PowerPoint PresentationOverviewRisk-adjustment in payoffsRisk-adjustment in ReturnsState-price BETA ModelSlide 6Slide 7Different Asset Pricing ModelsSlide 901:0501:05 Lecture 05 Lecture 05 State-price Beta ModelState-price Beta Model 1Fin 501: Asset PricingFin 501: Asset PricingLecture 05: State-price BETA ModelLecture 05: State-price BETA ModelProf. Markus K. Brunnermeier01:0501:05 Lecture 05 Lecture 05 State-price Beta ModelState-price Beta Model 2Fin 501: Asset PricingFin 501: Asset PricingOverviewOverview•Risk-adjustment in payoffs•Risk-adjustment in returns•State price beta model•Different specific asset pricing models01:0501:05 Lecture 05 Lecture 05 State-price Beta ModelState-price Beta Model 3Fin 501: Asset PricingFin 501: Asset PricingRisk-adjustment in payoffsRisk-adjustment in payoffsp = E[mxj] = E[m]E[x] + Cov[m,x]Since 1=E[mR], the risk free rate is Rf = 1/E[m] p = E[x]/Rf + Cov[m,x]Remarks: (i) If risk-free rate does not exist, Rf is the shadow risk free rate(ii) In general Cov[m,x] < 0, which lowers price and increases return) E[m(Rj-Rf)]=001:0501:05 Lecture 05 Lecture 05 State-price Beta ModelState-price Beta Model 4Fin 501: Asset PricingFin 501: Asset PricingRisk-adjustment in ReturnsRisk-adjustment in ReturnsE[mRj]=1 Rf E[m]=1) E[m(Rj-Rf)]=0E[m]{E[Rj]-Rf} + Cov[m,Rj]=0E[Rj] – Rf = - Cov[m,Rj]/E[m] (2)also holds for portfolios hNote: •risk correction depends only on Cov of payoff/return with discount factor.•Only compensated for taking on systematic risk not idiosyncratic risk.01:0501:05 Lecture 05 Lecture 05 State-price Beta ModelState-price Beta Model 5Fin 501: Asset PricingFin 501: Asset Pricing<X>c1State-price BETA ModelState-price BETA Modelshrink axes by factormm*R*p=1(priced with m*)R*= m*let underlying asset be x=(1.2,1)01:0501:05 Lecture 05 Lecture 05 State-price Beta ModelState-price Beta Model 6Fin 501: Asset PricingFin 501: Asset PricingState-price BETA ModelState-price BETA ModelE[Rj] – Rf = - Cov[m,Rj]/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[Rj] – Rf = - Cov[R*,Rh]/E[R*] (2’)Define h := Cov[R*,Rh]/ Var[R*] for any portfolio h Regression Rhs = h + h (R*)s + s with Cov[R*,]=E[]=001:0501:05 Lecture 05 Lecture 05 State-price Beta ModelState-price Beta Model 7Fin 501: Asset PricingFin 501: Asset PricingState-price BETA ModelState-price BETA Model(2) for R*: E[R*]-Rf=-Cov[R*,R*]/E[R*] =-Var[R*]/E[R*](2) for Rh: E[Rh]-Rf=-Cov[R*,Rh]/E[R*] = - h Var[R*]/E[R*]E[RE[Rhh] - R] - Rf f = = hh E[R E[R**- R- Rff]]where h := Cov[R*,Rh]/Var[R*]very general – but what is Rvery general – but what is R** in reality in reality??01:0501:05 Lecture 05 Lecture 05 State-price Beta ModelState-price Beta Model 8Fin 501: Asset PricingFin 501: Asset PricingDifferent Asset Pricing ModelsDifferent Asset Pricing Modelspt = E[mt+1 xt+1] )where mt+1=f(¢,…,¢) f(¢) = asset pricing modelGeneral Equilibrium f(¢) = MRS / Factor Pricing Model a+b1 f1,t+1 + b2 f2,t+1CAPM a+b1 f1,t+1 = a+b1 RME[RE[Rhh] - R] - Rf f = = hh E[R E[R**- R- Rff]]where h := Cov[R*,Rh]/Var[R*]CAPMR* = RM = return of market portfolio01:0501:05 Lecture 05 Lecture 05 State-price Beta ModelState-price Beta Model 9Fin 501: Asset PricingFin 501: Asset PricingDifferent Asset Pricing ModelsDifferent Asset Pricing Models•TheoryAll economics and modeling is determined by mt+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