FIN501 Asset Pricing Lecture 06 Mean-Variance & CAPM (1) Markus K. Brunnermeier LECTURE 06: MEAN-VARIANCE ANALYSIS & CAPM FIN501 Asset Pricing Lecture 06 Mean-Variance & CAPM (2) Overview 1. Introduction: Simple CAPM with quadratic utility functions (from beta-state price equation) 2. Traditional Derivation of CAPM – Demand: Portfolio Theory for given – Aggregation: Fund Separation Theorem prices/returns – Equilibrium: CAPM 3. Modern Derivation of CAPM – Projections – Pricing Kernel and Expectation Kernel 4. Testing CAPM 5. Practical Issues – Black-Litterman FIN501 Asset Pricing Lecture 06 Mean-Variance & CAPM (3) Recall State-price Beta model Recall: 퐸 푅ℎ − 푅푓 = 훽ℎ퐸 푅∗ − 푅푓 cov 푅∗,푅ℎ Where 훽ℎ ≔ var 푅∗ very general – but what is 푅∗ in reality? FIN501 Asset Pricing Lecture 06 Mean-Variance & CAPM (4) Simple CAPM with Quadratic Expected Utility 1. All agents are identical 휕1푢 • Expected utility 푈 푥0, 푥1 = 푠 휋푠푢 푥0, 푥푠 ⇒ 푚 = 퐸 휕0푢 2 • Quadratic 푢 푥0, 푥1 = 푣0 푥0 − 푥1 − 훼 • ⇒ 휕1푢 = −2 푥1,1 − 훼 , … , −2 푥푆,1 − 훼 • Excess return cov 푚, 푅ℎ 푅푓cov 휕 푢, 푅ℎ 퐸 푅ℎ − 푅푓 = − = − 1 퐸 푚 퐸 휕0푢 푅푓cov −2 푥 − 훼 , 푅ℎ 2cov 푥 , 푅ℎ = − 1 = 푅푓 1 퐸 휕0푢 퐸 휕0푢 • Also holds for market portfolio ℎ 푓 ℎ 퐸 푅 − 푅 cov 푥1, 푅 푚푘푡 푓 = 푚푘푡 퐸 푅 − 푅 cov 푥1, 푅 FIN501 Asset Pricing Lecture 06 Mean-Variance & CAPM (5) Simple CAPM with Quadratic Expected Utility ℎ 푓 ℎ 퐸 푅 − 푅 cov 푥1, 푅 푚푘푡 푓 = 푚푘푡 퐸 푅 − 푅 cov 푥1, 푅 2. Homogenous agents + Exchange economy 푚 ⇒ 푥1 = aggr. endowment and is perfectly correlated with 푅 퐸 푅ℎ − 푅푓 cov 푅푚푘푡, 푅ℎ = 퐸 푅푚푘푡 − 푅푓 var 푅푚푘푡 cov 푅ℎ,푅푚푘푡 Since 훽ℎ = var 푅푚푘푡 Market Security Line 퐸 푅ℎ = 푅푓 + 훽ℎ 퐸 푅푚푘푡 − 푅푓 푚푘푡 ∗ 푓 푎+푏1푅 NB: 푅 = 푅 푓 in this case 푏1 < 0 ! 푎+푏1푅 FIN501 Asset Pricing Lecture 06 Mean-Variance & CAPM (6) Overview 1. Introduction: Simple CAPM with quadratic utility functions 2. Traditional Derivation of CAPM – Demand: Portfolio Theory for given – Aggregation: Fund Separation Theorem prices/returns – Equilibrium: CAPM 3. Modern Derivation of CAPM – Projections – Pricing Kernel and Expectation Kernel 4. Testing