Home mortgages are an example of an annuity stream.

Auto leases, certain bond payments, and

amortizing loans are annuities, so it's actually fairly common in practice.

Now, if we wanted to find the present value of these cash flows,

we know how to do that, right?

We can brute force it.

We can take each cash flow and discount it back to today.

All right, so imagine I had a second cash flow.

The second cash flow here I could take those CF divided by 1 plus R squared,

that would bring it back to today.

I could take this cash flow CF over 1 plus R to the T minus 1.

That would bring it, and I do that for all the cash flows, and

then I could add them up here.

And that would give me the present value.

But that's a bit burdensome, especially when T is big.

So, what I'd like to show you is a shortcut or a simple formula

to compute the present value of this cash flow stream, and here it is.

We take the cash flow, CF.

We divide it by the discount rate and

multiply it by this term in parentheses here.

Now if I move the R over here I can re-express the present

value of the annuity formula as just the annuity cash flow

times this term here which is called an annuity factor.

Okay, that will give me the present value of this cash flow stream.