Question

Future Value of an Annuity for Various Compounding Periods

Find the future values of the following ordinary annuities.

1. FV of $200 each 6 months for 5 years at a nominal rate of 8%, compounded semiannually. Do not round intermediate calculations. Round your answer to the nearest cent.

   $  

2. FV of $100 each 3 months for 5 years at a nominal rate of 8%, 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 5-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 5 years. Why does this occur?

   Choices:
   The nominal deposits into the annuity in part (b) are greater than the nominal deposits into the annuity in part (a).
   The annuity in part (a) is compounded less frequently; therefore, more interest is earned on interest.
   The annuity in part (a) is compounded more frequently; therefore, more interest is earned on interest.
   The annuity in part (b) is compounded less frequently; therefore, more interest is earned on interest.
   The annuity in part (b) is compounded more frequently; therefore, more interest is earned on interest.Item 3

Answer 1

Future Value of annuity formula:

Where,
FV = Future Value of Annuity
A = Annuity or Deposit
i = rate of interest in decimal form
n = number of years
a = number of compounding in a year

a) Compounded semi-annually (a = 2).

Given:

A = $200
i = 8% or 0.08
n = 5 years
a = 2

Substituting the values in the formula, we get:

b) Compounded quarterly (a = 4)

Given:

A = $100
i = 8% or 0.08
n = 5 years
a = 4

Substituting the values in the formula, we get:

c) Answer: The annuity in part (b) is compounded more frequently; therefore, more interest is earned on interest.

Reason :

Total amount invested in (a) = $200 * 2 times a year * 5 years
= 200 * 2* 5
=$2000

Total amount invested in (b) = $100 * 4 times a year * 5 years
= 100 * 4 * 5
= $2000
Both (a) and (b) invested same amount, but (b) has future value greater than (a). This is because (b) is compounded more frequently.