OverviewProjectionsPowerPoint Presentation…Projections…Slide 5Expected Value and Co-Variance……Expected Value and Co-VarianceSlide 8New Notation (LeRoy & Werner)Pricing Kernel kq……Pricing Kernel - Examples……Pricing Kernel – UniquenessExpectations Kernel keMean Variance FrontierFrontier Returns…Slide 16Slide 17Slide 18…Frontier ReturnsMinimum Variance PortfolioMean-Variance Efficient ReturnsZero-Covariance Frontier ReturnsSlide 23Slide 24Slide 25Beta Pricing……Beta PricingCapital Asset Pricing Model……Capital Asset Pricing Model09:3209:32 Lecture 07 Lecture 07 Mean-Variance Analysis and CAPM Mean-Variance Analysis and CAPM (Derivation with Projections)(Derivation with Projections)Fin 501: Asset PricingFin 501: Asset PricingOverviewOverview•Simple CAPM with quadratic utility functionsSimple CAPM with quadratic utility functions(derived from state-price beta model)(derived from state-price beta model)•Mean-variance preferences–Portfolio Theory–CAPM (traditional derivation)•With risk-free bond•Zero-beta CAPM•CAPM (modern derivation)–Projections–Pricing Kernel and Expectation Kernel09:3209:32 Lecture 07 Lecture 07 Mean-Variance Analysis and CAPM Mean-Variance Analysis and CAPM (Derivation with Projections)(Derivation with Projections)Fin 501: Asset PricingFin 501: Asset PricingProjectionsProjections•States s=1,…,S with s >0 •Probability inner product• -norm (measure of length)09:3209:32 Lecture 07 Lecture 07 Mean-Variance Analysis and CAPM Mean-Variance Analysis and CAPM (Derivation with Projections)(Derivation with Projections)Fin 501: Asset PricingFin 501: Asset Pricing)shrinkaxesx xyyx and y are -orthogonal iff [x,y] = 0, I.e. E[xy]=009:3209:32 Lecture 07 Lecture 07 Mean-Variance Analysis and CAPM Mean-Variance Analysis and CAPM (Derivation with Projections)(Derivation with Projections)Fin 501: Asset PricingFin 501: Asset Pricing……Projections…Projections…• Z space of all linear combinations of vectors z1, …,zn•Given a vector y 2 RS solve•[smallest distance between vector y and Z space]09:3209:32 Lecture 07 Lecture 07 Mean-Variance Analysis and CAPM Mean-Variance Analysis and CAPM (Derivation with Projections)(Derivation with Projections)Fin 501: Asset PricingFin 501: Asset PricingyyZE[ zj]=0 for each j=1,…,n (from FOC)? z yZ is the (orthogonal) projection on Zy = yZ + ’ , yZ 2 Z,  ? z……ProjectionsProjections09:3209:32 Lecture 07 Lecture 07 Mean-Variance Analysis and CAPM Mean-Variance Analysis and CAPM (Derivation with Projections)(Derivation with Projections)Fin 501: Asset PricingFin 501: Asset PricingExpected Value and Co-Variance…Expected Value and Co-Variance…squeeze axis by x(1,1)[x,y]=E[xy]=Cov[x,y] + E[x]E[y][x,x]=E[x2]=Var[x]+E[x]2 ||x||= E[x2]½09:3209:32 Lecture 07 Lecture 07 Mean-Variance Analysis and CAPM Mean-Variance Analysis and CAPM (Derivation with Projections)(Derivation with Projections)Fin 501: Asset PricingFin 501: Asset Pricing……Expected Value and Co-Variance Expected Value and Co-Variance E[x] = [x, 1]=09:3209:32 Lecture 07 Lecture 07 Mean-Variance Analysis and CAPM Mean-Variance Analysis and CAPM (Derivation with Projections)(Derivation with Projections)Fin 501: Asset PricingFin 501: Asset PricingOverviewOverview•Simple CAPM with quadratic utility functionsSimple CAPM with quadratic utility functions(derived from state-price beta model)(derived from state-price beta model)•Mean-variance preferences–Portfolio Theory–CAPM (traditional derivation)•With risk-free bond•Zero-beta CAPM•CAPM (modern derivation)–Projections–Pricing Kernel and Expectation Kernel09:3209:32 Lecture 07 Lecture 07 Mean-Variance Analysis and CAPM Mean-Variance Analysis and CAPM (Derivation with Projections)(Derivation with Projections)Fin 501: Asset PricingFin 501: Asset PricingNew Notation (LeRoy & Werner)New Notation (LeRoy & Werner)•Main changes (new versus old)– gross return: r = R– SDF:  = m– pricing kernel: kq = m*– Asset span: M = <X>– income/endowment: wt = et09:3209:32 Lecture 07 Lecture 07 Mean-Variance Analysis and CAPM Mean-Variance Analysis and CAPM (Derivation with Projections)(Derivation with Projections)Fin 501: Asset PricingFin 501: Asset PricingPricing Kernel kPricing Kernel kqq……• M space of feasible payoffs.•If no arbitrage and  >>0 there exists SDF  2 RS,  >>0, such that q(z)=E( z).•  2 M – SDF need not be in asset span. •A pricing kernel is a kq 2 M such that for each z 2 M, q(z)=E(kq z).•(kq = m* in our old notation.)09:3209:32 Lecture 07 Lecture 07 Mean-Variance Analysis and CAPM Mean-Variance Analysis and CAPM (Derivation with Projections)(Derivation with Projections)Fin 501: Asset PricingFin 501: Asset Pricing……Pricing Kernel - Examples…Pricing Kernel - Examples…•Example 1:–S=3,s=1/3 for s=1,2,3, –x1=(1,0,0), x2=(0,1,1), p=(1/3,2/3).– Then k=(1,1,1) is the unique pricing kernel. •Example 2:–S=3,s=1/3 for s=1,2,3, –x1=(1,0,0), x2=(0,1,0), p=(1/3,2/3).–Then k=(1,2,0) is the unique pricing kernel.09:3209:32 Lecture 07 Lecture 07 Mean-Variance Analysis and CAPM Mean-Variance Analysis and CAPM (Derivation with Projections)(Derivation with Projections)Fin 501: Asset PricingFin 501: Asset Pricing……Pricing Kernel – UniquenessPricing Kernel – Uniqueness•If a state price density exists, there exists a unique pricing kernel.–If dim(M) = m (markets are complete), there are exactly m equations and m unknowns–If dim(M) · m, (markets may be incomplete) For any state price density (=SDF)  and any z 2 M E[(-kq)z]=0 =(-kq)+kq ) kq is the ``projection'' of  on M .•Complete markets ), kq= (SDF=state price density)09:3209:32 Lecture 07 Lecture 07 Mean-Variance Analysis and CAPM Mean-Variance Analysis and CAPM (Derivation with Projections)(Derivation with Projections)Fin 501: Asset PricingFin 501: Asset PricingExpectations Kernel kExpectations Kernel kee•An expectations kernel is a vector ke2 M –Such that E(z)=E(ke z) for each z 2 M.•Example –S=3, s=1/3, for s=1,2,3, x1=(1,0,0), x2=(0,1,0). –Then the unique $ke=(1,1,0).$•If  >>0, there exists a unique expectations kernel.•Let e=(1,…, 1) then for any z2 M•E[(e-ke)z]=0•ke is the “projection” of e on M•ke = e if bond can be replicated (e.g. if markets are complete)09:3209:32 Lecture 07 Lecture 07 Mean-Variance