Question

Ben invested $20,000 into a money market account and took out $5,000 at the end of year 5. He found out at the end of 10 years that he had $50,000 in the account. What is the annual interest rate Ben had earned on this investment? Please give the solution (include calculation) and explain why.

Answer 1

Formula :

Amount (A) = P(1 + r)ⁿ

where P = Principle , r = interest rate = r and n = time.

Now here P = principle = initial amount invested = 20,000

Hence After 5 years this Principle will become (A)

A = 20,000(1 + r)⁵

Now he will withdraw 5000.

Now After 5 years amount of investment left = 20,000(1 + r)⁵ - 5000.

Now this amount will be invested for = 10 - 5 = 5 years and will become 50,000

Hence A = P(1 + r)ⁿ

Now A= 50,000 , n = 5 and P = 20,000(1 + r)⁵ - 5000

=> (20,000(1 + r)⁵ - 5000)(1+ r)⁵ = 50,000

Let (1 + r)⁵ = x

=> (20,000x - 5000)x = 50,000

=> 4x² - x - 10 = 0

=> x = 1.71

Hence (1 + r)⁵ = 1.71

=> (1 + r) = 1.113

=> r = 1.113 - 1 = 0.113 + 11.3%

Hence Annual interest rate Ben has earned on his investment = 11.3%