In the last module, we saw interest-only mortgage back securities and principal-only mortgage back securities. And we saw how we can construct these from an underlying pool of mortgages or from an underlying pass through. In this short module, we're just going to discuss the risks of interest-only and principal-only securities. We will see how they have very different exposures to prepayments, for example. So if you recall, this is our diagram for an example of how a principal-only and interest-only mortgage backed security might be rated. We start off with a pool of 10,000 individual mortgages. Each of those mortgages has been issued by some broker, some bank to a home owner. these mortgages are then pooled together to form the collateral for the principal-only, interest-only mortgage backed security. so out of this pool of loans, we can create a principal-only tranche if you like and an interest-only tranche if you like. And in the last module, we saw how you could actually create these, these securities. I just want to spend a little bit of time talking about the risk of principal-only and interest-only mortgage backed securities. I'm just going to measure risk here by the concept of duration. Duration is a simple concept that's used throughout the fixed income markets for describing risk. If you haven't seen it before, that's fine. We're just going to use the definition of it here. This definition states that the duration of a cash flow is a weighted average of the times at which each component of the cash flow is received. And this is a standard measure of the risk of a cash flow. And it should be clear by the way that the principal stream has a longer duration than the interest stream. And I mentioned this in the last module, but just to remind ourselves. The interest payment Ik, if you recall, is equal to c times Mk minus 1. So this is interest from a level payment mortgage that is payed at time k. Then the principle that is payed at time k is equal to b minus c times m k minus 1. So clearly, in the earlier parts of the mortgage, Mk minus 1 is going to be larger. And so the interest payments will be larger. In the later parts of the mortgage, Mk minus 1. The outstanding principal will be smaller. And so, in that case, the interest payments will be smaller and the principal payments will be larger. It should therefore be clear that the principal stream has a longer duration than the interest stream. And this be, this is because the larger principal payments take place near the end of the mortgage. Whereas with the interest stream, the larger interest payments take place at the beginning of the mortgage. I mentioned that the duration of the cash flow is the weighted average of the times. Well, what are these weights? While these weights are given to us down here. So for example, let [UNKNOWN] the duration of the principle stream, then it is give by this quantity here. So I can write this as being equal to the sum from k equals 1 to n of Wk times k. Where Wk is equal to 1 over 12 times V0. And if you recall, V0 is the value of the principal stream of times 0. So Wk equals 1 over 12 times v0 times pk divided by 1 plus r to the power of k. We divide by 12 just to convert duration into annual units, rather than monthly units, because these ks are expressed monthly. So, for example, if k equals 6, that refers to month 6. So these are the weights. So notice that what we're doing is. We're saying that the duration is the weighted hours of the times at which each of the components in the cash flow is received. The particular cash flow at time k has value equal to this piece here that I'm circling. So, the weight Wk is proportional to the value of the cash flow received that time. And in fact, it's equal to the value of that cash flow divided by 1 over 12 times v 0. Recall then, that v0 is equal to the sum nk equals 1 of the Pk's divided by 1 plus r to the power of k. So, this is the value today. The present value of the principal payment stream. So, in fact, these weights, they actually sum to 1. If I ignore the factor of 1 over 12 here, which is just converting the months into, into years. So this is the duration of the principal stream. It tells us how long on average we have to wait until we receive the cash flows from that stream. The idea is that the longer the duration, the more risky the cash flow is because there's more uncertainty over interest rates and, and the timing of those cash flows. So generally, a longer duration is viewed as being equivalent to a riskier stream of cash flows. Similarly, we can also compute the duration DI of the interest-only stream as follows. DI equals 1 over 12W0 times the summation from k equals 1 to n of k times Ik divided by 1 plus r to the power of k. Now, we can actually think of this as being a weighted average of the times at which the payments occur in the interest-only stream. So we can write this as being a summation from k equals 1 to n of little wk times k, where wk is equal to 1 over 12 times w0, Ik divided by 1 plus R to the power of K. So I can think of the Di as being a weighted average of the times at which the payments in the interest only stream occur. This extra factor, 1 over 12 here, this just used to convert the duration into units of years rather than months. So I have this expression here. I also know what Ik is equal to. It's equal to B minus pk. So I can substitute B minus Bk in for Ik and get the second equation. And then, break it up to get this expression here. We saw what Dp was on the previous slide. And if I want to, I could actually substitute in for B as well. I know the value for B. I could substitute in here as well and simplify the expression down further. And so, I can get the duration of the interest-only mortgage backed security as well. In practice, of course, prepayments do occur to this point we have assumed they do not occur, but this is not realistic. Pass throughs do experience prepayments and the principle-only and interest-only cash flows must reflect these prepayments correctly. This is actually straightforward to do, although, I would mention that one should always be aware of the legal documentation in these securities [COUGH] but this is straightforward. The interest payment in period k is simply as before, c times Mk minus 1, where Mk minus 1 is the mortgage balance at the end of period k minus 1. In practice, of course, prepayments do occur. To this point, we've assumed they do not occur, but this is not realistic. Passthroughs do experience prepayments, and the principal-only and interest-only cash flows must reflect these prepayments correctly, but this is straightforward. The interest payment in period k is simply, as before, c times Mk minus 1 or Mk minus 1 is the mortgage balance at the end of period k minus 1. Mk, the mortgage balance at the end of period k must now be calculated iteratively on a path by path basis. So Mk is equal to Mk minus 1 minus the scheduled principle payment at time k minus any prepayments that take place at time k. So because the prepayments that take place at time k are random, it therefore means that Mk will also be rounded and that's what I mean by on a path by path basis. So the outstanding principle at time k will depend on the how the uncertainty in the economy has resolved between time periods 0 and k. Finally, the risk profiles of principal-only and interest-only securities are very different from one another. And this is one of the reasons I want to discuss principal-only and interest-only mortgage box securities. Even though, they are both constructed from the same underline pull of mortgages, they actually have very different risk profiles indeed. And in fact, they can be very risky securities. The principal-only investor would clearly like prepayments to increase. Now, why is that the case? Well, if you think about it, the principal-only investor is entitled to receive the principle stream from the underlined mortgages. That investor would prefer those principle payments to occur sooner rather than later. This just reflects the fact that money has a time value. And so, therefore, the principal-only investor would prepayments to increase. On the other hand, the interest-only investor wants prepayments to decrease. The interest-only investor earns only the interest payments, so interest payments are a function of the outstanding principle. The higher the outstanding principle at any point in time, the higher the corresponding interest payment at that point in time. So all other things being equal, the interest-only investor would like prepayments to decrease. And in fact, in an extreme case to see this, imagine that the entire principal pool repays immediately. So imagine an interest-only investor who owns the interest, who owns the interest-only stream, and suppose the underlying mortgages are prepay immediately. Well then, the interest-only investor will get nothing, because all of the principal will have been repaid, and so the outstanding interest on that principal will be 0. There will be no principal remaining, so no one interest will be paid, and so the interest-only investor will get nothing. So in the extreme case, you could see how an interest-only investor would receive nothing from this fixed income security. So I hope this makes clear to you that the principle only and interest only securities have very different behaviors. In fact, the interest only security is that rare fixed income security whose price tends to follow the general level of interest rates. When rates fall, the value of the interest-only security tends to decrease. And when interest rates increase, the expected cash flow increases due to few prepayments, but the discount factor decreases. The net effect can be a rise or fall in the value of the interest-only security.