Eco 525: Financial Economics I LectureLecture 05:05: MeanMean--VarianceVariance AnalysisAnalysis && CapitalCapital AssetAsset PricingPricing ModelModel ((CAPMCAPM)) Prof. Markus K. Brunnermeier 16:14 Lecture 05 Mean-Variance Analysis and CAPM Slide 05-1 Eco 525: Financial Economics I OverviewOverview •• SimpleSimple CAPMCAPM withwith quadraticquadratic utilityutility functionsfunctions (derived from state-price beta model) • Mean-variance preferences – Portfolio Theory – CAPM (intuition) •CAPM – Projections – Pricing Kernel and Expectation Kernel 16:14 Lecture 05 Mean-Variance Analysis and CAPM Slide 05-2 Eco 525: Financial Economics I RecallRecall StateState--priceprice BetaBeta modelmodel Recall:Recall: E[RE[Rh]] -- RRf == ββh E[RE[R*-- RRf]] where βh := Cov[R*,Rh] / Var[R*] veryvery generalgeneral –– butbut whatwhat isis RR* inin realityreality?? 16:14 Lecture 05 Mean-Variance Analysis and CAPM Slide 05-3 Eco 525: Financial Economics I SimpleSimple CAPMCAPM withwith QuadraticQuadratic ExpectedExpected UtilityUtility 1. All agents are identical • Expected utility U(x0, x1) = ∑s πs u(x0, xs) ⇒ m= ∂1u / E[∂0u] 2 • Quadratic u(x0,x1)=v0(x0) - (x1- α) ⇒ ∂1u = [-2(x1,1- α),…, -2(xS,1- α)] •E[Rh] – Rf = - Cov[m,Rh] / E[m] f h = -R Cov[∂1u, R ] / E[∂0u] f h = -R Cov[-2(x1- α), R ] / E[∂0u] f h = R 2Cov[x1,R ] / E[∂0u] • Also holds for market portfolio m f f m •E[R] – R = R 2Cov[x1,R ]/E[∂0u] ⇒ 16:14 Lecture 05 Mean-Variance Analysis and CAPM Slide 05-4 Eco 525: Financial Economics I SimpleSimple CAPMCAPM withwith QuadraticQuadratic ExpectedExpected UtilityUtility 2. Homogenous agents + Exchange economy m ⇒ x1 = agg. endowment and is perfectly correlated with R E[RE[Rh]=]=RRf ++ ββh {{E[RE[Rm]]--RRf}} MarketMarket SecuritSecurityy LinLinee * f M f N.B.: R =R (a+b1R )/(a+b1R ) in this case (where b1<0)! 16:14 Lecture 05 Mean-Variance Analysis and CAPM Slide 05-5 Eco 525: Financial Economics I OverviewOverview •• SimpleSimple CAPMCAPM withwith quadraticquadratic utilityutility functionsfunctions (derived(derived fromfrom statestate--priceprice betabeta model)model) • Mean-variance analysis – Portfolio Theory (Portfolio frontier, efficient frontier, …) – CAPM (Intuition) •CAPM – Projections – Pricing Kernel and Expectation Kernel 16:14 Lecture 05 Mean-Variance Analysis and CAPM Slide 05-6 Eco 525: Financial Economics I Definition:Definition: MeanMean--VarianceVariance DominanceDominance && EfficientEfficient FrontierFrontier • Asset (portfolio) A mean-variance dominates asset (portfolio) B if μA ≤ μB and σA < σΒ or if μA>μBwhile σA ≤σB. • Efficient frontier: loci of all non-dominated portfolios in the mean-standard deviation space. By definition, no (“rational”) mean-variance investor would choose to hold a portfolio not located on the efficient frontier. 