Lecture 05: MeanVariance Analysis & Capital Asset Pricing Model (CAPM)


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1 Lecture 05: MeanVariance Analysis & Capital Asset Pricing Model (CAPM) Prof. Markus K. Brunnermeier Slide 051
2 Overview Simple CAPM with quadratic utility functions (derived from stateprice beta model) Meanvariance preferences Portfolio Theory CAPM (intuition) CAPM Projections Pricing Kernel and Expectation Kernel Slide 052
3 Recall Stateprice 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 053
4 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 054
5 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 055
6 Overview Simple CAPM with quadratic utility functions (derived from stateprice beta model) Meanvariance analysis Portfolio Theory (Portfolio frontier, efficient frontier, ) CAPM (Intuition) CAPM Projections Pricing Kernel and Expectation Kernel Slide 056
7 Definition: MeanVariance Dominance & Efficient Frontier Asset (portfolio) A meanvariance 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 nondominated portfolios in the meanstandard deviation space. By definition, no ( rational ) meanvariance investor would choose to hold a portfolio not located on the efficient frontier. Slide 057
8 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 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 058
9 Illustration of 2 Asset Case For certain weights: w 1 and (1w 1 ) μ p = w 1 E[r 1 ]+ (1w 1 ) E[r 2 ] σ 2 p = w 12 σ 12 + (1w 1 ) 2 σ w 1 (1w 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 059
10 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 0510
11 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 0511
12 For 1 < ρ 1,2 < 1: E[r 2 ] E[r 1 ] σ 1 σ 2 Efficient Frontier: Two Imperfectly Correlated Risky Assets Slide 0512
13 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 0513
14 Efficient Frontier with Eco 525: Financial Economics I n risky assets and one riskfree asset The Efficient Frontier: One Risk Free and n Risky Assets Slide 0514
15 MeanVariance 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 0515
16 Optimal Portfolio: Two Fund Separation Price of Risk = = highest Sharpe ratio Optimal Portfolios of Two Investors with Different Risk Aversion Slide 0516
17 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. riskpremium Slide 0517
18 The CAPM with a riskfree 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 halfline 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 0518
19 The Capital Market Line Eco 525: Financial Economics I CML r M M r f j σ M σ p Slide 0519
20 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 0520
21 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 stateprice beta model? R * = R M Slide 0521
22 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 0522
23 Overview Simple CAPM with quadratic utility functions (derived from stateprice beta model) Meanvariance preferences Portfolio Theory CAPM (Intuition) CAPM (modern derivation) Projections Pricing Kernel and Expectation Kernel Slide 0523
24 Projections States s=1,,s with π s >0 Probability inner product πnorm (measure of length) Slide 0524
25 y shrink axes y x x x and y are πorthogonal iff [x,y] π = 0, I.e. E[xy]=0 Slide 0525
26 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 0526
27 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 0527
28 Expected Value and CoVariance 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 0528
29 Expected Value and CoVariance E[x] = [x, 1]= Slide 0529
30 Overview Simple CAPM with quadratic utility functions (derived from stateprice beta model) Meanvariance preferences Portfolio Theory CAPM (Intuition) CAPM (modern derivation) Projections Pricing Kernel and Expectation Kernel Slide 0530
31 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 0531
32 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 0532
33 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 0533
34 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[(μk q )z]=0 μ=(μk q )+k q k q is the projection'' of μ on M. Complete markets, k q =μ (SDF=state price density) Slide 0534
35 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[(ek 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 0535
36 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 0536
37 Frontier Returns Frontier returns are the returns of frontier payoffs with nonzero prices. x graphically: payoffs with price of p=1. Slide 0537
38 M = R S = R 3 MeanVariance Payoff Frontier ε k q MeanVariance Return Frontier p=1line = returnline (orthogonal to k q ) Slide 0538
39 MeanVariance (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 0539
40 MeanVariance (Payoff) Frontier efficient (return) frontier 0 (1,1,1) k q standard deviation expected return inefficient (return) frontier Slide 0540
41 Frontier Returns Eco 525: Financial Economics I (if agent is riskneutral) Slide 0541
42 Minimum Variance Portfolio Take FOC w.r.t. λ of Hence, MVP has return of Slide 0542
43 MeanVariance Efficient Returns Definition: A return is meanvariance 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 riskfree asset can be replicated Mean variance efficient returns correspond to λ 0. Pricing kernel (portfolio) is not meanvariance efficient, since Slide 0543
44 ZeroCovariance Frontier Returns Take two frontier portfolios with returns and C The portfolios have zero covariance if For all λ λ 0 μ exists μ=0 if riskfree bond can be replicated Slide 0544
45 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 0545
46 Illustration of ZC Portfolio M = R 2 and S=3 (1,1,1) arbitrary portfolio p Recall: Slide 0546
47 Illustration of ZC Portfolio (1,1,1) arbitrary portfolio p ZC of p Slide 0547
48 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 ZCportfolio of λ, Slide 0548
49 Beta Pricing Taking expectations and deriving covariance _ If riskfree asset can be replicated, betapricing equation simplifies to Problem: How to identify frontier returns Slide 0549
50 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
51 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
52 Outdated material follows Traditional derivation of CAPM is less elegant Not relevant for exams Slide
53 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
54 The first FOC can be written as: Slide
55 Noting that e T w p = w T pe, using the first foc, the second foc can be written as premultiplying first foc with 1 (instead of e T ) yields Solving both equations for λ and γ Slide
56 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
57 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
58 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
59 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
60 Figure 63 The Set of Frontier Portfolios: Mean/Variance Space Slide 0560
61 Figure 64 The Set of Frontier Portfolios: Mean/SD Space Slide 0561
62 Figure 65 The Set of Frontier Portfolios: Short Selling Allowed Slide 0562
63 Characterization of Efficient Portfolios (No RiskFree Assets) Definition 6.2: Efficient portfolios are those frontier portfolios which are not meanvariance 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 0563
64 Zero Covariance Portfolio ZeroCov Portfolio is useful for ZeroBeta 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 0564
65 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=BCA 2 Slide 0565
66 For zero covariance portfolio ZC(p) For graphical illustration, let s draw this line: Slide 0566
67 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 0567
68 E ( r ) p A/C MVP E[r ( ZC(p)] ZC(p) ZC (p) on frontier (1/C) Var ( r ) Figure 66 The Set of Frontier Portfolios: Location of the ZeroCovariance Portfolio Slide 0568
69 ZeroBeta CAPM (no riskfree 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 0569
70 All investors hold meanvariance 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 0570
71 ZeroBeta 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 0571
72 The Standard CAPM (with riskfree 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 0572
73 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 0573
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