Mean-Variance Optimization

10232012 p kolm 21 in this case the portfolios

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Unformatted text preview: e do not allow short selling? RISK AND PORTFOLIO MANAGEMENT WITH ECONOMETRICS, VER. 10/23/2012. © P. KOLM. 19 Recall that for all minimum-variance portfolios w ∗ = λΣ−1ι + γΣ−1μ Let us define the portfolio Σ−1μ Σ−1μ wd = = B ι ′Σ−1μ Then (for you: show this)2 w ∗ = λAwg + γBwd (two distinct portfolios) with C − μ0B λ= , Δ μ0A − B γ= , and Δ Δ ≡ AC − B 2 where A = ι ′Σ−1ι, B = ι ′Σ−1μ, and C = μ′Σ−1μ RISK AND PORTFOLIO MANAGEMENT WITH ECONOMETRICS, VER. 10/23/2012. © P. KOLM. 20 Mean-Variance with a Riskless Asset Previously, we considered a portfolio of just risky assets. Now let us look at a portfolio of risky assets and one riskless asset. What changes in this situation? Problem: 1 min w ′Σw w 2 s.t. w ′(μ − rf ⋅ ι) = μ0 − rf The solution is w = γΣ−1 (μ − rf ⋅ ι ) (risky assets) w 0 = 1 − w ′ι (riskless asset) We can use the constraint as before to solve for γ , obtaining γ= μ0 − rf (μ − rf ι)′Σ−1(μ − rf ι) RISK AND PORTFOLIO MANAGEMENT WITH ECONOMETRICS, VER. 10/23/2012. © P. KOLM. 21 In this case, the portfolio’s expected return and variance are give...
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This document was uploaded on 02/17/2014 for the course COURANT G63.2751.0 at NYU.

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