Asset Pricing
Zheng Zhenlong
Chapter 9:
Factor pricing models
Asset Pricing
Zheng Zhenlong
Contents
•
•
•
•
•
•
Introduction
CAPM
ICAPM
Comments on the CAPM and ICAPM
APT
APT vs. ICAPM
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Brief introduction
•
uct 1 

 a  bft 1
uct 
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Brief introduction
• More directly, the essence of asset pricing is that there are
special states of the world in which investors are especially
concerned that their portfolios not do badly.
• The factors are variables that indicate that these “bad states”
have occurred.
• Any variable that forecasts asset returns (“changes in the
investment opportunity set”) or macroeconomic variables is a
candidate factor.
• Such as :term premium, dividend/price ratio, stock returns
Should factors be unpredictable
over time?
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• Factors that proxy for marginal utility growth, though they
don’t have to be totally unpredictable, should not be highly
predictable. If one chooses highly predictable factors, the
model will counterfactually predict large interest rate
variation.
u  ct 1 
1
f
u(ct )   R Et [u(ct 1 )] 

  t 1
f
u  ct 
R
• In practice, this consideration means that one should choose
the right units: Use GNP growth rather than level, portfolio
returns rather than prices or price/dividend ratios, etc.
The derivations of factor pricing
model
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• Determine one particular list of factors that can proxy for
marginal utility growth
• Prove that the relation should be linear.
• Remark: all factor models are derived as specializations
of the consumption-based model.
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guard against fishing
• One should call for better theories or derivations, more
carefully aimed at limiting the list of potential factors and
describing the fundamental macroeconomic sources of
risk, and thus providing more discipline for empirical
work.
Capital Asset Pricing Model (CAPM)
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mt 1  a  bRtW1
• RtW1 wealth portfolio return.
• In expected return / beta language,
 
   
E Ri    i ,RW E RW  
• CAPM can be derived from consumption-based model
by different assumption.
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Different assumption
•
•
•
•
1) two-period quadratic utility
2) exponential utility and normal returns,
3) Infinite horizon, quadratic utility and i.i.d. returns
4) Log utility.
• Same assumption: no labor income
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Two-period quadratic utility，no
labor income
• Investors have quadratic preferences and only live two
periods,

U ct , ct 1   0.5 ct  c
  0.5E c
 2
t 1
c
• marginal rate of substitution is thus
mt 1

 2

c
 ct 1 


u  ct 1 



u  ct 
c
  ct 
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• the budget constraint is
ct 1  Wt 1
Wt 1  RtW
1 Wt  ct 
RtW
1 
N
i
w
R
 i t 1
i 1
N
w
i 1
i
1
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mt 1  
c  RtW1 Wt  ct 
 c  ct 

 c
c  ct

• Just as
mt 1  at  bt R
W
t 1
 Wt  ct 
c  ct
RtW1
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Exponential utility, normal
distributions, no labor income
• If consumption only in the last period and is normally
distributed, we have
 ac



Eu c   E  e 
• a is the coefficient of absolute risk aversion.
ù
Eé
u
c
= -e
(
)
ê
ú
ë
û
ù+ a 2 / 2 s 2 (c )
- aE é
c
ê
ëú
û
(
)
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• the budget constraint is
c  y R  yR
f
f
W  y f  y1
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•
ù= - e
E é
u
c
(
)
ê
ú
ë
û
- aé
y f R f + y ¢E (R )ù
+ a2 / 2 y ¢
ê
ú
ë
û
y   1
(
) å
y
E R   R f
a

ER  R f  a y  a cov R, RW
E  RW   R f  a 2 ( RW )

Quadratic value function, dynamic
programming
U  uct   EtV Wt 1 
• first order condition
pt uct   Et V Wt 1 xt 1 
• So,
mt 1
V  Wt 1 

u  ct 
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• suppose the value function were quadratic,
V Wt 1   


W
2
t 1
W 

2
• Then,
mt 1  
W 
u   ct 
  Wt  ct   W
 
 Rt 1

u  ct 


• Some addition assumptions:
– The value function only depends on wealth.
– The value function is quadratic. It needs the following
assumptions: the interest rate is constant, returns are iid,
no labor income.
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the existence of value function (Proof )
• Suppose investors last forever, and have the standard
sort of utility function

U  Et   j u ct  j 
j 0
• Define the value function as the maximized value of the
utility function in this environment.
V Wt  

max
ct ,ct 1...,wt ,wt 1...
Et   j u ct  j 
j 0
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• Value functions allow you to express an infinite
period problem as a two period problem
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Why is the value function
quadratic?
• Remark: quadratic utility function leads to a quadratic
value function in this environment
• Specify:
2

u ct   0.5 ct  c

• Guess:

