Question

Find the future values of the following ordinary annuities: FV of $800 paid each 6 months for 5 years at a nominal rate of 11% compounded semiannually. Do not round intermediate calculations. Round your answer to the nearest cent. $

FV of $400 paid each 3 months for 5 years at a nominal rate of 11% compounded quarterly. Do not round intermediate calculations. Round your answer to the nearest cent. $

These annuities receive the same amount of cash during the 5-year period and earn interest at the same nominal rate, yet the annuity in part b ends up larger than the one in part a. Why does this occur?

Answer 1

Calculating the Future value of Ordinary annuities:-

a). FV of $800 paid each 6 months for 5 years at a nominal rate of 11% compounded semiannually

Where, C= Periodic Payments = $800

r = Periodic Interest rate = 11%/2 = 5.5%

n= no of periods = 5 years*2 = 10

Future value = $10,300.28

b). FV of $400 paid each 3 months for 5 years at a nominal rate of 11% compounded quarterly

Where, C= Periodic Payments = $400

r = Periodic Interest rate = 11%/4 = 2.75%

n= no of periods = 5 years*4 = 20

Future value = $10,478.96

- The Value of Quarterly compounding with quarterly payments have higher Future Value than Semi-annually compounding with Semi-annually payments annuities. As when the frequency of payment as well as Interest compounding is higher in a year, the future value is higher than that of a lower frequency.

This happens because of Interest on Interest, in compounding Interest is earned on Interest which means Interest income is accumulated with deposits after a compounding period and later in next compounding period you don't only earn interest on deposits but also on Interest.

Thus, with higher frequnecy of compounding higher interest on interest is earned.

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