhaps transactions costs de-link consumption and asset
returns at high frequencies, and some diagnostic evidence suggests that the consumption
behavior necessary to save the consumption model is too wild to be believed.
However, the derivations make clear that the CAPM and ICAPM are not alternatives to
the consumption-based model, they are special cases of that model. In each case mt+1 =
ОІu0 (ct+1 )/u0 (ct ) still operates. We just added assumptions that allowed us to substitute other
variables in place of ct . One cannot adopt the CAPM on the belief that the consumption
based model is wrong. If you think the consumption-based model is wrong, the economic
justiп¬Ѓcation for the alternative factor models evaporates.
The only plausible excuse for factor models is a belief that consumption data are un-
satisfactory. However, while asset return data are well measured, it is not obvious that the
S&P500 or other portfolio returns are terriп¬Ѓc measures of the return to total wealth. вЂњMacro
factorsвЂќ used by Chen, Roll and Ross (1986) and others are distant proxies for the quanti-
ties they want to measure, and macro factors based on other NIPA aggregates (investment,
output, etc.) suffer from the same measurement problems as aggregate consumption.
In large part, the вЂњbetter performanceвЂќ of the CAPM and ICAPM relative to consumption-
based models comes from throwing away content. Again mt+1 = Оґu0 (ct+1 )/u0 (ct ) is there
in any CAPM or ICAPM. The CAPM and ICAPM make predictions concerning consump-
tion data that are wildly implausible, not only of admittedly poorly measured aggregate con-
sumption data but any imaginable perfectly measured individual consumption data as well.
For example, equation (9.129) says that the standard deviation of the wealth portfolio return
equals the standard deviation of consumption growth. The latter is about 1% per year. All the
miserable failures of the log-utility consumption-based model apply equally to the log util-
ity CAPM. Finally, most models take the market price of risk as a free parameter. Of course
it isnвЂ™t; it is related to risk aversion and consumption volatility and is very hard to justify as
such.
Ex-post returns
The log utility model also allows us for the п¬Ѓrst time to look at what moves returns ex-post
as well as ex-ante. Recall that, in the log utility model, we have

1 ct+1
RW = (129)
.
t+1
ОІ ct

160
SECTION 9.3 COMMENTS ON THE CAPM AND ICAPM

Thus, the wealth portfolio return is high, ex-post, when consumption is high. This holds at
every frequency: If stocks go up between 12:00 and 1:00, it must be because (on average) we
all decided to have a big lunch. This seems silly. Aggregate consumption and asset returns are
likely to be de-linked at high frequencies, but how high (quarterly?) and by what mechanism
are important questions to be answered. In any case, this is another implication of the log
utility CAPM that is just thrown out.
In sum, the poor performance of the consumption-based model is an important nut to
chew on, not just a blind alley or failed attempt that we can safely disregard and go on about
Identity of state variables
The ICAPM does not tell us the identity of the state variables zt , and many authors use
the ICAPM as an obligatory citation to theory on the way to using factors composed of
ad-hoc portfolios, leading Fama (1991) to characterize the ICAPM as a вЂњп¬Ѓshing license.вЂќ
The ICAPM really isnвЂ™t quite such an expansive license. One could do a lot to insist that the
factor-mimicking portfolios actually are the projections of some identiп¬Ѓable state variables on
to the space of returns, and one could do a lot to make sure the candidate state variables really
are plausible state variables for an explicitly stated optimization problem. For example, one
could check that investment-opportunity set state variables actually do forecast something.
The п¬Ѓshing license comes as much from habits of applying the theory as from the theory
itself.
General equilibrium models
The CAPM and other models are really general equilibrium models. Looking at the
derivation through general-equilibrium glasses, we have speciп¬Ѓed a set of linear technologies
with returns Ri that do not depend on the amount invested. Some derivations make further
assumptions, such as an initial capital stock, and no labor or labor income.
The CAPM is obviously very artiп¬Ѓcial. Its central place really comes from its long string
of empirical successes rather than its theoretical purity. The theory was extended and multiple
factors anticipated long before they became empirically popular.
Portfolio intuition
I have derived all the models as instances of the consumption-based model. The more tra-
ditional portfolio intuition for multifactor models is also useful. The intuition (and historical
development) comes from looking past consumption to its determinants in sources of income
or news.
The CAPM simpliп¬Ѓes matters by assuming that the average investor only cares the per-
formance of his investment portfolio. Most of us have jobs, so events like recessions hurt the
majority of investors. People with jobs will prefer stocks that donвЂ™t fall in recessions, even if
their market betas, mean returns, and standard deviations are the same as stocks that do fall
in recessions. Demanding such stocks, they drive down the corresponding expected returns.
Thus, we expect expected returns to depend on additional betas that capture labor market

