Chapter 4.2, Problem 23E

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

Continuous Compounding This is a continuation of Exercise 22. In this exercise, we examine the relationship between APR and the APY when interest is compounded continuously-in other words, at every instant. We will see by means of an example that the relationship is

$\text{Yearly}\text{growth}\text{factor}=e^{\text{APR}}$, $\left(4.1\right)$

and so

$\text{APY}=e^{\text{APR}}-1$ $\left(4.2\right)$

if both the APR and the APY are in decimal form and interest is compounded continuously. Assume that the APR is $10\%$, or 0.1 as a decimal.

a. The yearly growth factor for continuous compounding is just the limiting value of the function given by the formula in part b of Exercise 22. Find that limiting value to four decimal places.

b. Compute $e^{\text{APR}}$ with an APR of 0.1 (as a decimal).

c. Use your answers to parts a and b to verify that Equation (4.1) holds in the case where the APR is $10\%$.

Note: On the basis of part a, one conclusion is that there is a limit to the increase in the yearly growth factor (and hence in the APY) as the number of compounding periods increases. We might have expected the APY to increase without limit for more and more frequent compounding.

22. 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 percentage 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.