Question

You are saving for the college eduction of your two children. They are two years apart in age; one will begin college 17 years from today and other child will begin in 19 years from today. You estimate the college expenses to be $40000 per year per child. Given r = 8% EAR. College expenses are paid at the beginning of each school year. How much money must you deposit in an account each year to fund your children's education? Your deposits begin one year from today. You will make the last deposit at the end of the 15th year. Assume Five years of college.

Answer 1

Part A

PV of Annuity Due:
Annuity is series of cash flows that are deposited / withdrawn at regular intervals for specific period of time at the beginnign fo the year

PV of Annuity Due = Cash Flow + [ Cash Flow * [ 1 - [(1+r)^-(n-1)]] /r ]
r - Int rate per period = 8% or 0.08
n - No. of periods = 5 Years

Particulars	Amount
Cash Flow	$          40,000.00
Int Rate	8.000%
Periods	5

PV of Annuity Due = [ Cash Flow + Cash Flow * [ 1 - [(1+r)^-(n-1)]] / r ]
= [ $ 40000 + $ 40000 * [ 1 - [(1+0.08)^-4] ] / 0.08 ]
= [ $ 40000 + $ 40000 * [ 1 - [(1.08)^-4] ] / 0.08 ]
= [ $ 40000 + $ 40000 * [ 1 - [0.735] ] / 0.08 ]
= [ $ 40000 + $ 40000 * [0.265] ] / 0.08 ]
= [ $ 40000 + $ 132485.07 ]
= $ 172485.07

Part B
we need $ 172485.07 for each child at the beginning of 17 th year and 19 th year from today

Present value of 17th year annuity

Future value = presetn value * future value factor

present value = future value * 1/future value factor

future alue factor = (1+r)^n

r = rate of interest = 8 %

n = no. of years compounding = 17-1 = 16 (we need amount at the 17 th year beginning , so amount compounded for 16 years)

PV = $ 172485.07 * 1/(1.08)^16

PV = $ 172485.07 * 0.2919

PV = $ 50346.75

Present value of 19th year annuity

r = rate of interest = 8 %

n = no. of years compounding = 19-1 = 18 (we need amount at the 19 th year beginning , so amount compounded for 18 years)

PV = $ 172485.07 * 1/(1.08)^18

PV = $ 172485.07 * 0.2502

PV = $ 43164.22

Total present value of amount we need today = $ 50346.75 + $ 43164.22 = $ 93510.97

Part C

amout to be deposited in account each year for 15 years

PV of Annuity:

Annuity is series of cash flows that are deposited at regular intervals for specific period of time.

PV of Annuity = Cash Flow * [ 1 - [(1+r)^-n]] /r
r - Int rate per period = 8%
n - No. of periods = 15 years (deposit begin one year from today )

Particulars	Amount
PV Annuity	$          93,510.97
Int Rate	8.0000%
Periods	15

Cash Flow = PV of Annuity / [ 1 - [(1+r)^-n]] /r
= $ 93510.97 / [ 1 - [(1+0.08)^-9]] /0.08
= $ 93510.97 / [ 1 - [(1.08)^-9]] /0.08
= $ 93510.97 / [ 1 - 0.3152 ] /0.08
= $ 93510.97 / [0.6848 / 0.08 ]
= $ 93510.97 / 8.5595
= $ 10924.84

if we deposit $ 10924.84 each year for the next 15 years we can have the amount required college fee $ 40000 each year for 5 years at 17th year from today for one child and 19th year from today for another child .

please comment if any further assistance is required.