Question

First Simple Bank pays 10 percent simple interest on its investment accounts. If First Complex Bank pays interest on its accounts compounded annually, what rate should the bank set if it wants to match First Simple Bank over an investment horizon of 9 years?

Answer 1

To match a 10 percent simple interest rate over 9 years with compounded interest, the compounded annually interest rate would need to be approximately 7.54%.

Step-by-step explanation:

Matching Simple and Compound Interest

When First Simple Bank offers 10 percent simple interest, this means that the interest earned year-over-year does not compound; it remains a constant amount based on the initial principal. Conversely, with compound interest, the interest earned is added to the principal, and in subsequent years, interest is earned on the new total. Thus, the effective rate of interest is higher with compounding.

To match the simple interest rate of 10 percent over 9 years with an account that offers compounded annual interest, we'd need to find the equivalent rate. Using the formula A = P(1 + r/n)^(nt), where A is the final amount, P is the initial principal, r is the annual interest rate, n is the number of times interest is compounded per year, and t is time in years, we can solve for the rate that would make the final amount equal for both accounts over the same period.

For simple interest, the accumulated amount over 9 years for a principal P would be P (1 + 0.10 * 9). For compound interest, we would solve P (1 + r)^9 = P (1 + 0.10 * 9). Simplifying, we'd get (1 + r)^9 = 1.90. To find the equivalent rate for compounding annually, r, we would take the ninth root (1.90)^(1/9) and subtract 1. The compounded annually rate would thus be approximately 7.54%