( c ) Mary has an initial wealth of $ 1 . She can invest an amount x from it in non – dividend – paying stocks, y in bonds and the rest in cash. Consider three different scenarios as follows where cash always yields zero return:

Scenario 1 : There is a 30 % chance that the values of stocks and bonds do not change at all.

Scenario 2 : There is a 35 % chance that the value of stocks doubles but value of bonds goes down by 30% .

Scenario 3 : There is a 35 % chance that the value of stocks halves but value of bonds goes up by 50 % .

(i) State the amount invested in cash under the three scenarios. ( 1 mark )

(ii) Write down the expressions for Mary’s final wealth, denoted by W 1 , W 2 and W 3 , respectively for each of the three scenarios as a function of x and y . ( 3 marks )

(iii) Suppose Mary’s utility function is U ( W ) = W – 1 4 W 2 Write down the expression of Mary’s expected utility, as a function of W 1 , W 2 and W 3 . ( 2 marks )

(iv) Assuming borrowing and lending at the risk – free rate is possible, find the values of x and y that will maximize Mary’s expected utility.

Solution

(i) The amount invested in cash under the three scenarios is $1 – x – y.

(ii) The expressions for Mary’s final wealth in each of the three scenarios are as follows:

For Scenario 1:

For Scenario 2:

For Scenario 3:

(iii) Mary’s expected utility, as a function of W_(1), W_(2) and W_(3), is given by:

(iv) To find the values of x and y that will maximize Mary’s expected utility, we need to take the derivative of the expected utility function with respect to x and y, set them equal to zero, and solve for x and y. This is a calculus problem and requires knowledge of partial derivatives. The exact solution will depend on the specific form of the utility function and the probabilities of the different scenarios.