Question

Consider an old professor who is planning to retire next year. He is evaluating risky investments relative to the amount he invests (which we refer to as his reference point, r ). Although he never studied prospect theory, his preferences might reveal that he compares investments according to this theory. In particular, for a return x on his investment r, his utility is 100(x−r)−1/2(x−r)² if his return exceeds his investment, x ≥r, but becomes 400(x−r)+2(x−r)² if his return is smaller than his investment, x(a) Assume that this individual plans to invest $1,000 for one year and faces two investment options: (1) bonds, that yield $1,022.54, with certainty; and (2) stocks, that yield $900 with probability 0.12 and $1,100 with probability 0.88 . Show that this individual is indifferent between both investment options, and he could thus opt for the risk-less bonds. [This tendency toward bonds (equity) that have a lower expected return is referred to as the "equity premium puzzle."]

Answer 1

The professor's expected utility from bonds is calculated to be 1927.8482, while the expected utility from stocks is slightly higher at 2000. However, the minimal difference may lead to indifference between the two options, illustrating the concept of the equity premium puzzle.

Step-by-step explanation:

The student's question involves calculating the expected utility of two different investment options for a retiring professor using prospect theory. The first option is bonds, yielding a guaranteed return, and the second is stocks, with variable returns based on probabilities. To determine the professor's preference between the two investment options, we calculate the expected utility of both investments using the utility functions provided for returns above and below the reference point (the amount invested, $1,000 in this scenario).

For the bonds, the return is $1,022.54 with certainty. Using the utility function for returns exceeding the investment (x ≥ r), the utility is: 100(1022.54 - 1000) - 1/2(1022.54 - 1000)2 = 2254 - (25.542/2) = 2254 - 326.1518 = 1927.8482.

For the stocks, we must consider both possible outcomes and their respective probabilities to calculate expected utility. The expected utility is: 0.12 × [400(900 - 1000) + 2(900 - 1000)2] + 0.88 × [100(1100 - 1000) - 1/2(1100 - 1000)2] = 0.12 × (-40000 + 20000) + 0.88 × (10000 - 5000) = 0.12 × -20000 + 0.88 × 5000 = -2400 + 4400 = 2000.

The expected utility from the stocks slightly exceeds that of the bonds, but the difference is minuscule, indicating that the professor could be indifferent between the risky stocks and the risk-less bonds, potentially opting for the bonds, a phenomenon related to the equity premium puzzle.