Chapter 13 Factor pricing modelFan LongzhenIntroduction•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 11'+++=t t f b a m 1+t fCapital 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.wbR a m +=w RExponential 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 eE c U E α−−=2/)()(22))((c c E e c U E σαα+−−=f R f y y w Ry R y c ff f ''+=+=yy R E y R y f f e c U E Σ++−−='2/)]('[2))((ααExponential 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.)()(2wf w R R R E ασ=−Quadratic value function, Dynamicprogramming-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.Why is the value function quadratic•Good economists are unhappy about a utility function that have wealth in it.•Suppose investors last forever, and have the standard sort of utility function •Investors start with wealth which earns a random return andhave no other source of income;•Suppose further that interest rate are constant and stock returns are iid over time.•Define the value function as the maximized value of the utility function in this environment•∑∞=+=0)(j j t jt c u E U β0w w R {}11;);(..)()(''110,...,...,,max11==−==++∞=+∑++vtt tw tt t w t t j j t jt w w c c t R R c W R W t s c u E W V t t t t ωωβ，Why is the value function quadratic--continued•Without the assumption of no labor income, a constant interest rate, and I.I.d returns come in, the value maybe depend on the environment.For example, if D/P indicates returns would be high for a while, the investor might be happier and have a high value.•Value functions allow you to express an infinite-period problem as a two-period problem. Breaking up the maximization into the first period and the remaining periods, as follows•Or{}{}⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧⎥⎦⎤⎢⎣⎡+=∑∞=+++++++11,...,,...,,,)()()(maxmax2121jjtjtwwccttwctcuEEcuWVttttttββ{}{})()()(1,max++=tttwctWVEcuWVttβLinearizing any model •Goal of linear model: derive variables that drive the discount factor; derive a linear relation between discount factor and these variables;•Following gives three standard tricks to obtain a linear model;Linearizing any model-Talyorexpansion•From •We have)(11++=ttfgm))())((('))((11111+++++−+≈ttttttttfEffEgfEgmComments on the CAPM and ICAPM•Is CAPM conditional or unconditional? Are the parameter changes as conditional information changes ?•The two-period quadratic utility-based deviation results in a conditional CAPM, since the parameters change over time.•The log utility CAPM hold both conditionally or unconditionally.•Should CAPM price options? The quadratic utility CAPM and log utility CAPM should apply to all payoffs.•Why linearize? Why not take the log utility model whichshould price any asset? Turn it into cannot price no normally distributed payoff and must be applied at short horizons.•it is simple to use regression to estimate in CAPM.•Now with GMM approach, nolinear discount factor model is easy to estimate.tt b a ,W R m /1=W t t t t R b a m 11+++=γβ,Comments on the CAPM and ICAPM---continued •Identify the factors: it is a art!。