CAPM 5. Practical Issues – Black-Litterman FIN501 Asset Pricing Lecture 06 Mean-Variance & CAPM (7) Definition: Mean-Variance Dominance & Efficient Frontier • Asset (portfolio) A mean-variance dominates asset (portfolio) B if 휇퐴 ≥ 휇퐵 and 휎퐴 < 휎퐵 or if 휇퐴 > 휇퐵 while 휎퐴 ≤ 휎퐵. • 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. FIN501 Asset Pricing Lecture 06 Mean-Variance & CAPM (8) Expected Portfolio Returns & Variance • Expected returns (linear) 푗 ℎ ℎ ℎ 푗 ℎ – 휇 ≔ 퐸 푟 = 풘 ′흁, where each 푤 = 푗 푗 ℎ Everything is in returns • Variance (like all prices =1) 2 ′ – 휎ℎ ≔ var 푟ℎ = 풘 푉풘 휎2 휎 푤 푤 푤 1 12 1 = 1 2 2 푤 휎21 휎2 2 2 2 2 2 = 푤1 휎1 + 푤2 휎2 + 2푤1푤2휎12 ≥ 0 FIN501 Asset Pricing Lecture 06 Mean-Variance & CAPM (9) Illustration of 2 Asset Case • For certain weights: 푤1 and 1 − 푤1 휇ℎ = 푤1휇1 + 1 − 푤1 휇2 2 2 2 2 2 휎ℎ = 푤1 휎1 + 1 − 푤1 휎2 + 2푤1 1 − 푤1 휌12휎1휎2 2 (Specify 휎ℎ and one gets weights and 휇ℎ’s) • Special cases [푤1 to obtain certain 휎ℎ] ±휎ℎ−휎2 – 휌12 = 1 ⇒ 푤1 = 휎1−휎2 ±휎ℎ+휎2 – 휌12 = −1 ⇒ 푤1 = 휎1+휎2 FIN501 Asset Pricing Lecture 06 Mean-Variance & CAPM (10) ±휎ℎ−휎2 For 휌12 = 1 ⇒ 푤1 = 휎1−휎2 휎ℎ = 푤1휎1 + 1 − 푤1 휎2 휇2 − 휇1 휇ℎ = 푤1휇1 + 1 − 푤1 휇2 = 휇1 + ±휎ℎ − 휎1 휎2 − 휎1 휇2 휇ℎ 휇1 휎1 휎ℎ 휎2 Lower part is irrelevant 휇2 − 휇1 휇ℎ = 휇1 + −휎ℎ − 휎1 휎2 − 휎1 The Efficient Frontier: Two Perfectly Correlated Risky Assets FIN501 Asset Pricing Lecture 06 Mean-Variance & CAPM (11) ±휎푝+−휎2 For 휌12 = −1 ⇒ 푤1 = 휎1+휎2 휎ℎ = 푤1휎1 − 1 − 푤1 휎2 휎2 휎1 휇2 − 휇1 휇ℎ = 푤1휇1 + 1 − 푤1 휇2 = 휇1 + 휇2 ± 휎푝 휎1 + 휎2 휎1 + 휎2 휎1 + 휎2 휇 휇 −휇 2 slope: 2 1 휎1+휎2 휎2 휎1 intercept: 휇1+ 휇2 휎1+휎2 휎1+휎2 휇 −휇 slope: − 2 1 휇1 휎1+휎2 휎1 휎2 The Efficient Frontier: Two Perfectly Negative Correlated Risky Assets FIN501 Asset Pricing Lecture 06 Mean-Variance & CAPM (12) For 휌12 ∈ −1,1 E[r2] E[r1] s1 s2 The Efficient Frontier: Two Imperfectly Correlated Risky Assets FIN501 Asset Pricing Lecture 06 Mean-Variance & CAPM (13) For 휎1 = 0 휇2 휇ℎ 휇1 휎1 휎ℎ 휎2 The Efficient Frontier: One Risky and One Risk-Free Asset FIN501 Asset Pricing Lecture 06 Mean-Variance & CAPM (14) Efficient frontier with n risky assets • A frontier portfolio is one which displays minimum variance among all feasible portfolios with the same expected portfolio return. 