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

Bank A offers a standard savings account with 4.2% p.a. (per annum).

Bank B offers the following alternative:

There is a basis interest rate of 1% p.a. and 40% participation on the performance of the S&P500. The maximum interest rate is capped at 10% p.a. (If the S&P increases by 4%, there is a bonus of 1.6% so that the total return is 2.6% p.a. If the return of the S&P is 30%, the plan has a total return of 10%. Note, if the S&P has a negative return, the interest rate remains at 1%.)

Bank C offers the following alternative:

There is a basis interest rate of 2.8% p.a. and 20% participation on the performance of the S&P500. The maximum interest rate is capped at 8% p.a.

Suppose the S&P has 1000 points at t=0. At t=1 it can have {900, 990, 1000, 1020, 1030, 1040, 1100, 1200, 1300, 1400} 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 (total) return of the plan on the Y-axis.)

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