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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 w s . t . ( z- R 1) w = - R (1) 2. Forming the Lagrangian L = 1 2 w w- ( - R- ( z- R 1) w ) w L = w- ( z- R 1) = 0 w * = - 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) - 1 ( z- R 1) = - R z - 1 z- R z - 1 1- R 1 - 1 z- R 2 1 - 1 a = - R (3) 1 AEB 6182 Agricultural Risk Analysis and Decision Making Professor Charles B. Moss Lecture XXII Fall 2010 [ ] E r...
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- Fall '08