chapter 9: factor pricing models
DESCRIPTION
Chapter 9: Factor pricing models. Contents. Introduction CAPM ICAPM Comments on the CAPM and ICAPM APT APT vs. ICAPM. Brief introduction. Brief introduction. - PowerPoint PPT PresentationTRANSCRIPT
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Asset Pricing Zheng Zhenlong
Chapter 9:Factor pricing models
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Asset Pricing Zheng ZhenlongContents
• Introduction• CAPM• ICAPM• Comments on the CAPM and ICAPM• APT• APT vs. ICAPM
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Asset Pricing Zheng ZhenlongBrief introduction
•
1
1
tt
t fbacu
cu
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Asset Pricing Zheng ZhenlongBrief 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
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Asset Pricing Zheng Zhenlong
Should factors be unpredictable over time?
• 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.
• 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.
11 1
1( ) [ ( )] tft t t tf
t
u cu c R E u c
u c R
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Asset Pricing Zheng Zhenlong
The derivations of factor pricing model
• 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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Asset Pricing Zheng Zhenlongguard 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.
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Asset Pricing Zheng Zhenlong
Capital Asset Pricing Model (CAPM)
• wealth portfolio return.• In expected return / beta language,
• CAPM can be derived from consumption-based model by different assumption.
1 1W
t tm a bR WtR 1
WRi
i RERE W,
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Asset Pricing Zheng ZhenlongDifferent 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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Asset Pricing Zheng Zhenlong
Two-period quadratic utility, no labor income
• Investors have quadratic preferences and only live two periods,
• marginal rate of substitution is thus
21
21 5.05.0,
ccEccccU tttt
111
ttt
t t
c cu cm
u c c c
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Asset Pricing Zheng Zhenlong
• the budget constraint is
11 tt Wc
ttWtt cWRW 11
N
i
iti
Wt RwR
111
N
iiw
1
1
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Asset Pricing Zheng Zhenlong
• Just as
11 1
Wt t t t t W
t tt tt
c R W c W ccm Rc c c cc c
1 1W
t t t tm a b R
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Asset Pricing Zheng Zhenlong
Exponential utility, normal distributions, no labor income
• If consumption only in the last period and is normally distributed, we have
• a is the coefficient of absolute risk aversion.
aceEcuE
( ) ( ) ( )2 2/ 2aE c a cE u c e séù- +êúëûé ù= -ê úë û
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Asset Pricing Zheng Zhenlong
• the budget constraint is
RyRyc ff
1yyW f
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Asset Pricing Zheng Zhenlong
• ( ) ( ) ( )2/ 2ffa y R y E R a y yE u c e
é ù¢ ¢- + +ê úë û åé ù= -ê úë û
a
RREyf
1
Wf RRayaRRE ,cov
2 ( )W f WE R R a R
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Asset Pricing Zheng Zhenlong
Quadratic value function, dynamic programming
• first order condition
• So,
1 ttt WVEcuU
11 ttttt xWVEcup
11
tt
t
V Wm
u c
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Asset Pricing Zheng Zhenlong
• suppose the value function were quadratic,
• Then,
• 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.
211 2
WWWV tt
1 1
t t Wt t
t t
W cWm Ru c u c
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Asset Pricing Zheng Zhenlong
the existence of value function (Proof )• Suppose investors last forever, and have the standard sort of
utility function
• Define the value function as the maximized value of the utility function in this environment.
jtj
jt cuEU
0
jtj
jtwwcct cuEWV
tttt
0......,, 1,1
max
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Asset Pricing Zheng Zhenlong
• Value functions allow you to express an infinite period problem as a two period problem
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Asset Pricing Zheng ZhenlongWhy is the value function quadratic?
• Remark: quadratic utility function leads to a quadratic value function in this environment
• Specify:
• Guess:
• Thus,
25.0 cccu tt
211 5.0 WWWV tt
2 2
1max 0.5 0.5t
t t tcV W c c E W W
ttWtt cWRWts 11..
