Mean-Variance Analysis Jan Schneider McCombs School of Business University of Texas at Austin Jan Schneider Mean-Variance Analysis
Beta Representation of the Risk Premium risk premium E t [Rt t+τ ] R1 t+τ pt = Cov t[rt t+τ, Mpt t+τ ] E t [Mpt t+τ = ] divide by the price of M t+τ pt Cov t[rt t+τ, Mt ] E t [ Mt ] = ft if a risk-free asset exists this equation holds also for the traded discount factor: Mt, Rt+τ Mt ] Mt ] E t [ Mt ] Rt+τ 1 pt = Cov t[ E t [ plug into equation above: E t [ Mt ] = E t [R t+1 ] R1 t+τ pt = Cov t[r t+1, Var t [ Var t[ Mt ] E t [ Mt ] Rt+τ 1 pt Mt ] Mt ] ( ) E t [ Mt ] Rt+τ 1 pt Jan Schneider Mean-Variance Analysis 1 / 17
Unconditional Beta Representation E[R t+1 ] 1 E[1/R1 t+τ ] pt = Cov[R t+1, R Mt+1 ] Var[R Mt+1 ] ( E[R Mt+1 ] ) 1 E[1/R1 t+τ ] pt 1 E[P ft ] if the risk-free asset exists Jan Schneider Mean-Variance Analysis 2 / 17
Maximum Sharpe Ratio E t [Rt t+τ ] R1 t+τ pt Var t [Rt t+τ ] = R1 t+τ Corr pt t[mpt t+τ, Rt t+τ ] Var t [Mpt t+τ ] R 1+τ ft 1 pt = E t[rt t+τ ] Var t [ t ] if a risk-free asset exists R1 t+τ pt Var t [Mpt t+τ ] 1 pt Vart [M t+1 ] since M t+τ t = M t+τ pt + M t+τ p t Jan Schneider Mean-Variance Analysis 3 / 17
Mean-Variance Efficient Frontier I H t... H t+τ 1 self-financing portfolio portfolio payoff P Ht+τ + D Ht+τ is mean-variance efficient q t to t + τ if P H (q t) = P H (q t) E[P Ht+τ + D Ht+τ q t] = E[P H t+τ + D H t+τ q t] } = Var[P Ht+τ + D Ht+τ q t] Var[P H t+τ + D H t+τ q t] Jan Schneider Mean-Variance Analysis 4 / 17
Mean-Variance Efficient Frontier II Suppose H is mean-variance efficient. Suppose H satisfies conditions on the previous slide. Then: [ ] [ ] H t qt] = E PH t+τ + D H t+τ PHt+τ + D Ht+τ P H (q t) qt = E P H (q t) qt [ ] [ ] PHt+τ + D Ht+τ PH P H (q t) qt Var t+τ + D H t+τ P H (q t) qt E[ and Var[ Ht q t] = Var we have: = E[ Ht q t] = Var[ H t qt] mean-variance efficiency in payoffs mean-variance efficiency in returns Jan Schneider Mean-Variance Analysis 5 / 17
Portfolios of the Discount Factor and the Unity Payoff I Consider the market of all self-financing portfolios between q t and t + τ Define F t+τ q t = { Y : Y = am t+τ pt + b1 t+τ pt for some a, b R } Consider an arbitrary traded payoff Y. Projecting Y on F : Y = Y F + Y F Jan Schneider Mean-Variance Analysis 6 / 17
Portfolios of the Discount Factor and the Unity Payoff II some properties of Y : = 0 = 0 1. E t [Y ] = E t [Y F ] E t [Y F ] = E t [Y F 1 t+τ p t ] + E t [Y F 1 t+τ pt ] = 0 2. Cov t [Y F, Y F ] = 0 Cov t [Y F, Y F ] = E t [Y F Y F ] E t [Y F ] E t [Y F ] = 0 = 0 = 0 3. Y F 0 = Var t [Y F ] > 0 Var t [Y F ] = E t [YF 2 ] > 0 if Y F 0 4. P Yt = P YF t P Yt = E t [M t+τ pt corresponding decomposition for returns: Yt = Y P Yt = Y F + Y F P Yt = Y F P YF t (Y F + Y F )] = E t [Mpt t+τ Y F ] = P YF t + Y F P Yt R YF t+1 z F Jan Schneider Mean-Variance Analysis 7 / 17
Portfolios of the Discount Factor and the Unity Payoff III All payoff in F are mean variance efficient. Proof. Choose an arbitrary traded payoff Y. Then we can decompose Y as on the previous slide. Since P Y = P YF and since Y F only adds noise, Y cannot be mean-variance efficient if Y F 0 Jan Schneider Mean-Variance Analysis 8 / 17
