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In the last module, we saw interest-only mortgage back securities and

Â principal-only mortgage back securities. And we saw how we can construct these

Â from an underlying pool of mortgages or from an underlying pass through.

Â In this short module, we're just going to discuss the risks of interest-only and

Â principal-only securities. We will see how they have very different

Â exposures to prepayments, for example. So if you recall, this is our diagram for

Â an example of how a principal-only and interest-only mortgage backed security

Â might be rated. We start off with a pool of 10,000

Â individual mortgages. Each of those mortgages has been issued

Â by some broker, some bank to a home owner.

Â these mortgages are then pooled together to form the collateral for the

Â principal-only, interest-only mortgage backed security.

Â so out of this pool of loans, we can create a principal-only tranche if you

Â like and an interest-only tranche if you like.

Â And in the last module, we saw how you could actually create these, these

Â securities. I just want to spend a little bit of time

Â talking about the risk of principal-only and interest-only mortgage backed

Â securities. I'm just going to measure risk here by

Â the concept of duration. Duration is a simple concept that's used

Â throughout the fixed income markets for describing risk.

Â If you haven't seen it before, that's fine.

Â We're just going to use the definition of it here.

Â This definition states that the duration of a cash flow is a weighted average of

Â the times at which each component of the cash flow is received.

Â And this is a standard measure of the risk of a cash flow.

Â And it should be clear by the way that the principal stream has a longer

Â duration than the interest stream. And I mentioned this in the last module,

Â but just to remind ourselves. The interest payment Ik, if you recall,

Â is equal to c times Mk minus 1. So this is interest from a level payment

Â mortgage that is payed at time k. Then the principle that is payed at time

Â k is equal to b minus c times m k minus 1.

Â So clearly, in the earlier parts of the mortgage, Mk minus 1 is going to be

Â larger. And so the interest payments will be

Â larger. In the later parts of the mortgage, Mk

Â minus 1. The outstanding principal will be

Â smaller. And so, in that case, the interest

Â payments will be smaller and the principal payments will be larger.

Â It should therefore be clear that the principal stream has a longer duration

Â than the interest stream. And this be, this is because the larger

Â principal payments take place near the end of the mortgage.

Â Whereas with the interest stream, the larger interest payments take place at

Â the beginning of the mortgage. I mentioned that the duration of the cash

Â flow is the weighted average of the times.

Â Well, what are these weights? While these weights are given to us down

Â here. So for example, let [UNKNOWN] the

Â duration of the principle stream, then it is give by this quantity here.

Â So I can write this as being equal to the sum from k equals 1 to n of Wk times k.

Â Where Wk is equal to 1 over 12 times V0. And if you recall, V0 is the value of the

Â principal stream of times 0. So Wk equals 1 over 12 times v0 times pk

Â divided by 1 plus r to the power of k. We divide by 12 just to convert duration

Â into annual units, rather than monthly units, because these ks are expressed

Â monthly. So, for example, if k equals 6, that

Â refers to month 6. So these are the weights.

Â So notice that what we're doing is. We're saying that the duration is the

Â weighted hours of the times at which each of the components in the cash flow is

Â received. The particular cash flow at time k has

Â value equal to this piece here that I'm circling.

Â So, the weight Wk is proportional to the value of the cash flow received that

Â time. And in fact, it's equal to the value of

Â that cash flow divided by 1 over 12 times v 0.

Â Recall then, that v0 is equal to the sum nk equals 1 of the Pk's divided by 1 plus

Â r to the power of k. So, this is the value today.

Â The present value of the principal payment stream.

Â So, in fact, these weights, they actually sum to 1.

Â If I ignore the factor of 1 over 12 here, which is just converting the months into,

Â into years. So this is the duration of the principal

Â stream. It tells us how long on average we have

Â to wait until we receive the cash flows from that stream.

Â The idea is that the longer the duration, the more risky the cash flow is because

Â there's more uncertainty over interest rates and, and the timing of those cash

Â flows. So generally, a longer duration is viewed

Â as being equivalent to a riskier stream of cash flows.

Â Similarly, we can also compute the duration DI of the interest-only stream

Â as follows. DI equals 1 over 12W0 times the summation

Â from k equals 1 to n of k times Ik divided by 1 plus r to the power of k.

