Question

The present value of perpetuity of $600 paid at the end of each year plus the present value of a perpetuity of $800 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. solution with details

Answer 1

Given that,

Interest rate = 6% p.a. compounded quarterly,

So yearly rate = (1+r/n)^n - 1, where n is compounding frequency = 4 quarters in a year

So, Yearly rate = (1+0.06/4)^4 - 1 = 6.14%

So, present value of a perpetuity us calculated using formula:

PV = periodic payment/periodic rate

So, For a perpetuity of $600 paid at the end of each year

PV = 600/0.0614 = $9777.79

Similarly, for PV of a perpetuity of $800 paid at the end of every 5 years

5 year rate = (1+yearly rate)^5 - 1 = (1+0.0614)^5 - 1 = 34.69%

PV = 800/0.3469 = $2306.44

So, Total PV of both the perpetuity is 9777.79+2306.44 = $12084.23

So for a perpetuity of payment at the end of 25 years,

25 years rate = (1+yearly rate)^25 - 1 = (1+0.0614)^25 - 1 = 343.20%

=> Periodic payment = 12084.23*3.432 = $41473.63