Consider a Prospect Theory utility maximizer with value function defined over gains and losses relative to a reference point 0:

v(x) = x, for x 0 and 2x for x <0.

For simplicity, we assume () = in the question. He has some money invested, and each day the value of his investment goes up by $3000 with probability 1/4 and down by $1000 with probability 3/4. The probability of "up" and "down" on the second day is independent of what happened on the first day.

Suppose first that he has the choice of checking the performance of his portfolio either at the end of each day, or at the end of the second day. However, even if he chooses to check at the end of each day, he still cannot change his portfolio after the first day. His expected value is additive across days, so that if he checks at the end of each day, his total expected value equals his expected value from the first day plus his expected value from the second day. But if he checks only at the end of the second day, his total expected value is his expected value from the sum of both days' outcomes. That is, he experiences his gains or losses whenever he checks, regardless whether it is at the end of each day or only at the end of both days.

a. Which will he prefer to check his portfolio performance at the end of each day or at the end of the second day? Explain algebraically and intuitively.

Suppose he faces the same choice, but that if he decides to check at the end of each day, he can pull all of his money out of the stock market at the end of the first day if he wishes. Further, suppose that even if he decides to check at the end of each day, his value is still determined by his total gains or losses over both days, with reference point 0.

1. What would his investment decision be at the end of the first day, if he finds that his stocks have gone up by $3000? Explain algebraically and intuitively.
2. What would his investment decision be at the end of the first day, if he finds that his stocks have gone down by $1000?
3. Given the investment decision in (b) and (c), will he prefer to check at the end of each day or at the end of the second day?