Starting a year from now and making 10 yearly payments, Pierre would like to put into a retirement account enough money so that, starting 11 years from now, he can withdraw $30,000 per year until he dies. Pierre is confident that he can earn 8% per year on his money for the next 10 years, but he is only assuming that he will be able to get 5% per year after that. (a) How much does Pierre need to pay each year for the first 10 years in order to make the planned withdrawals? (b) Pierre's will states that, upon his death, any money left in his retirement account is to be donated to the Princeton Mathematics Department. If he dies immediately after receiving his 17th payment, how much will the Princeton Mathematics Department inherit?
(a) To calculate how much Pierre needs to pay each year for the first 10 years, we need to calculate the present value of the $30,000 withdrawals for 10 years starting 11 years from now.
Using the formula for the present value of an annuity:
PV = PMT * ((1 - (1 + r)^-n) / r)
Where PV is the present value, PMT is the annual payment, r is the interest rate, and n is the number of years.
PV = $30,000 * ((1 - (1 + 0.05)^-10) / 0.05)
PV = $30,000 * ((1 - 0.61391) / 0.05)
PV = $30,000 * (0.38609 / 0.05)
PV = $30,000 * 7.7218
PV = $231,654
Now, we need to calculate how much Pierre needs to pay each year for the first 10 years to accumulate $231,654 after 10 years at 8% interest.
Using the formula for the future value of an ordinary annuity:
FV = PMT * ((1 + r)^n - 1) / r
Where FV is the future value, PMT is the annual payment, r is the interest rate, and n is the number of years.
$231,654 = PMT * ((1 + 0.08)^10 - 1) / 0.08
$231,654 = PMT * (2.15892 - 1) / 0.08
$231,654 = PMT * 1.15892 / 0.08
$231,654 = PMT * 14.4865
PMT = $231,654 / 14.4865
PMT = $16,005.22
Therefore, Pierre needs to pay $16,005.22 each year for the first 10 years in order to make the planned withdrawals.
(b) If Pierre dies immediately after receiving his 17th payment, the Princeton Mathematics Department will inherit the remaining money in the retirement account. We need to calculate the future value of the remaining payments at 5% interest for 3 years (17 to 20).
Using the future value formula:
FV = $30,000 * ((1 + 0.05)^3 - 1) / 0.05
FV = $30,000 * (1.1576 - 1) / 0.05
FV = $30,000 * 0.1576 / 0.05
FV = $30,000 * 3.152
FV = $94,555.44
Therefore, the Princeton Mathematics Department will inherit $94,555.44 if Pierre dies immediately after receiving his 17th payment.