Question

1. Deposits of X are made at the end of every month for 10 years in order to accumulate an annuity paying 100 at the end of every year forever, with the first payment being paid at the end of year 11. Find X if nominal interest rate is 6% convertible quarterly.

Answer 1

Effective Interest Rate = [{1+(Nominal Interest Rate/Number of times compounded in a year)}^Number of times compounded in a year] - 1

= [{1+(0.06/4)}^4]-1 = 1.06136-1 = 0.06136 = 6.136%

PV of perpetuity = Annuity/Effective Interest Rate = 100/0.06136 = $1629.726

Therefore, at the end of year 10, there should be a balance of $1629.726

FV of Annuity = P*[{(1+i)^n}-1]/i

where, FV of annuity = 1629.726, i = Quarterly Interest Rate = 0.06/4 = 0.015, n = Number of quarters = 10*4 = 40

1629.726 = P*[{(1+0.015)^40}-1]/0.015

24.4589 = P*[0.814]

Therefore, Annuity = P = 24.4589/0.814 = $30.03

Amount to be deposited every QUARTER = $30.03

Therefore, X = Amount to be deposited every month = 30.03/3 = $10.01