Fin 501: Asset Pricing 01:36 Lecture 05State-price Beta Model1 Lecture 05: State-price BETA Model Prof. Markus K. Brunnermeier

Fin 501: Asset Pricing 01:36 Lecture 05State-price Beta Model2 Overview Risk-adjustment in payoffs Risk-adjustment in returns State price beta model Different specific asset pricing models

Fin 501: Asset Pricing 01:36 Lecture 05State-price Beta Model3 Risk-adjustment in payoffs p = E[mx j ] = E[m]E[x] + Cov[m,x] Since 1=E[mR], the risk free rate is R f = 1/E[m] p = E[x]/R f + Cov[m,x] Remarks: (i)If risk-free rate does not exist, R f is the shadow risk free rate (ii)In general Cov[m,x] < 0, which lowers price and increases return ) E[m(R j -R f )]=0

Fin 501: Asset Pricing 01:36 Lecture 05State-price Beta Model4 Risk-adjustment in Returns E[mR j ]=1R f E[m]=1 ) E[m(R j -R f )]=0 E[m]{E[R j ]-R f } + Cov[m,R j ]=0 E[R j ] – R f = - Cov[m,R j ]/E[m](2) also holds for portfolios h Note: risk correction depends only on Cov of payoff/return with discount factor. Only compensated for taking on systematic risk not idiosyncratic risk.

Fin 501: Asset Pricing 01:36 Lecture 05State-price Beta Model5 c1c1 State-price BETA Model shrink axes by factor m m* R* p=1 (priced with m * ) R * =  m * let underlying asset be x=(1.2,1)

Fin 501: Asset Pricing 01:36 Lecture 05State-price Beta Model6 State-price BETA Model E[R j ] – R f = - Cov[m,R j ]/E[m](2) also holds for all portfolios h and we can replace m with m * Suppose (i) Var[m * ] > 0 and (ii) R * =  m * with  > 0 E[R j ] – R f = - Cov[R *,R h ]/E[R * ](2’) Define  h := Cov[R *,R h ]/ Var[R * ] for any portfolio h Regression R h s =  h +  h (R * ) s +  s with Cov[R *,  ]=E[  ]=0

Fin 501: Asset Pricing 01:36 Lecture 05State-price Beta Model7 State-price BETA Model (2) for R * : E[R * ]-R f =-Cov[R *,R * ]/E[R * ] =-Var[R * ]/E[R * ] (2) for R h : E[R h ]-R f =-Cov[R *,R h ]/E[R * ] = -  h Var[R * ]/E[R * ] E[R h ] - R f =  h E[R * - R f ] where  h := Cov[R *,R h ]/Var[R * ] very general – but what is R * in reality ?

Fin 501: Asset Pricing 01:36 Lecture 05State-price Beta Model8 Different Asset Pricing Models p t = E[m t+1 x t+1 ] ) where m t+1 =f( ¢,…, ¢ ) f( ¢ ) = asset pricing model General Equilibrium f( ¢ ) = MRS /  Factor Pricing Model a+b 1 f 1,t+1 + b 2 f 2,t+1 CAPM a+b 1 f 1,t+1 = a+b 1 R M E[R h ] - R f =  h E[R * - R f ] where  h := Cov[R *,R h ]/Var[R * ] CAPM R * = R M = return of market portfolio

Fin 501: Asset Pricing 01:36 Lecture 05State-price Beta Model9 Different Asset Pricing Models Theory  All economics and modeling is determined by m t+1 = a + b’ f  Entire content of model lies in restriction of SDF Empery  m * (which is a portfolio payoff) prices as well as m (which is e.g. a function of income, investment etc.)  measurement error of m * is smaller than for any m  Run regression on returns (portfolio payoffs)! (e.g. Fama-French three factor model)