FIN501 Asset Pricing Lecture 06 Mean-Variance & CAPM

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

• 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)

Recall HARA class of preferences FIN501 Asset Pricing Lecture 06 Mean-Variance & CAPM (30) Two Fund Separation

Price of Risk = = highest Sharpe ratio

Optimal Portfolios of Two Investors with Different Risk Aversion FIN501 Asset Pricing Lecture 06 Mean-Variance & CAPM (31) Mean-Variance Preferences

휕푈 휕푈 • 푈 휇ℎ, 휎ℎ with > 0, 2 < 0 휕휇ℎ 휕휎ℎ 휌 – Example: 퐸 푊 − var 푊 2

• Also in expected utility framework – Example 1: Quadratic utility function (with portfolio return 푅) • 푈 푅 = 푎 + 푏푅 + 푐푅2 2 2 2 • vNM: 퐸 푈 푅 = 푎 + 푏퐸 푅 + 푐퐸 푅 = 푎 + 푏휇ℎ + 푐휇ℎ + 푐휎ℎ = 푔 휇ℎ, 휎ℎ – Example 2: CARA Gaussian 푖 푖 • asset returns jointly normal ⇒ 푖 푤 푟 normal 휌 • If 푈 is CARA ⇒ certainty equivalent is 휇 − 휎2 ℎ 2 ℎ FIN501 Asset Pricing Lecture 06 Mean-Variance & CAPM (32) Overview

• 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 FIN501 Asset Pricing Lecture 06 Mean-Variance & CAPM (34) The CAPM with a risk-free bond

• 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, 푟푓). 퐸 푅푚푘푡 −푅 • The slope of Capital Market Line (CML): 푓 휎푚푘푡 푚푘푡 퐸 푅 − 푅푓 퐸 푅ℎ = 푅푓 + 휎ℎ 휎푚푘푡 FIN501 Asset Pricing Lecture 06 Mean-Variance & CAPM (35) The Capital Market Line

CML

M rM

rf j

s M s p FIN501 Asset Pricing Lecture 06 Mean-Variance & CAPM (36) The Security Market Line

E(r)

SML

E(ri)

E(rM)

rf slope SML = (E(ri)-rf) /b i

b M=1 b i b FIN501 Asset Pricing Lecture 06 Mean-Variance & CAPM (37) Overview

• States 푠 = 1, … , 푆 with 휋푠 > 0 • Probability inner product

푥, 푦 휋 = 휋푠푥푠푦푠 = 휋푠푥푠 휋푠푦푠 푠 푠

• 휋-norm 푥 = 푥, 푥 휋 (measure of length) i. 푥 > 0 ∀푥 ≠ 0 and 푥 = 0 if 푥 = 0 ii. 휆푥 = 휆 푥 iii. 푥 + 푦 ≤ 푥 + 푦 ∀푥; 푦 ∈ ℝ푆 FIN501 Asset Pricing Lecture 06 Mean-Variance & CAPM (39)

) shrink axes y y

x x

x and y are 휋-orthogonal iff 푥, 푦 휋 = 0, i.e. 퐸 푥푦 = 0 FIN501 Asset Pricing Lecture 06 Mean-Variance & CAPM (40) …Projections…

• 풵 space of all linear combinations of vectors 푧1, … , 푧푛 • Given a vector 푦 ∈ ℝ푆 solve 2 min 퐸 푦 − 훼푗푧푗 훼∈ℝ푛 푗 푗 푗 푗 • FOC: 푠 휋푠 푦푠 − 푗 훼 푧푠 푧 = 0 풵 푗 푗 풵 – Solution 훼 ⇒ 푦 = 푗 훼 푧 , 휖 ≔ 푦 − 푦

• [smallest distance between vector y and space] Z FIN501 Asset Pricing Lecture 06 Mean-Variance & CAPM (41) …Projections

y e

퐸 휀푧푗 = 0 for each 푗 = 1, … , 푛 (FOC) 휀 ⊥ 푧 푦푧 is the (orthogonal) projection on 풵 푦 = 푦풵 + 휀′, 푦풵 ∈ 풵, 휀 ⊥ 푧 FIN501 Asset Pricing Lecture 06 Mean-Variance & CAPM (42) Expected Value and Co-Variance… squeeze axis by 휋푠 (1,1)

