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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
ι ′Σ−1μ Then (for you: show this)2 w ∗ = λAwg + γBwd
(two distinct portfolios)
with C − μ0B
Δ μ0A − B
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?
min w ′Σw
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.
- Fall '14