Slides21-2010 - Derivation of the Expected Value-Variance...

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Derivation of the Expected Value-Variance Frontier without a Risk-free Asset : Lecture XXI Charles B. Moss October 12, 2010 Charles B. Moss () Efficient Frontier without Risk-free Asset October 12, 2010 1 / 20
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1 Expected Value-Variance Versus Direct Utility Maximization Comparing Specifcations Numerically 2 Mathematical derivation without a Risk-Free asset Lagrange ±ormulation Gradients oF the Variance Matrix Stock PortFolio Example Charles B. Moss () Efficient Frontier without Risk-free Asset October 12, 2010 2 / 20
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Expected Value-Variance Versus Direct Utility Maximization Due to various fnancial economic models such as the Capital Asset Pricing Model that we will discuss in our discussion oF market models, the fnance literature relies on the use oF mean-variance decision rules rather than direct utility maximization. There is a practical aspect For stock-brokers who may want to give clients alternatives between efficient portFolios rather than attempting to directly elicit each individuals utility Function. Kroll, Levy, and Markowitz examines the acceptability oF the Mean-Variance procedure whether the expected utility maximizing choice is contained in the Mean-Variance efficient set. We assume that the decision maker is Faced with allocating a stock portFolio between various investments. Charles B. Moss () Efficient Frontier without Risk-free Asset October 12, 2010 3 / 20
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Comparing Specifcations Numerically Two approaches for making this problem are to choose between the set of investments to maximize expected utility max x E [ U ( x )] s . t . n X i =1 x i =1 x i 0 (1) The second alternative is to map out the efficient Mean-Variance space by solving max x c 0 x s . t . x 0 Ω x t x i 0 (2) Charles B. Moss () Efficient Frontier without Risk-free Asset October 12, 2010 4 / 20
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A better formulation of the problem is max x c 0 x ρ 2 x 0 Ω x s . t . x i 0 (3) where ρ is the Arrow-Pratt absolute risk aversion coefficient. Table 1 presents the optimal shares of the portfolio based on direct
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Slides21-2010 - Derivation of the Expected Value-Variance...

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