Prof. Markus K. Brunnermeier

Transcription

1 Lecture 07: Mean-Variance Analysis & Capital Asset Pricing Model (CAPM CAPM) Prof. Markus K. Brunnermeier Slide 07-1

2 Overview 1. Simple CAPM with quadratic utility functions (derived from state-price beta model) 2. Mean-variance preferences Portfolio Theory CAPM (intuition) 3. CAPM Projections Pricing Kernel and Expectation Kernel Slide 07-2

3 Recall State-price t 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 07-3

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 11 1,1 - α),,, -2(x S1 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 h 1 - α), R ]/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 07-4

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 07-5

6 Overview 1. Simple CAPM with quadratic utility functions (derived from state-price beta model) 2. Mean-variance analysis Portfolio Theory (Portfolio frontier, efficient frontier, ) CAPM (Intuition) 3. CAPM Projections Pricing Pii Kernel and de Expectation Kernel Slide 07-6

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 i frontier: loci of all non-dominated d portfolios in the mean-standard deviation space. By definition, iti no ( rational ) mean-variance investor would choose to hold a portfolio not located on the efficient i frontier. Slide 07-7

8 Expected dp Portfolio Returns & Variance Expected returns (linear) P h j μ := E[r = w j μ where each w = hj p [ p ] jμ j, j Pj hj Variance µ µ [ ] σ 2 σ w1 σp 2 := V ar[r p] = w V w = (w 1 w 2 ) σ 21 σ2 2 1 w 2 µ = (w 1 σ1 2 + w 2 σ 21 w 1 σ 12 + w 2 σ2) 2 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 07-8

9 Illustration ti of 2 Asset Case For certain weights: w 1 and (1-w 1 ) μ p = w 1 E[r 1 ]+ (1-w 1 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 07-9

10 For ρ 1,2 = 1: ρ 1,2 σ p = w 1 σ 1 +(1 w 1 )σ 2 Hence, w = ±σ p σ 2 1 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 07-10

11 For ρ 1,2 = -1: ρ 1,2 Fin 501: Asset Pricing σ p = w 1 σ 1 (1 w 1 )σ 2 Hence, w = ±σ p+σ 2 1 σ 1 +σ 2 μ p = w 1 μ 1 +(1 w 1 )μ 2 E[r 2 ] slope: μ 2 μ 1 σ 1 +σ 2 σ p σ 2 σ 1 +σ 2 μ 1 + σ 1 σ 1 +σ 2 μ 2 slope: μ 2 μ 1 σ E[r 1 ] σ 1 +σ 2 p σ 1 σ 2 Efficient Frontier: Two Perfectly Negative Correlated Risky Assets Slide 07-11

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 07-12

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 07-13

14 Efficient i frontier with n risky assets A frontier portfolio is one which displays minimum variance among all feasible portfolios with the same expected portfolio E[r] return. min w 1 2 w T Vw ( λ) ( ) s.t. w T e N = E w E( ~ i r i) = E i= 1 N γ w T 1 = 1 wi = 1 i =11 σ Slide 07-14

15 LL w L = Vw λe γ1 =0 L λ = E w T e = 0 L γ γ =1 w T 1=0 The first FOC can be written as: Vw p = λe + γ1 or w 1 V 1 p = λv 1 e + γv 1 1 e T w p = λ(e( T V 1 e) ) + γ(e T V 1 1) Slide 07-15

16 Noting that e T w p = w T pe, using the first foc, the second foc can be written as E[ r E[r T T 1 T 1 p ] = e w p = λ (e V e) +γ (e V 1) {z } :=B {z } =:A pre-multiplying first foc with 1 (instead of e T ) yields 1 T w p = wp T 1=λ(1 T V 1 e)+γ(1 T V 1 1) = =λ (1 T V 1 e) +γ (1 T V 1 1) {z } =:A {z } =:C Solving both equations for λ and γ λ = CE A D and γ = B AE D where D = BC A 2. Slide 07-16

17 Hence, w p = λ V -1 e + γ V -1 1 becomes CE A B AE = V e + V 1 1 w p D D (vector) (vector) 1 λ (scalar) γ (scalar) = [ B( V 1) A( V e) ] + [ C ( V e ) A ( V 1 )] 1 1 D [ ]E D Result: Portfolio weights are linear in expected portfolio return w p = g + h E If E = 0, wp = g If E = 1, wp = g + h Hence, g and g+h are portfolios Slide on the frontier.

