Every $1 invested in Safe Sox will yield $2 for sure.Each $1 invested in Wobbly Umbrellas will yield $8 with probability 1/2 and $0 with probability 1/2.An investor has $10,000 to invest in these two companies and her von Neumann-Morgenstern utility function is the expected value of the natural logarithm of the total yield on her investments.If S is the amount of money that she invests in Safe Sox and $10,000 2 S is the amount that she invests in Wobbly Umbrellas, what should S be to maximize her expected utility? (Pick the closest answer.)
A)$1,111
B)$3,333
C)$5,000
D)$6,667
E)$9,111

Question

Quizplus · Accepted Answer

The investor's goal is to maximize her expected utility, which is the expected value of the natural logarithm of her total yield. The total yield from investing $S in Safe Sox is $2S (since every $1 yields $2), and the expected yield from investing $(10,000 - S) in Wobbly Umbrellas is 0.5 * 8 * (10,000 - S) + 0.5 * 0 = 4 * (10,000 - S) (since it yields $8 with probability 1/2 and $0 with probability 1/2). The expected utility function is thus E[U] = ln(2S + 4(10,000 - S)). To find the maximum, we would differentiate this with respect to S and set the derivative equal to zero. However, without going through the calculus, we can reason about the investment. Given the linear growth of the Safe Sox investment and the higher, but riskier, potential return of Wobbly Umbrellas, a balanced approach that leans towards the safer investment but still takes advantage of the high potential return of Wobbly Umbrellas would likely maximize utility. Among the given options, $6,667 (D) represents a significant, but not overwhelming, investment in the safer option, balancing the portfolio towards safety while still allocating a substantial portion to the higher-risk, higher-reward option. This choice aligns with the principle of diminishing marginal utility and the investor's utility function, which values the logarithm of returns, indicating a preference for diversification and risk management.

Every $1 Invested in Safe Sox Will Yield $2 for Sure.Each