161
CHAPTER 9 FACTOR PRICING MODELS

conditions.
The traditional ICAPM intuition works the same way. Even jobless investors have long
horizons. Thus, they will prefer stocks that do well when news comes that future returns are
lower. Demanding more of such stocks, they depress expected returns. Thus, expected re-
turns come to depend on covariation with news of future returns, not just covariation with the
current market return. The ICAPM remained on the theoretical shelf for 20 years mostly be-
cause it took that long to accumulate empirical evidence that returns are, in fact, predictable.
It is vitally important that the extra factors affect the average investor. If an event makes
investor A worse off and investor B better off, then investor A buys assets that do well when
the event happens, and investor B sells them. They transfer the risk of the event, but the
price or expected return of the asset is unaffected. For a factor to affect prices or expected
returns, the average investor must be affected by it, so investors collectively bid up or down
the price and expected return of assets that covary with the event rather than just transfer the
risk without affecting equilibrium prices.
As you can see, this traditional intuition is encompassed by consumption. Bad labor
market outcomes or bad news about future returns are bad news that raise the marginal utility
of wealth, which equals the marginal utility of consumption.

9.4 Arbitrage Pricing Theory (APT)

The APT: If a set of asset returns are generated by a linear factor model
N
X
Лњ
i i
ОІ ij fj + Оµi
R = E(R ) +
j=1

Лњ
E(Оµi ) = E(Оµi fj ) = 0.

Then (with additional assumptions) there is a discount factor m linear in the factors m =
a + b0 f that prices the returns.

The APT starts from a statistical characterization. There is a big common component
to stock returns: when the market goes up, most individual stocks also go up. Beyond the
market, groups of stocks move together such as computer stocks, utilities, small stocks, value
stocks and so forth. Finally, each stockвЂ™s return has some completely idiosyncratic movement.
This is a characterization of realized returns, outcomes or payoffs. The point of the APT is to
start with this statistical characterization of outcomes, and derive something about expected
returns or prices.
The intuition behind the APT is that the completely idiosyncratic movements in asset

162
SECTION 9.4 ARBITRAGE PRICING THEORY (APT)

returns should not carry any risk prices, since investors can diversify them away by holding
portfolios. Therefore, risk prices or expected returns on a security should be related to the
securityвЂ™s covariance with the common components or вЂњfactorsвЂќ only.
The job of this section is then 1) to describe a mathematical model of the tendency for
stocks to move together, and thus to deп¬Ѓne the вЂњfactorsвЂќ and residual idiosyncratic compo-
nents, and 2) to think carefully about what it takes for the idiosyncratic components to have
zero (or small) risk prices, so that only the common components matter to asset pricing.
There are two lines of attack for the second item. 1) If there were no residual, then we
could price securities from the factors by arbitrage (really, by the law of one price, but the
current distinction between law of one price and arbitrage came after the APT was named.)
Perhaps we can extend this logic and show that if the residuals are small, they must have
small risk prices. 2) If investors all hold well-diversiп¬Ѓed portfolios, then only variations in
the factors drive consumption and hence marginal utility.
Much of the original appeal and marketing of the APT came from the п¬Ѓrst line of attack,
the idea that we could derive pricing implications without the economic structure required
of the CAPM, ICAPM, or any other model derived as a specialization of the consumption-
based model. In this section, I will п¬Ѓrst try to see how far we can in fact get with purely law
of one price arguments. I will conclude that the answer is, вЂњnot very far,вЂќ and that the most
satisfactory argument for the APT is in fact just another specialization of the consumption-
based model.

9.4.1 Factor structure in covariance matrices

I deп¬Ѓne and examine the factor decomposition

xi = О±i + ОІ 0 f + Оµi ; E(Оµi ) = 0, E(fОµi ) = 0
i

The factor decomposition is equivalent to a restriction on the payoff covariance matrix.

