Question

Future Value of an Annuity for Various Compounding Periods

Find the future values of the following ordinary annuities.

1. FV of $800 each 6 months for 9 years at a nominal rate of 12%, compounded semiannually. Do not round intermediate calculations. Round your answer to the nearest cent.

   $  

2. FV of $400 each 3 months for 9 years at a nominal rate of 12%, compounded quarterly. Do not round intermediate calculations. Round your answer to the nearest cent.

   $  

3. The annuities described in parts a and b have the same amount of money paid into them during the 9-year period, and both earn interest at the same nominal rate, yet the annuity in part b earns more than the one in part a over the 9 years. Why does this occur?

Answer 1

a. The future value is computed as shown below:

Future value = Annuity payment x [ [ (1 + r)ⁿ – 1 ] / r ]

r is computed as follows:

= 12% / 2 (Since the interest is compounded semi annually, hence divided by 2)

= 6% or 0.06

n is computed as follows:

= 9 x 2 (Since the interest is compounded semi annually, hence multiplied by 2)

= 18

So, the amount will be computed as follows:

= $ 800 x [ [ (1 + 0.06)¹⁸ - 1 ] / 0.06 ]

= $ 800 x 30.90565255

= $ 24,724.52 Approximately

b. The future value is computed as shown below:

Future value = Annuity payment x [ [ (1 + r)ⁿ – 1 ] / r ]

r is computed as follows:

= 12% / 4 (Since the interest is compounded quarterly, hence divided by 4)

= 3% or 0.03

n is computed as follows:

= 9 x 4 (Since the interest is compounded quarterly, hence multiplied by 4)

= 36

So, the amount will be computed as follows:

= $ 400 x [ [ (1 + 0.03)³⁶ - 1 ] / 0.03 ]

= $ 400 x 63.27594427

= $ 25,310.38 Approximately

c. The difference in the amount is on account of compounding times in a year. The more the number of compounding, the greater is the amount. Hence the amount in part b is greater than the amount in part a.

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