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 $30000 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 withdraw? 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?

To calculate the annual payment Pierre needs to make for the first 10 years, we can use the present value of an ordinary annuity formula.

a) Present Value of an Ordinary Annuity formula:

PV = PMT x [1 - (1 + r)^(-n)] / r

Where:
PV = Present Value (amount Pierre needs to pay each year)
PMT = Payment amount
r = Interest rate per period
n = Number of periods

For the first 10 years, the interest rate is 8% and the number of periods is 10. The payment amount is unknown.

Using the formula, we can solve for PMT:

PV = PMT x [1 - (1 + 0.08)^(-10)] / 0.08

PV = PMT x [1 - 1.08^(-10)] / 0.08

PV = PMT x [1 - 0.463] / 0.08

PV = PMT x 5.3375

Therefore, Pierre needs to pay PV / 5.3375 each year for the first 10 years in order to make the planned withdrawal.

b) If Pierre dies immediately after receiving his 17th payment, there will be 11 years remaining for his retirement account to grow at a 5% interest rate. We can calculate the future value of an ordinary annuity for the remaining 11 years.

Future Value of an Ordinary Annuity formula:

FV = PMT x [(1 + r)^n - 1] / r

Where:
FV = Future Value (amount that the Princeton Mathematics Department will inherit)
PMT = Payment amount
r = Interest rate per period
n = Number of periods

For the remaining 11 years, the interest rate is 5% and the number of periods is 11. The payment amount is $30,000.

Using the formula, we can solve for FV:

FV = $30,000 x [(1 + 0.05)^11 - 1] / 0.05

FV = $30,000 x [5.537 - 1] / 0.05

FV = $30,000 x 4.537 / 0.05

FV = $30,000 x 90.74

Therefore, the Princeton Mathematics Department will inherit $2,722,200.

To calculate how much Pierre needs to pay each year for the first 10 years, we can use the concept of an annuity.

a) The annuity formula is:

P = (A * (1 - (1 + r)^(-n))) / r

Where:
P = Payment per year
A = Desired annual withdrawal amount ($30,000)
r = Annual interest rate (8% or 0.08)
n = Number of years for which the withdrawals are made (from year 11 until death)

Let's plug in the values and calculate:

P = (30000 * (1 - (1 + 0.08)^(-n))) / 0.08

We know that Pierre will receive payments for 11 years from year 11 until his death. So, let's substitute n = 11:

P = (30000 * (1 - (1 + 0.08)^(-11))) / 0.08
P ≈ $2196.30

Pierre needs to pay approximately $2196.30 each year for the first 10 years in order to make the planned withdrawals.

b) Given that Pierre dies immediately after receiving his 17th payment, we need to calculate the remaining amount in his retirement account.

To find the future value of Pierre's retirement account after 17 payments, we can use the future value of an ordinary annuity formula:

FV = P * ((1 + r)^n - 1) / r

Where:
FV = Future value
P = Payment per year
r = Annual interest rate (5% after the 10th year, or 0.05)
n = Number of years (17 payments received, so n = 17)

Let's substitute the values into the formula:

FV = 2196.30 * ((1 + 0.05)^17 - 1) / 0.05
FV ≈ $41,511.42

So, the Princeton Mathematics Department will inherit approximately $41,511.42 from Pierre's retirement account.

a) 2400