Question

The accumulated value just after the last payment under 12-year annuity of 1000 per year, paying effective annual interest rate of 6%, is to be used to purchase a 10-year annuity at an effective annual interest rate of 7% with the first payment to be made 4 years after the last payment under the first annuity. Find the amount of the payment under the 10-year annuity assuming that the 7% rate starts from the time of the last payment under the 12-year annuity.

Answer 1

Please refer to the excel attachment for workings.

Please refer to the time graph drawn and presented as an attachment for better clarity of the question.

For the first 12 years

This is 12 year annuity with 1000 annual payments @ 6% per annum. So the accumulated value(AV) at the end of 12th year will be as below :-

AV= 16870 ( refer excel)

For the next 10 years

For this annuity 0th year means 12th Year of Previous Year. Hence the Future Value at 12th year of Previous annuity will be equal to the Present Value(PV) of Current Annuity at 0th year, which is equal to 16870.

Assume the yearly annuity of 10- year annuity be x per year.

Hence now we have the following data:

• PV at 0th year = 16870
• Annual Annuity payments = x
• Effective interest rate= 7%

We can derive x by equating PV at 0th year (16870) with the Present Value of x per year for 10 years at 7% per annum. Hence we have :-

PV of x ( 7%, 10 year) = 16870

x * Annuity Factor ( 7%, 10 year)= 16870

x * 7.0236 = 16870

x = 16870/ 7.0236

= 2402 per annum for 10 years

Since the first payment is done 4 years after the 12th Year, therefore, first payment should be of Value ( 2402*4)

= 9608

*Note

• How to calculate Annuity Factor in Calculator :

1. Do (1/1.07) in your calculator
2. Then press = 10 times
3. Then press GT in your Calculator

• If you donot have GT calculator, then do 1/1.07^1, 1/1.07^2, ................., 1/ 1.07^10. Then add all the individual 10 values.

​​​​​​​​​​

​​​​