Question

1. A fund is built with annual deposits increasing by 1 from 1 to 10 and then decreasing by 1 to $0 at an annual effective interest rate of 5%. At the end of 19 years, the fund is used to purchase a 8-year annuity with level payment $X at an annual effective interest rate of 3% with the first payment 20 years from today.

Calculate X

2. Annie wants to accumulate $60500 in a fund at the end of 20 years. She plans to deposit $800+tX at the end of year t (t = 1,2,...,10) and $1500 at the end of last 10 years. The fund earns an annual effective interest rate of 6%.

Calculate X

Please show all work and do not use excel, thanks!

Answer 1

Part 1

Computation of Accumualted Value of Deposits at the end of 19 years

Time Period	Amount	Years deposited (n)	Accumulation Factor @ 1.05^n	Accumulated Amount
1	$1.00	19	2.53	$2.53
2	$2.00	18	2.41	$4.82
3	$3.00	17	2.29	$6.87
4	$4.00	16	2.18	$8.72
5	$5.00	15	2.08	$10.40
6	$6.00	14	1.98	$11.88
7	$7.00	13	1.89	$13.23
8	$8.00	12	1.80	$14.40
9	$9.00	11	1.71	$15.39
10	$10.00	10	1.63	$16.30
11	$9.00	9	1.55	$13.95
12	$8.00	8	1.48	$11.84
13	$7.00	7	1.41	$9.87
14	$6.00	6	1.34	$8.04
15	$5.00	5	1.28	$6.40
16	$4.00	4	1.22	$4.88
17	$3.00	3	1.16	$3.48
18	$2.00	2	1.10	$2.20
19	$1.00	1	1.05	$1.05
TOTAL	$166.25

The the present value of the annuity should be equal to $166.25

166.25 = (1.03^(-20) ) * X * (CPVF(3%,8))

166.25 = 0.55 * X * 7.02

X = $43.06

Part 2

The depsoits will be (800 + X); (800 + 2X) ; ..... (800 + 10X); $1,500 from year 11 to 20

The accumulation factor for 10 years @ 6% is 13.1808 and 1.10^10 = 1.790 and the Increasing accumulation factor for 6% for 10 years is 36.9624 which can be calculated by geometric series,

For the first 10 years, the accumulation will be (800*13.1808*1.790) + (X * 36.9624 * 1.790)

=$18,874.91 + 66.16X

For the last 10 years the accumulation will be 1500 * 13.1808

=$ 19,771.20

Computation of X

$60,500 = $18,874.91 + 66.16X + $19,771.20

X = 330.31