Future value in 30 years

Question

Anna has just become eligible to participate in her company’s retirement plan; she is excited as her company matches her contributions dollar for dollar in this plan. The plan averages an annual return of 7% interest compounded monthly. Josie is 35 years old and plans to retire at age 65. She receives her pay at the beginning of each month and contributes 10% of her gross monthly salary of $2,500 into her retirement plan. What is the total amount that she will have when she retires?  

Answer 1

This is a variant of a simple Annuity question. We want the find the present value of the annuity and turn it into a Future value.

Anna is 35 will retire at 65 she has 30 years of Monthly payments = 360 payments. She will put in

10% of her Gross Monthly salary or $250. The company will match it so she will be having $500 deposited into the account every month.

So what is the PV of a stream of Payments Of $500 for 360 months. For An Annuity,

PV=Payment*ß(i,N) Where ß is the Multiplier. It is an Increasing function of N and Decreasing funing of i and is given as ß=(1/i)(1-(1/(1+i))^N) her i is &% compounded Monthly or 7%/12. =7/1200=.05833333

So ß= (1200/7)(1- (1/(1+7/1200))³⁶⁰) =(1200/7)(1-.123205) =(.876794)(1200/7) = 150.30756

So PV = 500*150.30756 = 75,153.78.

That may not seem like alot but &5,,153 accruing at 7% for 30 years should be about 8 times that (from the rule of 72's) We can use the monthly rate of 7/1200 and compound this for 360 periods and get

FV=PV*(1+7/1200)³⁶⁰ =609,985.41. Not too shabby. As J.P. Morgan once said, Compound interest is the 8th wonder of the world