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 of the total present value represented by the first set of N payments and the third set of N payments combined.

the % of the present value that corresponds to the first 9 payments (N) =  47.57% of the annuity's present value.

the % of the present value that corresponds to the first 27 payments (3N) =  90.86% of the annuity's present value.

Explanation:

we must use the present value of an annuity formula:

PV = annual payment x annuity factor

14,113 = 1,000 x annuity factor

annuity factor = 14,113 / 1,000 = 14.133

we know that the interest rate is 6.3%, now using an annuity calculator we can determine that the total number of periods is 36. The exact factor is 14.11322, but we can round to 14.113

the first set would represent 36/4 = 9 years

the % of the present value that corresponds to the first 9 payments (N) = PV = 1,000 x 6.71376 (PV annuity factor, 6.3%, 9 periods) = 6,713.76. This corresponds to 6,713.76 / 14,113 = 47.57% of the annuity's present value.

the % of the present value that corresponds to the first 27 payments (3N)  = PV = 1,000 x 12.82329 (PV annuity factor, 6.3%, 27 periods) = 12,823.29. This corresponds to 12,823.29 / 14,113 = 90.86% of the annuity's present value.