Question

Steve Hitchcock is 40 years old today and he wishes to accumulate $518,000 by his 65 th birthday so he can retire to his summer place on Lake Hopatcong. He wishes to accumulate this amount by making equal deposits on his 40 th through his 64 th birthdays. What annual deposit must Steve make if the fund will earn 9% interest compounded annually? (Round factor values to 5 decimal places, e.g. 1.25124 and final answer to 0 decimal places, e.g. 458,581.)

Answer 1

Here, the deposits will be same every year, so it is an annuity. And the deposits start at the beginning of each year, so it is an annuity due. For annual deposits, we will use the future value of annuity due formula as per below:

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

where, FVAD is future value of annuity due = $518000, P is the periodical amount, r is the rate of interest = 9% and n is the time period = 65 - 40 = 25

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

$518000 = (1 + 9%) * P * ((1 + 9%)²⁵ - 1 / 9%)

$518000 = (1 + 0.09) * P * ((1 + 0.09)²⁵ - 1 / 0.09)

$518000 = (1.09) * P * ((1.09)²⁵ - 1 / 0.09)

$518000 = (1.09) * P * ((8.6230806604 - 1) / 0.09)

$518000 = (1.09) * P * (7.6230806604 / 0.09)

$518000 = (1.09) * P * 84.7008962267

$518000 = P * $92.32397688

P = $518000 / 92.32397688

P = $5611

So, annual deposits are of $5611.

We can solve it from the future value tables also. Below is the procedure:

Since the deposits are equal every year, so it is an annuity. And the deposits start at the beginning of each year, so it is an annuity due. For annual deposits, we will use the future value of annuity due table (FVAD) table as per below:

Future value = P * FVAD (9%, 25 Years)

where, FVAD (9%, 25 years) is the value of $1 of annuity due at 9% after 25 years. Its value from the table is 92.32398

Now, putting this value in the above equation, we get,

$518000 = P * 92.32398

P = $518000 / 92.32398

P = $5611

So, annual deposits are of $5611.