Lecture12-2004

# 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 x x x x x x x x = = + + + = + + + = 8119 11366 6 298 8 014 10 10 10 10 7 0 10 8119 11366 6 298 8 014 7 000 1000 1 2 3 4 1 2 3 4 1 2 3 4 . . . . . . . . . . . . . . . . . The portfolio choice problem can then be stated as max ( ) ' x f x x x 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 x x st x Y µ > Freund showed that the expected utility of a normally distributed gamble given negative exponential preferences could be written as [ ] E U x x x [ ] ( ) ( ) = µ ρ σ 2 2 Extending this formulation to vector form, the problem can be expressed as: max ' ' x x x x x µ ρ .
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