푥, 푦 = 퐸 푥푦 = cov 푥, 푦 + 퐸 푥 퐸 푦 x 푥, 푥 = 퐸 푥2 = var 푥 + 퐸 푥 2 푥 푥 = 퐸 푥2

푥

푥 = 푥 + 푥 FIN501 Asset Pricing Lecture 06 Mean-Variance & CAPM (43) …Expected Value and Co-Variance

• 푥 = 푥 + 푥 where – 푥 is a projection of 푥 onto 1 – 푥 is a projection of 푥 onto 1 ⊥

• 퐸 푥 = 푥, 1 휋 = 푥 , 1 휋 = 푥 1,1 휋 = 푥 scalar • var 푥 = 푥 , 푥 휋 = var[푥 ] slight abuse of notation – 휎푥 = 푥 휋 • cov 푥, 푦 = cov 푥 , 푦 = 푥 , 푦 휋 • Proof: 푥, 푦 휋 = 푥 , 푦 휋 + 푥 , 푦 휋 – 푦 , 푥 휋 = 푦 , 푥 휋 = 0, 푥, 푦 휋 = 퐸 푦 퐸 푥 + cov[푥 , 푦 ] FIN501 Asset Pricing Lecture 06 Mean-Variance & CAPM (44) Overview

• 푋 space of feasible payoffs. • If no arbitrage and 휋 ≫ 0 there exists SDF 푚 ∈ ℝ푆, 푚 ≫ 0, such that 푞 푧 = 퐸 푚푧 . • 푚 ∈ ℝ푆– SDF need not be in asset span. • A pricing kernel is a 푚∗ ∈ 푋 such that for each 푧 ∈ 푋 , 푞 푧 = 퐸 푚∗푧 FIN501 Asset Pricing Lecture 06 Mean-Variance & CAPM (46) …Pricing Kernel - Examples…

• Example 1: 1 – 푆 = 3, 휋푠 = 3 1 2 – 푥 = 1,0,0 , 푥 = 0,1,1 and 푝 = , 1 2 3 3 – Then 푚∗ = 1,1,1 is the unique pricing kernel. • Example 2: 1 2 – 푥 = 1,0,0 , 푥 = 0,1,0 , 푝 = , 1 2 3 3 – Then 푚∗ = 1,2,0 is the unique pricing kernel. FIN501 Asset Pricing Lecture 06 Mean-Variance & CAPM (47) …Pricing Kernel – Uniqueness

• If a state price density exists, there exists a unique pricing kernel. – If dim 푋 = 푆 (markets are complete), there are exactly 푚 equations and 푚 unknowns – If dim 푋 < 푆, (markets may be incomplete) For any state price density (=SDF) 푚 and any 푧 ∈ 푋 퐸 푚 − 푚∗ 푧 = 0 푚 = 푚 − 푚∗ + 푚∗ ⇒ 푚∗is the “projection” of 푚 on 푋 • Complete markets ⇒ 푚∗ = 푚 (SDF=state price density) FIN501 Asset Pricing Lecture 06 Mean-Variance & CAPM (48) Expectations Kernel 푘∗

• An expectations kernel is a vector 푘∗ ∈ 푋 – Such that 퐸 푧 = 퐸 푘∗푧 for each 푧 ∈ 푋 • Example 1 – 푆 = 3, 휋푠 = , 푥 = 1,0,0 , 푥 = 0,1,0 3 1 2 – Then the unique 푘∗ = 1,1,0 • If 휋 ≫ 0, there exists a unique expectations kernel. • Let 퐼 = 1, … , 1 then for any 푧 ∈ 푋 퐸 퐼 − 푘∗ 푧 = 0

– 푘∗is the “projection” of 퐼 on 푋 ∗ – 푘 = 퐼 if bond can be replicated (e.g. if markets are complete) FIN501 Asset Pricing Lecture 06 Mean-Variance & CAPM (49) Mean Variance Frontier

• Definition 1: 푧 ∈ 푋 is in the mean variance frontier if there exists no 푧′ ∈ 푋 such that 퐸 푧′ = 퐸 푧 , 푞 푧′ = 푞 푧 and var 푧′ < var 푧