The APT models the tendency of asset payoffs (returns) to move together via a statistical
factor decomposition
M
X
ОІ ij fj + Оµi = О±i + ОІ 0 f + Оµi .
i
(130)
x = О±i + i
j=1

The fj are the factors, the ОІ ij are the betas or factor loadings and the Оµi are residuals.
ВЈ usual, I use the В¤0
As same letter without subscripts to denote a vector, for example f =
f1 f2 ... fK . A discount factor m, pricing factors f in m = b0 f and this factor
decomposition (or factor structure) for returns are totally unrelated uses of the word вЂњfactor.вЂќ

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CHAPTER 9 FACTOR PRICING MODELS

I didnвЂ™t invent the terminology! The APT is conventionally written with xi = returns, but it
ends up being much less confusing to use prices and payoffs.
It is a convenient and conventional simpliп¬Ѓcation to fold the factor means into the п¬Ѓrst,
Лњ
constant, factor and write the factor decomposition with zero-mean factors f в‰Ў f в€’ E(f).
M
X
Лњ
i i
ОІ ij fj + Оµi . (131)
x = E(x ) +
j=1

Remember that E(xi ) is still just a statistical characterization, not a prediction of a model.
We can construct the factor decomposition as a regression equation. Deп¬Ѓne the ОІ ij as
regression coefп¬Ѓcients, and then the Оµi are uncorrelated with the factors by construction,
Лњ
E(Оµi fj ) = 0.

The content вЂ” the assumption that keeps (9.131) from describing any arbitrary set of returns
вЂ” is an assumption that the Оµi are uncorrelated with each other.

E(Оµi Оµj ) = 0.

(More general versions of the model allow some limited correlation across the residuals but
the basic story is the same.)
The factor structure is thus a restriction on the covariance matrix of payoffs. For example,
if there is only one factor, then
ВЅ2
ПѓОµi if i = j
Лњ Лњ
cov(xi , xj ) = E[(ОІ i f + Оµi )(ОІ j f + Оµj )] = ОІ i ОІ j Пѓ2 (f) + .
0 if i 6= j

Thus, with N = number of securities, the N (N в€’ 1)/2 elements of a variance-covariance
matrix are described by N betas, and N + 1 variances. A vector version of the same thing is
пЈ®2 пЈ№
Пѓ1 0 0
пЈЇ пЈє
2
cov(x, x0 ) = ОІОІ 0 Пѓ 2 (f ) + пЈ° 0 Пѓ 2 0 пЈ» .
..
.
0 0
With multiple (orthogonalized) factors, we obtain

cov(x, x0 ) = ОІ 1 ОІ 0 Пѓ2 (f1 ) + ОІ 2 ОІ 0 Пѓ2 (f2 ) + . . . + (diagonal matrix)
1 2

In all these cases, we describe the covariance matrix a singular matrix ОІОІ 0 (or a sum of a few
such singular matrices) plus a diagonal matrix.
If we know the factors we want to use ahead of time, say the market (value-weighted
portfolio) and industry portfolios, or size and book to market portfolios, we can estimate
a factor structure by running regressions. Often, however, we donвЂ™t know the identities of
the factor portfolios ahead of time. In this case we have to use one of several statistical

164
SECTION 9.4 ARBITRAGE PRICING THEORY (APT)

techniques under the broad heading of factor analysis (thatвЂ™s where the word вЂњfactorвЂќ came
from in this context) to estimate the factor model. One can estimate a factor structure quickly
by simply taking an eigenvalue decomposition of the covariance matrix, and then setting
small eigenvalues to zero.

9.4.2 Exact factor pricing

With no error term,
Лњ
xi = E(xi )1 + ОІ 0 f .
i

implies
Лњ
p(xi ) = E(xi )p(1) + ОІ 0 p(f )
i

and thus

m = a + b0 f ; p(xi ) = E(mxi )

E(Ri ) = Rf + ОІ 0 О».
i

using only the law of one price.

