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Lecture21-2010

# Lecture21-2010 - 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 I. Mean-Variance Versus Direct Utility Maximization A. Due to various financial economic models such as the Capital Asset Pricing Model that we will discuss in our discussion of market models, the finance literature relies on the use of mean-variance decision rules rather than direct utility maximization. B. 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. C. Kroll, Levy, and Markowitz examines the acceptability of the Mean-Variance procedure whether the expected utility maximiz- ing choice is contained in the Mean-Variance eﬃcient set. D. We assume that the decision maker is faced with allocating a stock portfolio between various investments. 1. Two approaches for making this problem are to choose be- tween the set of investments to maximize expected utility max x E [ U ( x )] s . t . n i =1 x i = 1 x i 0 (1) 1

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AEB 6182 Agricultural Risk Analysis and Decision Making Professor Charles B. Moss Lecture XXI Fall 2010 Table 1: Comparison of Expected Utility Maximization and Expected Value- Variance Utility Texas Average Standard Function California Carpenter Chrysler Conelco Gulf Return Deviation Direct Expected Utility Portfolios exp ( x ) 44.3 34.7 0.2 5.5 15.3 22.4 27.3 x 0 . 1 33.2 36.0 13.6 17.2 23.3 32.3 x 0 . 5 42.2 34.4 23.4 25.9 49.4 ln ( x ) 37.9 34.8 11.1 16.2 23.1 29.4 Expected Value-Variance Portfolios exp ( x ) 39.4 38.6 5.0 17.0 22.5 27.0 x 0 . 1 28.5 43.4 8.6 8.6 23.1 30.0 x 0 . 5 41.8 32.1 26.1 25.7 47.3 ln ( x ) 32.9 41.8 7.4 18.7 22.9 28.9 2. The second alternative is to map out the eﬃcient Mean-Variance space by solving max x
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Lecture21-2010 - Derivation of the Expected Value-Variance...

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