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Asset Pricing Zheng Zhenlong
•
1 1ˆ ( )W Wt t t t tc c E R W c W R
2
1
211
1ˆ
Wt
tWt
Wt
t REWREWREcc
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Asset Pricing Zheng Zhenlong
•
212 ˆ5.0ˆ5.0
WcWREccWV tt
Wttt
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Asset Pricing Zheng ZhenlongLog utility, no labor income
•
tjt
j jt
tjtjt
j t
jtjt
Wt cc
ccEc
cucu
Ep
111
1 1 1 11
1 1
/ 1 1 1/ 1
WtW t t t t
t Wt t t t t
u cp c c cRp c c u c m
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Asset Pricing Zheng Zhenlong
• 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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Asset Pricing Zheng ZhenlongHow 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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Asset Pricing Zheng ZhenlongTaylor approximation
• The most obvious way to linearize the model is by a Taylor approximation
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11 )(
ttttttt
tt
fEffEgfEgfgm
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Asset Pricing Zheng ZhenlongContinuous time limit
• If the discrete time is short enough, we can apply the continuous time result as an approximation
• For a short discrete time interval,
tfg tt ,
22
2
5.0,tt
tt df
fgdf
ftfgdt
tgd
tit
it
tt
t
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it
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it
it
t
dfp
dpEf
tfgtfg
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dpEdtrdtpD
pdpE
,,
1
1 1 1 , ;1 ( , )( ) cov ( , )( )
( , )i f i f
t t t t t t i f t tg f tE R R R f
g f t f
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Asset Pricing Zheng ZhenlongNormal distribution in discrete time
• Stein’s lemma : If f and R are bivariate normal, g(f) is differentiable and ,then fgE
RffgERfg ,cov,cov
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Asset Pricing Zheng Zhenlong
• Remark: If m=g(f), if f and a set of the payoffs priced by m are normally distributed returns, and if , then there is a linear model m=a+bf that prices the normally distributed returns.
fgE
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Asset Pricing Zheng Zhenlong
( ) ( )( )( ) ( ) ( )( ) ( ) ( ) ( )
( ) ( ) ( ){ }( )( ) ( ) ( ) ( ){ }( )
( ) ( )( )1 1
cov ,cov ,
( )
t t t t
p E mx E g f xE g f E x g f xE g f E x E g ff x
E E g f E g ff E f x
E E g f E g f E f E g ff x
E m x E a bf x+ +
= =é ù é ù= +ê ú ê úë û ë ûé ù é ù¢= +ê ú ê úë û ë û
é ù é ù¢= + -ê ú ê úë û ë ûé ù é ù é ù¢ ¢= - +ê ú ê ú ê úë û ë û ë û
= = +
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Asset Pricing Zheng Zhenlong
• Similar,it allows us to derive an expected return-beta model using the factors
1 1 1
1 1 1
, ;
cov ,
cov ,
i f it t t t t
f it t t t t t
f ft i f t t
E R R R m
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R
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Asset Pricing Zheng Zhenlong
Two period CAPM• Stein’s lemma allows us to substitute a
normal distribution assumption for the quadratic assumption in the two period CAPM.
• Assuming RWand Ri are normally distributed, we have:
1 11
( ) ( ( ))( ) ( )
Wt t t t
tt t
u c u R W cm
u c u c
11 1 1 1
( ) [ ( )]cov ( , ) [ ]cov ( , )( )
Wi i Wt t t t t
t t t t t tt
W c u R W cR m E R Ru c
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Asset Pricing Zheng ZhenlongLog 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
• If RW is normally distributed, E(1/RW2) does not exist. The Stein’s lemma condition is violated.
21 1 11
1( ) ( ) cov ( , )i f i Wt t t t tW
t
E R R E R RR
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Asset Pricing Zheng Zhenlong
Intertemporal Capital Asset Pricing Model (ICAPM)
• The ICAPM generates linear discount factor models
• in which the factors are “state variables” for the investor’s consumption-portfolio decision.
11 tt fbam
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Asset Pricing Zheng Zhenlong
• the value function depends on the state variables
• so we can write 11, tt zWV
ttW
ttWt zWV
zWVm,, 11
1
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Asset Pricing Zheng Zhenlong
• Start from
• We have
ttWt
t zWVe ,
...
