Lecture 05: Mean-Variance Analysis & Capital Asset Pricing Model (CAPM) Prof. Markus K. Brunnermeier Slide 05-1
Overview Simple CAPM with quadratic utility functions (derived from state-price beta model) Mean-variance preferences Portfolio Theory CAPM (intuition) CAPM Projections Pricing Kernel and Expectation Kernel Slide 05-2
Recall State-price Beta model Recall: E[R h ] - R f = β h E[R * - R f ] where β h := Cov[R *,R h ] / Var[R * ] very general but what is R * in reality? Slide 05-3
Simple CAPM with Quadratic Expected Utility 1. All agents are identical Expected utility U(x 0, x 1 ) = s π s u(x 0, x s ) m= 1 u / E[ 0 u] Quadratic u(x 0,x 1 )=v 0 (x 0 ) - (x 1 - α) 2 1 u = [-2(x 1,1 - α),, -2(x S,1 - α)] E[R h ] R f = - Cov[m,R h ] / E[m] = -R f Cov[ 1 u, R h ] / E[ 0 u] = -R f Cov[-2(x 1 - α), R h ] / E[ 0 u] = R f 2Cov[x 1,R h ] / E[ 0 u] Also holds for market portfolio E[R m ] R f = R f 2Cov[x 1,R m ]/E[ 0 u] Slide 05-4
Simple CAPM with Quadratic Expected Utility 2. Homogenous agents + Exchange economy x 1 = agg. endowment and is perfectly correlated with R m E[R h ]=R f + β h {E[R m ]-R f } Market Security Line N.B.: R * =R f (a+b 1 R M )/(a+b 1 R f ) in this case (where b 1 <0)! Slide 05-5
Overview Simple CAPM with quadratic utility functions (derived from state-price beta model) Mean-variance analysis Portfolio Theory (Portfolio frontier, efficient frontier, ) CAPM (Intuition) CAPM Projections Pricing Kernel and Expectation Kernel Slide 05-6
Definition: Mean-Variance Dominance & Efficient Frontier Asset (portfolio) A mean-variance dominates asset (portfolio) B if μ A μ B and σ A < σ Β or if μ A >μ B while σ A σ B. 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. Slide 05-7
Expected Portfolio Returns & Variance Expected returns (linear) μ p := E[r p ]=w j μ j,whereeachμ j = Variance P hj j hj µ σp 2 := Var[r p] = w 0 σ 2 Vw=(w 1 w 2 ) 1 σ 12 σ 21 σ2 2 = (w 1 σ 2 1 + w 2 σ 21 w 1 σ 12 + w 2 σ 2 2) µ w1 w 2 µ w1 w 2 = w1 2 σ2 1 + w2 2 σ2 2 +2w 1w 2 σ 12 0 since σ 12 σ 1 σ 2. recall that correlation coefficient [-1,1] Slide 05-8
Illustration of 2 Asset Case For certain weights: w 1 and (1-w 1 ) μ p = w 1 E[r 1 ]+ (1-w 1 ) E[r 2 ] σ 2 p = w 12 σ 12 + (1-w 1 ) 2 σ 22 + 2 w 1 (1-w 1 )σ 1 σ 2 ρ 1,2 (Specify σ 2 p and one gets weights and μ p s) Special cases [w 1 to obtain certain σ R ] ρ 1,2 = 1 w 1 = (+/-σ p σ 2 ) / (σ 1 σ 2 ) ρ 1,2 = -1 w 1 = (+/- σ p + σ 2 ) / (σ 1 + σ 2 ) Slide 05-9
For ρ 1,2 = 1: σ p = w 1 σ 1 +(1 w 1 )σ 2 Hence, w 1 = ±σp σ2 σ 1 σ 2 μ p = w 1 μ 1 +(1 w 1 )μ 2 E[r 2 ] μ p E[r 1 ] σ 1 σ p σ 2 Lower part with is irrelevant The Efficient Frontier: Two Perfectly Correlated Risky Assets Slide 05-10
For ρ 1,2 = -1: σ p = w 1 σ 1 (1 w 1 )σ 2 μ p = w 1 μ 1 +(1 w 1 )μ 2 Hence, w 1 = ±σ p+σ 2 σ 1 +σ 2 E[r 2 ] σ 2 σ 1 +σ 2 μ 1 + σ 1 σ 1 +σ 2 μ 2 E[r 1 ] slope: μ 2 μ 1 σ 1 +σ 2 σ p slope: μ 2 μ 1 σ 1 +σ 2 σ p σ 1 σ 2 Efficient Frontier: Two Perfectly Negative Correlated Risky Assets Slide 05-11
For -1 < ρ 1,2 < 1: E[r 2 ] E[r 1 ] σ 1 σ 2 Efficient Frontier: Two Imperfectly Correlated Risky Assets Slide 05-12
For σ 1 = 0 E[r 2 ] μ p E[r 1 ] σ 1 σ p σ 2 The Efficient Frontier: One Risky and One Risk Free Asset Slide 05-13
Efficient Frontier with Eco 525: Financial Economics I n risky assets and one risk-free asset The Efficient Frontier: One Risk Free and n Risky Assets Slide 05-14
Mean-Variance Preferences U(μ p, σ p ) with U U μ p > 0, σ < 0 p 2 quadratic utility function (with portfolio return R) U(R) = a + b R + c R 2 vnm: E[U(R)] = a + b E[R] + c E[R 2 ] = a + b μ p + c μ p2 + c σ p 2 = g(μ p, σ p ) asset returns normally distributed R= j w j r j normal if U(.) is CARA certainty equivalent = μ p - ρ A /2σ 2 p (Use moment generating function) Slide 05-15
Optimal Portfolio: Two Fund Separation Price of Risk = = highest Sharpe ratio Optimal Portfolios of Two Investors with Different Risk Aversion Slide 05-16