18 Characterization ti of Frontier Portfolios Proposition 6.1: The entire set of frontier portfolios fli 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 07-18

19 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: 10:13 Lecture Slide 07-19

20 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 07-20

21 Figure 6-3 The Set of Frontier Portfolios: Mean/Variance Space Slide 07-21

22 Figure 6-4 The Set of Frontier Portfolios: Mean/SD Space Slide 07-22

23 Figure 6-5 The Set of Frontier Portfolios: Short Selling Allowed Slide 07-23

24 Efficient Frontier with risk-free asset The Efficient Frontier: One Risk Free and n Risky Assets Slide 07-24

25 Efficient Frontier with risk-free asset min 1 w 2 w T Vw s.t. w T e +(1 w T 1)r f = E[r p ] FOC: w p = λv 1 (e r f 1) f Multiplying by (e r f 1) T and solving for λ yields λ = E[r p ] r f (e r f 1) T V 1 (e r f 1) w p = V 1 (e r f 1) {z } n 1 H = E[r p ] r f H 2 q B 2Ar f + Crf 2 where Slide is a n

26 Efficient i frontier with risk-free asset Result 1: Excess return in frontier excess return Cov[r q,r p ] = w T q Vw p T = w T (e r 1) E[r p] r f q f {z } H 2 E[r q ] r f (E[r q] r f )([E[r p ] r f ) = H 2 Var[r p,r p ] = (E[r p] r f ) 2 H 2 E[r q ] r f = Cov[r q,r p ] (E[r p ] r f ) V ar[r p ] {z } :=β q,p Holds for any frontier portfolio p, in particular the market Slide portfol 07-26

27 Efficient Frontier with risk-free asset Result 2: Frontier is linear in (E[r], σ)-space V ar[r p, r p ] = (E[r p ] r f ) 2 p H 2 E[r p ] = r f + Hσ p f H = E[r p] r f σ p where H is the Sharpe ratio Slide 07-27

28 Two 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 07-28

29 Two Fund Separation Fin 501: Asset Pricing Price of Risk = = highest Sharpe ratio Optimal Portfolios of Two Investors with Different Risk Aversion Slide 07-29

30 Mean-Variance Preferences U(μ p, σ p ) with Example: U μ p E[W ] γ 2 Var[W ] > 0, U σ < 0 2 σ p Also in expected utility framework 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 2 p = 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 07-30

31 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 07-31

32 The CAPM with a risk-free bond Fin 501: Asset Pricing 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 ). E [ R M ] R f The slope of Capital Market Line (CML):. σ M E[ R p ] = R f + E[ R M σ ] M R f σ p Slide 07-32

33 The Capital Market Line Fin 501: Asset Pricing CML r M M r f j σ M σ p Slide 07-33

34 The Security Market Line Fin 501: Asset Pricing 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 07-34

35 Overview 1. Simple CAPM with quadratic utility functions (derived from state-price beta model) 2. Mean-variance preferences Portfolio Theory CAPM (Intuition) 3. CAPM (modern derivation) Projections Pricing Kernel and Expectation Kernel Slide 07-35

36 Projections States s=1,,s with π s >0 Probability inner product π-norm (measure of flength) Slide 07-36

37 y shrink axes y x x x and y are π-orthogonal iff [x,y] π = 0, I.e. E[xy]=0 Slide 07-37

38 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 07-38

39 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 07-39

40 Expected Value and Co-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 07-40

41 Expected dv Value and Co-Variance E[x] = [x, 1]= Slide 07-41

42 Overview 1. Simple CAPM with quadratic utility functions (derived from state-price beta model) 2. Mean-variance preferences Portfolio Theory CAPM (Intuition) 3. CAPM (modern derivation) Projections Pricing Kernel and Expectation Kernel Slide 07-42

43 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 07-43

44 Pricing Kernel k q M space of ffeasible 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 = m * q in our old notation.) Slide 07-44

45 Pricing Pii 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 (1,1,1) 1 1) is the unique pricing kernel. Example 2: S=3,π s =1/3 1/3for s=1,2,3, 123 x 1 =(1,0,0), x 2 =(0,1,0), p=(1/3,2/3). Then k=(1,2,0) (120)is the unique pricing i kernel. Slide 07-45

46 Pricing Pii 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 07-46

47 Expectations ti 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 07-47

48 Mean Variance Frontier Fin 501: Asset Pricing 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 07-48

49 Frontier Returns Frontier returns are the returns of frontier payoffs with non-zero prices. x graphically: payoffs with price of p=1. Slide 07-49

50 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 ) q Slide 07-50

51 Mean-Variance (Payoff) ff)frontier (111) (1,1,1) standard deviation 0 k q expected return NB: graphical illustrated of expected returns and standard deviation changes if bond is not in payoff span. Slide 07-51

52 Mean-Variance (Payoff) ff)frontier efficient (return) frontier (111) (1,1,1) standard deviation 0 k q expected return inefficient (return) frontier Slide 07-52

53 Frontier Returns (if agent is risk-neutral) Slide 07-53

54 Minimum i Variance Portfolio Take FOC w.r.t. λ of Hence, MVP has return of Slide 07-54

55 Illustration of MVP Fin 501: Asset Pricing Expected return of MVP M = R 2 and S=3 (1,1,1) Minimum i standard d deviation Slide 07-55