,,
,,
tttW
ttWz
tttW
ttWWt
t
t
dzzWVzWV
WdW
zWVzWVWdtd
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Asset Pricing Zheng Zhenlong
Define the coefficient of relative risk aversion,
Then we obtain the ICAPM,
ttW
ttWWt zWV
zWWVrra,
,
ti
t
it
tW
tWz
t
tit
it
tf
tit
it
it
it
t dzp
dpEVV
WdW
pdpErradtrdt
pD
pdpE
,
,
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Asset Pricing Zheng Zhenlong
• Thus, in discrete time
11
111
,cov
,cov
tittzt
t
tittt
fitt
zR
WWRrraRRE
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Asset Pricing Zheng Zhenlong
9.3 Comments on the CAPM and ICAPM
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Asset Pricing Zheng Zhenlong
Is the CAPM conditional or unconditional?
•
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Asset Pricing Zheng Zhenlong
•
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Asset Pricing Zheng Zhenlong
• 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.
• 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.
1
11
1
1111 tWt
tWt
t RR
ERR
E
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Asset Pricing Zheng ZhenlongShould 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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Asset Pricing Zheng ZhenlongWhy bother linearizing a model?
•
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Asset Pricing Zheng ZhenlongWhat 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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Asset Pricing Zheng ZhenlongImplicit consumption-based models
•
ttt cucum /11
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Asset Pricing Zheng Zhenlong
• The log utility model also allows us for the first time to look at what moves returns ex-post as well as ex-ante.
• 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 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 our business.
t
tWt c
cR
11
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Asset Pricing Zheng ZhenlongIdentity of state variables
• The ICAPM does not tell us the identity of the state variables zt , leading Fama (1991) to characterize the ICAPM as a ‘‘fishing license.’’
• The ICAPM 并非全无道理。关键是要在选择变量时要遵守纪律 .
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Asset Pricing Zheng Zhenlong
Portfolio Intuition and Recession State Variables
• The covariance (or beta) of with measures how much a marginal increase in affects the portfolio variance. Modern asset pricing starts when we realize that investors care about portfolio returns, not about the behavior of specific assets. That is the central insight in CAPM.
• The ICAPM adds long investment horizons and time-varying investment opportunities to this picture. People are unhappy when news comes that future returns are lower, they will thus prefer stocks that do well on such news.
• Most current theorizing and empirical work, while citing the ICAPM, really considers another source of additional risk factors: Investors have jobs. Or they own houses and shares of small businesses. People with jobs will prefer stocks that don’t fall in recessions.
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Asset Pricing Zheng ZhenlongArbitrage 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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Asset Pricing Zheng Zhenlong
• The APT models the tendency of asset payoffs (returns) to move together via a statistical factor decomposition
• Define
• So,
iiiij
M
jiji
i fafax 1
fEff ~
ij
M
jij
ii fxEx
~1
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Asset Pricing Zheng Zhenlong
•
0~;0 jii fEE
0jiE
2
2
cov ,
0 i
i ji i j j
i j
x x E f f
if i jf
if i j
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Asset Pricing Zheng Zhenlong
• 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.
• With multiple (orthogonalized) factors, we obtain
000000
,cov 22
21
2
fxx
trixdiagonalma
ffxx
2
2221
211,cov
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Asset Pricing Zheng Zhenlong
• 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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Asset Pricing Zheng ZhenlongExact factor pricing
• ( )1i i
ix E x fb¢= + %
fppxExp iii ~1
iff
ifi RfpRRRE
~
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Asset Pricing Zheng Zhenlong
Approximate APT using the law of one price
• 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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Asset Pricing Zheng Zhenlong
• Suppose
• Again take prices of both sides, i
iii fxEx
~1
ii
ii mEfppxExp ~1
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Asset Pricing Zheng Zhenlong
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Asset Pricing Zheng ZhenlongLimiting arguments
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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.
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Asset Pricing Zheng ZhenlongLaw of one price arguments fail
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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.
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Asset Pricing Zheng Zhenlong
the law of one price: arbitrage and Sharpe ratios
• 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.
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Asset Pricing Zheng Zhenlong
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Asset Pricing Zheng ZhenlongTheorem
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Asset Pricing Zheng ZhenlongAPT vs. ICAPM
• In the ICAPM there is no presumption that factors f in a pricing model describe the covariance matrix of returns. The factors do not have to be orthogonal or i.i.d. either. High in time-series regressions of the returns on the factors may imply factor pricing (APT), but again are not necessary (ICAPM). Factors such as industry may describe large parts of returns’ 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 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.
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Asset Pricing Zheng Zhenlong