Equilibrium leads to CAPM 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 Slide 05-17
The CAPM with a risk-free bond Eco 525: Financial Economics I 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,r f ). The slope of Capital Market Line (CML):. E[ R p ] = R f + E[ R M σ ] M R f σ p E[ R M ] R f σ M Slide 05-18
The Capital Market Line Eco 525: Financial Economics I CML r M M r f j σ M σ p Slide 05-19
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 σ p 2 p = αμ M 2 = α σ + (1 α) μ As α varies we trace a locus which - passes through M (- and through j) - cannot cross the CML (why?) 2 M + (1 α) - hence must be tangent to the CML at M dμ p Tangency = = slope of the locus at M = dσ p α=1 = slope of CML = j 2 σ 2 j + 2α (1 α) σ jm μ M r f σ M Slide 05-20
slope of locus dμ p = = σ M dσ p α=1 = (μ M μ j )σ M σ 2 M σ jm = μ M r f at α =1 σ p = σ M slope of tangent! E[r j ]=μ j = r f + σ jm σ 2 M (μ M r f ) Do you see the connection to earlier state-price beta model? R * = R M Slide 05-21
The Security Market Line Eco 525: Financial Economics I E(r) E(r i ) SML E(r M ) r f slope SML = (E(r i )-r f ) /β i β M =1 β i β Slide 05-22
Overview Simple CAPM with quadratic utility functions (derived from state-price beta model) Mean-variance preferences Portfolio Theory CAPM (Intuition) CAPM (modern derivation) Projections Pricing Kernel and Expectation Kernel Slide 05-23
Projections States s=1,,s with π s >0 Probability inner product π-norm (measure of length) Slide 05-24
y shrink axes y x x x and y are π-orthogonal iff [x,y] π = 0, I.e. E[xy]=0 Slide 05-25
Projections Z space of all linear combinations of vectors z 1,,z n Given a vector y R S solve [smallest distance between vector y and Z space] Slide 05-26
Projections y ε y Z E[ε z j ]=0 for each j=1,,n (from FOC) ε z y Z is the (orthogonal) projection on Z y = y Z + ε, y Z Z, ε z Slide 05-27
Expected Value and Co-Variance Variance squeeze axis by (1,1) x [x,y]=e[xy]=cov[x,y] + E[x]E[y] [x,x]=e[x 2 ]=Var[x]+E[x] 2 x = E[x 2 ] ½ Slide 05-28
Expected Value and Co-Variance E[x] = [x, 1]= Slide 05-29
Overview Simple CAPM with quadratic utility functions (derived from state-price beta model) Mean-variance preferences Portfolio Theory CAPM (Intuition) CAPM (modern derivation) Projections Pricing Kernel and Expectation Kernel Slide 05-30
New (LeRoy( & Werner) Notation Main changes (new versus old) gross return: r = R SDF: μ = m pricing kernel: k q = m * Asset span: M = <X> income/endowment: w t = e t Slide 05-31
Pricing Kernel k q M space of feasible payoffs. If no arbitrage and π >>0 there exists SDF μ R S, μ >>0, such that q(z)=e(μ z). μ M SDF need not be in asset span. A pricing kernel is a k q M such that for each z M, q(z)=e(k q z). (k q = m * in our old notation.) Slide 05-32
Pricing Kernel - Examples Example 1: S=3,π s =1/3 for s=1,2,3, x 1 =(1,0,0), x 2 =(0,1,1), p=(1/3,2/3). Then k=(1,1,1) is the unique pricing kernel. Example 2: S=3,π s =1/3 for s=1,2,3, x 1 =(1,0,0), x 2 =(0,1,0), p=(1/3,2/3). Then k=(1,2,0) is the unique pricing kernel. Slide 05-33
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) μ and any z M E[(μ-k q )z]=0 μ=(μ-k q )+k q k q is the projection'' of μ on M. Complete markets, k q =μ (SDF=state price density) Slide 05-34
Expectations Kernel k e An expectations kernel is a vector k e M Such that E(z)=E(k e z) for each z M. Example S=3, π s =1/3, for s=1,2,3, x 1 =(1,0,0), x 2 =(0,1,0). Then the unique k e =(1,1,0). If π >>0, there exists a unique expectations kernel. Let e=(1,, 1) then for any z M E[(e-k e )z]=0 k e is the projection of e on M k e = e if bond can be replicated (e.g. if markets are complete) Slide 05-35
