Question

You have $30,637.17 in a brokerage account, and you plan to deposit an additional $4,000 at the end of every future year until your account totals $270,000. You expect to earn 13% 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

Solution:
	Years to reach goals of $270,000	13 years
Working Notes:
	We have balance of $30,637.17 and $4000 payment annuity till reach $270,000, how many years it will takes will calculated by combination of following two equation of future values:
	Future value of annuity = p x ((1+r)^t - 1)/r
	this will help to get all the future values of $4000 annual payments .
	Future value of deposit in brokerage account = deposit (1+r)^t
	this will help to get future value in t years of balance in brokerage account.
	Where, t = no. of years to reach goals of $270,000 = t
	r= annual rate of returns = 13%
	P= payments received per year =$4000
	deposit = $30,637.17
	so,
	270,000 = Future value of annuity + Future value of deposit in brokerage account
	270,000 = p x ((1+r)^t - 1)/r + deposit (1+r)^t
	270,000 = 4000 x ((1+0.13)^t - 1)/0.13 + $30,637.17 (1+0.13)^t
	270,000 = (4000 x (1+0.13)^t)/0.13 - (4000/0.13) + $30,637.17 (1+0.13)^t
	270,000 = 30,769.230769 x (1+0.13)^t) - 30,769.230769 + $30,637.17 (1+0.13)^t
	270,000 + 30,769.230769 = 61,406.400769 ( 1 + 0.13)^t
	4.89801   = (1.13)^t
	taking log on both side
	Log(4.89801)   =   Log (1.13)^t
	Log (4.89801) = t x Log(1.13)
	t= Log (4.89801) / Log(1.13)	putting log values
	t= .690019667/.053078443
	t= 12.99999816	Log (4.89801) = .690019667
	t=13 years	Log(1.13) = .053078443
lets check	270,000 = Future value of annuity + Future value of deposit in brokerage account
	270,000 = p x ((1+r)^t - 1)/r + deposit (1+r)^t
	270,000 = 4000 x ((1+0.13)^t - 1)/0.13 + $30,637.17 (1+0.13)^t
	putting t-= 13 years
	270,000 = 4000 x ((1+0.13)^13 - 1)/0.13 + $30,637.17 (1+0.13)^13
	270,000 = $ 119,938.8032+ 150,061.1988
	270,000 = 270,000.002
	270,000 = 270,000
hence,	our above computation is correct
Please feel free to ask if anything about above solution in comment section of the question.