Suppose that there are no idiosyncratic terms Оµi . This is called an exact factor model.
Now look again at the factor decomposition,
Лњ
xi = E(xi )1 + ОІ 0 f . (132)
i

It started as a statistical decomposition. But it also says that the payoff xi can be synthesized
as a portfolio of the factors and a constant (risk-free payoff). Thus, the price of xi can only
depend on the prices of the factors f,
Лњ
p(xi ) = E(xi )p(1) + ОІ 0 p(f). (133)
i

The law of one price assumption lets you take prices of right and left sides.
If the factors are returns, their prices are 1. If the factors are not returns, their prices are
free parameters which can be picked to make the model п¬Ѓt as well as possible. Since there
are fewer factors than payoffs, this procedure is not vacuous. (Recall that the prices of the
factors are related to the О» in expected return beta representations. О» is determined by the
expected return of a return factor, and is a free parameter for non-return factor models.)
We are really done, but the APT is usually stated as вЂњthere is a discount factor linear
in f that prices returns Ri ,вЂќ or вЂњthere is an expected return-beta representation with f as

165
CHAPTER 9 FACTOR PRICING MODELS

factors.вЂќ Therefore, we should take a minute to show that the rather obvious relationship
(9.133) between prices is equivalent to discount factor and expected return statements.
Assuming only the law of one price, we know there is a discount factor m linear in
factors that prices the factors. We usually call it xв€— , but call it f в€— here to remind us that it
ВЈ В¤
Лњ 0 the factors including the constant. As with xв€— ,
Л†
prices the factors. Denote f = 1 f
Л† Л†Л† Л† Л† Л†
f в€— = p(f )0 E(f f 0 )в€’1 f = a + b0 f satisп¬Ѓes p(f ) = E(f в€— f ) and p(1) = E(f в€— ). If the
discount factor prices the factors, it must price any portfolio of the factors; hence f в€— prices
all payoffs xi that follow the factor structure (9.132).
We could now go from m linear in the factors to an expected return-beta model using the
above theorems that connect the two representations. But there is a more direct and elegant
connection. Start with (9.133), specialized to returns xi = Ri and of course p(Ri ) = 1. Use
p(1) = 1/Rf and solve for expected return as
h i
Лњ = Rf + ОІ 0 О».
0
i f f
E(R ) = R + ОІ i в€’R p(f) i

The last equality deп¬Ѓnes О». Expected returns are linear in the betas, and the constants (О») are
related to the prices of the factors. In fact, this is the same deп¬Ѓnition of О» that we arrived at
above connecting m = b0 f to expected return-beta models.

9.4.3 Approximate APT using the law of one price

Attempts to extend the exact factor model to an approximate factor pricing model when
errors are вЂњsmall,вЂќ or markets are вЂњlarge,вЂќ still only using law of one price.
For п¬Ѓxed m, the APT gets better and better as R2 or the number of assets increases.
However, for any п¬Ѓxed R2 or size of market, the APT can be arbitrarily bad.
These observations mean that we must go beyond the law of one price to derive factor
pricing models.

Actual returns do not display an exact factor structure. There is some idiosyncratic or
residual risk; we cannot exactly replicate the return of a given stock with a portfolio of a few
large factor portfolios. However, the idiosyncratic risks are often small. For example, factor
model regressions of the form (9.130) often have very high R2 , especially when portfolios
rather than individual securities are on the left hand side. And the residual risks are still
idiosyncratic: Even if they are a large part of an individual securityвЂ™s variance, they should
be a small contributor to the variance of well diversiп¬Ѓed portfolios. Thus, there is reason to
hope that the APT holds approximately, especially for reasonably large portfolios. Surely, if
the residuals are вЂњsmallвЂќ and/or вЂњidiosyncratic,вЂќ the price of an asset canвЂ™t be вЂњtoo differentвЂќ
from the price predicted from its factor content?

166
SECTION 9.4 ARBITRAGE PRICING THEORY (APT)

To think about these issues, start again from a factor structure, but this time put in a
residual,

Лњ
xi = E(xi )1 + ОІ 0 f + Оµi
i

Again take prices of both sides,

Лњ
p(xi ) = E(xi )p(1) + ОІ 0 p(f ) + E(mОµi )
i

Now, what can we say about the price of the residual p(Оµi ) = E(mОµi )?
Figure 23 illustrates the situation. Portfolios of the factors span a payoff space, the line
from the origin through ОІ 0 f in the Figure. The payoff we want to price, xi is not in that space,
i
since the residual Оµi is not zero. A discount factor f в€— that is in the f payoff space prices the
factors. The set of all discount factors that price the factors is the line m perpendicular to
f в€— . The residual Оµi is orthogonal to the factor space, since it is a regression residual, and to
f в€— in particular, E(f в€— Оµi ) = 0. This means that f в€— assigns zero price to the residual. But the
other discount factors on the m line are not orthogonal to Оµi , so generate non-zero price for
the residual Оµi . As we sweep along the line of discount factors m that price the f, in fact, we
generate every price from в€’в€ћ to в€ћ for the residual. Thus, the law of one price does not nail
down the price of the residual Оµi and hence the price or expected return of xi .

