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Example 9.2a An investor with capital x can invest any amount be- tween 0 and x; if y is invested then y is either won or lost, with respective probabilities p and 1-p. If p > 1/2, how much should be invested by an investor having a log utility function?

Example 9.2a An investor with capital x can invest any amount be- tween 0 and x; if y is invested then y is either won or lost, with respective probabilities p and 1-p. If p > 1/2, how much should be invested by an investor having a log utility function?

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Solve example 9.2a but using the exponential utility function u(x) = 1 − e−bx, b > 0. Use the same method and steps

Example 9.2a An investor with capital x can invest any amount be- tween 0 and x; if y is invested then y is either won or lost, with respective probabilities p and 1 - p. If p > 1/2, how much should be invested by an investor having a log utility function? Solution. Suppose the amount ax is invested, where 0 ≤ ≤ 1. Then the investor's final fortune, call it X, will be either x + ax or x - ax Valuing Investments by Expected Utility 173 with respective probabilities p and 1-p. Hence, the expected utility of this final fortune is plog((1+a)x)+ (1 - p) log((1-a)x) = plog(1 + a) + plog(x) + (1 − p) log(1 − a) + (1 − p) log(x) = log(x) + plog(1 + a) + (1 − p) log(1-a). To find the optimal value of a, we differentiate plog(1 + a) + (1 - p) log(1-a) to obtain d da Setting this equal to zero yields (plog(1+a)+(1-p) log(1-a)) = P 1+α p-ap=1-p+a-ap or a=2p-1. Hence, the investor should always invest 100(2p-1) percent of her present fortune. For instance, if the probability of winning is .6 then the investor should invest 20% of her fortune; if it is .7, she should invest 40%. (When p ≤ 1/2, it is easy to verify that the optimal amount to in- vest is 0.) 0
Transcribed Image Text:Example 9.2a An investor with capital x can invest any amount be- tween 0 and x; if y is invested then y is either won or lost, with respective probabilities p and 1 - p. If p > 1/2, how much should be invested by an investor having a log utility function? Solution. Suppose the amount ax is invested, where 0 ≤ ≤ 1. Then the investor's final fortune, call it X, will be either x + ax or x - ax Valuing Investments by Expected Utility 173 with respective probabilities p and 1-p. Hence, the expected utility of this final fortune is plog((1+a)x)+ (1 - p) log((1-a)x) = plog(1 + a) + plog(x) + (1 − p) log(1 − a) + (1 − p) log(x) = log(x) + plog(1 + a) + (1 − p) log(1-a). To find the optimal value of a, we differentiate plog(1 + a) + (1 - p) log(1-a) to obtain d da Setting this equal to zero yields (plog(1+a)+(1-p) log(1-a)) = P 1+α p-ap=1-p+a-ap or a=2p-1. Hence, the investor should always invest 100(2p-1) percent of her present fortune. For instance, if the probability of winning is .6 then the investor should invest 20% of her fortune; if it is .7, she should invest 40%. (When p ≤ 1/2, it is easy to verify that the optimal amount to in- vest is 0.) 0
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