• Definition 2: Let ℰ be the space generated by 푚∗and 푘∗ – Decompose 푧 = 푧휀 + 휀 with 푧ℰ ∈ ℰ and 휀 ⊥ ℰ – Hence, 퐸 휀 = 퐸 휀푘∗ = 0, 푞 휀 = 퐸 휀푚∗ = 0 cov 휀, 푧휀 = 퐸 휀푧휀 = 0, since 휀 ⊥ ℰ 휀 – var 푧 = var 푧 + var 휀 (price of 휀 is zero, but positive variance) • 푧 is in mean variance frontier z . ) 2 E – Every 푧 ∈ ℰ is in mean variance frontier. FIN501 Asset Pricing Lecture 06 Mean-Variance & CAPM (50) Frontier Returns…

• Frontier returns are the returns of frontier payoffs with non-zero prices. [Note: R indicates Gross return]

푘∗ 푘∗ 푅 ∗ = = 푘 푞 푘∗ 퐸 푚∗ 푚∗ 푚∗ 푅 ∗ = = 푚 푞 푚∗ 퐸 푚∗푚∗ • If 푧 = 훼푚∗ + 훽푘∗ then 훼푞 푚∗ 훽푞 푘∗ 푅 = 푅 ∗ + 푅 ∗ 푧 훼푞 푚∗ + 훽푞 푘∗ 푚 훼푞 푚∗ + 훽푞 푘∗ 푘 휆 1−휆

• graphically: payoffs with price of p=1. FIN501 Asset Pricing Lecture 06 Mean-Variance & CAPM (51)

푋 = RS = R3

Mean-Variance Payoff Frontier e

푚∗

Mean-Variance Return Frontier p=1-line = return-line (orthogonal to 푚∗) FIN501 Asset Pricing Lecture 06 Mean-Variance & CAPM (52)

Mean-Variance (Payoff) Frontier

(1,1,1) standard deviation

0 푚∗ expected return

NB: graphical illustrated of expected returns and standard deviation changes if bond is not in payoff span. FIN501 Asset Pricing Lecture 06 Mean-Variance & CAPM (53)

Mean-Variance (Payoff) Frontier

efficient (return) frontier

(1,1,1) standard deviation

0 푚∗ expected return

inefficient (return) frontier FIN501 Asset Pricing Lecture 06 Mean-Variance & CAPM (54) …Frontier Returns (if agent is risk-neutral) ∗ ∗ • If 푘 = 훼푚 , frontier returns ≡ 푅푘∗ • If푘∗ ≠ 훼푚∗, frontier returns can be written as: 푅휆 = 푅푘∗ + 휆 푅푚∗ − 푅푘∗ • Expectations and variance are 퐸 푅휆 = 퐸 푅푘∗ + 휆 퐸 푅푚∗ − 퐸 푅푘∗ var 푅휆 = 2 = var 푅푘∗ + 2휆cov 푅푘∗, 푅푚∗ − 푅푘∗ + 휆 var 푅푚∗ − 푅푘∗ • If risk-free asset exists, these simplify to: 퐸 푅푚∗ − 푅푓 퐸 푅휆 = 푅푓 + 휆 퐸 푅푚∗ − 푅푓 = 푅푓 ± 휎 푅휆 휎 푅푚∗ 2 var 푅휆 = 휆 var[푅푚∗] , 휎 푅휆 = 휆 휎 푅푚∗ FIN501 Asset Pricing Lecture 06 Mean-Variance & CAPM (55) Minimum Variance Portfolio

• Take FOC w.r.t. 휆 of var 푅휆 = var 푅푘∗ + 2휆cov 푅푘∗, 푅푚∗ − 푅푘∗ 2 + 휆 var 푅푚∗ − 푅푘∗

• Hence, MVP has return of 푅푘∗ + 휆0 푅푚∗ − 푅푘∗ cov 푅푘∗, 푅푚∗ − 푅푘∗ 휆0 = − var 푅푚∗ − 푅푘∗ FIN501 Asset Pricing Lecture 06 Mean-Variance & CAPM (56) Illustration of MVP

