Question

1) Yaron has $25,000 he would like to invest. He is considering two possible investments. With Company A, the investment will pay back his $25,000 plus a return of $11,100 with probability 0.8. But there is also a 0.2 probability that Company A will go bankrupt and not pay back any of his money. With Company B, the investment will pay back his $25,000 plus a return of $37,500 with probability 0.5, but also probability 0.5 that it will go bankrupt and not pay back anything. (Note this is different than examples in class and the text in that Yaron begins with W = 25,000, rather than W = 0. Therefore, the possible outcomes are the investment plus the return, or W=0 if the company goes bankrupt.) a) Draw the decision tree for Yaron’s investment choice. b) If Yaron were risk neutral (which means he behaves as if he maximizes expected value), which investment should he choose? c) Suppose Yaron’s total initial wealth W is $25,000. That is, he is basically planning to gamble with his entire wealth. If Yaron maximizes expected utility and his utility is given by U(W) = 2√?, which investment should he choose? In this case, is Yaron risk averse, risk neutral, or risk seeking? Explain d) Suppose again that Yaron’s total initial wealth W is $25,000, but now his utility is given by U(W) = W². Which investment should he choose? In this case, is Yaron risk averse, risk neutral, or risk seeking? Explain.

Answer 1

b
for maximizing the expected value, the best investment is Company B as the probability weighted expected return for B is higher than A.

c
U(W)= SQRT(W)
Here, we see that Yaron's utility is increasing at a much lower rate than the increase in total wealth. Hence, he needs to have much higher wealth to increase his total utility as compared to a risk neutral investor. Assuming risk will increase with the increasing wealth, the perceived risk also increase at a much higher rate than the increase in expected utility. This is a typical behavior of a risk averse inventor. Another way of looking at this is for a risk averse investor, the curve of total utility to total wealth is concave, as is the case with square root function used to describe the utility in this case. A risk averse investor shall prefer a more certain return generating investment, even if the total return is lower. in this case, Yaron is most likely to chose Company A since the probability of success is higher.

d
U(W)= W^2
Here, we see that Yaron's utility is increasing at a much higher rate than the increase in total wealth. Hence, he will have a much higher total utility for the same amount of wealth as compared to a risk neutral investor. Assuming risk will increase with the increasing wealth, the perceived risk will increase at a much lower rate than the increase in expected utility. This is a typical behavior of a risk seeking inventor. Another way of looking at this is for a risk seeking investor, the curve of total utility to total wealth is convex, as is the case with square function used to describe the utility in this case. A risk seeking investor shall prefer a more risky return generating investment. i.e. he will prefer a high risk high return scenario. In this case, Yaron is most likely to chose Company B since it has higher risk and higher payoff.