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Portfolio Theory (PDF) Portfolio Theory Lecture 14: Portfolio Theory MIT 18.S096 Dr. Kempthorne Fall 2013 MIT 18.S096 Portfolio Theory 1 Markowitz Mean-Variance Optimization Mean-Variance Optimization with Risk-Free Asset Von Neumann-Morgenstern Utility Theory Portfolio Theory Portfolio Optimization Constraints Estimating Return Expectations and Covariance Alternative Risk Measures Outline 1 Portfolio Theory Markowitz Mean-Variance Optimization Mean-Variance Optimization with Risk-Free Asset Von Neumann-Morgenstern Utility Theory Portfolio Optimization Constraints Estimating Return Expectations and Covariance Alternative Risk Measures MIT 18.S096 Portfolio Theory 2 Markowitz Mean-Variance Optimization Mean-Variance Optimization with Risk-Free Asset Von Neumann-Morgenstern Utility Theory Portfolio Theory Portfolio Optimization Constraints Estimating Return Expectations and Covariance Alternative Risk Measures Markowitz Mean-Variance Analysis (MVA) Single-Period Analyisis m risky assets: i = 1; 2;:::; m Single-Period Returns: m−variate random vector 0 R = [R1; R2;:::; Rm] Mean and Variance/Covariance of Returns: 2 α1 3 2 Σ1;1 ··· Σ1;m 3 . E[R] = α = 6 . 7 ; Cov[R] = Σ = 6 .... 7 4 . 5 4 . 5 αm Σm;1 ··· Σm;m Portfolio: m−vector of weights indicating the fraction of portfolio wealth held in each asset Pm w = (w1;:::; wm): i=1 wi = 1: 0 Pm Portfolio Return: Rw = w R = i=1 wi Ri a r.v. with 0 αw = E[Rw] = w α 2 0 σw = var[Rw] = w Σw MIT 18.S096 Portfolio Theory 3 Markowitz Mean-Variance Optimization Mean-Variance Optimization with Risk-Free Asset Von Neumann-Morgenstern Utility Theory Portfolio Theory Portfolio Optimization Constraints Estimating Return Expectations and Covariance Alternative Risk Measures Markowitz Mean Variance Analysis Evaluate different portfolios w using the mean-variance pair of the 2 portfolio: (αw; σw) with preferences for Higher expected returns αw Lower variance varw Problem I: Risk Minimization: For a given choice of target mean return α0; choose the portfolio w to 1 0 Minimize: 2 w Σw 0 Subject to: w α = α0 0 w 1m = 1 Solution: Apply the method of Lagrange multipliers to the convex optimization (minimization) problem subject to linear constraints: MIT 18.S096 Portfolio Theory 4 Markowitz Mean-Variance Optimization Mean-Variance Optimization with Risk-Free Asset Von Neumann-Morgenstern Utility Theory Portfolio Theory Portfolio Optimization Constraints Estimating Return Expectations and Covariance Alternative Risk Measures Risk Minimization Problem Define the Lagrangian 1 0 0 0 L(w; λ1; λ2) = 2 w Σw+λ1(α0 −w α)+λ2(1−w 1m) Derive the first-order conditions @L @w = 0m = Σw − λ1α − λ21m @L = 0 = α − w0α @λ1 0 @L = 0 = 1 − w01 @λ2 m Solve for w in terms of λ1; λ2: −1 −1 w0 = λ1Σ α + λ2Σ 1m Solve for λ1; λ2 by substituting for w: 0 0 −1 0 −1 α0 = w0α = λ1(α Σ α) + λ2(α Σ 1m) 0 0 −1 0 −1 1 = w01m = λ1(α Σ 1m) + λ2(1mΣ 1m) α a b λ =) 0 = 1 with 1 b c λ2 0 −1 0 −1 0 −1 a = (α Σ α), b = (α Σ 1m), and c = (1mΣ 1m) MIT 18.S096 Portfolio Theory λ a b −1 α aα b 5 Markowitz Mean-Variance Optimization Mean-Variance Optimization with Risk-Free Asset Von Neumann-Morgenstern