Lecture12-2004 - The Farm Portfolio Problem: Part I Lecture...

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The Farm Portfolio Problem: Part I Lecture XII I. Deriving the EV Frontier A. The discussion over the past two weeks has touched on the equivalence between the mean-variance approach and direct utility maximization. Now, I want to further develop the EV approach. B. Let us begin with the traditional portfolio model. Assume that we want to minimize the variance associated with attaining a given level of income. To specify this problem we assume a variance matrix: Ω= 924 41 458 52 202 22 135 22 458 52 76129 452 99 72 55 202 22 452 99 49011 109 09 135 22 72 25 109 09 284 17 .... ... . .. . . and a constraint matrix which depicts (1) the income constraint and (2) the portfolio constraint Ax b x x x x xx x x xxxx = = ++ + = + = 8119 11366 6 298 8 014 10 70 8119 11366 6 298 8 014 7 000 1000 1 2 3 4 12 3 4 123 4 . . . . . . . The portfolio choice problem can then be stated as max ( ) ' x f xxx st Ax b = = . C. In this initial formulation we find that the optimum solution is x = . . . . 11613 10666 39508 59546 which yields a variance of 228.25. II. A Diversion into Computer Usage GAMS Program sets icrops crops /1*4/ consts constraints /income,portf/; alias(icrops,jcrops);
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AEB 6182– Agricultural Risk Analysis and Decision Making Fall 2004 Professor Charles Moss 2 table var(icrops,jcrops) variance covariance matrix 1 2 3 4 1 924.41 458.52 202.22 135.22 2 458.52 761.29 452.99 72.55 3 202.22 452.99 490.11 109.09 4 135.22 72.55 109.09 284.17 table line(icrops,const) constraints income portf 1 8.119 1.0 2 11.366 1.0 3 6.298 1.0 4 8.014 1.0 parameter b(const) /income 7.00, portf 1.00/; variables x(icrops) level of each crop rhs(consts) level of each right hand side risk; positive variables x; equations vari variance lconst linear constraints; vari. . sum((icrops,jcrops),var(icrops,jcrops)*x(icrops)*x(jcrops)) =e= risk; lconst(consts). . sum(icrops,line(icrops,consts)*x(icrops)) =e= b(consts); model risk1 using /all/; solve risk1 using nlp minimizing risk; III. Starting with the basic model of portfolio choice: min ' ' * x xx st x Y µ > Freund showed that the expected utility of a normally distributed gamble given negative exponential preferences could be written as [] EU x x x () =− ρ σ 2 2 Extending this formulation to vector form, the problem can be expressed as: max ' ' x x x −Ω .
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This note was uploaded on 07/15/2011 for the course AEB 6182 taught by Professor Weldon during the Fall '08 term at University of Florida.

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Lecture12-2004 - The Farm Portfolio Problem: Part I Lecture...

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