Annuity related question

Question

Use the future value formula to find the indicated value.

FV=11,000​; i=0.02​; PMT=$500​; n ?

Answer 1

We Know Present Value = Payment * (1/r)(1- (1/(1+r))^N Equation 1

We Also Know Future Value = Present Value *(1+r)^N Equation 2

Equation 1 Can be rewritten as

PV = PMT(1/r)((1+r)^N-1)/(1+r)^N Call this Equation 1a

Now divide both sides of equation 2 by (1+r)^N and Get

FV*(1/(1+r))^N= PV = (1/(1+r))^N((1+r)^N-1)*(1/r)+PMT

There is a (1/(1+r))^N on both sides of the equation so we can divide them out

and get

FV =(PMT/r)*((1+r)^N-1)

Now divide both sides by (PMT/r) and get

rFV/PMT = (1+r)^N-1

add 1 to both sides and get

rFV/PMT + 1= (1+r)^N

With r = .02, PMT = 500 and FV = 11,000

we get .02*11,000/500 = (1.02)^N

or 1.44 = (1.02)^N

Take natural Logarithms of both sides and get

ln(1.44) = Nln(1.02) or N = ln(1.44)/ln(1.02) = 18.414 years.

As A Check, The PV of 11,000 discounted for 18.414 years at 2% is $7,638.84.

So this is the amount you could borrow if you wanted to make 500 dollar payments and would make those payments for 18.4 periods.

If you have any questions on discounting cash flows oe anything invoilovin finance or economics please reach out

Best,

Lenny