푋 = ℝ2 and 푆 = 3 Expected return of MVP

Minimum standard deviation (1,1,1) FIN501 Asset Pricing Lecture 06 Mean-Variance & CAPM (57) 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 퐸 푅휆 ≥ 퐸 푅휆0

• If risk-free asset can be replicated – Mean variance efficient returns correspond to 휆0. – Pricing kernel (portfolio) is not mean-variance efficient, 퐸 푚∗ 1 퐸 푅 ∗ = < = 푅 since 푚 퐸 푚∗ 2 퐸 푚∗ 푓 FIN501 Asset Pricing Lecture 06 Mean-Variance & CAPM (58) Zero-Covariance Frontier Returns

• Take two frontier portfolios with returns 푅휆 = 푅푘∗ + 휆 푅푚∗ − 푅푘∗ and 푅휇 = 푅푘∗ + 휇 푅푚∗ − 푅푘∗

• cov 푅휇, 푅휆 = var 푅푘∗ + 휆 + 휇 cov 푅푘∗, 푅푚∗ − 푅푘∗ + 휆휇var 푅푚∗ − 푅푘∗

• The portfolios have zero co-variance if var 푅 ∗ + 휆cov 푅 ∗, 푅 ∗ − 푅 ∗ 휇 = − 푘 푘 푚 푘 cov 푅푘∗, 푅푚∗ − 푅푘∗ + 휆var 푅푚∗ − 푅푘∗

• For all 휆 ≠ 휆0, 휇 exists – 휇 = 0 if risk-free bond can be replicated FIN501 Asset Pricing Lecture 06 Mean-Variance & CAPM (59) Illustration of ZC Portfolio…

푋 = ℝ2 and 푆 = 3

arbitrary portfolio p (1,1,1) Recall: cov 푥, 푦 = 푥 , 푦 휋 FIN501 Asset Pricing Lecture 06 Mean-Variance & CAPM (60) …Illustration of ZC Portfolio

Green lines do not necessarily cross.

arbitrary portfolio p (1,1,1)

ZC of p FIN501 Asset Pricing Lecture 06 Mean-Variance & CAPM (61) Beta Pricing…

• Frontier Returns (are on linear subspace). Hence 푅훽 = 푅휇 + 훽 푅휆 − 푅휇 • Consider any asset with payoff 푥푗 휀 – It can be decomposed in 푥푗 = 푥푗 + 휀푖 휀 휀 – 푞 푥푗 = 푞 푥푗 and 퐸 푥푗 = 퐸 푥푗 , since 휀 ⊥ ℰ 휀 휀푗 – Return of 푥푗 is 푅푗 = 푅푗 + 푞 푥푗 – Using above and assuming 휆 ≠ 휆0 and 휇 is ZC-portfolio of 휆, 휀푗 푅푗 = 푅휇 + 훽푗 푅휆 − 푅휇 + 푞 푥푗 FIN501 Asset Pricing Lecture 06 Mean-Variance & CAPM (62) …Beta Pricing

• Taking expectations and deriving covariance

• 퐸 푅푗 = 퐸 푅휇 + 훽푗 퐸 푅휆 − 퐸 푅휇

cov 푅휆,푅푗 • cov 푅휆, 푅푗 = 훽푗var 푅휆 ⇒ 훽푗 = var 푅휆 휀푗 – Since 푅휆 ⊥ 푞 푥푗 • If risk-free asset can be replicated, beta-pricing equation simplifies to

퐸 푅푗 = 푅푓 + 훽푗 퐸 푅휆 − 푅푓

• Problem: How to identify frontier returns FIN501 Asset Pricing Lecture 06 Mean-Variance & CAPM (63) Capital Asset Pricing Model…

• CAPM = market return is frontier return – Derive conditions under which market return is frontier return – Two periods: 0,1.

