Question

For 50000, Smith purchases a 36-payment annuity-immediate with monthly payments. Assume an effective annual interest rate of 12.68%. For each of the following cases find the unknown amount X.

(a) The first payment is X and each subsequent payment is 50 more than the previous one.

(b) The first payment is X and each subsequent payment until the 18th pay- ment (and including the 18th payment) is 0.2% larger than the previous one. After the 18th payment, each subsequent payment is 0.2% smaller than the previous one.

Answer 1

	Payment for annuity	50,000
	Monthly interest=(12.68/12)%=	0.0105667
	Present Value (PV) of Cash Flow:
	(Cash Flow)/((1+i)^N)
	i=Discount Rate=0.0105667
	N=Month of Cash Flow
(a)	Subsequent payment 50 more than previous one
	First payment=X=
	Total Number of Payments=36
	Present Value of Payment =
	X+(X+50)/1.0105667+(X+100)/(1.0105667^2)+…….(X+(35*50))/(1.0105667^35)
	X+X/(1.0105667)+X/(1.0105667^2)+………+X/(1.0105667^35)+50/(1.0105667)+(2*50)/(1.0105667^2)+……(35*50)/(1.0105667^35)
	The above equation is sum of two parts
	Part A:X+X/(1.0105667+X/(1.0105667^2)+………+X/(1.0105667^35)
	Part A;X(1+1/1.0105667+1/(1.0105667^2)+……..1/(1.0105667^35)
	Part B:50/(1.0105667)+(2*50)/(1.0105667^2)+……(35*50)/(1.0105667^35)
	Part A=Balance amount =50000-24659.05852=          25,341 Part A Present Value of Constant payment of 1 for 35 months                            30.13 (Using Excel PV function with Rate=0.0105667,Nper=36 PMT=-1, Type=1(Beginning of period payments) X*30.13=          25,341 X=25341/30.13=          841.05 X=841.05