Annuities and loans

Question

suppose you invest $150 a month for four years into an account earning 9% compound monthly. After 4 years, you leave the money, without making additional deposits, in the account for another 20 years. How much will you have in the end?

I can’t figure out this problem it’s been frustrating me all day if someone could help explain it to me I’d love it!

thank you!

Answer 1

These types of problems are derived from an iterative addition of interest bearing money. In the case, the $150 is added at the beginning of each month and then at the end of the month, interest is earned. This money is then the basis for the next month. The monthly interest rate is .09/12 = .0075.

Start of Month End of Month

Month 1 150 150(1.0075)

Month 2 150(1.0075) + 150 (150(1.0075)+150)(1.0075) = 150(1.0075)² + 150(1.0075)

if you follow this trend, at the end of the 4 year/48 month period, the sum of money will be

Sum = 150(1.0075)⁴⁸ + 150(1.0075)⁴⁷ + ........... + 150(1.0075)² + 150(1.0075)

In order to sum the series, multiply it by (1.0075) to obtain

(1.0075)Sum = 150(1.0075)⁴⁹ + 150(1.0075)⁴⁸ + ........ + 150(1.0075)³ + 150(1.0075)²

Then, (1.0075)Sum -Sum = 150(1.0075)⁴⁹ - 150(1.0075) Then the sum is

Sum = { 150(1.0075)⁴⁹ - 150(1.0075) }/(1.0075 - 1) = $8,692.82 Now for the next 20 years at

the same interest rate with nothing more deposited, we have

Total = 8,692.82(1.0075)²⁴⁰ = $52,236.47 There are standard formulas for this but I

thought I would show where it comes from!