Question

Please explain carefully step by step.

So you can retire early, you have decided to start saving $500 a month starting one month from now. You plan to retire as soon as you can accumulate $1 million. If you can earn 5 percent (per year) on your savings, how many years will it be before you can retire?

Answer 1

Rate = 0.05 / 12 = 0.004167 ( since it is monthly payment, we divide the rate by 2)

Future value of annutiy = Annuity[ ( 1 + r)ⁿ - 1] / r

1,000,000 = 500 * [ ( 1 + 0.004167)ⁿ - 1] / 0.004167

2,000 = ( 1 + 0.004167)ⁿ - 1] / 0.004167

8.334 =  (1 + 0.004167)ⁿ - 1

9.334 = ( 1.004167)ⁿ

We take the natural log on both sides

LN 9.334 = n Ln 1.004167

2.233664 = n 0.004167

536.0365 = n

these are 536.0365 months

44.67 years

Keys to use in a financial calculator:

2nd I/Y 12, PMT = -500, I/Y = 5, FV = 1,000,000, CPT N

You will get N as 537.178

Dividing this value by 12 will give you 44.67 years