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In the last module, we discussed prepayment risk and the simplest kind of

Â mortgage-backed security that is a mortgage passthrough.

Â In this module, we're going to discuss two new types of mortgage-backed

Â securities. The first type is a principle only MBS

Â and the second type is an interest only MBS.

Â We're going to see how these securities can be constructed from an underlying

Â pool of mortgages. We will see that the principal only MBS

Â is constructed from the principal payments of the underlying pool.

Â And that the interest only MBS is constructed from the interest payments in

Â the underlying pool of mortgages. So in this module we're going to discuss

Â the construction of a principal only and interest only mortgage backed security.

Â We're going to construct these mortgage backed securities from an underlying pool

Â of mortgages. So in this example here we've got a total

Â of 10,000 mortgages. These mortgages correspond to 10,000

Â separate homeowners, each of whom has a mortgage on their house.

Â So these mortgages are going to be pooled together into a pool of mortgages.

Â And from these mortgages we're going to construct two separate mortgage backed

Â securities. One is called a principal only mortgage

Â backed security. And the other is called an interest only

Â mortgage backed security. So if you like these 10,000 mortages form

Â the collateral of the principle only, interest only mortgage back securities.

Â Again we're going to assume our deterministic world with no defaults, no

Â prepayments and so on. And that's fine because what we want to

Â do here is to just get an idea of how these securities work in practice and

Â what their risks might be. All right.

Â So we saw in an earlier module that we know m k minus 1.

Â The outstanding principle on a mortgage at time k minus 1.

Â So what we have on he, mind, and this slide is just a single mortgage, where

Â the outstanding principle at time k minus 1 is given to us by m k minus 1.

Â Well, the interest on this outstanding principle is going to be denoted by Ik

Â and that's equal to c times Mk minus one. If you recall c is the coupon rate of the

Â mortgage. This is the rate that the homeowner must

Â pay on their mortgage every month. So, it also means that we can interpret

Â the case payment, as paying Pk. And Pk equals b minus c times Mk minus 1.

Â So again what we have in mind here is a level payment mortgage, where a fixed

Â payment of b dollars is paid in every period for the entire duration of the

Â mortgage. So this b dollars, some of it goes

Â towards paying the interest in that period, and that's c times mk minus 1.

Â And the remainder, b minus the interest, ecmk minus 1, is paying the principle.

Â So if you recall our earlier expression from mk in an earlier module, we saw that

Â mk, the outstanding mortgage principle of time k Is given to us by this quantity.

Â So using this expression here in 7 we can actually calculate Pk.

Â Pk equals b minus c times this expression here evaluated at k minus 1.

Â Because we have k minus 1 here. So if I substitute for k minus 1 instead

Â of k here. I get everything down there.

Â So these two quantities are equal, except k has now gone to k mines 1.

Â So therefore Pk equals b minus c times this.

Â Now it's just straightforward algebra to tidy up this expression, and I get the

Â value Pk. So this is the amount of b, or if you

Â like, this is the number of dollars that goes towards paying the principal in time

Â period k. If I wanted to, I could compute the

Â present value of these principal payments.

Â Let's call this present value V0, so then V0 is equal to the sum from k equals 1 up

Â as far as n of Pk divided by 1 plus 4 to the power of k, for if you recall, r is

Â the risk free interest rate. Think of R, if you like, and this is just

Â a loose approximation, but you can think of R as being the borrowing rate for the

Â bank that wrote the mortgage in the first place, or that gave the mortgage to the

Â homeowner in the first place. So from the bank's perspective, you can

Â think of the fair value of the principal stream, the Pk's from k equals 1 to n, as

Â being equal to the sum of these pk's [UNKNOWN] discounted.

Â If you do that and if you, if you evaluate that expression.

Â Remember we know what the Pks are so we can actually write this as being B minus

Â c times m zero times the sumation n k equals 1 of 1 plus of c to the k minus 1

Â divided by one plus r to the k. Now I can easily take out a fact of one

Â plus r over outside here, and make this a simple geometric sum which is easy to

Â calculate. If I do that I'm going to get this

Â expression up here. So this therefore is the present value,

Â V0, of the principle payment stream. And is given to us in terms of B, which

Â is the monthly payment on the mortgage. C, the coupon rate on the mortgage, M0

Â the initial mortgage principle, the fixed inch-, risk interest rate.

Â Or, and n the number of time periods in the mortgage.

Â Now, if we want to, we can actually compute the limit of V0 as c goes to r.

Â you see that both the numerator and the dneominator in this case go to zero so we

Â need to use the so called[UNKNOWN] to complete this limit while its

Â straightforward to do so. And we find that the limit of V0 as c

Â goes to r equals n times B minus rM0 divided by 1 plus r.

Â 6:03

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

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

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

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

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

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

Â here. Now it is clear that earlier mortgage

Â payments comprise of interest payments rather than principle payments.

Â Only later in the mortgage is this relationship reversed.

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

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

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

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

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

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

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

Â because it's c times the initial mortgage amount.

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

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

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

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

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

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

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

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

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

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

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

Â dollars. Well therefore, what we are seeing here,

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

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

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

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

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

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

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

Â infinity. You could actually check that the limit

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

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

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

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

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

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

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

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

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

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

Â How about an interest only mortgage backed security?

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

Â Again assuming there are no mortgage prepayments.

Â To do this we could compute the following sum.

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

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

Â k. Now we could calculate this, but in fact

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

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

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

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

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

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

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

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

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

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

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

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

Â algebra and get the expression like this for W0.

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

Â here. And this is as expected from 11 because

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

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

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

Â flows worksheet. This is still the single underlying

Â mortgage. You can actually see how we compute the

Â monthly interest payments and the principle repayments in each period.

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

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

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

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

Â is the period in which the payment takes place.

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

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

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

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

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

Â about the structure of level payment mortgages.

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

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

Â components have very different risk profiles.

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

Â