Question

You plan to make equal annual donations over a 6-year period (beginning 1 year from now) to a college for supporting poor students per year for 21 years, with the first scholarship to be awarded after 1 year from now. The cost of tuition at the college is $4000 per year and is expected to stay at that amount for 4 years. After that time (i.e., year 5), the tuition is expected to increase by 8% per year. If the college can invest the money at a rate of 10% per year, what size must the donations be? [Ans. A = $11,988]

Answer 1

ANSWER:

year	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21
tution fees per year with 8% increase from 5th year	4000	4000	4000	4000	4320	4665.6	5038.848	5441.95584	5877.312307	6347.497292	6855.297075	7403.720841	7996.018508	8635.699989	9326.555988	10072.68047	10878.4949	11748.7745	12688.67646	13703.77057	14800.07222

Now we will find the present worth of this cash flow at 10% rate.

pw = cash flow in year 1(p/f,i,n) + cash flow in year 2(p/f,i,n) + cash flow in year 3(p/f,i,n) + cash flow in year 4(p/f,i,n) + cash flow in year 5(p/f,i,n) + cash flow in year 6(p/f,i,n) + cash flow in year 7(p/f,i,n) + cash flow in year 8(p/f,i,n) + cash flow in year 9(p/f,i,n) + cash flow in year 10(p/f,i,n) + cash flow in year 11(p/f,i,n) + cash flow in year 12(p/f,i,n) + cash flow in year 13(p/f,i,n) + cash flow in year 14(p/f,i,n) + cash flow in year 15(p/f,i,n) + cash flow in year 16(p/f,i,n) + cash flow in year 17(p/f,i,n) + cash flow in year 18(p/f,i,n) + cash flow in year 19(p/f,i,n) + cash flow in year 20(p/f,i,n) + cash flow in year 21(p/f,i,n)

pw = 4,000(p/f,10%,1) + 4,000(p/f,10%,2) + 4,000(p/f,10%,3) + 4,000(p/f,10%,4) + 4,320(p/f,10%,5) + 4,665.6(p/f,10%,6) + 5,038.84(p/f,10%,7) + 5,441.95(p/f,10%,8) + 5,877.31(p/f,10%,9) + 6,347.49(p/f,10%,10) + 6,855.29(p/f,10%,11) + 7,403.72(p/f,10%,12) + 7,996.01(p/f,10%,13) + 8,365.7(p/f,10%,14) + 9,326.55(p/f,10%,15) + 10,072.68(p/f,10%,16) + 10,878.49(p/f,10%,17) + 11,748.77(p/f,10%,18) + 12,688.68(p/f,10%,19) + 13,703.77(p/f,10%,20) + 14,800.07(p/f,10%,21)

pw = 4,000 * 0.9091 + 4,000 * 0.8264 + 4,000 * 0.7513 + 4,000 * 0.683 + 4,320 * 0.6209 + 4,665.6 * 0.5645 + 5,038.84 * 0.5132 + 5,441.95 * 0.4665 + 5,877.31 * 0.4241 + 6,347.49 * 0.3855 + 6,855.29 * 0.3505 + 7,403.72 * 0.3186 + 7,996.01 * 0.2897 + 8,365.7 * 0.2633 + 9,326.55 * 0.2394 + 10,072.68 * 0.2176 + 10,878.49 * 0.1978 + 11,748.77 * 0.1799 + 12,688.68 * 0.1635 + 13,703.77 * 0.1486 + 14,800.07 * 0.1351

pw = 3,636.36 + 3,305.79 + 3,005.26 + 2,372.05 + 2,682.38 + 2,633.61 + 2,585.73 + 2,538.71 + 2,492.55 + 2,447.23 + 2,402.74 + 2,359.05 + 2,316.16 + 2,274.05 + 2,232.70 + 2,192.11 + 2,152.25 + 2,113.12 + 2,074.70 + 2,036.98 + 1,999.94

pw = 52,213.49

now we will find the annual equivalent amount.

n = 6

annual equivalent amount = pw(a/p,i,n) = 52,213.49(a/p,10%,6) = 52,213.49 * 0.2296 = 11,988.6

so the annual equivalent amount is $11,988