Report

Fin 501: Asset Pricing Overview • Simple CAPM with quadratic utility functions (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 Kernel 16:27 Lecture 07 Mean-Variance Analysis and CAPM (Derivation with Projections) Fin 501: Asset Pricing Projections • States s=1,…,S with ps >0 • Probability inner product • p-norm 16:27 Lecture 07 (measure of length) Mean-Variance Analysis and CAPM (Derivation with Projections) Fin 501: Asset Pricing ) shrink axes y x y x x and y are p-orthogonal iff [x,y]p = 0, I.e. E[xy]=0 16:27 Lecture 07 Mean-Variance Analysis and CAPM (Derivation with Projections) Fin 501: Asset Pricing …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] 16:27 Lecture 07 Mean-Variance Analysis and CAPM (Derivation with Projections) Fin 501: Asset Pricing …Projections y e yZ E[e zj]=0 for each j=1,…,n (from FOC) e? z yZ is the (orthogonal) projection on Z Z + e’ , yZ 2 Z, e ? z y = y 16:27 Lecture 07 Mean-Variance Analysis and CAPM (Derivation with Projections) Fin 501: Asset Pricing Expected Value and Co-Variance… squeeze axis by (1,1) x 16:27 Lecture 07 [x,y]=E[xy]=Cov[x,y] + E[x]E[y] [x,x]=E[x2]=Var[x]+E[x]2 ||x||= E[x2]½ Mean-Variance Analysis and CAPM (Derivation with Projections) Fin 501: Asset Pricing …Expected Value and Co-Variance E[x] = [x, 1]= 16:27 Lecture 07 Mean-Variance Analysis and CAPM (Derivation with Projections) Fin 501: Asset Pricing Overview • Simple CAPM with quadratic utility functions (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 Kernel 16:27 Lecture 07 Mean-Variance Analysis and CAPM (Derivation with Projections) Fin 501: Asset Pricing New Notation (LeRoy & Werner) • Main changes (new versus old) – gross return: – SDF: – pricing kernel: r=R m=m kq = m* – Asset span: – income/endowment: M = <X> wt = et 16:27 Lecture 07 Mean-Variance Analysis and CAPM (Derivation with Projections) Fin 501: Asset Pricing Pricing Kernel kq… • M space of feasible payoffs. • If no arbitrage and p >>0 there exists SDF m 2 RS, m >>0, such that q(z)=E(m z). • m 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.) 16:27 Lecture 07 Mean-Variance Analysis and CAPM (Derivation with Projections) Fin 501: Asset Pricing …Pricing Kernel - Examples… • Example 1: – S=3,ps=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,ps=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. 16:27 Lecture 07 Mean-Variance Analysis and CAPM (Derivation with Projections) Fin 501: Asset Pricing …Pricing 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) m and any z 2 M E[(m-kq)z]=0 m=(m-kq)+kq ) kq is the ``projection'' of m on M. • Complete markets ), kq=m (SDF=state price density) 16:27 Lecture 07 Mean-Variance Analysis and CAPM (Derivation with Projections) Fin 501: Asset Pricing Expectations Kernel ke • An expectations kernel is a vector ke2 M – Such that E(z)=E(ke z) for each z 2 M. • Example – S=3, ps=1/3, for s=1,2,3, x1=(1,0,0), x2=(0,1,0). – Then the unique $ke=(1,1,0).$ • • • • • If p >>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) 16:27 Lecture 07 Mean-Variance Analysis and CAPM (Derivation with Projections) Fin 501: Asset Pricing Mean Variance Frontier • Definition 1: z 2 M is in the mean variance frontier if there exists no z’ 2 M such that E[z’]= E[z], q(z')= q(z) and var[z’] < var[z]. • Definition 2: Let E the space generated by kq and ke. • Decompose z=zE+e, with zE2 E and e ? E. • Hence, E[e]= E[e ke]=0, q(e)= E[e kq]=0 Cov[e,zE]=E[e zE]=0, since e ? E. • var[z] = var[zE]+var[e] (price of e is zero, but positive variance) • If z in mean variance frontier ) z 2 E. • Every z 2 E is in mean variance frontier. 16:27 Lecture 07 Mean-Variance Analysis and CAPM (Derivation with Projections) Fin 501: Asset Pricing Frontier Returns… • Frontier returns are the returns of frontier payoffs with non-zero prices. • x • graphically: payoffs with price of p=1. 