Grandparents plan to open an account on their grandchild's birthday and contribute each month until she goes to college. How much must they contribute at the beginning of each month in an investment that pays 8%, compounded monthly, if they want the balance to be $160,000 at the end of 18 years?

They contribute $ at the beginning of each month

Explanation:

A constant payment for a specified period is called annuity. The future value of the annuity can be calculated using a required rate of return.

Formula for Future value of annuity is

F = P x ( [ 1 + I ]^N - 1 ) / I

I = interest rate = 8% / 12 = 0.67% per month

N = Number of periods = 18 years x 12  = 216 months

F = Future / final value = $160,000

P =Payment amount = ?

$160,000 = P x ( [ 1 + 0.67% ]^216 - 1 ) / 0.67%

$160,000 = P x ( [ 1.0067 ]^216 - 1 ) / 0.0067

$160,000 = P x ( [ 1.0067 ]^216 - 1 ) / 0.0067

$160,000 = P x 482.20

P = $160,000 / 482.2

p = $331.81

$332.74

Explanation:

We can use the annuity formula to calculate the monthly deposit. Since this problem requires a lot of compounding periods we must use the memory function of our calculator to get a more exact answer.

future value = payment x [(1 + i)ⁿ - 1] / i

$160,000 = payment x [(1 + 0.006666)²¹⁶ - 1] / 0.006666

$160,000 = payment x (1.006666²¹⁶ - 1) / 0.006666

$160,000 = payment x 3.2 / 0.006666

$160,000 = payment x 480.861

payment = $160,000 / 480.861 = $332.736 ≈ $332.74

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