Ordinary annuity

Question

if at age 35 you start saving up for retirement and your plan pays an APR of 6% and you want to have 1.5 million in 30 years how much should you deposit monthly?

Answer 1

There are standard formulas for this but let's calculate it using first principals.... Since the interest is compounded monthly, the rate will be .06/12 = .005

Let x be the amount of money that you deposit at the beginning of the month and make a table

Month Start of Month End of Month

1 x 1.005x

2 1.005x + x 1.0052x + 1.005x

3 1.005²x + 1.005x + x 1.005³x + 1.005²x + 1.005x

360 1.005³⁵⁹x + 1.005³⁵⁸x +...+ 1.005x + x 1.005³⁶⁰x + 1.005³⁵⁹x +....+ 1.005x

So at the end of 30 years or 360 months

let Sum = 1.005³⁶⁰x + 1.005³⁵⁹x +.......+ 1.005²x + 1.005x

the 1.005Sum = 1.005³⁶¹x + 1.005³⁶⁰x +.....+ 1.005³ + 1.005²x

Then 1.005Sum - Sum = 1.005³⁶¹x - 1.005x then Sum = {x(1.005361 - 1.005)}/(.005) = 1009.54x

Then x = 1,000,000/1009.54 = $990.55 invested each month.