Question

You are planning for your future needs and retirement. You want to receive $50,000 five years from today and retirement annuity of $100,000 per year for 25 years with the first payment 10 years from today. To pay for this, you will make 5 payments of $A per year beginning today and 10 annual payments of $A with the first payment 8 years from today. With an interest rate of 8%, what is the value of A?

Answer 1

The value of A is $69,041.75

Calculations and explanations:

The situation can be shown below and I have also computed the present value of all cash inflows required using the formula Present value = amount of cash flow*PVIF (present value interest factor). PVIF = 1/(1.08)^n

End of year	Cash inflow required	1+r	PVIF	PV = cash flow * PVIF
5	50,000	1.08	0.68058	34,029.16
10	100,000		0.46319	46,319.35
11	100,000		0.42888	42,888.29
12	100,000		0.39711	39,711.38
13	100,000		0.36770	36,769.79
14	100,000		0.34046	34,046.10
15	100,000		0.31524	31,524.17
16	100,000		0.29189	29,189.05
17	100,000		0.27027	27,026.90
18	100,000		0.25025	25,024.90
19	100,000		0.23171	23,171.21
20	100,000		0.21455	21,454.82
21	100,000		0.19866	19,865.57
22	100,000		0.18394	18,394.05
23	100,000		0.17032	17,031.53
24	100,000		0.15770	15,769.93
25	100,000		0.14602	14,601.79
26	100,000		0.13520	13,520.18
27	100,000		0.12519	12,518.68
28	100,000		0.11591	11,591.37
29	100,000		0.10733	10,732.75
30	100,000		0.09938	9,937.73
31	100,000		0.09202	9,201.60
32	100,000		0.08520	8,520.00
33	100,000		0.07889	7,888.89
34	100,000		0.07305	7,304.53
			Total	568,033.73612

Thus the present value of all cash inflows required is $568,033.73612

Now deposits are being made at the start of each year. So present value of deposits of the first 5 years= A + A/1.08 + A/1.08^2 + A/1.08^3 + A/1.08^4 [Equation 1]

Next set of deposits are being made from the end of 8th year onwards. Thus present value = A/1.08^8+A/1.08^9+..........A/1/08^17 [Equation 2]

Now Equation 1 + Equation 2 = $568,033.73612

or A + A/1.08 + A/1.08^2 + A/1.08^3 + A/1.08^4 + A/1.08^8+A/1.08^9+..........A/1/08^17 = $568,033.73612

Solving the above equation mathematically we get A as $69,041.75

Year	Amount	1+r	PVIF	PV = cash flow * PVIF
0	69,041.75	1.08	1.00000	69,041.74941
1	69,041.75		0.92593	63,927.54575
2	69,041.75		0.85734	59,192.17199
3	69,041.75		0.79383	54,807.56666
4	69,041.75		0.73503	50,747.74690
8	69,041.75		0.54027	37,301.10894
9	69,041.75		0.50025	34,538.06383
10	69,041.75		0.46319	31,979.68873
11	69,041.75		0.42888	29,610.82290
12	69,041.75		0.39711	27,417.42861
13	69,041.75		0.36770	25,386.50797
14	69,041.75		0.34046	23,506.02590
15	69,041.75		0.31524	21,764.83880
16	69,041.75		0.29189	20,152.62852
17	69,041.75		0.27027	18,659.84122
			Total	568,033.73612