# Lecture 05: Mean-Variance Analysis & Capital Asset Pricing Model (CAPM)

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1 Lecture 05: Mean-Variance Analysis & Capital Asset Pricing Model (CAPM) Prof. Markus K. Brunnermeier Slide 05-1

2 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

3 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

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 05-4

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 05-5

6 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

7 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

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 05-8

9 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 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

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 05-10

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 05-11

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 05-12

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 05-13

14 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

15 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

16 Optimal Portfolio: Two Fund Separation Price of Risk = = highest Sharpe ratio Optimal Portfolios of Two Investors with Different Risk Aversion Slide 05-16

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. risk-premium Slide 05-17

18 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

19 The Capital Market Line Eco 525: Financial Economics I CML r M M r f j σ M σ p Slide 05-19

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 05-20

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 state-price beta model? R * = R M Slide 05-21

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 05-22

23 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

24 Projections States s=1,,s with π s >0 Probability inner product π-norm (measure of length) Slide 05-24

25 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

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 05-26

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 05-27

28 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

29 Expected Value and Co-Variance E[x] = [x, 1]= Slide 05-29

30 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

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 05-31

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 05-32

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 05-33

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[(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

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[(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

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 05-36

37 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

38 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

39 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

40 Mean-Variance (Payoff) Frontier efficient (return) frontier 0 (1,1,1) k q standard deviation expected return inefficient (return) frontier Slide 05-40

41 Frontier Returns Eco 525: Financial Economics I (if agent is risk-neutral) Slide 05-41

42 Minimum Variance Portfolio Take FOC w.r.t. λ of Hence, MVP has return of Slide 05-42

43 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

44 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

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 05-45

46 Illustration of ZC Portfolio M = R 2 and S=3 (1,1,1) arbitrary portfolio p Recall: Slide 05-46

47 Illustration of ZC Portfolio (1,1,1) arbitrary portfolio p ZC of p Slide 05-47

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 ZC-portfolio of λ, Slide 05-48

49 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

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 pre-multiplying 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 6-3 The Set of Frontier Portfolios: Mean/Variance Space Slide 05-60

61 Figure 6-4 The Set of Frontier Portfolios: Mean/SD Space Slide 05-61

62 Figure 6-5 The Set of Frontier Portfolios: Short Selling Allowed Slide 05-62

63 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

64 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

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=BC-A 2 Slide 05-65

66 For zero co-variance portfolio ZC(p) For graphical illustration, let s draw this line: Slide 05-66

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 05-67

68 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

69 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

70 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

71 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

72 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

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 05-73

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