푖 푋 – Endowment: individual 푤1 at time 1, aggregate 푤 1 = 푤 1 + 푌 푋 푌 푋 푤 1 , where 푤 1 , 푤 1 are orthogonal and 푤 1 is the

orthogonal projection of 푤 1 on 푋 . 푋 – The market payoff is 푤 1 푋 푋 푤 1 – Assume 푞 푤 1 ≠ 0, let 푅푚푘푡 = 푋 , and assume that 푞 푤 1 푅푚푘푡 is not the minimum variance return. FIN501 Asset Pricing Lecture 06 Mean-Variance & CAPM (64) …Capital Asset Pricing Model

• If 푅0 is the frontier return that has zero covariance with 푅푚푘푡 then, for every security j,

• 퐸 푅푗 = 퐸 푅0 + 훽푗 퐸 푅푚푘푡 − 퐸 푅0 with cov 푅푗,푅푚푘푡 훽푗 = var 푅푚푘푡

• If a risk free asset exists, equation becomes, 퐸[푅푗] = 푅푓 + 훽푗 퐸 푅푚푘푡 − 푅푓

• N.B. first equation always hold if there are only two assets. FIN501 Asset Pricing Lecture 06 Mean-Variance & CAPM (65) Overview

• Testing of CAPM • Jumping weights – Domestic investments – International investment • Black-Litterman solution FIN501 Asset Pricing Lecture 06 Mean-Variance & CAPM (67) Testing the CAPM

• Take CAPM as given and test empirical implications

• Time series approach – Regress individual returns on market returns

푅푖푡 − 푅푓푡 = 훼 푖 + 훽푖푚 푅푚푡 − 푅푓푡 + 휀푖푡 – Test whether constant term 훼푖 = 0

• Cross sectional approach – Estimate betas from time series regression – Regress individual returns on betas

푅푖 = 휆훽푖푚 + 훼푖 – Test whether regression residuals 훼푖 = 0 FIN501 Asset Pricing Lecture 06 Mean-Variance & CAPM (68) Empirical Evidence

• Excess returns on high-beta stocks are low • Excess returns are high for small stocks – Effect has been weak since early 1980s • Value stocks have high returns despite low betas • Momentum stocks have high returns and low betas FIN501 Asset Pricing Lecture 06 Mean-Variance & CAPM (69) Reactions and Critiques

• Roll Critique – The CAPM is not testable because composition of true market portfolio is not observable • Hansen-Richard Critique – The CAPM could hold conditionally at each point in time, but fail unconditionally • Anomalies are result of “data mining” • Anomalies are concentrated in small, illiquid stocks • Markets are inefficient – “joint hypothesis test” FIN501 Asset Pricing Lecture 06 Mean-Variance & CAPM (70) Practical Issues

• Estimation – How do we estimate all the parameters we need for portfolio optimization?

• What is the market portfolio? – Restricted short-sales and other restrictions – International assets & currency risk

• How does the market portfolio change over time? – Empirical evidence – More in dynamic models FIN501 Asset Pricing Lecture 06 Mean-Variance & CAPM (71) Overview

• An investor seeking to use mean-variance portfolio construction has to – Estimate N means, – N variances, – N*(N-1)/2 co-variances

• Estimating means – For any partition of [0,T] with N points (∆t=T/N): 1 1 푝 −푝 퐸 푟 = ⋅ ⋅ 푁 푟 = 푇 0 (in log prices) Δ푡 푁 푖=1 푖⋅Δ푡 푇 – Knowing the first and last price is sufficient FIN501 Asset Pricing Lecture 06 Mean-Variance & CAPM (73) Estimating Means

• Let 푋푘 denote the logarithmic return on the market, with 푘 = 1, … , 푛 over a period of length ℎ – The dynamics to be estimated are: 푋푘 = 휇 ⋅ Δ + 휎 ⋅ Δ ⋅ 휖푘 where the 휖푘are i.i.d. standard normal random variables. – The standard estimator for the expected logarithmic mean rate of return is: 1 푛 where 휇 = ⋅ 푋푘 ℎ ℎ 1 is length of observation – 푛 number of observations The mean and variance of this estimator Δ = 푛/ℎ 1 푛 1 퐸 휇 = ⋅ 퐸 푋푘 = ⋅ 푛 ⋅ 휇 ⋅ Δ = 휇 ℎ 1 ℎ 푛 2 1 1 2 휎 푉푎푟 휇 = 2 ⋅ 푉푎푟 푋푘 = 2 ⋅ 푛 ⋅ 휎 ⋅ Δ = ℎ 1 ℎ ℎ – The accuracy of the estimator depends only upon the total length of the observation period (h), and not upon the number of observations (n). FIN501 Asset Pricing Lecture 06 Mean-Variance & CAPM (74) Estimating Variances