Utility Theory Portfolio Theory Portfolio Optimization Constraints Estimating Return Expectations and Covariance Alternative Risk Measures Risk Minimization Problem Variance of Optimal Portfolio with Return α0 With the given values of λ1 and λ2, the solution portfolio −1 −1 w0 = λ1Σ α + λ2Σ 1m has minimum variance equal to 2 0 σ0 = w0Σw0 2 0 −1 0 −1 2 0 −1 = λ1(α Σ α) + 2λ1λ2(α Σ 1m) + λ2(1mΣ 1m) λ 0 a b λ = 1 1 λ2 b c λ2 λ a b −1 α Substituting1 = 0 gives λ2 b c 1 α 0 a b −1 α σ2 = 0 0 = 1 cα2 − 2bα + a 0 1 b c 1 ac−b2 0 0 2 Optimal portfolio has variance σ0: parabolic in the mean return α0. MIT 18.S096 Portfolio Theory 6 Markowitz Mean-Variance Optimization Mean-Variance Optimization with Risk-Free Asset Von Neumann-Morgenstern Utility Theory Portfolio Theory Portfolio Optimization Constraints Estimating Return Expectations and Covariance Alternative Risk Measures Equivalent Optimization Problems Problem II: Expected Return Maximization: For a given choice 2 of target return variance σ0, choose the portfolio w to 0 Maximize: E(Rw) = w α 0 2 Subject to: w Σw = σ0 0 w 1m = 1 Problem III: Risk Aversion Optimization: Let λ ≥ 0 denote the Arrow-Pratt risk aversion index gauging the trade-ff between risk and return. Choose the portfolio w to 1 0 1 0 Maximize: E(Rw) − 2 λvar(Rw) = w α − 2 λw Σw 0 Subject to: w 1m = 1 N.B Problems I,II, and III solved by equivalent Lagrangians 2 Efficient Frontier:f(α0; σ0 ) = (E(Rw0 ); var(Rw0 ))jw0 optimalg 2 Efficient Frontier: traces of α0 (I), σ0 (II), or λ (III) MIT 18.S096 Portfolio Theory 7 Markowitz Mean-Variance Optimization Mean-Variance Optimization with Risk-Free Asset Von Neumann-Morgenstern Utility Theory Portfolio Theory Portfolio Optimization Constraints Estimating Return Expectations and Covariance Alternative Risk Measures Outline 1 Portfolio Theory Markowitz Mean-Variance Optimization Mean-Variance Optimization with Risk-Free Asset Von Neumann-Morgenstern Utility Theory Portfolio Optimization Constraints Estimating Return Expectations and Covariance Alternative Risk Measures MIT 18.S096 Portfolio Theory 8 Markowitz Mean-Variance Optimization Mean-Variance Optimization with Risk-Free Asset Von Neumann-Morgenstern Utility Theory Portfolio Theory Portfolio Optimization Constraints Estimating Return Expectations and Covariance Alternative Risk Measures Mean-Variance Optimization with Risk-Free Asset Risk-Free Asset: In addition to the risky assets (i = 1;:::; m) assume there is a risk-free asset (i = 0) for which R0 ≡ r0; i.e., E(R0) = r0; and var(R0) = 0: Portfolio With Investment in Risk-Free Asset Suppose the investor can invest in the m risky investment as well as in the risk-free asset. 0 Pm w 1m = i=1 wi is invested in risky assets and 1 − w1m is invested in the risk-free asset. If borrowing allowed, (1 − w1m) can be negative. 