Lecture22-2010

# Lecture22-2010 - Derivation of the Expected Value-Variance...

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Unformatted text preview: Derivation of the Expected Value-Variance Frontier with a Risk-Free Asset: Lecture XXII Charles B. Moss October 14, 2010 I. Introduction of a Risk-free Asset A. If a risk-free asset is introduced into the portfolio, the eﬃcient set of portfolios becomes a straight line between the risk-free asset and a tangency on the ES frontier as depicted in Figure 1. B. Mechanics 1. Setting up the portfolio problem min w 1 2 w Σ w s . t . (¯ z − R 1) w = μ − R (1) 2. Forming the Lagrangian L = 1 2 w Σ w − γ ( μ − R − (¯ z − R 1) w ) ∇ w L = Σ w − γ (¯ z − R 1) = 0 w ∗ = γ Σ − 1 (¯ z − R 1) (2) with w ∗ = 1 − 1 w defined as the amount of wealth invested in the risk-free asset if w ∗ > 0 or borrowed if w ∗ < 0. 3. Substituting this result into the constraint yields (¯ z − R 1) γ Σ − 1 (¯ z − R 1) = μ − R γ ¯ z Σ − 1 ¯ z − R ¯ z Σ − 1 1 − R 1 Σ − 1 ¯ z − R 2 1 Σ − 1 a = μ − R (3) 1 AEB 6182 Agricultural Risk Analysis and Decision Making...
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Lecture22-2010 - Derivation of the Expected Value-Variance...

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