Calculating Annuity (Account)

Problem 231: A traditional individual retirement account (IRA) is a special type of retirement account in which the money you invest is exempt from income taxes until you withdraw it. If you deposit $100 each month into an IRA earning 6% interest, how much will you have in the account after 20 years?

Solution: This type of accounts are called savings annuity. One can calculate the amount present in the annuity account with the following formula:

(1) $\begin{equation*}P_N = \frac{d\left(\left( 1+\frac{r}{k}\right)^{Nk} - 1\right)}{\left( \frac{r}{k}\right)}, \end{equation*}$

where $P_N$ is the balance in the account after N years, $d$ is the regular deposit (the amount you deposit each year, each month, etc.), $r$ is the annual interest rate in decimal form, $k$ is the number of compounding periods in one year.

For our problem, we are looking for $P_{20}$ , and we are given $d = $100, r = 0.06, k=12,$ and $N=20$ .

(2) $\begin{align*} P_N & = \frac{d\left(\left( 1+\frac{r}{k}\right)^{Nk}-1 \right)}{\left( \frac{r}{k}\right)} \\ P_{20} & = \frac{100\left(\left( 1+\frac{0.06}{12}\right)^{20(12)} - 1 \right)}{\left( \frac{0.06}{12}\right)} \\ & = \frac{100\left((1.005)^{240} - 1 \right)}{(0.005)} \\ & = 46,200. \end{align*}$

Thus, in 20 years you will have $46,200.

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