All m
ОІвЂ™if
m>0
Оµi
m: Пѓ2(m) < A
xi
f*

m

Figure 23. Approximate arbitrage pricing.

167
CHAPTER 9 FACTOR PRICING MODELS

Limiting arguments

We would like to show that the price of xi has to be вЂњclose toвЂќ the price ofОІ 0 f . One notion
i
of вЂњclose toвЂќ is that in some appropriate limit the price of xi converges to the price of ОІ 0 f.
i
вЂњLimitвЂќ means, of course, that you can get arbitrarily good accuracy by going far enough in
the direction of the limit (for every Оµ > 0 there is a Оґ....). Thus, establishing a limit result is
a way to argue for an approximation.
Here is one theorem that seems to imply that the APT should be a good approximation
for portfolios that have high R2 on the factors. I state the argument for the case that there is a
constant factor, so the constant is in the f space and E(Оµi ) = 0. The same ideas work in the
less usual case that there is no constant factor, using second moments in place of variance.
Theorem: Fix a discount factor m that prices the factors. Then, as var(Оµi ) в†’ 0,
p(xi ) в†’ p(ОІ 0 f ).
i

This is easiest to see by just looking at the graph. E(Оµi ) = 0 so var(Оµi ) = E(Оµi2 ) =
||Оµi ||2 . Thus, as the size of the Оµi vector in Figure 23 gets smaller, xi gets closer and closer to
ОІ 0 f. For any п¬Ѓxed m, the induced pricing function (lines perpendicular to the chosen m) is
i
continuous. Thus, as xi gets closer and closer to ОІ 0 f , its price gets closer and closer to ОІ 0 f.
i i
The factor model is deп¬Ѓned as a regression, so

var(xi ) = var(ОІ 0 f ) + var(Оµi )
i

Thus, the variance of the residual is related to the regression R2 .

var(Оµi )
= 1 в€’ R2
i)
var(x

The theorem says that as R2 в†’ 1, the price of the residual goes to zero.
We were hoping for some connection between the fact that the risks are idiosyncratic and
factor pricing. Even if the idiosyncratic risks are a large part of the payoff at hand, they
are a small part of a well-diversiп¬Ѓed portfolio. The next theorem shows that portfolios with
high R2 donвЂ™t have to happen by chance; well-diversiп¬Ѓed portfolios will always have this
characteristic.
Theorem: As the number of primitive assets increases, the R2 of well-diversiп¬Ѓed
portfolios increases to 1.
Proof: Start with an equally weighted portfolio
N
1X i
p
x= x.
N i=1

Going back to the factor decomposition (9.130) for each individual asset xi , the

168
SECTION 9.4 ARBITRAGE PRICING THEORY (APT)

factor decomposition of xp is
N N N N
1 XВЎ 1X 1X 0 1Xi
Вў 0
0
p i
Оµ = ap + ОІ p f + Оµp .
x= ai + ОІ i f + Оµ = ai + ОІ if +
N i=1 N i=1 N i=1 N i=1

The last equality deп¬Ѓnes notation О±p , ОІ p , Оµp . But
Гѓ !
N
1Xi
var(Оµp ) = var Оµ
N i=1

So long as the variance of Оµi are bounded, and given the factor assumption E(Оµi Оµj ) =
0,

lim var(Оµp ) = 0.
Nв†’в€ћ

Obviously, the same idea goes through so long as the portfolio spreads some weight
ВҐ
on all the new assets, i.e. so long as it is вЂњwell-diversiп¬Ѓed.вЂќ

These two theorems can be interpreted to say that the APT holds approximately (in the
usual limiting sense) for portfolios that either naturally have high R2 , or well-diversiп¬Ѓed
portfolios in large enough markets. We have only used the law of one price.