Mean Variance Frontier Definition 1: z M is in the mean variance frontier if there exists no z M such that E[z ]= E[z], q(z')= q(z) and var[z ] < var[z]. Definition 2: Let E the space generated by k q and k e. Decompose z=z E +ε, with z E E and ε E. Hence, E[ε]= E[ε k e ]=0, q(ε)= E[ε k q ]=0 Cov[ε,z E ]=E[ε z E ]=0, since ε E. var[z] = var[z E ]+var[ε] (price of ε is zero, but positive variance) If z in mean variance frontier z E. Every z E is in mean variance frontier. Slide 05-36
Frontier Returns Frontier returns are the returns of frontier payoffs with non-zero prices. x graphically: payoffs with price of p=1. Slide 05-37
M = R S = R 3 Mean-Variance Payoff Frontier ε k q Mean-Variance Return Frontier p=1-line = return-line (orthogonal to k q ) Slide 05-38
Mean-Variance (Payoff) Frontier 0 (1,1,1) k q standard deviation expected return NB: graphical illustrated of expected returns and standard deviation changes if bond is not in payoff span. Slide 05-39
Mean-Variance (Payoff) Frontier efficient (return) frontier 0 (1,1,1) k q standard deviation expected return inefficient (return) frontier Slide 05-40
Frontier Returns Eco 525: Financial Economics I (if agent is risk-neutral) Slide 05-41
Minimum Variance Portfolio Take FOC w.r.t. λ of Hence, MVP has return of Slide 05-42
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[r λ ] E[r λ0 ]. If risk-free asset can be replicated Mean variance efficient returns correspond to λ 0. Pricing kernel (portfolio) is not mean-variance efficient, since Slide 05-43
Zero-Covariance Frontier Returns Take two frontier portfolios with returns and C The portfolios have zero co-variance if For all λ λ 0 μ exists μ=0 if risk-free bond can be replicated Slide 05-44
Illustration of MVP Eco 525: Financial Economics I Expected return of MVP M = R 2 and S=3 (1,1,1) Minimum standard deviation Slide 05-45
Illustration of ZC Portfolio M = R 2 and S=3 (1,1,1) arbitrary portfolio p Recall: Slide 05-46
Illustration of ZC Portfolio (1,1,1) arbitrary portfolio p ZC of p Slide 05-47
Beta Pricing Frontier Returns (are on linear subspace). Hence Consider any asset with payoff x j It can be decomposed in x j = x je + ε j q(x j )=q(x je ) and E[x j ]=E[x je ], since ε E. Let r je be the return of x E j x Using above and assuming λ λ 0 and μ is ZC-portfolio of λ, Slide 05-48
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 Slide 05-49
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 w i 1 at time 1, aggregate where the orthogonal projection of on M is. The market payoff: Assume q(m) 0, let r m =m / q(m), and assume that r m is not the minimum variance return. Slide 05-50 50
Capital Asset Pricing Model If r m0 is the frontier return that has zero covariance with r m then, for every security j, E[r j ]=E[r m0 ] + β j (E[r m ]-E[r m0 ]), with β j =cov[r j,r m ] / var[r m ]. If a risk free asset exists, equation becomes, E[r j ]= r f + β j (E[r m ]- r f ) N.B. first equation always hold if there are only two assets. Slide 05-51 51
Outdated material follows Traditional derivation of CAPM is less elegant Not relevant for exams Slide 05-52 52
Deriving the Frontier Eco 525: Financial Economics I Definition 6.1: A frontier portfolio is one which displays minimum variance among all feasible portfolios with the same E( ). r~p min w 1 2 w T n risky assets Vw ( λ) s.t. w T e = E N w E( ~ r ) i= 1 i i = E ( γ) w T 1 = 1 N w i= 1 i = 1 Slide 05-53 53
The first FOC can be written as: Slide 05-54 54
Noting that e T w p = w T pe, using the first foc, the second foc can be written as pre-multiplying first foc with 1 (instead of e T ) yields Solving both equations for λ and γ Slide 05-55 55
Hence, w p = λ V -1 e + γ V -1 1 becomes Eco 525: Financial Economics I w p CE A B AE = V e + V D D 1 1 (vector) λ (scalar) γ (scalar) 1 (vector) = D [ 1 1 ( ) ( )] 1 1 B V 1 A V e + C( V e) A( V 1) 1 1 [ ]E D w p = g + h E (vector) (vector) (scalar) (6.15) linear in expected return E! If E = 0, If E = 1, wp = g wp = g + h Hence, g and g+h are portfolios on the frontier. Slide 05-56 56