피[푟] 1 • min 풘′푉풘 풘 2 ′ ℎ ℎ – 휆: 풘 흁 = 휇 , 푗 푤푗 피 푟푖 = 휇 ′ – 훾: 풘 ퟏ = 1, 푗 푤푗 = 1 s • Result: Portfolio weights are linear in expected portfolio return ℎ 푤ℎ = 퓰 + 퓱휇 ℎ – If 휇 = 0, 푤ℎ = 퓰 ℎ – If 휇 = 1, 푤ℎ = 퓰 + 퓱 • Hence, 퓰 and 퓰 + 퓱 are portfolios on the frontier FIN501 Asset Pricing Lecture 06 Mean-Variance & CAPM (15) 휕ℒ = 푉풘 − 휆흁 − 훾ퟏ = 0 휕푤 휕ℒ = 휇ℎ − 풘′흁 = 0 휕휆 휕ℒ = 1 − 풘′ퟏ = 0 휕훾 The first FOC can be written as: 푉풘 = 휆흁 + 훾ퟏ 풘 = 휆푉−1흁 + 훾푉−1ퟏ ′ ′ −1 ′ −1 흁 풘 = 휆 흁 푉 흁 + 훾 흁 푉 ퟏ skip FIN501 Asset Pricing Lecture 06 Mean-Variance & CAPM (16) ′ ′ st nd • Noting that 흁 풘ℎ = 푤ℎ, combining 1 and 2 FOC ′ ′ −1 ′ −1 휇ℎ = 흁 풘ℎ = 휆 흁 푉 흁 + 훾 흁 푉 ퟏ 퐵 퐴 • Pre-multiplying the 1st FOC by 1 yields ′ ′ ′ −1 ′ −1 ퟏ 풘ℎ = 풘ℎퟏ = 휆(ퟏ 푉 흁 + 훾 ퟏ 푉 ퟏ = 1 1 = 휆(ퟏ′푉−1흁)+ 훾 ퟏ′푉−1ퟏ 퐴 퐶 • Solving for 휆, 훾 퐶휇ℎ − 퐴 퐵 − 퐴휇ℎ 휆 = , 훾 = 퐷 퐷 퐷 = 퐵퐶 − 퐴2 skip FIN501 Asset Pricing Lecture 06 Mean-Variance & CAPM (17) −1 −1 • Hence, 풘ℎ = 휆푉 흁 + 훾푉 ퟏ becomes 퐶휇ℎ − 퐴 퐵 − 퐴휇ℎ 풘 = 푉−1흁 + 푉−1ퟏ ℎ 퐷 퐷 1 1 = 퐵 푉−1ퟏ − 퐴 푉−1흁 + 퐶 푉−1흁 − 퐴 푉−1ퟏ 휇ℎ 퐷 퐷 퓰 퓱 • Result: Portfolio weights are linear in expected portfolio ℎ return 풘ℎ = 퓰 + 퓱휇 ℎ – If 휇 = 0, 풘ℎ = 퓰 ℎ – If 휇 = 1, 풘ℎ = 퓰 + 퓱 • Hence, 퓰 and 퓰 + 퓱 are portfolios on the frontier skip FIN501 Asset Pricing Lecture 06 Mean-Variance & CAPM (18) Characterization of Frontier Portfolios • Proposition: The entire set of frontier portfolios can be generated by ("are convex combinations" 퓰 of) and 퓰 + 퓱. • Proposition: The portfolio frontier can be described as convex combinations of any two frontier portfolios, not just the frontier portfolios 퓰 and 퓰 + 퓱. • Proposition: Any convex combination of frontier portfolios is also a frontier portfolio. skip FIN501 Asset Pricing Lecture 06 Mean-Variance & CAPM (19) …Characterization of Frontier Portfolios… • For any portfolio on the frontier, 휎2 휇ℎ = 퓰 + 퓱휇ℎ ′푉 퓰 + 퓱휇ℎ with 퓰 and 퓱 as defined