Question

Christopher invested $11,000 into a fund earning 5.50% compounded monthly. She plans to withdraw $900 from the fund at the end of every quarter. If the first annuity withdrawal is to be made 2 years from now, how long will it take for the fund to be depleted? how many years and months?

Answer 1

1) Total fund will be available after 2 years Just before first annuity withdrawl :

Amonut = P * (1+r)ⁿ

Where,

P = $ 11,000

r = 5.5/12   = 0.45833%

n = 12*2 = 24 months

So, Amount = 11000 * (1+0.0045833)²⁴

Amount = 11000 * 1.1159966

Amount = $ 12,275.96

This amount of $ 12,275.96 we be available just before his first withdrwal of $ 900.

2) After 2 year, christopher will withdrawl $ 900 every quarter.

Since he will withdraws first amount of $ 900 from the starting of period that means our remaining balance after withdrawl will be 12,275.96 - $900 = $ 11,375.96

Now, as we can see that christopher will withdraw quarterly payment, whereas the interest is compounding monthly, so we will convert our monthly interest rate of 0.45833% into effective quarterly rate.

So, Effective quarterly rate = (1+r)ⁿ - 1     where n = no. of months in one quarter, r = monthly interest rate

Effective quarterly rate = (1+0.0045833)³ - 1

Effective quarterly rate = 1.013813 - 1

Effective quarterly rate = = 0.013813 or 1.3813%

Now,

Present value of annuity = P * [1-(1+r)^-n]/r

Where,  

P = 900

r = 0.013813

n = ??

Present value of annuity = 11375.96

(1+r)^-n = 1/(1+r)ⁿ

So,

11375.96 = 900 * [1-(1+0.013813)^-n]/0.013813

11375.96/900 = [1-(1.013813)^-n]/0.013813

12.63996*0.013813 = [1-(1.013813)^-n]

0.174596 = 1-(1.013813)^-n

(1.013813)^-n = 1 - 0.174596

1 / (1.013813)ⁿ = 0.825404

1/0.825401 = (1.013813)ⁿ

1.2115279 = (1.013813)ⁿ

So, n = 13.99

So, n = 14.00 (rounded off)

It means, it will take 14 quaters for the fund to be depleted.

Christopher will withdrawl 15 times, the amount of $ 900. and it will take 14 quarters or 42 months to deplete the fund after the first withdrawl.