56 Mean-Variance Efficient i 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 i returns correspond to λ 0. Pricing kernel (portfolio) is not mean-variance efficient, since Slide 07-56

57 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 07-57

58 Illustration of ZC Portfolio M = R 2 and S=3 (1,1,1) arbitrary yportfolio p Recall: Slide 07-58

59 Illustration of ZC Portfolio Green lines do not necessarily cross. (1,1,1) arbitrary yportfolio p ZC of p Slide 07-59

60 Bt Beta Pricing Pii 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 )ande[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 07-60

61 Beta Bt Pricing Pii Taking expectations and deriving covariance _ If risk-free asset can be replicated, beta-pricing equation simplifies to Problem: How to identify frontier returns Slide 07-61

62 Capital Asset Pricing i 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 07-62

63 Capital Asset Pricing i 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 f) N.B. first equation always hold if there are only two assets. Slide 07-63

64 Outdated t d material follows Traditional derivation of CAPM is less elegant Not relevant for exams Slide 07-64

65 Characterization ti of Frontier Portfolios Proposition 6.1: The entire set of frontier portfolios fli 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 07-65

66 Characterization ti of Frontier Portfolios Proposition 6.1: The entire set of frontier portfolios can be generated by ("are convex combinations" of) g and g+h. Proof: To see this let q be an arbitrary frontier portfolio with as its expected return. E ( ) r~q Consider portfolio weights (proportions) π ( ) ( ) g = 1 E r~ q and π ~ g+ h = E r q then, as asserted, [ 1 E ( r~ ) ] q g + E ( r~ q )( g + h ) = g + he ( r~ q ) = wq. Slide 07-66

67 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. Proof: To see this, let p 1 and p 2 be any two distinct frontier portfolios; since the frontier portfolios are E[r p1 ] E[r p2 ] different. e Let q be an arbitrary a frontier portfolio, o o, with expected return equal to E[r q ]. Since E[r p1 ] E[r p2 ], there must exist a unique number such that E[r q ]=α α E[r p1 ]+(1 - α) E[r p2 ] (6.16) Now consider a portfolio of p 1 and p 2 with weights α, 1-α, respectively, as determined by (6.16). We must show that w q = α w p1 +(1-α)w p2. Slide 07-67

68 Proof of Proposition 6.2 (continued) ( 1 α) w = α[ g + he( r~ )] + ( 1 α) [ g he( )] αw ~ p + + r 1 p2 p1 p2 = g + h = g + he [ αe( r~ ) ( ) ( )] p + 1 α E r~ 1 p2 ( r~ ), by construction q = w q, since q is a frontier portfolio. Slide 07-68

69 Proposition 6.3 : Any convex combination of frontier portfolios fli is also a frontier portfolio. fli Proof : Let ( w1... w N ), define N frontier portfolios ( w th i represents the vector defining the composition of the i portfolios) and let α i, i = 1,..., N be real numbers such that N. Lastly, let denote the expected return i= 1αi = 1 E( r i ) of portfolio with weights w i. The weights corresponding to a linear combination of the above N portfolios are : N α ( g he( )) N N αiwi = αi + ~ r i i= 1 i= 1 N = α N g + h E ( r ) α i i= 1 i= 1 N α ~ i r = g + h E( i ) i= 1 N iw i E( r) = αie( ~ r i ) Thus is a frontier portfolio with. i= 1 i r~ i i= 1 Slide 07-69

70 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

71 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

72 Figure 6-3 The Set of Frontier Portfolios: Mean/Variance Space Slide

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

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

75 Characterization of Efficient Portfolios (No Risk-Free Assets) Definition 6.2: Efficient portfolios are those frontier portfolios which h are not mean-variance dominated. d Lemma: Efficient portfolios are those frontier portfolios for which the expected return exceeds A/C, the expected return of the minimum variance portfolio. Slide

76 Proposition 6.4 : The set of efficient portfolios is a convex set. This does not mean, however, that the frontier of this set is convexshaped in the risk-return space. Proof : Suppose each of the N portfolios considered above was efficient; then r~ A /, for every portfolio i. ( ) C E i N N A αie( r~ i ) αi = i= 1 i= 1 C A C However ; thus, the convex combination is efficient as well. So the set of efficient portfolios, fli as characterized by their hi portfolio fli weights, ih is a convex set. Slide

77 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

78 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

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

80 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 07-80

81 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 07-81

82 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 07-82

83 All investors hold mean-variance efficient portfolios the market portfolio is convex combination of efficient i 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) ]} p q q (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~ ) ( r~ r~ r~ ( )) [ ( ) ( ( ))] j = E ZC M + βmj E M E ZC M (6.28) (6.29) Slide 07-83

84 Zero-Beta Bt 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 07-84