earlier. Multiplying all this out and some algebra yields: 퐶 퐴 2 1 휎2 휇ℎ = 휇ℎ − + 퐷 퐶 퐶 skip FIN501 Asset Pricing Lecture 06 Mean-Variance & CAPM (20) …Characterization of Frontier Portfolios… 퐴 i. the expected return of the minimum variance portfolio is ; 퐶 1 ii. the variance of the minimum variance portfolio is given by ; 퐶 퐶 퐴 2 1 iii. Equation 휎2 휇ℎ = 휇ℎ − + is a 퐷 퐶 퐶 1 퐴 – parabola with vertex , in the expected return/variance space 퐶 퐶 – hyperbola in the expected return/standard deviation space. skip FIN501 Asset Pricing Lecture 06 Mean-Variance & CAPM (21) 퐴 퐷 1 퐸 푟 = ± 휎2 − ℎ 퐶 퐶 퐶 Figure 6-3 The Set of Frontier Portfolios: Mean/Variance Space FIN501 Asset Pricing Lecture 06 Mean-Variance & CAPM (22) Figure 6-4 The Set of Frontier Portfolios: Mean/SD Space FIN501 Asset Pricing Lecture 06 Mean-Variance & CAPM (23) Figure 6-5 The Set of Frontier Portfolios: Short Selling Allowed FIN501 Asset Pricing Lecture 06 Mean-Variance & CAPM (24) Efficient Frontier with risk-free asset The Efficient Frontier: One Risk Free and n Risky Assets FIN501 Asset Pricing Lecture 06 Mean-Variance & CAPM (25) Efficient Frontier with risk-free asset 1 • min 풘′푉풘 풘 2 – s.t. 풘′흁 + 1 − 푤푇ퟏ 푟푓 = 휇ℎ – FOC −1 푓 • 풘ℎ = 휆푉 흁 − 푟 ퟏ ℎ 푓 푓 푇 휇 −푟 • Multiplying by 흁 − 푟 ퟏ yields 휆 = ′ 흁−푟푓ퟏ 푉−1 흁−푟푓ퟏ – Solution 푉−1 흁−푟푓ퟏ 휇ℎ−푟푓 • 풘 = , where 퐻 = 퐵 − 2퐴푟푓 + 퐶(푟푓)2 ℎ 퐻2 FIN501 Asset Pricing Lecture 06 Mean-Variance & CAPM (26) Efficient frontier with risk-free asset • Result 1: Excess return in frontier excess return 푓 퐸 푟푝 − 푟 cov 푟 , 푟 = 풘′ 푉풘 = 풘′ 흁 − 푟푓ퟏ ℎ 푝 ℎ 푝 ℎ 퐻2 푓 푓 퐸 푟ℎ − 푟 퐸 푟푝 − 푟 = 퐻2 푓 2 퐸 푟푝 − 푟 var 푟 = 푝 퐻2 푓 cov 푟ℎ, 푟푝 푓 퐸 푟ℎ − 푟 = 퐸 푟푝 − 푟 var 푟푝 훽ℎ,푝 (Holds for any frontier portfolio 푝, in particular the market portfolio) FIN501 Asset Pricing Lecture 06 Mean-Variance & CAPM (27) Efficient Frontier with risk-free asset • Result 2: Frontier is linear in 퐸 푟 , 휎 -space 2 퐸 푟ℎ − 푟푓 var 푟 = ℎ 퐻2 퐸 푟ℎ = 푟푓 + 퐻휎ℎ where 퐻 is the Sharpe ratio 퐸 푟ℎ − 푟푓 퐻 = 휎ℎ FIN501 Asset Pricing Lecture 06 Mean-Variance & CAPM (28) Overview 1.