The payment at the end of the fourth year has to be moved from year 4 to year 4.

So its future value is 1,000 (1 + 0,1), the whole raised to the power

of 0 which simply equals $1,000.

So the future value of the 4 year ordinary

annuity is 1,331 + 1,210 +

1,100 + 1,000, = $4,461.

To get a more general form let's denote

the future value of a ordinary annuity as FVA.

The periodic payment by PMT, and the interest rate as r.

Then we have FVA4 = PMT times (1 + r) raised to the power of 3

+ PMT times (1 + r) raised to the power of 2 + PMT times ( 1

+ r) raised to the the power of 1 + PMT times (1 + r) raised to the power of 0.

For a n-payment ordinary annuity we have FVA sub n =

PMT times (1 + r) raised to the power of n- 1 + PMT

times (1 + r) raised to the power n- 2 + so on.

+ PMT times (1+r) raised to the power of 0.

This can be simplified to FVA sub n = PMT times

((1+r) whole raised to the power of n- 1) / r.

What if you want to calculate the present value of an ordinary annuity.

We have already calculated the future value of an ordinary annuity.

Essentially we can merger a stream of cash flows into a single lump sum number.

We can now move the single lump sum from the future

back to the present by discarding it back by any others.

Denoting the present value today of an ordinary annuity as PVA sub 0.

We simply have PVA sub 0 = FVA sub

n / (1 + r) the whole raised to the power of n.

But we know that FVA sub n = PMT

times 1 + r to the power of n- 1 / r.

So, we now have PVA sub 0 = PMT / r times [1-1

/ (1+r) the whole raised to the power of n].

Let's look at an example of how you would use these formulas.

Say you have your heart set on buying this beautiful condo.

Its current price is $500,000 all of which you plan to borrow from the bank.

The bank will charge you an interest rate of 10% per year and the loan will be for

30 years.

Assuming that you make annual payments at the end of each year,

how much will you repay the bank each year?

To start of, I hope you recognize that this is an ordinary annuity

as the payments are made at the end of each year.

Since you borrow $500,000 today, PVA sub 0 = $500,000.

It is a 30-year loan with annual payments.

So, there will be 30 payments in this ordinary annuity, little, and

easy called 30.

r is given to be 10%.

We're interested in calculating the annual payment that is PMT.

We will use the formula PVA sub 0 = PMT / r times [1- 1

divided by (1 + r) raised to the power of n].

Plugging all known values, we have 500,000 = PMT / 0.10.