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 11 ' ++ += tt fbam 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 bRam += w R Two-period quadratic utility ? Investor have a quadratic preferences and live only for two periods; ]*)[( 2 1 *)( 2 1 ),( 2 1 2 1 ccEccccU tttt ????= ++ β w ttt w t t tt tt tt w t t t t t t Rba R cc cw cc c cc ccwR cc cc cU cU m 1 1 1 11 1 * )(* * *)( * * )(' )(' + + + ++ + += ? ? + ? ? = ? ?? = ? ? == ββ β ββ 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 eEcUE α? ?= 2/)()( 22 ))(( ccE ecUE σαα +? ?= f R fyyw RyRyc f ff ' ' += += yyREyRy ff ecUE Σ++? ?= '2/)]('[ 2 ))(( αα Exponential utility, Normal distributions--continued ? The optimal amount invested in risky asset is ? Sensibly, the investor invest more in risky assets if their expected returns are higher; ? Less if his risky aversion coefficient is higher; ? Less if assets are more risky. ? The amount invested in risky assets is independent of level of wealth, so we say absolute rather than relative( to wealth risk) aversion; ? Note also that these demand for risky assets are linear in expected returns. ? Inverting the first-order conditions, we obtain ? If all investors are identical, then the market portfolio is the same as the individual portfolio, and also gives the correlation of each return with α f RRE y ? Σ= ? )( 1 ),cov()( wf RRyRRE αα =Σ=? yΣ RyRyR ffm '+= 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. )()( 2 wfw RRRE ασ=? Quadratic value function, Dynamic programming ? Since investor like for more than two periods, we have to use multi period assumptions; ? Let us start by writing the utility function as this period consumption and next period’s wealth: ? His first-order condition is ? The discount factor is ? Suppose the value function is quadratic ? Then, we would have ? Or , once again ? ） 1 ()( + += ttt WVEcuU β ])('[ 11 ++ = tttt xWVEp β )(' )(' 1 1 t t t cu WV m + + =β 2 11 *)( 2 )( WWWV tt ??= ++ η W t t tt t t tt W t t t t R cu cW cu W cu WcWR cu WW m 1 1 * 1 1 )(' )( )(' * )(' *)( )(' + ++ + ? ? ? ? ? ? ? ?+ ? ? ? ? ? ? = ?? ?= ? ?= βηβη βηβη W tttt Rbam += +1 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. 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 and have 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 jt j t cuEU β 0 w w R {} 11;);(.. )()( '' 11 0,...,...,, max 11 ==?= = ++ ∞ = +∑ ++ v ttt w ttt w tt j jt j t wwcc t RRcWRWts cuEWV tttt ωω β ， 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 {} { } ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? += ∑ ∞ = +++ ++++ 0 11 ,...,,...,,, )()()( maxmax 2121 j jt j t wwcc tt wc t cuEEcuWV tttttt ββ {} { })()()( 1 , max + += ttt wc t WVEcuWV tt β Why is the value function quadratic-- continued ? For quadratic utility, the value function is also quadratic. ? Let us specify ? Guess ? The problem is ? Substituting the constant into the objective ? The first-order condition with respect to is ? 2* )( 2 1 )( cccu tt ??= 2* 11 )( 2 )( WWWV tt ??= ++ η {} )(.. *)( 2 *)( 2 1 )( 11 2 1 2 max tt W tt tt c t cWRWts WWEccWV t ?= ? ? ? ? ? ? ????= ++ + η β {} [ ] ? ? ? ? ? ? ?????= + 2 1 2 *)( 2 *)( 2 1 )( max WcWREccWV tt W tt c t t η β t c [ ]{ } W ttt W tt RWcWREcc 11 *) ? (* ? ++ ??=? βη Why is the value function quadratic-- continued ? Solving ? We obtain ? This is a linear function of ? Writing value function in terms of optimal value of c, we get ? This is a quadratic function of t c ? )(1 )(*)(* ? 2 1 2 11 W t t W t W t t RE WREWREc c + ++ + +? = βη βηβη t W 2 1 2 *])?([ 2 *)?( 2 1 )( WcWREccWV tt W ttt ?????