Law of one price arguments fail

Now, let me pour some cold water on these results. I п¬Ѓxed m and then let other things take
limits. The п¬‚ip side is that for any nonzero residual Оµi , no matter how small, we can pick a
discount factor m that prices the factors and assigns any price to xi ! As often in mathematics,
the order of вЂњfor allвЂќ and вЂњthere existsвЂќ matters a lot.
Theorem: For any nonzero residual Оµi there is a discount factor that prices the fac-
tors f (consistent with the law of one price) and that assigns any desired price in
(в€’в€ћ, в€ћ) to the payoff xi .

So long as ||Оµi || > 0, as we sweep the choice of m along the dashed line, the inner
product of m with Оµi and hence xi varies from в€’в€ћ to в€ћ. Thus, for a given size R2 < 1, or
a given п¬Ѓnite market, the law of one price says absolutely nothing about the prices of payoffs
that do not exactly follow the factor structure. The law of one price says that two ways of
constructing the same portfolio must give the same price. If the residual is not exactly zero,
there is no way of replicating the payoff xi from the factors and no way to infer anything
about the price of xi from the price of the factors.
I think the contrast between this theorem and those of the last subsection accounts for
most of the huge theoretical controversy over the APT. If you п¬Ѓx m and take limits of N or

169
CHAPTER 9 FACTOR PRICING MODELS

Оµ, the APT gets arbitrarily good. But if you п¬Ѓx N or Оµ, as one does in any application, the
APT can get arbitrarily bad as you search over possible m.
The lesson I learn is that the effort to extend prices from an original set of securities (f in
this case) to new payoffs that are not exactly spanned by the original set of securities, using
only the law of one price, is fundamentally doomed. To extend a pricing function, you need
to add some restrictions beyond the law of one price.

9.4.4 Beyond the law of one price: arbitrage and Sharpe ratios

We can п¬Ѓnd a well-behaved approximate APT if we impose the law of one price and a
restriction on the volatility of discount factors, or, equivalently, a bound on the Sharpe ratio
achievable by portfolios of the factors and test assets.

The approximate APT based on the law of one price fell apart because we could always
choose a discount factor sufп¬Ѓciently вЂњfar outвЂќ to generate an arbitrarily large price for an
arbitrarily small residual. But those discount factors are surely вЂњunreasonable.вЂќ Surely, we
can rule them out, reestablishing an approximate APT, without jumping all the way to fully
speciп¬Ѓed discount factor models such as the CAPM or consumption-based model
A natural п¬Ѓrst idea is to impose the no-arbitrage restriction that m must be positive.
Graphically, we are now restricted to the solid m line in Figure 23. Since that line only
extends a п¬Ѓnite amount, restricting us to strictly positive m0 s gives rise to п¬Ѓnite upper and
lower arbitrage bounds on the price of Оµi and hence xi . (The word arbitrage bounds comes
from option pricing, and we will see these ideas again in that context. If this idea worked, it
would restore the APT to вЂњarbitrage pricingвЂќ rather than вЂњlaw of one-pricing.вЂќ)
Alas, in applications of the APT (as often in option pricing), the arbitrage bounds are
too wide to be of much use. The positive discount factor restriction is equivalent to saying
вЂњif portfolio A gives a higher payoff than portfolio B in every state of nature, then the price
of A must be higher than the price of B.вЂќ Since stock returns and factors are continuously
distributed, not two-state distributions as I have graphed for п¬Ѓgure 23, there typically are no
strictly dominating portfolios, so adding m > 0 does not help.
A second restriction does let us derive an approximate APT that is useful in п¬Ѓnite markets
with R2 < 1. We can restrict the variance and hence the size (||m|| = E(m2 ) = Пѓ 2 (m) +
E(m)2 = Пѓ 2 (m) + 1/Rf2 ) of the discount factor. Figure 23 includes a plot of the discount
factors with limited variance, size, or length in the geometry of that Figure. The restricted
range of discount factors produces a restricted range of prices for xi . The restricted range
of discount factors gives us upper and lower price bounds for the price of xi in terms of the

170
SECTION 9.5 APT VS. ICAPM

factor prices. Precisely, the upper and lower bounds solve the problem

min ( or max) p(xi ) = E(mxi ) s.t. E(mf ) = p(f), m в‰Ґ 0, Пѓ2 (m) в‰¤ A.
{m} {m}