V Wt 1   0.5 Wt 1  W

 2
• Thus,
 2
 2

V Wt   max 0.5  ct  c   0.5 E Wt 1  W  
ct  

s.t.Wt 1  RtW
1 Wt  ct 
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•

cˆt  c   E  RtW1 (Wt  ct )  W   RtW1






W2
c   E RtW
W



E
R
1
t 1 Wt
ˆt 
c
2
1   E RtW
1


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•

V Wt   0.5 cˆt  c

 2

 0.5 E R
W
t 1
Wt  cˆt   W

 2
Log utility, no labor income
•

uct  j 

j ct
p  Et  
ct  j  Et  
ct  j 
ct

u ct 
ct  j
1 
j 1
j 1

W
t
W
t 1
R
j
u  ct 
ptW1  ct 1   / 1     1 ct 1 ct 1
1





ptW
 / 1    ct
 ct  u  ct 1  mt 1
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• Log utility has a special property that “income effects
offset substitution effects,” or in an asset pricing context
that “discount rate effects offset cash flow effects.”
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How to linearize the model?
• The twin goals of a linear factor model derivation are to
derive what variables derive the discount factor, and to
derive a linear relation between the discount factor and
these variables. This section covers three tricks that are
used to obtain a linear functional form.
• Taylor approximation
• the continuous time limit
• normal distribution
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Taylor approximation
• The most obvious way to linearize the model is by a Taylor
approximation
mt 1  g ( f t 1 )
 g Et  f t 1   g Et  f t 1  f t 1  Et  f t 1 
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Continuous time limit
• If the discrete time is short enough, we can apply the
continuous time result as an approximation
t  g  f t , t 
g
g  ft , t 
2 g
2
dt 
dt 
dft  0.5
df
t
t
f
f 2
 dpti 
 dpti d t
Dti
f
Et 
 pi 
  p i dt  rt dt   Et 
 pi 
t 
t
t
t


 dpti

1
g  f t , t 


Et 
df
t 
i

g f ,t 
f
 pt





• For a short discrete time interval,
1 g ( f , t )
Et ( R )  Rt  covt ( R , ft 1 )(
)  i , f ;t t f
g ( f , t ) f
i
t 1
f
i
t 1
Normal distribution in discrete
time
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• Stein’s lemma : If f and R are bivariate normal, g(f) is
differentiable and E g f    ,then
covg  f , R  Eg f cov f , R
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• Remark: If m=g(f), if f and a set of the payoffs priced
by m are normally distributed returns, and
if E g f    , then there is a linear model m=a+bf
that prices the normally distributed returns.
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(
)
p = E (m x ) = E g (f )x
ùE (x ) + cov ég (f ), x ù
= E é
g
f
(
)
ê
ú
ê
ú
ë
û
ë
û
ùE (x ) + E ég ¢(f )ùcov (f , x )
= E é
g
f
(
)
ê
ú
ê
ú
ë
û
ë
û
ù+ E ég ¢(f )ù(f - E ( f )) x
= E E é
g
f
(
)
ê
ú
ê
ú
ë
û
ë
û
ù- E ég ¢(f )ùE (f ) + E ég ¢(f )ùf
= E E é
g
f
(
)
ê
ú
ê
ú
ê
ú
ë
û
ë
û
ë
û
= E (m t + 1x ) = E (at + bt ft + 1 )x
({
({
})
(
)
}x )
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• Similar,it allows us to derive an expected return-beta model
using the factors
E  Rti1   Rt f  covt  Rti1 , mt 1 
 Rt f  Et  g   f t 1   covt  Rti1 , f t 1 
 Rt f  i , f ;t t f
Two period CAPM
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• Stein’s lemma allows us to substitute a normal
distribution assumption for the quadratic
assumption in the two period CAPM.
mt 1
u(ct 1 )
u( RtW1 (Wt  ct ))


u(ct )
u(ct )
• Assuming RWand Ri are normally distributed, we
have:
W

(
W

c
)
u
[
R
i
t
t
t 1 (Wt  ct )]
covt ( Rt 1 , mt 1 )  E[ 
]covt ( Rti1, RtW1 )
u(c t )
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Log utility CAPM
• Stein’s lemma cannot be applied to the log utility
CAPM because the market return cannot be
normally distributed. For log utility CAPM,
g(f)=1/RW, so
1
i
f
E ( Rt 1 )  R  Et ( W 2 ) covt ( Rti1 , RtW1 )
Rt 1
• If RW is normally distributed, E(1/RW2) does not
exist. The Stein’s lemma condition is violated.
Intertemporal Capital Asset Pricing
Model (ICAPM)
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• The ICAPM generates linear discount factor models
mt 1  a  bft 1
• in which the factors are “state variables” for the
investor’s consumption-portfolio decision.
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• the value function depends on the state variables
V Wt 1, zt 1 
• so we can write
mt 1
VW Wt 1 , zt 1 

VW Wt , zt 
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• Start from
t  e VW Wt , zt 
t
• We have
d t
WtVWW Wt , zt  dW
 dt 
t
VW Wt , zt  Wt
VWz Wt , zt 

dzt  ...
VW Wt , zt 
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Define the coefficient of relative risk aversion,
WVWW Wt , zt 
rrat  
VW Wt , zt 
Then we obtain the ICAPM,
i
 dpti  Dti

