Question

1. Your child is about to enter college a year from now. A local foundation provides scholarships (essentially interest-free loans) each year perpetually for students. Your child has won the scholarship just now. When your child joins college, he/she would receive a scholarship of $ 10,000 per year annually for 4 years. Your child is expected to repay the scholarship amount of $ 40,000 in 15 equal yearly installments, interest-free beginning a year after the expiration of his/her scholarship. The foundation seems to be giving an interest free loan. The market interest rate is 8% and would not change in the future.
   1. What is the PV of the scholarship?
   2. If the foundation invests a lump sum today to fund all future scholarships. What must be that investment today?
2. solve by formula method and not by excel method

Answer 1

Answer:
a)	PV of Scholarship
	Repayment of Scolarship of $40000 in 15 equal yearly instalments
	So amount of yearly instalment is $40000/15 = $2666.67
	PV = CF/(1+r)^n
	PV = Present value of all cash flows
	CF = Future cash flow in each period
	r = rate of interest
	n = no. of years

	Let us name the denominator in the avove formula i.e (1+r)^n as PV factor
	Year	FV	PV factor	PV
	1	-10000	1	-10000
	2	-10000	1.08	-10800		Here, the annual scholarship of $10000 starts immediately on
	3	-10000	1.17	-11664		joining i.e at the beginning of year 1 and so on for Year 2, 3
	4	-10000	1.26	-12597		and 4. Repayment starts a year after the expiriration of
	5	2666.67	1.47	3918		scholarship i.e at the end of Year 5. Hence the for calculation
	6	2666.67	1.59	4232		of PV factor, n for Year 1, 2, 3 & 4 are taken as 0,1,2 and 3. For
	7	2666.67	1.71	4570		repayment from the end of Year 5, n in the PV factorare taken
	8	2666.67	1.85	4936		as 5, 6 and so on.
	9	2666.67	2.00	5331
	10	2666.67	2.16	5757		So the PV of the scholarship is $61327
	11	2666.67	2.33	6218
	12	2666.67	2.52	6715
	13	2666.67	2.72	7252
	14	2666.67	2.94	7833
	15	2666.67	3.17	8459
	16	2666.67	3.43	9136
	17	2666.67	3.70	9867
	18	2666.67	4.00	10656
	19	2666.67	4.32	11509
	Total	0.00		61327

b)	For the foundation to invests a lump sum today to fund all future scholarships, we need to
	calculate what sum today if invested will generate a Future Value (FV) of $61327 in 19 years.
	Formula of Future Value is given hereunder:
	FV = Co x (1+r)^n
	FV = Future value of Cash Flow at the beginning
	Co = Cash flow at the beginning
	r = rate of interest
	n = no. of years
	Let us name the multiplier in the above formula i.e (1+r)^n as FV factor
	Year	FV factor
	1	1.08		Here the FV factor for n = 19 is computed as	4.32
	2	1.17
	3	1.26		So, $61327          =	Co x 4.32
	4	1.36
	5	1.47		So, the Cash flow required at the beginning is          $	14210
	6	1.59
	7	1.71
	8	1.85
	9	2.00
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