Chapter R.1, Problem 48E

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.

Belltown Bank offers a certificate of deposit at 3.75%, compounded annually. Shea Savings offers savings accounts with interest compounded quarterly. What is the minimum annual interest rate that Shea needs to pay to make its annual yield exceed that of Belltown?