 VWz ,t
dp
f
t dWt
 
Et  i   i dt  rt dt  rrat E i
 pt  pt
 pt Wt  VW ,t
 dpti 
E i dzt 
 pt

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• Thus, in discrete time
 i Wt 1 

Et R  R  rrat covt  Rt 1 ,
Wt 

 zt covt Rti1 , zt 1

i
t 1

f


Is the CAPM conditional or
unconditional?
•
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•
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• 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 with conditioning
information.
 1

 1

1  Et 
 RW Rt 1 
  1  E
 RW Rt 1 

 t 1

 t 1

• Finally it requires no specification of the investment
opportunity set, or no specification of technology.
• However, the expectations in the linearized log utility
CAPM are conditional.
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Should the CAPM price options?
• the quadratic utility CAPM and the nonlinear log utility
CAPM should apply to all payoffs: stocks, bonds,
options, contingent claims, etc.
• However, if we assume normal return distributions to
obtain a linear CAPM, we can no longer hope to price
options, since option returns are non-normally
distributed
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Why bother linearizing a model?
•
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What about the wealth portfolio?
• To own a (share of) the consumption stream, you have
to own not only all stocks,but all bonds, real estate,
privately held capital, publicly held capital (roads, parks,
etc.), and human capital.
• Clearly, the CAPM is a poor defense of common proxies
such as the value-weighted NYSE portfolio.
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Implicit consumption-based models
•
mt 1  uct 1  / uct 
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Ex-post returns
• The log utility model also allows us for the first time to
look at what moves returns ex-post as well as ex-ante.
RtW1 
ct 1
ct
• 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.
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Identity of state variables
•
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Arbitrage Pricing Theory (APT)
• The intuition behind the APT is that the completely
idiosyncratic movements in asset 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 security’s covariance with the
common components or “factors” only.
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• The APT models the tendency of asset payoffs (returns) to
move together via a statistical factor decomposition
M
x  ai   ij f j   i  ai  i f   i
i
j 1
• Define
~
f  f  E f

• So,
 
M
~
xi  E xi    ij f j   i
j 1
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•
 
 0
~
E  i   0; E  i f j  0
Ei j

cov  x i , x j   E   i f   i

  i  j 2 
 
2

  i if i  j
f 

0 if i  j
j

f  j 

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• Thus, with N= number of securities, the N(N-1)/2 elements of
a variance-covariance matrix are described by N betas, and
N+1 variances.
 12

cov x, x    2  f    0
 0

0
 22
0
0

0

• With multiple (orthogonalized) factors, we obtain
covx, x  11 2  f1    2  2 2  f 2 
   diagonalm trix
a 
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• If we know the factors we want to use ahead of time, we
can estimate a factor structure by running regressions.
• If we don’t, we use factor analysis to estimate the factor
model.
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Exact factor pricing
•
 
~
xi  E xi 1  i f
 
i
p x
 
 
 
~

 E x p1  i p f
i

 
~
E Ri  R f  i  R f p f  R f  i
Approximate APT using the law of
one price
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• 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. There is
reason to hope that the APT holds approximately, especially
for reasonably large portfolios.
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• Suppose
 
~
x  E x 1  i f   i
i
i
• Again take prices of both sides,
 
 
   
~
p x  E x p1  ip f  E m i
i
i
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Limiting arguments
•
 
 
var xi  vari f   var  i
 
 
var  i
2

1

R
var xi
Asset Pricing
Zheng Zhenlong
• These two theorems can be interpreted to say that the APT
holds approximately (in the usual limiting sense) for either
portfolios that naturally have high R2, or well-diversified
portfolios in large enough markets.
Asset Pricing
Zheng Zhenlong
Law of one price arguments fail
•
Asset Pricing
Zheng Zhenlong
• Remark: 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.
the law of one price: arbitrage and
Sharpe ratios
Asset Pricing
Zheng Zhenlong
• The approximate APT based on the law of one price fell apart
because we could always choose a discount factor sufficiently
“far out” to generate an arbitrarily large price for an
arbitrarily small residual.
• But those discount factors are surely “unreasonable.” Surely,
we can rule them out.
Asset Pricing
Zheng Zhenlong
•
 
m  E m 2   2 m   E m    2 m   1 / R 2f
2
Asset Pricing
Zheng Zhenlong
Theorem
•
Asset Pricing
Zheng Zhenlong
APT vs. ICAPM
• Factor structure can imply factor pricing (APT), but factor
pricing does not require a factor structure.
• High R2 in time-series regressions of the returns on the factors
may imply factor pricing (APT), but again are not necessary
(ICAPM).
Asset Pricing
Zheng Zhenlong
• 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 find
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.
Asset Pricing
Zheng Zhenlong
•
Asset Pricing
Zheng Zhenlong