Question

the present value of a perpetuity of 6500 paid at the end of each year plus the present value of a perpetuity of 8500 paid at the end of every 5 years is equal to the present value of annuity of k paid at the end of each year of 25 years. interest is 6% convertible quarterly. calculate k. please show and explain work

Answer 1

Effective Interest Rate or EAR = [{1+(APR/n)}^n]-1

Where, APR = Annual Interest Rate or Nominal Rate, n = Number of times compounded in a year

Therefore,

For 1 Year = [{1+(0.06/4)}^4]-1 = 0.06136355

For 5 Years = [{1+(0.06/4)}^(4*5)]-1 = 0.346855

For 25 Years = [{1+(0.06/4)}^(4*25)]-1 = 3.4320456

PV of Perpetuity of $6500 every year = 6500/EAR for 1 year = 6500/0.06136355 = $105926.075

PV of Perpetuity of $8500 every 5 years = 8500/EAR for 5 years = 8500/0.346855 = $24505.917

Total of Above PVs = 105926.075+24505.917 = $130431.992

PV of Perpetuity of $k every 25 years = k/EAR for 25 years

130431.992 = k/3.4320456

Therefore, k = 130431.992*3.4320456 = $447648.54