16:14 Lecture 05 Mean-Variance Analysis and CAPM Slide 05-7 Eco 525: Financial Economics I ExpectedExpected PortfolioPortfolio ReturnsReturns && VarianceVariance • Expected returns (linear) hj μp := E[rp]=wj μj ,whereeachμj = hj j •Variance P 2 2 σ1 σ12 w1 σp := Var[rp]=w0Vw=(w1 w2) σ σ2 w µ 21 2 ¶µ 2 ¶ 2 2 w1 =(w1σ1 + w2σ21 w1σ12 + w2σ2) w µ 2 ¶ 2 2 2 2 = w1σ1 + w2σ2 +2w1w2σ12 ≤ 0 since σ12 ≤−σ1σ2. recall that correlation coefficient ∈ [-1,1] 16:14 Lecture 05 Mean-Variance Analysis and CAPM Slide 05-8 Eco 525: Financial Economics I IllustrationIllustration ofof 22 AssetAsset CaseCase • For certain weights: w1 and (1-w1) μp = w1 E[r1]+ (1-w1) E[r2] 2 2 2 2 2 σ p = w1 σ1 + (1-w1) σ2 + 2 w1(1-w1)σ1 σ2 ρ1,2 2 (Specify σ p and one gets weights and μp’s) • Special cases [w1 to obtain certain σR] – ρ1,2 = 1 ⇒ w1 = (+/-σp – σ2) / (σ1 – σ2) – ρ1,2 = -1 ⇒ w1 = (+/- σp + σ2) / (σ1 + σ2) 16:14 Lecture 05 Mean-Variance Analysis and CAPM Slide 05-9 Eco 525: Financial Economics I σp = |w1σ1 +(1− w1)σ2| σp σ2 For ρ1,2 = 1: Hence, w1 = ±σ −σ 1− 2 μp = w1μ1 +(1− w1)μ2 E[r2] μp E[r1] σ1 σp σ2 Lower part with … is irrelevant The Efficient Frontier: Two Perfectly Correlated Risky Assets 16:14 Lecture 05 Mean-Variance Analysis and CAPM Slide 05-10 Eco 525: Financial Economics I σ +σ For ρ = -1:σp = |w1σ1 − (1 − w1)σ2| Hence, w = ± p 2 1,2 1 σ1+σ2 μp = w1μ1 +(1− w1)μ2 E[r ] 2 μ μ slope: 2− 1 σ σ1+σ2 p σ2 σ1 σ +σ μ1 + σ +σ μ2 μ μ 1 2 1 2 slope: − 2− 1 σ σ1+σ2 p E[r1] σ1 σ2 Efficient Frontier: Two Perfectly Negative Correlated Risky Assets 16:14 Lecture 05 Mean-Variance Analysis and CAPM Slide 05-11 Eco 525: Financial Economics I For -1 < ρ1,2 < 1: E[r2] E[r1] σ1 σ2 Efficient Frontier: Two Imperfectly Correlated Risky Assets 16:14 Lecture 05 Mean-Variance Analysis and CAPM Slide 05-12 Eco 525: Financial Economics I For σ1 = 0 E[r2] μp E[r1] σ1 σp σ2 The Efficient Frontier: One Risky and One Risk Free Asset 16:14 Lecture 05 Mean-Variance Analysis and CAPM Slide 05-13 Eco 525: Financial Economics I EfficientEfficient FrontierFrontier withwith nn riskyrisky assetsassets andand oneone riskrisk--freefree assetasset The Efficient Frontier: One Risk Free and n Risky Assets 16:14 Lecture 05 Mean-Variance Analysis and CAPM Slide 05-14 Eco 525: Financial Economics I MeanMean--VarianceVariance PreferencesPreferences ∂U ∂U •U(μ , σ ) with > 0, 2 < 0 p p ∂μp ∂σp – quadratic utility function (with portfolio return R) U(R) = a + b R + c R2 vNM: E[U(R)] = a + b E[R] + c E[R2] 2 2 = a + b μp + c μp + c σp = g(μp, σp) j j – asset returns normally distributed ⇒ R=∑j w r normal 2 • if U(.) is CARA ⇒ certainty equivalent = μp - ρA/2σ p (Use moment generating function) 16:14 Lecture 05 Mean-Variance Analysis and CAPM Slide 05-15 Eco 525: Financial Economics I OptimalOptimal Portfolio:Portfolio: TwoTwo FundFund SeparationSeparation Price of Risk = = highest Sharpe ratio Optimal Portfolios of Two Investors with Different Risk Aversion 16:14 Lecture 05 Mean-Variance Analysis and CAPM Slide 05-16 Eco 525: Financial Economics I EquilibriumEquilibrium leadsleads toto CAPMCAPM • Portfolio theory: only analysis of demand – price/returns are taken as given – composition of risky portfolio is same for all investors • Equilibrium Demand = Supply (market portfolio) • CAPM allows to derive – equilibrium prices/ returns. – risk-premium 16:14 Lecture 05 Mean-Variance Analysis and CAPM Slide 05-17 omics I Financial EconEco 525: TheThe CAPMCAPM withwith aa riskrisk--freefree bondbond • The market portfolio is efficient since it is on the efficient frontier. • All individual optimal portfolios are located