Question

A family wants to have a $130,000 college fund for their children at the end of 16 years. What contribution must be made at the end of each quarter if their investment pays 7.2%, compounded quarterly?

(a) State whether the problem relates to an ordinary annuity or an annuity due.

ordinary annuity due    

(b) Solve the problem. (Round your answer to the nearest cent.)

2

If $5500 is deposited at the end of each quarter in an account that earns 5% compounded quarterly, after how many quarters will the account contain $100,000? (Round your answer UP to the nearest quarter.)
quarters

Answer 1

1 (a)  Here, the deposits will be same every year, so it is an annuity. And since the deposits are made at the end of each year, so it is ordinary annuity.

(b) The future value of annuity is $130000. Here we will use the future value of annuity formula as per below:

FVA = P * ((1 + r)ⁿ - 1 / r)

where, FVA is future value of annuity = $130000, P is the periodical amount, r is the rate of interest = 7.2% compounded quarterly. So quarterly rate = 7.2% / 4 = 1.8% and n is the time period = 16 * 4 = 64

Now, putting these values in the above formula, we get,

$130000 = P * ((1 + 1.8%)⁶⁴ - 1 / 1.8%)

$130000 = P * ((1 + 0.018)⁶⁴- 1 / 0.018)

$130000 = P * ((1.018)⁶⁴- 1 / 0.018)

$130000 = P * ((3.13225991067- 1) / 0.018)

$130000 = P * (2.13225991067 / 0.018)

$130000 = P * 118.458883926

P = $130000 / 118.458883926

P = $1097.43

At the end of each quarter,a contribution of $1097.43 must be made.

2. The future value of annuity is $100000. Here we will use the future value of annuity formula as per below:

FVA = P * ((1 + r)ⁿ - 1 / r)

where, FVA is future value of annuity = $100000, P is the periodical amount = $5500, r is the rate of interest = 5% compounded quarterly. So quarterly rate = 5% / 4 = 1.25% and n is the time period

Now, putting these values in the above formula, we get,

$100000 = $5500 * ((1 + 1.25%)ⁿ- 1 / 1.25%)

$100000 = $5500 * ((1 + 0.0125)ⁿ- 1 / 0.0125)

$100000 / $5500 = ((1.0125)ⁿ- 1 / 0.0125)

18.18181818 = ((1.0125)ⁿ - 1 / 0.0125)

18.18181818 * 0.0125 = (1.0125)ⁿ - 1

0.2272727272 = (1.0125)ⁿ - 1

0.2272727272 + 1 = (1.0125)ⁿ

1.2272727272 = (1.0125)ⁿ

(1.0125)^16.5 = (1.0125)ⁿ

n = 16.5

So, n= 16.5 quarters.