Dora is thinking about investing in a risky asset. She is offered to invest in a

Question

Dora is thinking about investing in a risky asset. She is offered to invest in a mutual fund whose perfomance is tied to the overall economic conditions that are either GOOD or BAD. The likelihood of the good economic condition is70%. After one year, a share of the mutual fund will be worth $107 in the good state and $101 in the bad state of the world. Dora would like to own 1,000 shares of the mutual fund. Suppose that after a year her investment in the mutual fund is only source of wealth, and that everyone in the economy has a log utility funtion u(W)=ln(W),

(a) Calculate the expected value and variance of the year-end wealth as a result of investing the 1,000 shares of this mutual fund.

(b) Calculate the expected utility and centainty equivalent of the investment

(c) The mutual fund's current share price is $105.18. Can this price represent a fair price from Dora's pespective? Draw a graph to support your argumet

Explanation / Answer

Let's name our terms here:

pg = Probability of good gamble = 0.7, pb = 0.3,

$107(cg) with pg = 0.7 and $101(cb) with pb = 0.3.

Expected return is pg * cg + pb*cb = 107*0.7+101*0.3 = 105.2

This price represents a fair price from Dora's perspective as the Expected price 105.2 is more or equal than the amount that is bet which is 105.18, the current price of the bond.

Expected Utility is pg*u(cg) + pb*u(cb), where u is the utility which in this case is given by ln(w) or 1/w

0.7*(ln107)+0.3*(ln101) = 4.649 = 4.65

You will reject the gamble if u(107)>EU; You will be

indifferent to the gamble if u(107)=EU; You will accept the
gamble if u(107)<EU

4.67>4.65 as ln(107) = 4.67, Dora should not go ahead with this investment.

The certainity equivalent of a gamble: the sum of money, X, which, if received with certainty will yield the same utility as the gamble.

u(x) = Expected return:

u(x) = 4.67

pg*(cg) + pb*(cb) = 105.2

u(X) = 105.2

4.67(X) = 105.2

X = 22.52$

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