Question

5.05

eBook

You have $11,856.41 in a brokerage account, and you plan to deposit an additional $4,000 at the end of every future year until your account totals $260,000. You expect to earn 14% annually on the account. How many years will it take to reach your goal? Round your answer to the nearest whole number.

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.

Future value of 11,856.41=11,856.41*(1.14)^n

Also:

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

=4000[(1.14)^n-1]/0.14

Hence

260,000=11,856.41*(1.14)^n+4000[(1.14)^n-1]/0.14

260,000=11,856.41*(1.14)^n+28571.42857*[(1.14)^n-1]

260,000=11,856.41*(1.14)^n+28571.42857*(1.14)^n-28571.42857

(260,000+28571.42857)=(1.14)^n[28571.42857+11,856.41]

(1.14)^n=(260,000+28571.42857)/(28571.42857+11,856.41)

Taking log on both sides;

n*log 1.14=log 7.137938578

n=log 7.137938578/log 1.14

=15 years(Approx).