Fin 501: Asset Pricing Lecture 07: MeanMean--VarianceVariance Analysis & Capital Asset Pricing Model (CAPM) Prof. Markus K. Brunnermeier MeanMean--VarianceVariance Analysis and CAPM Slide 0707--11 Fin 501: Asset Pricing OiOverview 1. Simple CAPM with quadratic utility functions (derived from statestate--priceprice beta model) 2. MeanMean--variancevariance preferences – Portfolio Theory – CAPM (intuition) 3. CAPM – Projections – Pricing Kernel and Expectation Kernel MeanMean--VarianceVariance Analysis and CAPM Slide 0707--22 Fin 501: Asset Pricing RllSttRecall Statee--prpriBtice Beta mod dlel Recall: E[Rh] - Rf = βh E[R*- Rf] where βh := Cov[R*,Rh] / Var[R*] very general – but what is R* in reality? MeanMean--VarianceVariance Analysis and CAPM Slide 0707--33 Fin 501: Asset Pricing Simple CAPM with Quadratic Expected Utility 1. All agents are identical • Expected utility U(x 0, x1)=) = ∑s πs u(x0, xs) ⇒ m= ∂1u/E[u / E[∂0u] 2 • Quadratic u(x0,x1)=v0(x0) - (x1- α) ⇒ ∂1u = [-2(x111,1- α),…, -2(xS1S,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[] / 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] ⇒ MeanMean--VarianceVariance Analysis and CAPM Slide 0707--44 Fin 501: Asset Pricing Simple CAPM with Quadratic Expected Utility 2. Homogenous agents + Exchange economy m ⇒ x1 = agg. enddiflldihdowment and is perfectly correlated with R E[E[RRh]=]=RRf + βh {E[{E[RRm]]--RRf} Market Security Line * f M f N.B.: R =R (a+b1R )/(a+b1R ) in this case (where b1<0)! MeanMean--VarianceVariance Analysis and CAPM Slide 0707--55 Fin 501: Asset Pricing OiOverview 1. Simple CAPM with quadratic utility functions (derived from state-state-priceprice beta model) 2. MeanMean--variancevariance analysis – Portfolio Theory (Port fo lio f ronti er, effi ci ent f ronti er, …) – CAPM (Intuition) 3. CAPM – Projections – PPiiricing K erne l and dE Expectati on K ernel MeanMean--VarianceVariance Analysis and CAPM Slide 0707--66 Fin 501: Asset Pricing Deeo:efinition: Meanan--VVceoceariance Dominance & Efficient Frontier • Asset (portfolio) A mean-variance dominates asset (portfolio) B if μA ≥ μB and σA < σΒ or if μA>μBwhile σA ≤σB. • Effici ent f ronti er: lifllloci of all non-ditddominated portfolios in the mean-standard deviation space. BBy de finiti on, no (“ rati onal”) mean-variance investor would choose to hold a portfolio not locat ed on th e effi ci ent ftifrontier. MeanMean--VarianceVariance Analysis and CAPM Slide 0707--77 Fin 501: Asset Pricing EdPfliR&ViExpected Portfolio Returns & Variance • Expected returns (linear) hj μp := E[rp] = wj μj , where each wj = hj j •Variance P 2 2 σ1 σ12 w1 σp := V ar[rp] = w0V w = (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] MeanMean--VarianceVariance Analysis and CAPM Slide 0707--88 Fin 501: Asset Pricing Illust rati on of 2 A sse t Case • 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 (Sppyecify σ p and one gggets 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) MeanMean--VarianceVariance Analysis and CAPM Slide 0707--99 Fin 501: Asset Pricing For ρ = 1: σp = |w1σ1 +(1− w1)σ2| σp σ2 1,2 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 MeanMean--VarianceVariance Analysis and CAPM Slide 0707--1010 Fin 501: Asset Pricing 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[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 MeanMean--VarianceVariance Analysis and CAPM Slide 0707--1111 Fin 501: Asset Pricing For -1 < ρ1,2 < 1: E[r2] E[r1] σ1 σ2 Efficient Frontier: Two Imperfectly Correlated Risky Assets MeanMean--VarianceVariance Analysis and CAPM Slide 0707--1212 Fin 501: Asset Pricing For σ1 =0= 0 E[r2] μp E[r1] σ1 σp σ2 The Efficient Frontier: One Risky and One Risk Free Asset MeanMean--VarianceVariance Analysis and CAPM Slide 0707--1313 Fin 501: Asset Pricing Effici ent f ronti er wi th n ri sk y assets • A frontier portfolio is one which displays minimum variance among all feasible portfolios with the same expected portfolio E[r] return. 