Â Now, we can actually think of this as being a weighted average of the times at

Â which the payments occur in the interest-only stream.

Â So we can write this as being a summation from k equals 1 to n of little wk times

Â k, where wk is equal to 1 over 12 times w0, Ik divided by 1 plus R to the power

Â of K. So I can think of the Di as being a

Â weighted average of the times at which the payments in the interest only stream

Â occur. This extra factor, 1 over 12 here, this

Â just used to convert the duration into units of years rather than months.

Â So I have this expression here. I also know what Ik is equal to.

Â It's equal to B minus pk. So I can substitute B minus Bk in for Ik

Â and get the second equation. And then, break it up to get this

Â expression here. We saw what Dp was on the previous slide.

Â And if I want to, I could actually substitute in for B as well.

Â I know the value for B. I could substitute in here as well and

Â simplify the expression down further. And so, I can get the duration of the

Â interest-only mortgage backed security as well.

Â In practice, of course, prepayments do occur to this point we have assumed they

Â do not occur, but this is not realistic. Pass throughs do experience prepayments

Â and the principle-only and interest-only cash flows must reflect these prepayments

Â correctly. This is actually straightforward to do,

Â although, I would mention that one should always be aware of the legal

Â documentation in these securities [COUGH] but this is straightforward.

Â The interest payment in period k is simply as before, c times Mk minus 1,

Â where Mk minus 1 is the mortgage balance at the end of period k minus 1.

Â In practice, of course, prepayments do occur.

Â To this point, we've assumed they do not occur, but this is not realistic.

Â Passthroughs do experience prepayments, and the principal-only and interest-only

Â cash flows must reflect these prepayments correctly, but this is straightforward.

Â The interest payment in period k is simply, as before, c times Mk minus 1 or

Â Mk minus 1 is the mortgage balance at the end of period k minus 1.

Â Mk, the mortgage balance at the end of period k must now be calculated

Â iteratively on a path by path basis. So Mk is equal to Mk minus 1 minus the

Â scheduled principle payment at time k minus any prepayments that take place at

Â time k. So because the prepayments that take

Â place at time k are random, it therefore means that Mk will also be rounded and

Â that's what I mean by on a path by path basis.

Â So the outstanding principle at time k will depend on the how the uncertainty in

Â the economy has resolved between time periods 0 and k.

Â Finally, the risk profiles of principal-only and interest-only

Â securities are very different from one another.

Â And this is one of the reasons I want to discuss principal-only and interest-only

Â mortgage box securities. Even though, they are both constructed

Â from the same underline pull of mortgages, they actually have very

Â different risk profiles indeed. And in fact, they can be very risky

Â securities. The principal-only investor would clearly

Â like prepayments to increase. Now, why is that the case?

Â Well, if you think about it, the principal-only investor is entitled to

Â receive the principle stream from the underlined mortgages.

Â That investor would prefer those principle payments to occur sooner rather

Â than later. This just reflects the fact that money

Â has a time value. And so, therefore, the principal-only

Â investor would prepayments to increase. On the other hand, the interest-only

Â investor wants prepayments to decrease. The interest-only investor earns only the

Â interest payments, so interest payments are a function of the outstanding

Â principle. The higher the outstanding principle at

Â any point in time, the higher the corresponding interest payment at that

Â point in time. So all other things being equal, the

Â interest-only investor would like prepayments to decrease.

Â And in fact, in an extreme case to see this, imagine that the entire principal

Â pool repays immediately. So imagine an interest-only investor who

Â owns the interest, who owns the interest-only stream, and suppose the

Â underlying mortgages are prepay immediately.

Â Well then, the interest-only investor will get nothing, because all of the

Â principal will have been repaid, and so the outstanding interest on that

Â principal will be 0. There will be no principal remaining, so

Â no one interest will be paid, and so the interest-only investor will get nothing.

Â So in the extreme case, you could see how an interest-only investor would receive

Â nothing from this fixed income security. So I hope this makes clear to you that

Â the principle only and interest only securities have very different behaviors.

Â In fact, the interest only security is that rare fixed income security whose

Â price tends to follow the general level of interest rates.

Â When rates fall, the value of the interest-only security tends to decrease.

Â And when interest rates increase, the expected cash flow increases due to few

Â prepayments, but the discount factor decreases.

Â The net effect can be a rise or fall in the value of the interest-only security.

Â