16:27 Lecture 07 Mean-Variance Analysis and CAPM (Derivation with Projections) Fin 501: Asset Pricing M = RS = R3 Mean-Variance Payoff Frontier e kq Mean-Variance Return Frontier p=1-line = return-line (orthogonal to kq) 16:27 Lecture 07 Mean-Variance Analysis and CAPM (Derivation with Projections) Fin 501: Asset Pricing Mean-Variance (Payoff) Frontier (1,1,1) 0 16:27 Lecture 07 kq Mean-Variance Analysis and CAPM (Derivation with Projections) standard deviation expected return Fin 501: Asset Pricing Mean-Variance (Payoff) Frontier efficient (return) frontier (1,1,1) 0 standard deviation expected return kq inefficient (return) frontier 16:27 Lecture 07 Mean-Variance Analysis and CAPM (Derivation with Projections) …Frontier Returns 16:27 Lecture 07 Mean-Variance Analysis and CAPM (Derivation with Projections) Fin 501: Asset Pricing Fin 501: Asset Pricing Minimum Variance Portfolio • Take FOC w.r.t. l of • Hence, MVP has return of 16:27 Lecture 07 Mean-Variance Analysis and CAPM (Derivation with Projections) Fin 501: Asset Pricing Mean-Variance Efficient Returns • Definition: A return is mean-variance efficient if there is no other return with same variance but greater expectation. • Mean variance efficient returns are frontier returns with E[rl] ¸ E[rl0]. • If risk-free asset can be replicated – Mean variance efficient returns correspond to l · 0. – Pricing kernel (portfolio) is not mean-variance efficient, since 16:27 Lecture 07 Mean-Variance Analysis and CAPM (Derivation with Projections) Fin 501: Asset Pricing Zero-Covariance Frontier Returns • Take two frontier portfolios with returns and • C • The portfolios have zero co-variance if • For all l l0 m exists • m=0 if risk-free bond can be replicated 16:27 Lecture 07 Mean-Variance Analysis and CAPM (Derivation with Projections) Fin 501: Asset Pricing Illustration of MVP M = R2 and S=3 Expected return of MVP Minimum standard deviation (1,1,1) 16:27 Lecture 07 Mean-Variance Analysis and CAPM (Derivation with Projections) Fin 501: Asset Pricing Illustration of ZC Portfolio… M = R2 and S=3 arbitrary portfolio p (1,1,1) 16:27 Lecture 07 Recall: Mean-Variance Analysis and CAPM (Derivation with Projections) Fin 501: Asset Pricing …Illustration of ZC Portfolio arbitrary portfolio p (1,1,1) ZC of p 16:27 Lecture 07 Mean-Variance Analysis and CAPM (Derivation with Projections) Fin 501: Asset Pricing Beta Pricing… • Frontier Returns (are on linear subspace). Hence • Consider any asset with payoff xj – – – – – It can be decomposed in xj = xjE + ej q(xj)=q(xjE) and E[xj]=E[xjE], since e ? E. Let rjE be the return of xjE Rdddf Using above and assuming l lambda0 and m is ZC-portfolio of l, 16:27 Lecture 07 Mean-Variance Analysis and CAPM (Derivation with Projections) Fin 501: Asset Pricing …Beta Pricing • Taking expectations and deriving covariance • _ • If risk-free asset can be replicated, beta-pricing equation simplifies to • Problem: How to identify frontier returns 16:27 Lecture 07 Mean-Variance Analysis and CAPM (Derivation with Projections) Fin 501: Asset Pricing Capital Asset Pricing Model… • CAPM = market return is frontier return – Derive conditions under which market return is frontier return – Two periods: 0,1, – Endowment: individual wi1 at time 1, aggregate where the orthogonal projection of on M is. – The market payoff: – Assume q(m) 0, let rm=m / q(m), and assume that rm is not the minimum variance return. 16:27 Lecture 07 Mean-Variance Analysis and CAPM (Derivation with Projections) Fin 501: Asset Pricing …Capital Asset Pricing Model • If rm0 is the frontier return that has zero covariance with rm then, for every security j, • E[rj]=E[rm0] + bj (E[rm]-E[rm0]), with bj=cov[rj,rm] / var[rm]. • If a risk free asset exists, equation becomes, • E[rj]= rf + bj (E[rm]- rf) • N.B. first equation always hold if there are only two assets. 16:27 Lecture 07 Mean-Variance Analysis and CAPM (Derivation with Projections)