Limiting the variance of the discount factor is of course the same as limiting the maximum
Sharpe ratio (mean / standard deviation of excess return) available from portfolios of the
factors and xi . Recall that
E (Re ) Пѓ(m)
в‰¤ .
Пѓ(Re ) E(m)

Though a bound on Sharpe ratios or discount factor volatility is not a totally preference-
free concept, it clearly imposes a great deal less structure than the CAPM or ICAPM which
are essentially full general equilibrium models. Ross (1976) included this suggestion in his
original APT paper, though it seems to have disappeared from the literature since then in
the failed effort to derive an APT from the law of one price alone. Ross pointed out that
deviations from factor pricing could provide very high Sharpe ratio opportunities, which
seem implausible though not violations of the law of one price. SaГЎ-Requejo and I (2000)
dub this idea вЂњgood-dealвЂќ pricing, as an extension of вЂњarbitrage pricing.вЂќ Limiting Пѓ(m) rules
out вЂњgood dealsвЂќ as well as pure arbitrage opportunities.
Having imposed a limit on discount factor volatility or Sharpe ratio A, then the APT limit
does work, and does not depend on the order of вЂњfor allвЂќ and вЂњthere exists.вЂќ
Theorem: As Оµi в†’ 0 and R2 в†’ 1, the price p(xi ) assigned by any discount factor
m that satisп¬Ѓes E(mf) = p(f ), m в‰Ґ 0, Пѓ 2 (m) в‰¤ A approaches p(ОІ 0 f ).
i

9.5 APT vs. ICAPM

A factor structure in the covariance of returns or high R2 in regressions of returns on
factors can imply factor pricing (APT) but factors can price returns without describing their
covariance matrix (ICAPM).
Differing inspiration for factors.
The disappearance of absolute pricing.

The APT and ICAPM stories are often confused. Factor structure can imply factor pric-
ing (APT), but factor pricing does not require a factor structure. In the ICAPM there is no
presumption that factors f in a pricing model m = b0 f describe the covariance matrix of
returns. The factors donвЂ™t have to be orthogonal or i.i.d. either. High R2 in time-series re-
gressions of the returns on the factors may imply factor pricing (APT), but again are not

171
CHAPTER 9 FACTOR PRICING MODELS

necessary (ICAPM). The regressions of returns on factors can have low R2 in the ICAPM.
Factors such as industry may describe large parts of returnsвЂ™ variances but not contribute to
the explanation of average returns.
The biggest difference between APT and ICAPM for empirical work is in the inspiration
for factors. The APT suggests that one start with a statistical analysis of the covariance matrix
of returns and п¬Ѓnd portfolios that characterize common movement. The ICAPM suggests that
one start by thinking about state variables that describe the conditional distribution of future
asset returns and non-asset income. More generally, the idea of proxying for marginal utility
growth suggests macroeconomic indicators, and indicators of shocks to non-asset income in
particular.
The difference between the derivations of factor pricing models, and in particular an ap-
proximate law-of-one-price basis vs. a proxy for marginal utility basis seems not to have
had much impact on practice. In practice, we just test models m = b0 f and rarely worry
about derivations. The best evidence for this view is the introductions of famous papers.
Chen, Roll and Ross (1986) describe one of the earliest popular multifactor models, using
industrial production and inп¬‚ation as some of the main factors. They do not even present
a factor decomposition of test asset returns, or the time-series regressions. A reader might
well categorize the paper as a macroeconomic factor model or perhaps an ICAPM. Fama and
French (1993) describe the currently most popular multifactor model, and their introduction
describes it as an ICAPM in which the factors are state variables. But the factors are sorted
on size and book/market just like the test assets, the time-series R2 are all above 90%, and
much of the explanation involves вЂњcommon movementвЂќ in test assets captured by the factors.
A a reader might well categorize the model as much closer to an APT.
In the п¬Ѓrst chapter, I made a distinction between relative pricing and absolute pricing. In
the former, we price one security given the prices of others, while in the latter, we price each
security by reference to fundamental sources of risk. The factor pricing stories are interesting
in that they start with a nice absolute pricing model, the consumption-based model, and
throw out enough information to end up with relative models. The CAPM prices Ri given
the market, but throws out the consumption-based modelвЂ™s description of where the market
return came from.

