In the last module, we discussed prepayment risk and the simplest kind of mortgage-backed security that is a mortgage passthrough. In this module, we're going to discuss two new types of mortgage-backed securities. The first type is a principle only MBS and the second type is an interest only MBS. We're going to see how these securities can be constructed from an underlying pool of mortgages. We will see that the principal only MBS is constructed from the principal payments of the underlying pool. And that the interest only MBS is constructed from the interest payments in the underlying pool of mortgages. So in this module we're going to discuss the construction of a principal only and interest only mortgage backed security. We're going to construct these mortgage backed securities from an underlying pool of mortgages. So in this example here we've got a total of 10,000 mortgages. These mortgages correspond to 10,000 separate homeowners, each of whom has a mortgage on their house. So these mortgages are going to be pooled together into a pool of mortgages. And from these mortgages we're going to construct two separate mortgage backed securities. One is called a principal only mortgage backed security. And the other is called an interest only mortgage backed security. So if you like these 10,000 mortages form the collateral of the principle only, interest only mortgage back securities. Again we're going to assume our deterministic world with no defaults, no prepayments and so on. And that's fine because what we want to do here is to just get an idea of how these securities work in practice and what their risks might be. All right. So we saw in an earlier module that we know m k minus 1. The outstanding principle on a mortgage at time k minus 1. So what we have on he, mind, and this slide is just a single mortgage, where the outstanding principle at time k minus 1 is given to us by m k minus 1. Well, the interest on this outstanding principle is going to be denoted by Ik and that's equal to c times Mk minus one. If you recall c is the coupon rate of the mortgage. This is the rate that the homeowner must pay on their mortgage every month. So, it also means that we can interpret the case payment, as paying Pk. And Pk equals b minus c times Mk minus 1. So again what we have in mind here is a level payment mortgage, where a fixed payment of b dollars is paid in every period for the entire duration of the mortgage. So this b dollars, some of it goes towards paying the interest in that period, and that's c times mk minus 1. And the remainder, b minus the interest, ecmk minus 1, is paying the principle. So if you recall our earlier expression from mk in an earlier module, we saw that mk, the outstanding mortgage principle of time k Is given to us by this quantity. So using this expression here in 7 we can actually calculate Pk. Pk equals b minus c times this expression here evaluated at k minus 1. Because we have k minus 1 here. So if I substitute for k minus 1 instead of k here. I get everything down there. So these two quantities are equal, except k has now gone to k mines 1. So therefore Pk equals b minus c times this. Now it's just straightforward algebra to tidy up this expression, and I get the value Pk. So this is the amount of b, or if you like, this is the number of dollars that goes towards paying the principal in time period k. If I wanted to, I could compute the present value of these principal payments. Let's call this present value V0, so then V0 is equal to the sum from k equals 1 up as far as n of Pk divided by 1 plus 4 to the power of k, for if you recall, r is the risk free interest rate. Think of R, if you like, and this is just a loose approximation, but you can think of R as being the borrowing rate for the bank that wrote the mortgage in the first place, or that gave the mortgage to the homeowner in the first place. So from the bank's perspective, you can think of the fair value of the principal stream, the Pk's from k equals 1 to n, as being equal to the sum of these pk's [UNKNOWN] discounted. If you do that and if you, if you evaluate that expression. Remember we know what the Pks are so we can actually write this as being B minus c times m zero times the sumation n k equals 1 of 1 plus of c to the k minus 1 divided by one plus r to the k. Now I can easily take out a fact of one plus r over outside here, and make this a simple geometric sum which is easy to calculate. If I do that I'm going to get this expression up here. So this therefore is the present value, V0, of the principle payment stream. And is given to us in terms of B, which is the monthly payment on the mortgage. C, the coupon rate on the mortgage, M0 the initial mortgage principle, the fixed inch-, risk interest rate. Or, and n the number of time periods in the mortgage. Now, if we want to, we can actually compute the limit of V0 as c goes to r. you see that both the numerator and the dneominator in this case go to zero so we need to use the so called[UNKNOWN] to complete this limit while its straightforward to do so. And we find that the limit of V0 as c goes to r equals n times B minus rM0 divided by 1 plus r. 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. So we'll see that in the next module.
