Question

Rhoda, age​ 25, would like to start saving for retirement. She will be able to save​ $400 per month beginning immediately. She will retire once she has saved​ $1 million. Her investment portfolio will earn a rate of return of​ 8% compounded monthly during the entire time that she is saving. Approximately how many years will it take her to reach her savings​ goal?

Answer 1

Here, the deposits will be same every month, so it is an annuity. And the deposits start at the beginning of each year, so it is an annuity due. For calculating the required number of years, we will use the following future value of annuity due formula:

FVAD = (1 + r) * P * ((1 + r)ⁿ - 1 / r)

where, FVAD is future value of annuity due = $1000000, P is the periodical amount = $400, r is the rate of interest = 8% compounded monthly, so monthly rate = 8% / 12 = 0.666% and n is the time period

Now, putting these values in the above formula, we get,

$1000000 = (1 + 0.6667%) * $400 * ((1 + 0.6667%)ⁿ - 1 / 0.6667%)

$1000000 = (1 + 0.006667) * $400 * ((1 + 0.006667)ⁿ - 1 / 0.006667)

$1000000 = (1.006667) * $400 * ((1.006667)ⁿ - 1 / 0.006667)

$1000000 = $402.6667 * ((1.006667)ⁿ- 1) / 0.006667)

$1000000 / $402.6667 = ((1.006667)ⁿ - 1 ) / 0.006667)

$2487.52093664 = ((1.006667)ⁿ - 1 ) / 0.006667)

$2487.52093664 * 0.006667 = ((1.006667)ⁿ - 1 )

$16.5843020846 = (1.006667)ⁿ - 1

$16.5843020846 + 1 = (1.006667)ⁿ

$17.5843020846 = (1.006667)ⁿ

(1.006667)^431.5 = (1.006667)ⁿ

n = 431.5

So, it will take 431.5 months or 431.5 / 12 = 35.96 years for her to reach her savings goal.