1 min wTVw w 2 T N σ λ s.t. w e = E ⎛ ~ ⎞ () ⎜ ∑ wiE( ri ) = E⎟ ⎝ i=1 ⎠ N T ⎛ ⎞ ()γ w 1 = 1 ⎜ ∑ wi = 1⎟ ⎝ i=1 ⎠ Slide 0707--1414 Fin 501: Asset Pricing ∂ = Vw λe γ1=0 ∂wL − − ∂ = E wT e = 0 ∂λL − ∂ =1 wT 1=0 ∂Lγ − The first FOC can be written as: Vwp = λe + γ1or 1 1 wp = λV − e + γV − 1 T T 1 T 1 e wp = λ(e V − e) + γ(e V − 1) Slide 0707--1515 Fin 501: Asset Pricing T T NiNoting t hat e wp = w pe, usihfifhing the first foc, the second df foc can be written as T T 1 T 1 E[r˜p] = e wp = λ (e V − e) +γ (e V − 1) :=B =:A pre-multiplying first foc with | 1 { (instead z } of e |T) yields{z } T T T 1 T 1 1 wp = wp 1=λ(1 V − e)+γ(1 V − 1) = 1 T 1 T 1 1=λ (1 V − e) +γ (1 V − 1) =:A =:C Solving| both{z equations} for| λ and{z γ } CE A B AE λ = D− and γ = −D where D = BC A2. − Slide 0707--1616 Fin 501: Asset Pricing -1 -1 Hence, wp = λ V e + γ V 1 becomes CE − A B − AE w = V−1e + V−11 p D D (vector) (vector) λ (scalar) γ (scalar) 1 1 = [B(V−11)− A(V−1e)]+ [C(V−1e)− A(V−11)]E D D • Result: Portfolio weights are linear in expected w = g + h E portfolio return p If E = 0, wp = g Hence, g and g+h are portfolios Slide 0707--1717 If E = 1, wp = g + h on the frontier. Fin 501: Asset Pricing Charac ter iza tion of F ronti er P ortf oli os • Proposition 6.1: The entire set of frontier portflifolios can begenerated by ("are convex combinations" of) g and g+h. • Proposition 6.2. The portfolio frontier can be described as convex combinations of any two frontier portfolios, not just the frontier portfolios g and g+h. • Proposition 6.3 :Any convex combination of frontier portfolios is also a frontier portfolio. MeanMean--VarianceVariance Analysis and CAPM Slide 0707--1818 Fin 501: Asset Pricing …Characterization of Frontier Portfolios… 2 ~ ~ T ~ • For any portfolio on the frontier, σ ()E[rp ] = [g + hE(rp )] V [g + hE(rp )] with g and h as defined earlier. Multiplying all this out yields: 10:13 Lecture MeanMean--VarianceVariance Analysis and CAPM Slide 0707--1919 Fin 501: Asset Pricing …Characterization of Frontier Portfolios… • (i) the expected return of the minimum variance portfolio is A/C; • (ii) the variance of the minimum variance portfolio is given by 1/C; • (iii) equation (6.17) is the equation of a parabola with vertex (1/C, A/C) inthe expected return/i/variance space and of a hyperbola in the expected return/standard deviation space. See Figures 6.3 and 6.4. MeanMean--VarianceVariance Analysis and CAPM Slide 0707--2020 Fin 501: Asset Pricing Figure 6-3 The Set of Frontier Portfolios: Mean/Variance Space MeanMean--VarianceVariance Analysis and CAPM Slide 0707--2121 Fin 501: Asset Pricing Figure 6-4 The Set of Frontier Portfolios: Mean/SD Space MeanMean--VarianceVariance Analysis and CAPM Slide 0707--2222 Fin 501: Asset Pricing Figure 6-5 The Set of Frontier Portfolios: Short Selling Allowed MeanMean--VarianceVariance Analysis and CAPM Slide 0707--2323 Fin 501: Asset Pricing Efficient Frontier with risk-free asset The Efficient Frontier: One Risk Free and n Risky Assets Slide 0707--2424 Fin 501: Asset Pricing Efficient Frontier with risk -free asset 1 T minw 2w Vw T T s.t. w e +(1 w 1)r = E[rp] − f 1 FOC: wp = λV (e r 1) − − f T E[rp] rf Multiplying by (e–rf 1) and solving for λ yields λ = − (e r 1)T V 1(e r 1) − f − − f 1 E[rp] rf wp = V − (e rf 1) 2− − H n 1 × 2 where H = B 2 Ar f + Cr f Slide 0707- -25is25 a n | {z } − q Fin 501: Asset Pricing Effici ent f ronti er wi th ri sk -free asset • Result 1: Excess return in frontier excess return T Cov[rq,rp]=wq Vwp E[r ] r = wT (e r 1) p − f q − f H2 E[r ] r q − f (E[r ] r )([E[r ] r ) = | q{z− f } p − f H2 (E[r ] r )2 Var[r ,r ]= p − f p p H2 Cov[rq,rp] E[rq] rf = (E[rp] rf) − V ar[rp] − :=βq,p Holds for any frontier portfolio p, in particular the marketSlide portfo 0707--2626 l | {z } Fin 501: Asset Pricing Efficient Frontier with risk -free asset • Result 2: Frontier is linear in (E[r], σ)-space (E[r ] r )2 V ar[r , r ] = p − f p p H2 E[rp]=rf + Hσp E[r ] r H = p − f σp where H is the Sharpe ratio Slide 0707--2727 Fin 501: Asset Pricing TFdStiTwo Fund Separation • Doing it in two steps: – First solve frontier for n risky asset – Then solve tangency point • Advantage: – Same portfolio of n risky asset for different agents with different risk aversion – Useful for applying equilibrium argument (later) Slide 0707--2828 Fin 501: Asset Pricing Two Fund Separation Price of Risk = = highest Sharpe ratio Optimal Portfolios of Two Investors with Different Risk Aversion Slide 0707--2929 Fin 501: Asset Pricing Meanean--ViVariance P Pfreferences ∂U ∂U > 0, 2 < 0 ∂μp ∂σp •U(μp, σp) with E[W ] γ Var[W ] − 2 –Example: • Also in expected utility framework – 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) Slide 0707--3030 Fin 501: Asset Pricing Equilibrium leads to CAPM • Portfolio theory: only analysis of demand – price/returns are taken as given – composit ion of ri sk y portf oli o i s same f or all i nvestors • Equilibrium Demand = Supply (market portfolio) • CAPM allows to derive – equilibrium prices/ returns.