Question

Your father is about to retire, and he wants to buy an annuity that will provide him with $78,000 of income a year for 25 years, with the first payment coming immediately. The going rate on such annuities is 5.15%. How much would it cost him to buy the annuity today?

Answer 1

Here, the cash inflow will be same every year, so it is an annuity. And since the cash flows will start immediately or at the beginning of each year so it will be termed as an annuity due. We need to calculate the present value of annuity due by the following formula:

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

where, PVD is the present value of annuity due, P is the periodical amount = $78000, r is the rate of interest = 5.15% and n is the time period = 25

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

PVAD = $78000 * (1 - (1 / (1 + 5.15%)²⁵ / 5.15%) * (1 + 5.15%)

PVAD = $78000 * (1 - (1 / (1 + 0.0515)²⁵/ 0.0515) * (1 + 0.0515)

PVAD = $78000 * (1 - (1 / (1.0515)²⁵ / 0.0515) * (1.0515)

PVAD = $78000 * (1 - (1 / 3.50939235415) / 0.0515) * (1.0515)

PVAD = $78000 * ((1 - 0.28494961494) / 0.0515) * (1.0515)

PVAD = $78000 * (0.71505038505 / 0.0515) * (1.0515)

PVAD = $78000 * 13.8844734962 * 1.0515

PVAD = $1138762.86

So, it would cost $1138762.86 for him to buy the annuity today.