Portfolios of the Discount Factor and the Unity Payoff IV All payoff in E qt are mean variance efficient. Proof.Choose Y F. There exist a m-v efficient traded payoff Y such that P Yt = P Y t and E t [Y ] = E t [Y ] By the argument above Y is in F. Hence Y Y F. But Y Y is also orthogonal to F : E t [(Y Y )1pt t+τ ] + E t [(Y Y )1 t+τ p t ] = E t [Y Y ] = 0 E t [M t+τ pt (Y Y )] = P Yt P Y t = 0 Therefore, since Y Y F and Y Y / F, Y Y = 0. Hence we have for any market between q t and t + τ: Y is mean-variance efficient Y F t+τ q t Jan Schneider Mean-Variance Analysis 9 / 17
2 M-V Efficient Portfolios M-V Frontier any mean-variance efficient return is given by ht = h Mt + (1 h) = + h( Mt ) Jan Schneider Mean-Variance Analysis 10 / 17
Minimum-Variance Portfolio Var t [ ht variance of the return on the previous slide: ] = Var t [R t+τ ]+h 2 Var t [ minimum of the variance: Var t [ ht ] h Mt Rt+τ ]+2hCov t [R t+τ, Mt Rt+τ ] = 0 = h = Cov t[r t+τ, Mt Var t [ Mt If a risk-free asset is traded, then h = 0 and ht ] ] = ft. Jan Schneider Mean-Variance Analysis 11 / 17
Zero-Covariance Portfolio Covariance between two mean-variance efficient returns R h1 and R h2 : Cov t [ h 1 t, Rt+τ h 2 t ] = Var t[r t+τ ] + h 1 h 2 Var t [ Mt ] + (h 1 + h 2 )Cov t [R t+τ, Mt ] hence: Cov t [ h 1 t, Rt+τ h 2 t ] = 0 h 2 = Var t [ h 1 Var t [ Mt ] + h 1 Cov t [R t+τ, Mt ] + Cov t [, Mt ] ] Jan Schneider Mean-Variance Analysis 12 / 17
M-V Efficiency and Beta Representation Choose any traded return t : t = h 1 t Ft + β t ( h 2 t R +z F h 1 t ) = E t [Rt t+τ ] = E t [ h 1 t ] + β ( t Et [ h 2 t ] E t[ h 1 t ]) h 2 t ] = β t = Cov t[rt t+τ Cov t [Rt t+τ, h 2 t ] = β tvar t [, h 2 t ] Var t [ h 2 t ] hence: E t [ t ] = E t [ h 1 t ]+ Cov t[rt t+τ, Var t [ h 2 t ] h 2 t ] ( Et [ h 2 t ] E t[ h 1 t ]) zero-covariance portfolio, = R ft+1 if risk-free asset exists Jan Schneider Mean-Variance Analysis 13 / 17
Market Portfolio M-V Efficient = CAPM Suppose a risk-free asset exists and suppose the market portfolio is mean-variance efficient. Then we have: E t [Rt t+τ ] = ft + Cov t[rt t+τ, Var t [ mt ] mt ] ( Et [ mt ] ) ft This model of expected returns is known as the capital asset pricing model (CAPM). Jan Schneider Mean-Variance Analysis 14 / 17
Mean-Variance Efficient Returns Discount Factor Suppose h 1 t and h 2 t suppose Cov t [ h 1 t, Rt+τ h 2 t ] = 0 Then: M t+τ t = Proof: any return E t[m t+τ t t are mean-variance efficient 1 E t [ h 1 t ] (Rt+τ h 2 t E t [ ] = Et[Rt+τ t ] E t[ h 1 t ] = 1 ( Cov t[rt t+τ, h 2 t ] E t[rt t+τ h 2 t ] E t[rt t+τ h 2 t ])E t[ h 2 t ] E t[ E t [ h 1 t ]Var t[ h 1 t ] h 2 t ] ) Et[ ]E t[ h h 2 t ] 2 t ] Et[Rt+τ h 1 t ] E t[ h 1 t ]Vart[Rt+τ h 2 t ] Jan Schneider Mean-Variance Analysis 15 / 17
CAPM Discount Factor For example, if the CAPM holds: M t+τ t = 1 ft ( Rmt t+τ ft ) Et [Rmt t+τ ] ft Var t [ ft mt ] Jan Schneider Mean-Variance Analysis 16 / 17
Maximizing Sharpe Ratio Sharpe ratio of a portfolio H: E t [R Ht+1 ] R ft+1 SD t [R Ht+1 ] Derivative: = A a=1 since: h a = 1 h a (E t [R at+1 ] R ft+1 ) A A a=1 b=1 h ah b Cov[R at+1, R bt+1 ] (Sharpe ratio) ] Cov h t [R at+1, h b R bt+1 a b (E t[r at+1] R ft+1 )SD t[r Ht+1 ] (E[R Ht+1 ] R ft+1 ) 1 Var[R 2 Ht+1] 1 2 2 h b Cov t[r at+1, R bt+1 ] = Var t[r Ht+1 ] = Et[Rat+1] R ft+1 (E t[r Ht+1 ] R ft+1 )Var t[r Ht+1 ] 1 Cov t[r at+1, R Ht+1 ] Var t[r Ht+1 ] 1 2 b R Ht+1 Jan Schneider Mean-Variance Analysis 17 / 17