1Consider some implications of risk attitudes for investing. Consider a collection of investors whose preferences are represented by the expected utility function U(x₁, x₂) = ₁ x₁ + ₂ x₂.

These investors each place 100 into a fund. The manager of the fund has the opportunity to place each 100 in capital into one of two lotteries (often called in one of two assets). One lottery gives payoff 64 with probability 1/2 and payoff 225 with probability 1/2. The other gives payoff 0 with probability 3/4 and payoff 1024 with probability 1/4. The manager of the fund is scrupulously concerned with the expected utilities of the investors, and always invests in the lottery that gives the investors the highest expected utility.

1.2 Which lottery gives an investor the highest expected utility? How do the expected utilities of the two lotteries compare to the utility an investor would receive if she simply consumed her initial 100?

1.3 Now suppose the federal government is concerned that sometimes investors lose money. As a result, the government institutes an insurance scheme that reimburses investors for any losses they incur, up to a total reimbursement of 36. What expected utility do the investors now receive in each lottery? Which lottery will the manager choose? Are the investors better or worse off with the insurance scheme?

1.4 For each of the lotteries in the preceding question, calculate the actuarially fair premium for the insurance offered by the government. Suppose the investors had to pay this premium. Which lottery would they choose?