0 0 Portfolio: Rw = w R + (1 − w 1m)R0, where R = (R1;:::; Rm), has expected return and variance: 0 0 αw = w α + (1 − w 1m)r0 2 0 σw = w Σw Note: R0 has zero variance and is uncorrelated with R MIT 18.S096 Portfolio Theory 9 Markowitz Mean-Variance Optimization Mean-Variance Optimization with Risk-Free Asset Von Neumann-Morgenstern Utility Theory Portfolio Theory Portfolio Optimization Constraints Estimating Return Expectations and Covariance Alternative Risk Measures Mean-Variance Optimization with Risk-Free Asset Problem I': Risk Minimization with Risk-Free Asset For a given choice of target mean return α0; choose the portfolio w to 1 0 Minimize: 2 w Σw 0 0 Subject to: w α + (1 − w 1m)r0 = α0 Solution: Apply the method of Lagrange multipliers to the convex optimization (minimization): Define the Lagrangian 1 0 0 L(w; λ1) = 2 w Σw + λ1[(α0 − r0) − w (α − 1mr0)] Derive the first-order conditions @L @w = 0m = Σw − λ1[α − 1mr0] @L = 0 = (α − r ) − w0(α − 1 r ) @λ1 0 0 m 0 −1 Solve for w in terms of λ1: w0 = λ1Σ [α − 1mr0] 0 −1 and λ1 = (α0 − r0)=[(α − 1mr0) Σ (α − 1mr0)] MIT 18.S096 Portfolio Theory 10 Markowitz Mean-Variance Optimization Mean-Variance Optimization with Risk-Free Asset Von Neumann-Morgenstern Utility Theory Portfolio Theory Portfolio Optimization Constraints Estimating Return Expectations and Covariance Alternative Risk Measures Mean-Variance Optimization with Risk-Free Asset Available Assets for Investment: Risky Assets (i = 1;:::; m) with returns: R = (R1;:::; Rm) with E[R] = α and Cov[R] = Σ Risk-Free Asset with return R0 : R0 ≡ r0; a constant. Optimal Portfolio P: Target Return = α0 Invests in risky assets according to fractional weights vector: −1 w0 = λ1Σ [α − 1mr0], where (α0−r0) λ1 = λ1(P) = 0 −1 (α−1mr0) Σ (α−1mr0) 0 Invests in the risk-free asset with weight (1 − w01m) 0 0 Portfolio return: RP = w0R + (1 − w01m)r0 MIT 18.S096 Portfolio Theory 11 Markowitz Mean-Variance Optimization Mean-Variance Optimization with Risk-Free Asset Von Neumann-Morgenstern Utility Theory Portfolio Theory Portfolio Optimization Constraints Estimating Return Expectations and Covariance Alternative Risk Measures Mean-Variance Optimization with Risk-Free Asset 0 0 Portfolio return: RP = w0R + (1 − w01m)r0 Portfolio variance: 0 0 0 Var(RP ) = Var(w0R + (1 − w01m)r0) = Var(w0R) 0 2 0 −1 = w0Σw0 = (α0 − r0) =[(α − 1mr0) Σ (α − 1mr0)] MIT 18.S096 Portfolio Theory 12 Markowitz Mean-Variance Optimization Mean-Variance Optimization with Risk-Free Asset Von Neumann-Morgenstern Utility Theory Portfolio Theory Portfolio Optimization Constraints Estimating Return Expectations and Covariance Alternative Risk Measures Market Portfolio M The fully-invested optimal portfolio with 0 wM : wM 1m = 1: I.e. −1 wM = λ1Σ [α − 1mr0], where 0 −1 −1 λ1 = λ1(M) = 1mΣ [α − 1mr0] 0 Market Portfolio Return: RM = wM R + 0 · R0 0 −1 0 0 (α Σ [α−1mr0]) E(RM ) = E(wM R) = wM α = 0 −1 (1mΣ [α−1mr0]) 0 −1 [α−1mr0] Σ [α−1mr0]) = r0+ 0 −1 (1mΣ [α−1mr0]) 0 Var(RM ) = wM ΣwM 2 0 −1 (E(RM )−r0) [α−1mr0] Σ [α−1mr0]) = 0 −1 = 0 −1 2 [(α−1mr0) Σ (α−1mr0)] (1mΣ [α−1mr0]) MIT 18.S096 Portfolio Theory 13 Markowitz Mean-Variance Optimization Mean-Variance Optimization with Risk-Free Asset Von Neumann-Morgenstern Utility Theory Portfolio Theory Portfolio Optimization Constraints Estimating Return Expectations and Covariance Alternative Risk Measures Tobin's Separation Theorem: Every optimal portfolio invests in a combination of the risk-free asset and the Market Portfolio.
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