1. 5.11 Consider the investment example from Section 5.2 in which a retiree invests

$100,000 in a fund in order to reap the benefits for 20 years. The rate of return on the fund for the past year was 14%, and the retiree hopes for a yearly profit of $15,098 over the coming 20 years. If the rate of return remained at 14% for each year, then at the end of the xth year, the invested capital would be equal to f(x) = (1+r)xA − x−1 k=0(1 +r)kb for x = 1,...,20, where A = 100,000, r =0.14, and b = 15,098. However, the yearly rate of return fluctuates with an average value of 14%. If last year the rate of return was r%, then next year the rate of return will ber%, (1 + f )r%or(1− f)r%withrespectiveprobabilities p, 1 2

(1 − p), and 1 2

(1 − p). For each of the cases (p = 0.8, f = 0.1) and (p = 0.5, f =0.2), simulate a histogram of the distribution of the number of years during the 20-year period that the invested capital at the end of the year will fall below or on the curve of the function f (x).