Chapter R.1, Problem 47E

The annual interest rate r, when compounded more than once a year, results in a slightly higher yearly interest rate; this is called the annual (or effective) yield and denoted as Y. For example, $1000 deposited at 5%, compounded monthly for 1 yr (12 months), will have a value of $A=1000{\left(1+\frac{0.05}{12}\right)}^{12}=\$1051.16$. The interest earned is $51.16/$1000, or 0.05116, which is 5.116% of the original deposit. Thus, we say this account has a yield of $Y=0.05116$, or 5.116%. The formula for annual yield depends on the annual interest rate r and the compounding frequency n:

$Y={\left(1+\frac{r}{n}\right)}^n-1$.

For Exercises 41-48, find the annual yield as a percentage, to two decimal places, given the annual interest rate and the compounding frequency.

Stockman’s Bank will pay 4.2%, compounded annually, on a savings account. A competitor, Mesalands Savings, offers monthly compounding on savings accounts. What is the minimum annual interest rate that Mesalands needs to pay to make its annual yield exceed that of Stockman’s?