Question

The present value of a perpetuity of 6,500 paid at the end of each year plus the present value of a perpetuity of 8,500 paid at the end of every 5 years is equal to the present value of an annuity of k paid at the end of each year for 25 years. Interest is 6% convertible quarterly. Calculate k.

Answer 1

$10,340

Step-by-step explanation:

The computation of k is shown below;

Rate per quarter = 6% ÷ 4 = 1.5%

In the case when perpetuity paid every year, the effective rate is

= (1 + 1.5%)^4 - 1

= 6.136%

Now Effective rate in the case when perpetuity paid every 5 years

= (1 + 1.5%)^(4 × 5) - 1

= 34.68%

Now

The present value of Both perpetuities = $6,500 ÷ 6.13635506249994% + $8,500 ÷ 34.6855006550052%

= $130,431.99

Now

annuity =k

Number of Periods=25

effective rate = 6.13635506249994%

Annuity k =PV ÷ ((1 - (1 + r)^-n) ÷ r

= $130,431.99 ÷ ((1-(1 + 6.13635506249994%)^-25) ÷ 6.13635506249994%

= $10,335.84

= $10,340