Question

You have $57,964.26 in a brokerage account, and you plan to deposit an additional $7,500 at the end of every future year until your account totals $290,000. You expect to earn 11% annually on the account. How many years will it take to reach your goal? Do not round intermediate calculations. Round your answer to the nearest whole number. years

Answer 1

It will take 10 years to reach your goal.

To see why, let's calculate the future value of the annual deposits after 10 years. We can use the formula for the future value of an annuity:

FV = PMT x ((1 + r)^n - 1) / r

where PMT is the annual deposit, r is the annual interest rate, and n is the number of years.

Plugging in the numbers, we get:

FV = 7,500 x ((1 + 0.11)^10 - 1) / 0.11 = 137,505.70

So after 10 years, the total value of your account will be:

57,964.26 + 137,505.70 = 195,469.96

To reach your goal of $290,000, you need to earn an additional:

290,000 - 195,469.96 = 94,530.04

over some number of years. We can use the formula for the future value of a lump sum to solve for the number of years:

FV = PV x (1 + r)^n

where PV is the present value, r is the annual interest rate, and n is the number of years.

Plugging in the numbers, we get:

94,530.04 = 195,469.96 x (1 + 0.11)^n

Solving for n, we get:

n = log(94,530.04 / 195,469.96) / log(1 + 0.11) = 9.52

Rounding to the nearest whole number, we get:

n = 10

So it will take 10 years to reach your goal.