Question

Investor A has an initial wealth of $100 and a utility function of the form: U(w) = log(w) where w is her wealth at any time. Investment Z offers her a return of −18% or +20% with equal probability. i. What is her expected utility if she invests nothing in Investment Z? ii. What is her expected utility if she invests entirely in Investment Z? iii. What proportion a of her wealth should she invest in Investment Z to maximize her expected utility? What is her expected utility if she invests this proportion in Investment Z?

Answer 1

With this initial investment of 100, the investors utility if she invests nothing is 2. If she invests entirely her utility is 2.00432

From the available question, these are the the solutions from option (i) to (iii)

i.) We have utility defined as

U(w) = log w

w = 100

$Utility=log(100)$

= 2

ii) If she invests entirely in Z, utility:

$return*probability$

-18% x 50% =  -9%

20% x 50% = 10%

-9% + 10% = 1%

w = 100 + 1

= 101

$Utility = log(101)$

= 2.00432

This is her utility if she invests entirely in Z.

iii) This investor has these two choices:

• invest in z
• leave resources idle

If she invests in z she gets a 1% increase. Therefore her wealth increases or is maximum when she invests in Z.

Her utility if she invests this proportion in Z is the same as what was solved in (ii) above.

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