Now, if you recall our earlier expression for b, which is the fixed monthly period

on, monthly payment on the mortgage, then we can substitute for 9 inside up here in

8 to find V0. So here, this is the fair value, or the

present value today at time zero, of all the principle payments on the mortgage.

Assuming admittedly it's a deterministic world, no prepayments, no defaults.

In the case that r equals c, then we see that V0 collapses down to this expression

here. Now it is clear that earlier mortgage

payments comprise of interest payments rather than principle payments.

Only later in the mortgage is this relationship reversed.

Well, how do we know this? Well we saw earlier that Ik is equal to c

times Mk minus 1. So this is the interest payment that is

made in time period k, and that Pk, the principal payment that is made in time k

is equal to b minus the interest payment. So that's minus cMk minus 1.

Now if you think about it, early on in the mortgage, let's set k equal to 1.

Well then in that case Mk minus 1 is M0. It's the initial principal.

The initial mortgage amount, and so clearly we see that Ik is large then,

because it's c times the initial mortgage amount.

And Pk will be small because it's B minus c times the initial mortgage amount.

On the other hand, as time elapses and we move towards the end of the mortgage.

Well then M k minus 1 will be getting much smaller, and so c times Mk minus 1

will be much smaller. And therefore the interest Ik towards the

end of the mortgage will be quite small. On the other hand, remember we're paying

a fixed amount B in every period. And so the principal that is paid towards

the end of the mortgage would be B minus c times Mk minus 1, c times Mk minus 1

would be small, so the principle payment will be large.

So, therefore, what is happening, is that the fixed payment B, that is paid in

every period. So remember, we've got our periods, maybe

this is 360, maybe this is T equals 0. And in every payment we're paying B

dollars. Well therefore, what we are seeing here,

is that in the earlier time periods most of this B period, B is going to interest

and in later time periods most of the B is going towards principal.

So this is an important fact to keep in mind with mortgage payments.

Certainly level payment mortgages. Most of the payments are going to pay

interest early in the mortgage and most of the payments later in the mortgage are

going to pay down the principle. And in fact, it's quite interesting but

if you can imagine a mortgage where the number of time periods n goes to

infinity. You could actually check that the limit

of V0 in that case is equal to zero. And in fact, that's not hard to see, and

the reason it's not is because in the numerator, we've got r n times M0, so

certainly the numerator is going to infinity up here as n goes to infinity.

However in the denominator, we've a one plus r to the power of n.

This is also going to infinity, but because it's raised to the power of n,

the denominator is going to infinity much faster than the numerator and so V0

actually goes to 0. And so that's what happens.

We see that if we stretch out the, the, the duration of the mortgage, so if we

let n go to infinity, all of the payments, all of these fees goes towards

the interest and none of them goes towards paying the principal.

How about an interest only mortgage backed security?

Well we could also compute the present value W0 say of the interest stream.

Again assuming there are no mortgage prepayments.

To do this we could compute the following sum.

We could set W0 equal to the sum of the interest payments.

Sure to be discounted by the risk-free rate, which is one plus r to the power of

k. Now we could calculate this, but in fact

it's much easier to recognize that the sum of the principal-only payments and

the interest-only payments must equal the total value of the mortgage, F0.

Which we calculated in an earlier module. If you recall, F0 is the fair value of

all the payments in the mortgage. So F0, if you recall, was equal to the

sum of the B's, divided 1 plus r to the power of k.

And we actually calculated this expression earlier, and we found that F0

was equal to this. Well F0 must be equal to V0, the fair

value or present value of the principle stream, plus W0.

The present value today of the interest payment stream so F0 plus V0 equals W0.

We've calculated F0 before, we've calculated V0 in the previous slide, so

W0 is equal to F0 minus V0, and we find it's, using 12 and 10.

12 is here, 10 is here, we can actually use the two of these simplify, the

algebra and get the expression like this for W0.

It is also easy to check that when r goes to c, W0 is equal to this expression

here. And this is as expected from 11 because

we know that when r equals c, F0 equals M0.

And so we see that W0 equals F0 minus V0, this is V0 when r is equals c and that's

exactly what we've calculated up here. If we go back to our single mortgage cash

flows worksheet. This is still the single underlying

mortgage. You can actually see how we compute the

monthly interest payments and the principle repayments in each period.

We just used what we saw there. Ik is equal to c times Mk minus 1.

And Pk equals b minus c times Mk minus 1. Therefore, if we wanted to compute the

fair value of all of these payments. We could simply sum them all up, and

suitably discount each of these payments by one plus or to the power of k, where k

is the period in which the payment takes place.

Likewise, we could compute these, the fair value of all of these principle

payment by computing the sum of these weighted 1 over 1 plus r to the power of

k. And so, what we've done here, it's just a

simple single mortgage, but what, and it's a very, it's a deterministic world

with no prepayments and no defaults. But we've still learned an awful lot

about the structure of level payment mortgages.

And how we can split the monthly payment up into an interest component and a

principal component. We'll see in the next module how these

components have very different risk profiles.

They react very differently to changes in prepayments and so on.