• Consider the following estimator: 1 휎 2 = ⋅ 푛 푋2 ℎ 푖=1 푘

• The mean and variance of this estimator: 푛 1 1 ℎ 퐸 휎 2 = ⋅ 퐸 푋2 = ⋅ 푛 ⋅ 휇2 ⋅ Δ2 + 휎2 ⋅ Δ = 휎2 + 휇2 ⋅ ℎ 푘 ℎ 푛 푖=1 푛 푛 4 2 1 1 푛 2 2 ⋅ 휎 4 ⋅ 휇 ⋅ ℎ 푉푎푟 휎 2 = ⋅ 푉푎푟 푋2 = ⋅ 푉푎푟 푋2 = ⋅ 퐸 푋4 − 퐸 푋2 = + ℎ2 푘 ℎ2 푘 ℎ2 푘 푘 푛 푛2 푖=1 푖=1 – The estimator is biased b/c we did not subtract out the expected return from each realization. – Magnitude of the bias declines as n increases. – For a fixed h, the accuracy of the variance estimator can be improved by sampling the data more frequently. FIN501 Asset Pricing Lecture 06 Mean-Variance & CAPM (75) Estimating variances: Theory vs. Practice • For any partition of [0, 푇] with 푁 points (Δ푡 = 푇/푁): 푁 1 푉푎푟 푟 = ⋅ 푟 − 퐸 푟 2 → 휎2 푎푠 푁 → ∞ 푁 푖⋅Δ푡 푖=1

• Theory: Observing the same time series at progressively higher frequencies increases the precision of the estimate. • Practice: – Over shorter interval increments are non-Gaussian – Volatility is time-varying (GARCH, SV-models) – Market microstructure noise FIN501 Asset Pricing Lecture 06 Mean-Variance & CAPM (76) Estimating covariances: Theory vs. Practice • In theory, the estimation of covariances shares the features of variance estimation. • In practice: – Difficult to obtain synchronously observed time-series -> may require interpolation, which affects the covariance estimates. – The number of covariances to be estimated grows very quickly, such that the resulting covariance matrices are unstable (check condition numbers!). – Shrinkage estimators (Ledoit and Wolf, 2003, “Honey, I Shrunk the Covariance Matrix”) FIN501 Asset Pricing Lecture 06 Mean-Variance & CAPM (77) Unstable Portfolio Weights

• Are optimal weights statistically different from zero? – Properly designed regression yields portfolio weights – Statistical tests for significance of weight

• Example: Britton-Jones (1999) for international portfolio – Fully hedged USD Returns • Period: 1977-1966 • 11 countries – Results • Weights vary significantly across time and in the cross section • Standard errors on coefficients tend to be large FIN501 Asset Pricing Lecture 06 Mean-Variance & CAPM (78) Britton-Jones (1999)

1977-1996 1977-1986 1987-1996 weights t-stats weights t-stats weights t-stats Australia 12.8 0.54 6.8 0.20 21.6 0.66 Austria 3.0 0.12 -9.7 -0.22 22.5 0.74 Belgium 29.0 0.83 7.1 0.15 66 1.21 Canada -45.2 -1.16 -32.7 -0.64 -68.9 -1.10 Denmark 14.2 0.47 -29.6 -0.65 68.8 1.78 France 1.2 0.04 -0.7 -0.02 -22.8 -0.48 Germany -18.2 -0.51 9.4 0.19 -58.6 -1.13 Italy 5.9 0.29 22.2 0.79 -15.3 -0.52 Japan 5.6 0.24 57.7 1.43 -24.5 -0.87 UK 32.5 1.01 42.5 0.99 3.5 0.07 US 59.3 1.26 27.0 0.41 107.9 1.53 FIN501 Asset Pricing Lecture 06 Mean-Variance & CAPM (79) Black-Litterman Appraoch

• Since portfolio weights are very unstable, we need to discipline our estimates somehow – Our current approach focuses only on historical data

• Priors – Unusually high (or low) past return may not (on average) earn the same high (or low) return going forward – Highly correlated sectors should have similar expected returns – A “good deal” in the past (i.e. a good realized return relative to risk) should not persist if everyone is applying mean-variance optimization.

