Chapter 4.2, Problem 22E

Reminder Round all answers to two decimal places unless otherwise indicated.

APR and APY Recall that financial institutions sometimes report the annual interest rate that they offer on investments as the APR, often called the nominal interest rate. To indicate how an investment will actually grow, they advertise the annual percentage yield, or APY. In mathematical terms, this is the yearly percentage growth rate for the exponential function that models the account balance. In this exercise and the next, we study the relationship between the APR and the APY. We assume that the APR is $10\%$ or $0.1$ as a decimal.

To determine the APY when we know the APR, we need to know how often interest is compounded. For example, suppose for the moment that interest is compounded twice a year. Then to say that the APR is $10\%$ means that in half a year, the balance grows by $\frac{10}{2}\%$ (or $5\%$). In other words, the $\frac{1}{2}$-year age growth rate is $\frac{0.1}{2}$ (as a decimal). Thus, the $\frac{1}{2}$-year growth factor is $1+\frac{0.1}{2}$. To find the yearly growth factor, we need to perform a unit conversion: One year is 2 half-year periods, so the yearly growth factor is ${\left(1+\frac{0.1}{2}\right)}^2$, or $1.1025$.

a.	What is the yearly growth factor if interest is compounded four times a year?
b.	Assume that interest is compounded $n$ times each year. Explain why the formula for the yearly growth factor is ${\left(1+\frac{0.1}{n}\right)}^n$.
c.	What is the yearly growth factor if interest is compounded daily? Give your answer to four decimal places/