How do you solve this annuities problem?

Question

Question: Kenny wants to invest $250 every 3 months at 5.2%/a compounded quarterly. He would like to have at last $6500 at the end of his investment. How long will he need to make regular payments?

Answer 1

This is a Fun One.

Let’s start with some definitions

The Present value PV, of a stream of payments, A for T periods discounted at some rate,

I, is given as PV=A*K(I,T) where K is a multiplier which increases as the number of periods increases and decreases as the interest rate increases.

Formally,

K=(1/i)(1-(1/(1+i)^T) = (1/i)(1/(1+i)^T)((1+i)^T-1) where I is the PERIODIC INTEREST RATE and T is the NUMBER of PERIODS

So for a thirty year mortgage with a 12% rate we would have 360 monthly payments with a rate of 6%/15 or 0.5% monthly rate.

The Future Value of a current cash balance is defined as FV= PV((1+i)^T) Similarly we see the Pesent value of any cash amount in the Future can be brought back to present value as

PV(FV,I,T) = FV/((1+i)^T).

In Your Case we want to find some Future value, FV= 6500. We just don't know when. Whenever it is, it will be worth (6500/((1+i)^T) today

sooooo,

(6500)/((1+i))^T= AK If We Multiply both sides by (1+i)^T and divide both sides by A we get

FV/A= 6500/250 = (1+i)^T*K =(1/i)((1+i)^T/(1+i)^T((1+i)^T-1) = (1/i)((1+i)^T-1)

Now multiply both sides by i and get iFV/A =(1+i)^T -1 add 1 to both sides and get

iFV/A +1 = (1+i)^T

In your case, FV=6500, A=250 i= 5.2%/4 =1.3% so.

(.013)(6500/250) +1 = (1.013)^T =1.338 if we take natural logs of both sides we get

ln(1.338) = Tln(1.103) so T = ln(1.338)/ln(1.013) = 22.54 quarters or about 5 and a half years. If the interest rate were 0 it would have taken you 6.5 years to have 6500 dollars.

I hope this helps.

If you need help with anything please feel free to reach out.

Lenny D.