Question

You have $13,037.39 in a brokerage account, and you plan to deposit an additional $4,000 at the end of every future year until your account totals $200,000. You expect to earn 11% annually on the account. How many years will it take to reach your goal? Round your answer to two decimal places at the end of the calculations.

years

Answer 1

We use the formula:
A=P(1+r/100)^n
where
A=future value
P=present value
r=rate of interest
n=time period.

Hence future value of $13,037.39=$13,037.39*(1.11)^n

Also:

Future value of annuity=Annuity[(1+rate)^time period-1]/rate

=$4000[(1.11)^n-1]/0.11

Hence

200,000=$13,037.39*(1.11)^n+$4000[(1.11)^n-1]/0.11

200,000=$13,037.39*(1.11)^n+$36,363.64[(1.11)^n-1]

200,000=$13,037.39*(1.11)^n+$36,363.64(1.11)^n-36,363.64

(200,000+36,363.64)=(1.11)^n[13037.39+36,363.64]

(1.11)^n=(200,000+36,363.64)/[13037.39+36,363.64]

(1.11)^n=4.784589588

Taking log on both sides;

n*log 1.11=log 4.784589588

n=log 4.784589588/log 1.11

=15 years(Approx).