Question

At the age of 30, Alfred began investing $350.00 monthly into an investment account at an interest rate of 7.00%, compounded monthly. When he turned 45, the interest rate on this investment decreased to 5.00%, compounded monthly. Alfred plans to continue making monthly investments until he retires at 58.
a) How much money will Alfred have in his account when he retires? Show your work.
b) Alfred’s sister, Marianne, retires at the age of 60. Her portfolio is valued at $200 000.00, earning 5.00%, compounded monthly. If Marianne wants the money to last until she is 85 years old, what is the maximum she can withdraw each month? Show your work.

Answer 1

a)

Alfred began investing $350 every month at the age of 30 at a compounding interest rate of 7%

Interest rate is decreased to 5% at age of 45.

Therefore we need to calculate amount till age of 45 first

Compounding interest formula with regular contributions is

A=P(1+R/N)^NT+ {PMT(1+R/N)^NT -1}/R/N

where A = Future Value of investment
P = Principle amound invested (the original contribution)
PMT = Regular contributions (additional money added to investment)
r = Interest rate investment is earning
n = Number of times interest compounds** i.e. 12 = monthly, 4 = quarterly, 2 = semi-annually, 1 = annually
t = Number of years investment will be active

By substituting the values in the above formula we get

A=350(1+7%/12)^12*15+{350(1+7%/12)^12*15-1}/7/12

A=350(1.005833)^180+{350(1.005833)^180-1}/0.5833

A=350(2.85)+{350(2.85)-1}/0.5833

A=997.13+{997.13-1}/0.5833

A=997.13+170765

A=171762 i.e., at the age of 45 this amount will be with alfred.

This amount is re-invested with a interest rate of 5% Similar calculation is to be made with above formula taking P=171762 and PMT 350 and r=5% t=13

By solving we get A=$405261