At an annual effective interest rate of 6.3%, an annuity immediate with 4N level annual payments of 1,000 has a present value of 14,113. Determine the fraction (in percentage) of the total present value represented by the first set of N payments and the third set otâ…£ payments combined.

$8,949.22

Explanation:

PV = annual payment x PV annuity factor

PV annuity factor = 14,133 / 1,000 = 14.133

PV annuity factor = [1 - 1/(1 + 0.063)ⁿ ] / 0.063

14.133 x 0.063 = 1 - 1/(1 + 0.063)ⁿ

0.890379 = 1 - 1/(1 + 0.063)ⁿ

1/(1 + 0.063)ⁿ = 0.109621

1 / 0.109621 = 1.063ⁿ

9.12234 = 1.063ⁿ

n = log 9.12234 / log 1.063 = 0.96010624 / 0.0265333 = 36

the present value of the first 36/4 = 9 payments = $1,000 x 6.71376 (PV annuity factor, 9 periods, 6.3%) = $6,713.76

the present value of the third set of 9 payments = $6,713.76 / (1 + 6.3%)¹⁸ = $2,235.46

present value of the first and third sets = $8,949.22