Mean-Variance Optimization

10232012 p kolm 27 summary of discussion an investor

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Unformatted text preview: herefore write μp = λw Tang ′μ + (1 − λ)rf where λ is determined by the investor RISK AND PORTFOLIO MANAGEMENT WITH ECONOMETRICS, VER. 10/23/2012. © P. KOLM. 23 Another way of calculating the tangency portfolio It is easy to verify that the market portfolio can be determined by solving the maximal Sharpe ratio optimization problem: max w w ′μ − rf w ′Σw subject to w ′ι = 1 What geometric picture does this imply?3 RISK AND PORTFOLIO MANAGEMENT WITH ECONOMETRICS, VER. 10/23/2012. © P. KOLM. 24 The Capital Market Line (CML) RISK AND PORTFOLIO MANAGEMENT WITH ECONOMETRICS, VER. 10/23/2012. © P. KOLM. 25 Calculating the CML Here we will use Fama’s result that w Tang ≡ wM . We let w f = 1 − wM ′ι so that μp = λw M ′μ + (1 − λ)rf = rf + λ(w M ′μ − rf ) ≡ rf + λ(μM − rf ) and ( 2 σp = Var (rp ) = (1 − λ)2Var (rf ) + λ 2Var (rM ) + 2λ(1 − λ)Cov rf , rM = λ 2Var (rM ) ) 2 = λσM Since λ = σp σM we obtain μp = rf + σp σM = rf + σp (μM − rf ) μM...
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