Question

Derek plans to retire on his 65th birthday. However, he plans to work part-time until he turns 73.00. During these years of part-time work, he will neither make deposits to nor take withdrawals from his retirement account. Exactly one year after the day he turns 73.0 when he fully retires, he will wants to have $3,044,197.00 in his retirement account. He he will make contributions to his retirement account from his 26th birthday to his 65th birthday. To reach his goal, what must the contributions be? Assume a 7.00% interest rate.

Answer format: Currency: Round to: 2 decimal places.

Answer 1

First we need to calculate the present value of the investment grow in future (Exactly one year after the day Derek turns 73.0) to know the amount required at age of 65

Formula for present value calculation

PV = FV / (1+i) ^N

Where, FV is the future value of investment = $3,044,197.00

Present Value (PV) of the investment (at age of 65 years) =?

i = I/Y = interest rate per year or discount rate = 7.00% or 0.07

And N is time period = 73+1 – 65 = 9 years

Therefore,

PV = $3,044,197.00 / (1+7%) ^9

= $3,044,197.00 / (1+ 0.07) ^9

= $1,655,841.47

The value of the investment at age of 65 years should be $1,655,841.47

To reach his goal, Derek’s contributions could be calculated in following manner –

We can use Future value (FV) of annuity formula to calculate annual deposits

FV = PMT*{(1+i) ^n−1} / i

Where FV = $1,655,841.47 (the present value of fund required at age of 65 is now future value of the annual deposits required for investments)

PMT = Annual savings =?

n = N = number of payments = 40 (26 years to 65 years)

i = I/Y = interest rate per year = 7.0%

Therefore,

$1,655,841.47 = Annual savings *{(1+7%) ^40−1} / 7%

Annual savings = $8,294.34

Therefore Derek has to deposit $8,294.34 each year for 40 years to achieve his goal