Question

Equal end-of-period semiannual payments of $500, increasing by $100 with each subsequent payment, are made to a fund paying 10 percent compounded continuously. What will the fund amount to after 7 years? What is the present worth equivalent of the total set of payments? What is the equal semiannual equivalent amount of the payments?

Answer 1

First convert continuous compounded rate to semi-annual rate, as follows:

If the annual effective rate is i then continuous compounded rate is ln(1+i). So,

ln(1+i) =10%

1+i = e^10%

1 +i = 1.1052

i = 10.52%

Semi-annual rate becomes [(1+i)^(1/2)]-1 = [(1+10.52%)^(1/2)] -1 = 5.13%

Calculations:

	Constant payment (CP)	500.00
Monthly increment in payment	Payment gradient (G)	100.00
	N	14.00
	Semi-annual rate (I)	5.13%
Using PV function: PV(I, N, CP,0)	PV of constant payment (PV1)	4,909.34
G*(P/G, I, N)	PV of payment gradient (PV2)	5,590.85
PV1 + PV2	Total PV	10,500.19
Using PMT function: PMT(I,N,Total PV,0)	Equivalent semiannual payments	1,069.41
Using FV function: FV(I,N,CP,0)	FV of constant payment (FV1)	9,886.20
Using FV function: PV(I,N,0, PV2)	FV of payment gradient (FV2)	11,258.59
FV1 + FV2	Total FV	21,144.79

1). Fund balance after 7 years = 21,144.79

2). Present worth equivalent of total set of payments = 10,500,19

3). Equivalent semi-annual payments = 1,069.41