Characterization of Frontier Portfolios Proposition 6.1: The entire set of frontier portfolios can be generated 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. Slide 05-57 57
Characterization of Frontier Portfolios For any portfolio on the frontier, with g and h as defined earlier. 2 σ T ( E [ ~ r ]) = [ g + he( ~ r )] V [ g + he( ~ r )] p p p Multiplying all this out yields: Slide 05-58 58
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) in the expected return/variance space and of a hyperbola in the expected return/standard deviation space. See Figures 6.3 and 6.4. Slide 05-59 59
Figure 6-3 The Set of Frontier Portfolios: Mean/Variance Space Slide 05-60
Figure 6-4 The Set of Frontier Portfolios: Mean/SD Space Slide 05-61
Figure 6-5 The Set of Frontier Portfolios: Short Selling Allowed Slide 05-62
Characterization of Efficient Portfolios (No Risk-Free Assets) Definition 6.2: Efficient portfolios are those frontier portfolios which are not mean-variance dominated. Lemma: Efficient portfolios are those frontier portfolios for which the expected return exceeds A/C, the expected return of the minimum variance portfolio. Slide 05-63
Zero Covariance Portfolio Zero-Cov Portfolio is useful for Zero-Beta CAPM Proposition 6.5: For any frontier portfolio p, except the minimum variance portfolio, there exists a unique frontier portfolio with which p has zero covariance. We will call this portfolio the "zero covariance portfolio relative to p", and denote its vector of portfolio weights by ZC p. ( ) Proof: by construction. Slide 05-64
collect all expected returns terms, add and subtract A 2 C/DC 2 and note that the remaining term (1/C)[(BC/D)-(A 2 /D)]=1/C, since D=BC-A 2 Slide 05-65
For zero co-variance portfolio ZC(p) For graphical illustration, let s draw this line: Slide 05-66
Graphical Representation: line through p (Var[r p ], E[r p ]) AND 2 2 C A 1 MVP (1/C, A/C) (use σ ( r~ p ) = E( r~ p ) + ) for σ 2 (r) = 0 you get E[r ZC(p) ] D C C Slide 05-67
E ( r ) p A/C MVP E[r ( ZC(p)] ZC(p) ZC (p) on frontier (1/C) Var ( r ) Figure 6-6 The Set of Frontier Portfolios: Location of the Zero-Covariance Portfolio Slide 05-68
Zero-Beta CAPM (no risk-free asset) (i) agents maximize expected utility with increasing and strictly concave utility of money functions and asset returns are multivariate normally distributed, or (ii) each agent chooses a portfolio with the objective of maximizing a derived utility function of the form 2 U( e, σ ), U1 > 0, U2 < 0, U concave. (iii) common time horizon, (iv) homogeneous beliefs about e and σ 2 Slide 05-69
All investors hold mean-variance efficient portfolios the market portfolio is convex combination of efficient portfolios is efficient. Cov[r p,r q ] = λ E[r q ]+ γ (q need not be on the frontier) (6.22) Cov[r p,r ZC(p) ] = λ E[r ZC(p) ] + γ=0 Cov[r p,r q ] = λ {E[r q ]-E[r ZC(p) ]} Var[r p ] = λ {E[r p ]-E[r ZC(p) ]}. E E Divide third by fourth equation: ( r~ q ) = E( r~ ZC( M )) + βmq[ E( r~ M ) E( ~ r ZC( M ))] ( r~ ) ( ( )) [ ( ) ( ( ))] j = E r~ ZC M + β ~ Mj E r~ M E r ZC M (6.28) (6.29) Slide 05-70
Zero-Beta CAPM mean variance framework (quadratic utility or normal returns) In equilibrium, market portfolio, which is a convex combination of individual portfolios E[r q ] = E[r ZC(M) ]+ β Mq [E[r M ]-E[r ZC(M) ]] E[r j ] = E[r ZC(M) ]+ β Mj [E[r M ]-E[r ZC(M) ]] Slide 05-71
The Standard CAPM (with risk-free asset) FOC: Multiplying by (e r f 1) T and solving for λ yields w p ( e r ) 1 = V 1 n x n n x 1 n x 1 f E( ~ r ) p H a number r f where H = B - 2Ar f + Cr f 2 (6.30) Slide 05-72
NB:Derivation in DD is not correct. Rewrite first equation and replace G using second equation. Holds for any frontier portfolio, in particular the market portfolio. Slide 05-73