Question

1. Your younger brother will start college in five years. He has just informed your parents that he wants to go to State University, which will cost $22,000 per year for four years (cost assumed to come at the beginning of each year). Anticipating his ambitions, your parents started investing $3,000 per year (at the END of each year) five years ago and will continue to do so for five more years. How much more will your parents have to invest at the END of each year for the next five years to have the necessary fund for his education? Use 12% as the appropriate interest rate throughout this problem.

Answer 1

Formula: The present value of an annuity due (PV)

PV = {C× [1-(1+r)^-n]/r}×(1+r)

PV = Present value (The cumulative amount available at Present).
C= Periodic cash flow.
r =effective interest rate for the period.
n = number of periods.

PV = {22000× [1-(1+0.12)^-4]/0.12}×(1+0.12)
PV = 74,840.

PV = Present value (The cummulative amount needed for college) = $74,840.
C= Periodic cash flow. = 22000
r =effective interest rate for the period. =0.12
n = number of periods. = 4

Formula: The Future Value of an ordinary annuity (FV)

FV= C× {[(1+r)^n]-1}/r

FV = Future value (The cummulative amount available in Future)
C= Periodic cash out flow.
r =effective interest rate for the period.
n = number of periods.

FV= 3000× {[(1+0.12)^10]-1}/0.12
FV = 52,646

FV = 52,646 (The cummulative amount available for college).
C= Periodic cash outflow. 3000
r =effective interest rate for the period. 12%
n = number of periods. 10

The extra funds needed in future= 74,840-52,646= 22,194

FV= C× {[(1+r)^n]-1}/r
22194= C× {[(1+0.12)^5]-1}/0.12
C = 3493.55

FV = 22194 (The cummulative amount available in Future)
C= Periodic cash out flow.
r =effective interest rate for the period. 12%
n = number of periods. 5

The extra cash need per year = $3493.55(approx).