Consider example 9.2a. If p < 1/2, will it change the derivation of α? Answer: Yes/No. 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 respectiveprobabilities p and 1 - p. If p > 1/2, how much should be invested byan investor having a log utility function?Solution. Suppose the amount ax is invested, where 0 ≤ ≤ 1. Thenthe investor's final fortune, call it X, will be either x + ax or x - axValuing Investments by Expected Utility 173with respective probabilities p and 1-p. Hence, the expected utility ofthis final fortune isplog((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 differentiateplog(1 + a) + (1 - p) log(1-a)to obtainddaSetting this equal to zero yields(plog(1+a)+(1-p) log(1-a)) =P1+αp-ap=1-p+a-ap or a=2p-1.Hence, the investor should always invest 100(2p-1) percent of herpresent fortune. For instance, if the probability of winning is .6 then theinvestor should invest 20% of her fortune; if it is .7, she should invest40%. (When p ≤ 1/2, it is easy to verify that the optimal amount to in-vest is 0.)0

Given Information: Consider the given example 9.2a

Answered: Consider example 9.2a. If p < 1/2, will…

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Consider example 9.2a. If p < 1/2, will it change the derivation of α? Answer: Yes/No.

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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Question

Consider example 9.2a. If p < 1/2, will it change the derivation of α? Answer: Yes/No.

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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