Suppose a risk neutral agent has $100,000 today that he wants to save for one year. Compare the following two savings plans.

Bank A offers a standard savings account with 4% p.a.

Bank B offers the following alternative:

There is a basis interest rate of 1% p.a. and 50% participation on the performance of the S&P500. The maximum interest rate is capped at 7% p.a. (E.g. If the S&P increases by 6%, there is a bonus of 3% so that the total return is 4% p.a. If the return of the S&P is 20%, the plan has a return of 7%. Note, if the S&P has a negative return, the interest rate remains at 1% p.a.)

Suppose the S&P has 1000 points at t=0. At t=1 it can have {900, 990, 1000, 1020, 1040, 1100, 1120, 1200, 1260, 1300} points with equal probability.

(a) Draw the payoff of alternative B as a function of the S&P (with the S&P performance on the X-axis, and the return of the plan on the Y-axis.)

(a) The agent maximizes the expected amount at t=1. Which plan is better? How much more can the agent spend in expectation at t=1, if he chooses the better one?