Question

Imagine that you graduate from college and your first employer offers a retirement fund that is optional for you to join. After one year on the job, you can contribute a monthly sum to the fund and earn an average return equal to the Standard and Poor’s 500, which is 10%. Assume that you are 22 when you graduate (and 23 when you start contributing), and you plan on working until you are 67, and you can contribute $500 per month to the fund. Please show work with formulas, not Excel.

a. How much would you accumulate by the time you retire, i.e. at 67?

b. If you wait until you are 30 to begin contributing to the fund, how much will your monthly contributions have to be for you to obtain the same future savings as in part a?

Hint: Note that here, 23 is when you start contributing, until 67, so the total years is (67 – 23 + 1) = 45 years. Like if you start contributing from year 1 to year 10, the total years is (10 – 1 + 1) = 10 years.

Answer 1

a]

Future value of annuity = P * [(1 + r)ⁿ - 1] / r,

where P = periodic payment. This is $500

r = periodic rate of interest. This is (10%/12). We divide by 12 since we need to convert the annual rate into monthly rate)

n = number of periods. This is 45 * 12 = 540 (there are 45 years, or 540 months in the investment period)

Future value of annuity = $500 * [(1 + (10%/12))⁵⁴⁰ - 1] / (10%/12)

Future value of annuity = $5,241,250.86

Amount you will accumulate by the time you retire =  $5,241,250.86

b]

Future value of annuity = P * [(1 + r)ⁿ - 1] / r,

where P = periodic payment. We need to calculate this.

r = periodic rate of interest. This is (10%/12). We divide by 12 since we need to convert the annual rate into monthly rate)

n = number of periods. This is 38 * 12 = 456 (there are 38 years, or 456 months in the investment period)

$5,241,250.86 = P * [(1 + (10%/12))⁴⁵⁶ - 1] / (10%/12)

P = $5,241,250.86 * (10%/12) / [(1 + (10%/12))⁴⁵⁶ - 1]

P = $1,015.68

To obtain the same future savings as in part a, monthly contributions have to be $1,015.68