• Black Litterman Approach – Begin with “CAPM prior” – Add views on assets or portfolios – Update estimates using Bayes rule FIN501 Asset Pricing Lecture 06 Mean-Variance & CAPM (80) Black-Litterman Model: Priors

• Suppose the returns of 푁 risky assets (in vector/matrix notation) are 푟 ∼ 풩(휇, Σ)

• CAPM: The equilibrium risk premium on each asset is given by: Π = 훾 ⋅ Σ ⋅ 푤푒푞 – 훾 is the investors coefficient of risk aversion. – 푤푒푞 are the equilibrium (i.e. market) portfolio weights.

• The investor is assumed to start with the following Bayesian prior (with imprecision): 휇 = Π + 휖푒푞 where 휖푒푞 ∼ 푁 0, 휏 ⋅ Σ – The precision of the equilibrium return estimates is assumed to be proportional to the variance of the returns. – 휏 is a scaling parameter FIN501 Asset Pricing Lecture 06 Mean-Variance & CAPM (81) Black-Litterman Model: Views

• Investor views on a single asset affect many weights. • “Portfolio views” – Investor views regarding the performance of K portfolios (e.g. each portfolio can contain only a single asset) – P: K x N matrix with portfolio weights – Q: K x 1 vector of views regarding the expected returns of these portfolios • Investor views are assumed to be imprecise: 푃 ⋅ 휇 = 푄 + 휖푣 where 휖푣 ∼ 푁(0, Ω) – Without loss of generality, Ω is assumed to be a diagonal matrix – 휖푒푞 and 휖푣 are assumed to be independent FIN501 Asset Pricing Lecture 06 Mean-Variance & CAPM (82) Black-Litterman Model: Posterior

• Bayes rule: 푓(휃, 푥) 푓 푥 휃 ⋅ 푓(휃) 푓 휃 푥 = = 푓 푥 푓(푥)

• Posterior distribution: – If 푋1, 푋2 are normally distributed as: 푋 휇 Σ Σ 1 ∼ 푁 1 , 11 12 푋2 휇2 Σ21 Σ22 – Then, the conditional distribution is given by −1 −1 푋1|푋2 = 푥 ∼ 푁 휇1 + Σ12Σ22 푥 − 휇2 , Σ11 − Σ12Σ22 Σ21 FIN501 Asset Pricing Lecture 06 Mean-Variance & CAPM (83) Black-Litterman Model: Posterior

• The Black-Litterman formula for the posterior distribution of expected returns 퐸 푅 푄 = 휏 ⋅ Σ −1 + 푃′ ⋅ Ω−1 ⋅ 푃 −1 ⋅ 휏 ⋅ Σ −1 ⋅ Π + 푃′ ⋅ Ω−1 ⋅ 푄

var 푅 푄 = 휏 ⋅ Σ −1 + 푃′ ⋅ Ω−1 ⋅ 푃 −1 FIN501 Asset Pricing Lecture 06 Mean-Variance & CAPM (84) Black Litterman: 2-asset Example

• Suppose you have a view on the equally 1 1 weighted portfolio 휇 + 휇 = 푞 + 휀푣 2 1 2 2 • Then 1 −1 푞 퐸 푅 푄 = 휏 ⋅ Σ −1 + ⋅ 휏 ⋅ Σ −1 ⋅ Π + 2Ω 2Ω

1 −1 var 푅 푄 = 휏 ⋅ Σ −1 + 2Ω FIN501 Asset Pricing Lecture 06 Mean-Variance & CAPM (85) Advantages of Black-Litterman

• Returns are adjusted only partially toward the investor’s views using Bayesian updating – Recognizes that views may be due to estimation error – Only highly precise/confident views are weighted heavily. • Returns are modified in way that is consistent with economic priors – Highly correlated sectors have returns modified in the same direction. • Returns can be modified to reflect absolute or relative views. • Resulting weight are reasonable and do not load up on estimation error.