9.6 Problems

1. Suppose the investor only has a one-period horizon. He invests wealth W at date zero,
and only consumes with expected utility Eu(c) = Eu(W ) in period one. Derive the
quadratic utility CAPM in this case. (This is an even simpler derivation. The Lagrange
multiplier on initial wealth W now becomes the denominator of m in place of u0 (c0 )).
2. Express the log utility CAPM in continuous time to derive a discount factor linear in
wealth.
3. Figure 23 suggests that m > 0 is enough to establish a well-behaved approximate APT.

172
SECTION 9.6 PROBLEMS

The text claims this is not true. Which is right?
4. Can you use any excess return for the market factor in the CAPM, or must it be the
market less the riskfree rate?

173
PART II
Estimating and evaluating asset
pricing models

174
SECTION 9.6 PROBLEMS

Our п¬Ѓrst task in bringing an asset pricing model to data is to estimate the free parameters; the
ОІ and Оі in m = ОІ(ct+1 /ct )в€’Оі , or the b in m = b0 f. Then we want to evaluate the model. Is
it a good model or not? Is another model better?
Statistical analysis helps us to evaluate a model by providing a distribution theory for
numbers such as parameter estimates that we create from the data. A distribution theory
pursues the following idea: Suppose that we generate artiп¬Ѓcial data over and over again from
a statistical model. For example, we could specify that the market return is an i.i.d. normal
random variable, and a set of stock returns is generated by Rt = О±i + ОІ i Rem + Оµi . After
ei
t t
picking values for the mean and variance of the market return and the О±i , ОІ i , Пѓ 2 (Оµi ), we could
ask a computer to simulate many artiп¬Ѓcial data sets. We can repeat our statistical procedure
in each of these artiп¬Ѓcial data sets, and graph the distribution of any statistic which we have
estimated from the real data, i.e. the frequency that it takes on any particular value in our
artiп¬Ѓcial data sets.
In particular, we are interested in a distribution theory for the estimated parameters, to give
us some sense of how much the data really has to say about their values; and for the pricing
errors, which helps us to judge whether pricing errors are just bad luck of one particular
historical accident or if they indicate a failure of the model. We also will want to generate
distributions for statistics that compare one model to another, or provide other interesting
evidence, to judge how much sample luck affects those calculations.
All of the statistical methods I discuss in this part achieve these ends. They give methods
for estimating free parameters; they provide a distribution theory for those parameters, and
they provide distributions for statistics that we can use to evaluate models, most often a
quadratic form of pricing errors in the form О±0 V в€’1 О±.
Л† Л†
I start by focusing on the GMM approach. The GMM approach is a natural п¬Ѓt for a
discount factor formulation of asset pricing theories, since we just use sample moments in
the place of population moments. As you will see, there is no singular вЂњGMM estimate and
test.вЂќ GMM is a large canvas and a big set of paints and brushes; a п¬‚exible tool for doing
all kinds of sensible (and, unless youвЂ™re careful, not-so-sensible) things to the data. Then
I consider traditional regression tests (naturally paired with expected return-beta statements
of factor models) and their maximum likelihood formalization. I emphasize the fundamental
similarities between these three methods, as I emphasized the similarity between p = E(mx),

175
CHAPTER 9 FACTOR PRICING MODELS

expected return-beta models, and mean-variance frontiers. A concluding chapter highlights
some of the differences between the methods, as I contrasted p = E(mx) and beta or mean-
variance representations of the models.

176
Chapter 10. GMM in explicit discount
factor models
The basic idea in the GMM approach is very straightforward. The asset pricing model pre-
dicts

E(pt ) = E [m(datat+1 , parameters) xt+1 ] . (134)

The most natural way to check this prediction is to examine sample averages, i.e. to calculate
T T
1X 1X
pt and [m(datat+1 , parameters) xt+1 ] . (135)
T t=1 T t=1

GMM estimates the parameters by making the sample averages as close to each other as
possible. It seems natural, before evaluating a model, to pick parameters that give it its best
chance. GMM then works out a distribution theory for the estimates. This distribution theory
is a generalization of the simplest exercise in statistics: the distribution of the sample mean.
Then, it suggests that we evaluate the model by looking at how close the sample averages
of price and discounted payoff are to each other, or equivalently by looking at the pricing
errors. It gives a statistical test of the hypothesis that the underlying population means are in
fact zero.

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