chapter 9: factor pricing models

Asset Pricing Zheng Zhenlong

Chapter 9:Factor pricing models

Asset Pricing Zheng ZhenlongContents

• Introduction• CAPM• ICAPM• Comments on the CAPM and ICAPM• APT• APT vs. ICAPM

Asset Pricing Zheng ZhenlongBrief introduction

•

1

1

tt

t fbacu

cu

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


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


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.

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.


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,

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


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


• the budget constraint is

11 tt Wc

ttWtt cWRW 11

N

i

iti

Wt RwR

111

N

iiw

1

1


• 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


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


• the budget constraint is

RyRyc ff

1yyW f


• ( ) ( ) ( )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


Quadratic value function, dynamic programming

• first order condition

• So,

1 ttt WVEcuU

11 ttttt xWVEcup

11

tt

t

V Wm

u c


• 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


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


• Value functions allow you to express an infinite period problem as a two period problem

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..


•

1 1ˆ ( )W Wt t t t tc c E R W c W R

2

1

211

1ˆ

Wt

tWt

Wt

t REWREWREcc


•

212 ˆ5.0ˆ5.0

WcWREccWV tt

Wttt

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


• 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.”

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

Asset Pricing Zheng ZhenlongTaylor approximation

• The most obvious way to linearize the model is by a Taylor approximation

1111

11 )(

ttttttt

tt

fEffEgfEgfgm

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

tit

it

tf

tit

it

it

it

t

dfp

dpEf

tfgtfg

dp

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

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


• 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


( ) ( )( )( ) ( ) ( )( ) ( ) ( ) ( )

( ) ( ) ( ){ }( )( ) ( ) ( ) ( ){ }( )

( ) ( )( )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

= = +


• 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

R E g f R f

R


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

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


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


• the value function depends on the state variables

• so we can write 11, tt zWV

ttW

ttWt zWV

zWVm,, 11

1


• Start from

• We have

ttWt

t zWVe ,

...

,,

,,

tttW

ttWz

tttW

ttWWt

t

t

dzzWVzWV

WdW

zWVzWVWdtd


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

,

,


• Thus, in discrete time

11

111

,cov

,cov

tittzt

t

tittt

fitt

zR

WWRrraRRE


9.3 Comments on the CAPM and ICAPM


Is the CAPM conditional or unconditional?

•


•


• 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

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

Asset Pricing Zheng ZhenlongWhy bother linearizing a model?

•

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.

Asset Pricing Zheng ZhenlongImplicit consumption-based models

•

ttt cucum /11


• 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

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 .


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.

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.


• 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


•

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


• 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


• 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.

Asset Pricing Zheng ZhenlongExact factor pricing

• ( )1i i

ix E x fb¢= + %

fppxExp iii ~1

iff

ifi RfpRRRE

~


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.


• Suppose

• Again take prices of both sides, i

iii fxEx

~1

ii

ii mEfppxExp ~1

Asset Pricing Zheng ZhenlongLimiting arguments

•

ii

i fx varvarvar

21varvar R

xi

i


• 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 ZhenlongLaw of one price arguments fail

•


• 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

• 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.


•

22222 /1 fRmmEmmEm

Asset Pricing Zheng ZhenlongTheorem

•

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.

chapter 9: factor pricing models

Documents

essence of asset pricing

factor pricing models

factor models

predictable factors

quadratic preferences

normal returns

marginal utility growthprove

candidate factor