Chapter 3.2, Problem 1ED

Determine the value after $1$ year of a $\$1,000$ CD purchased from each of the banks in Table $1$. Which CD offers the greatest return? Which offers the least return?

If a principal P is invested at an annual rate r compounded m times a year, then the amount after 1 year is

$A=P{\left(1+\frac{r}{m}\right)}^m$

The simple interest rate that will produce the same amount A in 1 year is called the annual percentage yield $\left(\text{APY}\right)\text{.}$ To find the $\text{APY}$, we proceed as follows:

$\begin{array}{l}\left(\begin{array}{l}\text{amount at}\\ \text{simple interest}\\ \text{ after 1 year}\end{array}\right)=\left(\begin{array}{l}\text{amount at}\\ \text{compound interest }\\ \text{after 1 year}\end{array}\right)\\ P\left(1+\text{APY}\right)=P{\left(1+\frac{r}{m}\right)}^m\text{ Divide both sides by }P.\\ 1+\text{APY=}{\left(1+\frac{r}{m}\right)}^m\text{ Isolate APY on the left side}\text{.}\\ \text{APY=}{\left(1+\frac{r}{m}\right)}^m-1\end{array}$

If interest is compounded continuously, then the amount after 1 year is $A=P{e}^r.$ So to find the annual percentage yield, we solve the equation

$P\left(1+\text{APY}\right)=P{e}^r$

for $\text{APY}$, obtaining $\text{APY}$ = $e^r-1.$ We summarize our results in Theorem 3