Question

You need to accumulate $10,000. To do so, you plan to make deposits of $1,400 per year - with the first payment being made a year from today - into a bank account that pays 11% annual interest. Your last deposit will be less than $1,400 if less is needed to round out to $10,000. How many years will it take you to reach your $10,000 goal? Do not round intermediate calculations. Round your answer up to the nearest whole number.

Answer 1

Annual Deposits = $ 1400, Interest Rate = 11 %, Target Future Value = FV = $ 10000

Let the required number of years be n

Therefore, 1400 x (1.11)^(n-1) + 1400 x (1.11)^(n-2) +............+ 1400 = 10000

1400 x [{(1.11)^(n) - 1}/{1.11-1}] = 10000

Using hit and trial method to solve the above equation we find out that when n=5, FV = $ 8718.92 and when n = 6 , FV = $ 11078

This implies that the required number of years is n= 6 as the last deposit need not be equal to $ 1400 and is in fact equal to the shortfall between the target FV and the total FV value of the annual depoists each worth $ 1400 for 5 years

FV of 5 Deposits = 1400 x (1.11)^(5) + 1400 x (1.11)^(4) + 1400 x (1.11)^(3) + 1400 x (1.11)^(2) + 1400 x (1.11) = $ 9678.003 ~ $ 9678

Therefore, required value of deposit at the end of Year 6 = 10000 - 9678 = $ 322

Hence, the number of years required to accumulate $ 10000 is 6 years with the last deposit being equal to $ 322.