= + η β t W Log utility ? The point of CAPM is to avoid to use consumption data, to use wealth or return on wealth instead. ? Log utility is another special cases that allow this substitution. Log utility is much more plausible than quadratic utility. Rubinstein (1976) introduce log utilty CAPM. ? Suppose that the investor has log utility ? The return on the wealth portfolio is proportional to consumption growth ? )ln()( ccu = t jj jt jt t j tjt t jt j t W t cc c c Ec cu cu Ep ∑∑ ∞ = ∞ = + + + + ? === 11 1)(' )(' β β ββ )(' )('1 )1/( )1)1/(( 1 111 1 + +++ + = ? +? = + = t t t t W t t W t W t cu cu c c p cp R βββ ββ Log utility--continued ? So the discount factor equals the inverse of the wealth portfolio return ? Note that log utility is the only assumption so far, we don’t assume constant interest rate, I.I.d. returns, or the absence of labor income. W t t R m 1 1 1 + + = Linearizing 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-Talyor expansion ?From ? We have )( 11 ++ = tt fgm ))())((('))(( 11111 +++++ ?+≈ tttttttt fEffEgfEgm Linearizing any model---continuous-time ? In continuous-time, we can derive exact linearization; ? Write the nonlinear discount factor as ? The basic pricing equation in continuous time for asset ? For a short discrete tome interval ),( tfg tt =Λ 2 2 2 2 1),( tt t t df f g df f tfg dt t g d ? ? + ? ? + ? ? =Λ i ( ) ? ? ? ? ? ? ? ? ? ? ?= ? ? ? ? ? ? ? ? Λ Λ ?=?+ ? ? ? ? ? ? ? ? t i t i t t t t t i t i t t f t i t i t i t i t t df p dp E f tfg tfg d p dp Edtrdt p D p dp E , ),( 1 f ttfi t t i tt f t i tt f tfg tfg fRRRE λβ ;,111 ) ),( ),( 1 )(,(cov)( ≈ ? ? ?≈? +++ Consumption-based model ? We have ? And hence ? For a short discrete time interval γδ ?? =Λ t t t ce 2 2 )1( t t t t t t c dc c dc dt d ++??= Λ Λ γγγδ ? ? ? ? ? ? ? ? =?+ ? ? ? ? ? ? ? ? t t i t i t t f t i t i t i t i t t c dc p dp Edtrdt p D p dp E γ c ttci t t i tt f t i tt c c RRRE ? ? + ++ ≈≈? λβγ ;, 1 11 ),(cov)( Intertemporal capital asset pricing model ? ICAPM generates linear discount factor models ? The factors are states variables for investor’s consumption-portfolio decision; ? The states variables are the variables that determine how well the investors can do in his maximization: current wealth is a state variable, additional variables describe the conditional distribution of asset returns the gent will face in the future,or shifts in the investment opportunity set. In mutiple-good models, relative price changes are also state variables. ? Optimal consumption is a function of the state variables, ? We can use this fact again to substitute out consumption ? 11 ' ++ += tt fbam )( tt zgc = ))((' ))((' 1 1 t t t zgu zgu m + + =β Intertemporal capital asset pricing model--continued ? Value function depend on state variables ? So we can write ? For simple, let us start with continuous-time model, ? We have ? the elasticity of marginal value with respect to wealth is also called the coefficient of relative risk aversion ),( 11 ++ tt zWV ),(/),( 111 ttWttWt zWVzWVm +++ =β ),( ttW t t zWVe δ? =Λ sderivativeonddz zWV zWV W dW zWV zWVW dtd t ttW ttWz ttW ttWWt tt sec ),( ),( ),( ),( / +++?=ΛΛ δ ),( ),( ttW ttWW t zWV zWWV rra ?= Intertemporal capital asset pricing model-continued ? We often simply approximate the continuous-time result as ? We can substitute covariance with the wealth portfolio in place of covariance with wealth ---shocks to the two are the same, and we can use factor-mimicking portfolio for other factors as well. ? Factor-mimicking portfolio are interesting for portfolio advice, because they give the purest way of hedging against of profiting from state variable risk exposure. ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? =?+ i t i t t tW tWz i t i t t t t f t i t i t i t i t p dp dzE V V p dp W dW Erradtrdt p D p dp E , , ),(cov),(cov)( 11111 +++++ ?+?≈? t i ttt i ttt f t i tt zRWRrraRRE zt λ Comments 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 which should 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 ba , W Rm /1= W tttt Rbam 11 ++ += γβ, Comments on the CAPM and ICAPM---continued ? Identify the factors: it is a art!