Suppose now that we wish to compute the present value of the mortgage assuming a deterministic world. By the way what do I mean by deterministic world? Well, what I mean is that there is no uncertainty in the world. We're using a very stylized situation here. There's no randomness. There is no possibility of default. Prepayments though of course these are very important features in practice. Well, in that situation we can actually compute the present value of a level payment mortgage. If we assume a risk-free interest rate of r per period. We obtain that the fair mortgage value is and we're going to use F zero to denote the fair mortgage value. Then the fair mortgage value is equal to the sum of the Bs divided by one plus r to the power of k. From k equals one to the power of n. Now, just to keep in mind r you can think of r as being the borrowing rate for the banks. The banks that write the mortgages or that lend the money out to the homeowners. Presumably they can borrow at r, and for all intents and purposes here we can imagine r to be risk-free interest rate. R in general would certainly not be equal to c. C is the coupon rate or if you like the interest rate that the homeowner must pay on their mortgage. R is the interest rate that the banks use to discount their payments. In this case we're going to get F zero equals to the sum of the Bs over one plus r to the power of k. Again, this is a geometric series. We can easily calculate this term. Then we can substitute in for B, using our expression on the previous slide. If we do that, we'll get this expression here. Expression number five. Note that if r equals c, then actually F0 equals M0. Because if r equals c this term will cancel out with this term. This term will cancel out with this term. So not surprisingly, we get F0 equals M0. That's exactly as we would expect. In general however, r is less than c and that is because the banks who write the mortgages or lend the money out to the homeowners must charge a larger rate of interest c to account for the possibility of default, prepayment, servicing fees on the mortgages, they must make some profit, they must also account for payment uncertainty and so on. In general r will be less than c. The difference here, between r and c accounts for the difference between F0, the fair value of the mortgage from the bank's perspective and M0, the amount of money that the homeowner is being lent in the first place. So in some sense you can think of F0 minus M0 as being the amount of money that the bank is earning from the mortgage, but that money must be used to handle these effects here. The possibility of default, prepayment, servicing fees, and so on as I mentioned already. By the way, we're only working here with level payment mortgages but you could do similar kinds of analysis with other types of mortgages including adjustable rate mortgages or arms and so on. I also want to mention the fact that we can actually decompose the payment B that is paid in every period into an interest component and the principal component. This is very easy to do. Since we know Mk minus one. We can compute the interest. Let's call that Ik. In every period. So Ik is equal to c times Mk minus one. After all Mk minus one is the outstanding principle at the end of period k minus one. C times Mk minus one, is the interest that is due on that principle in time period k. This is the interest that is paid in time period k. Therefore, that means we can interpret the kth payment which is B, as paying Pk equals B minus cMk minus one of the remaining principle. Ik is the interest that is paid at period k, and Pk is the principal that is paid at period k. Of course note that Ik plus Pk is equal to B. Which must be the case, because we paid B dollars in every period. That B dollars must be split up between principal and interest and the way to split up is given to us up here and here for Pk. At any time period k as I said we can easily break down the payment B into scheduled principal payment and a scheduled interest payment Ik. We're actually going to use this observation later to create principal-only and interest-only mortgage backed securities. These are an interesting class of mortgage backed securities. We will see that when we actually use a pool of mortgages to create these new securities. We're actually creating new securities that have very different risk profiles. But we will return to that in a later module. In the spreadsheet that goes with these modules and mortgage backed securities, there are three worksheets. The first worksheet is called single mortgage cash flows. It simply shows you how a single mortgage, a single level payment mortgage works. So that we can see everything here on the same, on the one sheet, on the one screen. I've assumed that there's just 18 periods in the mortgage. In reality there might be 240 or 360. If I change it to 360 for example, we will see that the actual spreadsheet adjusts appropriately. We see that we get 360 months appearing. But just so that we can see anything. See everything in the same screenshot. I'm going to assume that there's just 18 periods. We start off with a mortgage loan of $20000. The mortgage rate is five percent. Now this is an annual rate. This is not c. This is an annual rate. But we can easily convert it into a monthly rate which is what we do here, when we calculate the monthly payment. This is B. This monthly payment in cell C3 is our B from the slides. We see how to calculate B using C2, which is the monthly rate. As well of course C1 which is M0. The initial mortgage loan. What we have here, is we have the 18 months. We see the beginning monthly balance. This is the outstanding principle on the mortgage at the beginning of each month. We see the monthly payment is $1155.61. It's the same monthly payment in every period. As we saw on the final slide, we can break this payment down into a monthly interest payment, and a scheduled principal payment. That's what these quantities are here. Note that the monthly interest and the scheduled principle, always sum up to the monthly payment. In any cell here, you'll see that these numbers always sum up to the corresponding cell in column E. Monthly interest plus scheduled principal repayment is equal to the total monthly payment. Then of course, in order to get the ending mortgage balance. We simply subtract the scheduled principal repayment from the monthly payment. Of course after 18 periods, the outstanding mortgage balance is zero at the end. You can play with the spreadsheet if you like. As I said in practice, you typically have a much longer term of the loan. Instead of 18 periods you might have 240 for a 20 year mortgage, or a 180 for a 15 year mortgage, or 360 for a 30 year mortgage. Another interesting observation to make, is that in the earlier part of the mortgage. The monthly interest payments are larger than they are in the latter part of the mortgage. Up here, you see in the earlier months of the mortgage the interest payments around $80, $75, and so on. But they're much smaller later in the mortgage. On the other hand, the principal repayments in the earlier part of the mortgage are smaller. In this case around the $1075, $1080 versus later in the mortgage when they're $1141, $1146, $1150. The impact isn't so obvious here. But if I switch to say a 360 period mortgage. You'll notice this effect. You'll see that this effect is much greater. Here and let's make it say a $200000 mortgage. Here, what you'll notice is the monthly payment is $1073. But most of that payment is going to pay interests. Look at that. Of that $1073 in the earlier part of the mortgage life, most of that is going to pay interest and only a smaller fraction of it maybe on the odd of 25 percent is going to pay principle. However, if I scroll down towards the end of the mortgage which is a long way down because it's 360 months. Then you will see that now only a small part of the monthly payment is going to pay monthly interest, and that's because the outstanding principle is much smaller in these time periods. A much smaller interest amount is due. A much larger fraction of the monthly payment is going to pay the principal. This observation is important. It's worthwhile knowing for anybody who's thinking about taking out a level payment mortgage, that in the earlier part of the mortgage life, most of their monthly payments will actually be going towards paying interest, and only a small fraction would be going towards paying principle. That would be reversed towards the end of the life of the mortgage. This as I mentioned earlier will also have a significant impact when we create principle only, and interest only, mortgage backed securities and see how they are constructed. This differing behavior of the monthly interest payments versus the scheduled principal payments will have an impact on the characteristic of interest only, and principal-only, mortgage backed securities