on the half-line originating at point (0,rf). ERR[]− • The slope of Capital Market Line (CML): M f σ M . ERR[]M− f ERR[]p= f + σ p σ M 16:14 Lecture 05 nMea-Variance Analysis and CAPM Slide 05-18 Eco 525: Financial Economics I TheThe CapitalCapital MarketMarket LineLine CML M rM rf j σ M σ p 16:14 Lecture 05 Mean-Variance Analysis and CAPM Slide 05-19 Eco 525: Financial Economics I Proof of the CAPM relationship [old traditional derivation] • Refer to previous figure. Consider a portfolio with a fraction 1 - α of wealth invested in an arbitrary security j and a fraction α in the market portfolio μ p = αμM (+ 1−α μ )j 2 2 2 2 2 σ α=p σ( +M 1 −α ) σj 2 + ( α 1 −jM α ) σ As α varies we trace a locus which - passes through M (- and through j) - cannot cross the CML (why?) - hence must be tangent to the CML at M Tangency = d μ p = slope of the locus at M = | μ r α=1 M − f dσp = slope of CML = σM 16:14 Lecture 05 Mean-Variance Analysis and CAPM Slide 05-20 Eco 525: Financial Economics I at α =1 σp = σM slope of dμp (μM μj)σM μM rf slope of ==α=1 = 2− = − tangent! locus dσp σ σ σM | M − jM σjM E[rj]=μj = rf + 2 (μM − rf ) σM * Do you see the connection to earlier state-price beta model? R = RM 16:14 Lecture 05 Mean-Variance Analysis and CAPM Slide 05-21 Eco 525: Financial Economics I TheThe SecuritySecurity MarketMarket LineLine E(r) SML E(ri) E(rM) rf slope SML = (E(ri)-rf) /β i β M=1 β i β 16:14 Lecture 05 Mean-Variance Analysis and CAPM Slide 05-22 Eco 525: Financial Economics I OverviewOverview •• SimpleSimple CAPMCAPM withwith quadraticquadratic utilityutility functionsfunctions (derived(derived fromfrom statestate--priceprice betabeta model)model) • Mean-variance preferences – Portfolio Theory –CAPM (Intuition) • CAPM (modern derivation) – Projections – Pricing Kernel and Expectation Kernel 16:14 Lecture 05 Mean-Variance Analysis and CAPM Slide 05-23 Eco 525: Financial Economics I ProjectionsProjections • States s=1,…,S with πs >0 • Probability inner product • π-norm (measure of length) 16:14 Lecture 05 Mean-Variance Analysis and CAPM Slide 05-24 Eco 525: Financial Economics I ⇒ shrink y axes y xx x and y are π-orthogonal iff [x,y]π = 0, I.e. E[xy]=0 16:14 Lecture 05 Mean-Variance Analysis and CAPM Slide 05-25 Eco 525: Financial Economics I ……ProjectionsProjections…… • Z space of all linear combinations of vectors z1, …,zn • Given a vector y ∈ RS solve • [smallest distance between vector y and Z space] 16:14 Lecture 05 Mean-Variance Analysis and CAPM Slide 05-26 Eco 525: Financial Economics I ……ProjectionsProjections y ε yZ E[ε zj]=0 for each j=1,…,n (from FOC) ε⊥ z yZ is the (orthogonal) projection on Z Z Z ∈ Z ⊥ 16:14y = Lecture y + 05 ε’, y Mean, ε -Variancez Analysis and CAPM Slide 05-27 Eco 525: Financial Economics I ExpectedExpected ValueValue andand CoCo--VarianceVariance…… squeeze axis by (1,1) x [x,y]=E[xy]=Cov[x,y] + E[x]E[y] [x,x]=E[x2]=Var[x]+E[x]2 ||x||= E[x2]½ 16:14 Lecture 05 Mean-Variance Analysis and CAPM Slide 05-28 Eco 525: Financial Economics I ……ExpectedExpected ValueValue andand CoCo--VarianceVariance E[x] = [x, 1]= 16:14 Lecture 05 Mean-Variance Analysis and CAPM Slide 05-29 Eco 525: Financial Economics I OverviewOverview