Question

If you invest $100/month for your child’s education at a nominal 6% rate of return with monthly compounding, how long will it take to save enough to pay for the tuition of $30,000? (answer in months)

Answer 1

FV of Annuity :

Annuity is series of cash flows that are deposited at regular intervals for specific period of time. Here deposits are made at the end of the period. FV of annuity is future value of cash flows deposited at regular intervals grown at specified int rate or Growth rate to future date.

FV of Annuity = CF [ (1+r)^n - 1 ] / r
r - Int rate per period
n - No. of periods

Particulars	Amount
FV of Annuity	$      30,000.00
Int Rate	0.5000%
Cash flow	100

FV of Annuity = Cash Flow * [ [(1+r)^n ] - 1 ] /r
$ 30000 = $ 100 * [ [ ( 1 + 0.005 ) ^ n ] - 1 ] /0.005
$ 30000 = $ 100 * [ [ ( 1.005 ) ^ n ] - 1 ] /0.005
[ [ ( 1.005 ) ^ n ] - 1 ] = 1.5
( 1.005 ) ^ n = 2.5
Take Log on both sides

Log ( ( 1.005 ) ^ n ) = Log ( 2.5 )
Log ( a ^ b ) = b * Log ( a )

n * Log (1.005) = Log (2.5 )
n * 0.0022 = 0.3979
n * 0.0022 = 0.3979
n = 183.7159
It takes 183.72 ( 184) Months to reach $ 30000