复旦大学 研究生投资学讲义 CHPT13- Factor pricing model--CAPM

Chapter 13 Factor pricing model

Fan Longzhen

Introduction

•The consumption-based model as a complete answer to most asset pricing question in principle, does not work well in practice;

•This observation motivates effects to tie the discount factor m to other data;

•Linear factor pricing models are most popular models of this sort in finance;

•They dominate discrete-time empirical work.

Factor pricing models

•Factor pricing models replace the consumption-based expression for marginal utility growth with a linear model of the form

•The key question: what should one use for factors 1

1'+++=t t f b a m 1

+t f

Capital asset pricing model (CAPM)•

CAPM is the model , is the wealth portfolio return.•

Credited Sharpe (1964) and Linterner (1965), is the first, most famous, and so far widely used model in asset pricing.•

Theoretically, a and b are determined to price any two assets, such as market portfolio and risk free asset.•

Empirically, we pick a,b to best price larger cross section of assets;•

We don’t have good data, even a good empirical definition for wealth portfolio, it is often deputed by a stock index;•

We derive it from discount factor model by •

(1)two-periods, exponential utility, and normal returns; •

(2) infinite horizon, quadratic utility, and normal returns;•

(3) log utility •(4) by seeing several derivations, you can see how one assumption can be traded for another. For example, the CAPM does not require normal distributions, if one is willing to swallow quadratic utility instead.w

bR a m +=w R

Exponential utility, Normal distributions

•We present a model with consumption only in the last period, utility is

•

If consumption is normally distributed, we have •

Investor has initial wealth w, which invest in a set of risk-free assets with return and a set of risky assets paying return R.•Let y denote the mount of wealth w invested in each asset, the budget constraint is •Plugging the first constraint into the utility function, we obtain

][)]([c e

E c U E α−−=2/)()(22))((c c E e c U E σαα+−−=f R f y y w R

y R y c f

f f ''+=+=y

y R E y R y f f e c U E Σ++−−='2/)]('[2))((αα

Exponential utility, Normal distributions--continued •Applying the formula to market return itself, we have

•The model ties price of market risk to the risk aversion coefficient.)

()(2w

f w R R R E ασ=−

Quadratic value function, Dynamic

programming-continued

•(1) the value function only depends on wealth. If other variables enter the value function, m would depend on other variables. The ICAPM, allows other variables in the value function, and obtain more factors.•(other variable can enter the function, so long as they do not affect marginal utility value of wealth.)

•(2) the value function is quadratic, we wanted the marginal value function is linear.