In the next series of modules, we're going to discuss mortgage mathematics and mortgage backed securities. We're going to look at mortgage backed securities because they will give us an example of the process of securitization. That is the process by which new securities are created from pools of underlying loans or mortgages. So, we're going to begin, in this module, with basic mortgage mathematics and an introduction to the mortgage markets. We called it, according to SIFMA, the Securities Industry and Financial Markets Association, that in the third quarter of 2012, the total outstanding amount of US bonds was $35.3 trillion. Now, if you look at this, you can see that the mortgage market actually accounted for 23.3% of this total. So, the mortgage markets are therefore huge and they played a big role in the financial crisis of 2008 and 2009. And that's one of the reasons we're going to talk about mortgages and mortgage backed securities over the next few modules. And it's interesting to understand what they are and how they're constructed and some of the mechanics behind the basic or most standard types of mortgage backed securities. Mortgage backed securities are a particular class of what are called asset backed securities. These are assets backed by underlying pools of securities, such as mortgages, auto loans, credit card receivable,s student loans, and so on. The process by which ABS, or mortgage backed securities, are created is often called securitization. Here's a picture of how securitization works in the context of mortgages and mortgage backed securities. What we have here are 10,000 different mortgages, each of these mortgages corresponds with a different homeowner. What we do is we combine these 10,000 mortgages into one big pool of mortgages, so these 10,000 mortgages form the collateral for the mortgage backed securities that we will create. Here, we have what are called tranches, Tranche A, Tranche B, Tranche C, and Tranche D, and Tranche E. We won't worry about these right now, we'll see an example of this later on, but basically what we're getting at is the following idea. We combine these 10,000 mortgages into a pool of mortgages or a pool of loans. And then from this large pool, we can construct a series of different securities. Each of these securities are labeled Tranche A down to Tranche E, and the payments, the mechanism, the risk characteristics of each of these securities are very different, even though they are all built from the same underlying pool of loans or mortgages. So, this is the process of securitization. Now, you might ask the following question, why bother with securitization? So, why securitize? Well, a standard answer to this is that by securitizing, we are enabling the sharing or spreading of risk. So, it is, in order to share risk, if you like, any one of these individual mortgages might be risky by itself. Maybe the owner of the home will default and not pay. So, any one mortgage by itself might be too risky for a small bank to hold, so instead, what they can do is they can pull all of these mortgages together and then sell them off to investors who are willing to bare that risk. Not only that, but by securitizing them and tranching them up like this, so that the securities have different types of characteristics, different risks profiles, you can actually share different types of risks to different types of investors. And so, that's the main idea behind securitization. The goal is to share risk or sell it on to investors who are willing to hold that risk. We will look at some examples of mortgage backed securities, but first, we must consider the mathematics of the underlying mortgages. Now, there are many different types of mortgages, both here in the US and in different parts of the world. We're going to consider just level-payment mortgages. Level-payment mortgages are mortgages where a constant payment is paid every month until the end of the mortgage, so that's a level-payment mortgage. But there are other types of mortgages, for example, adjustable-rate mortgages are mortgages where the mortgage rate is reset periodically. And in fact, these kinds of mortgages actually play quite a big role in the subprime crisis. So as I said, we're only going to consider level-payment mortgages, but that's fine, it's important to note though that mortgage backed securities may be constructed out of other mortgage types, as well. The construction of mortgage backed securities, as I said in the previous slide, is an example of securitization. And the same ideas apply to asset backed securities, more generally. And so, that's one of the goals of these modules and mortgage backed securities. It's just to show how the process of securitization might work. How you can combine pools of loans, be they from mortgages or other markets, credit cards, or auto loans, for example, how you can combine these pools of loans and create new securities out of them. That's a very big part of the financial industry. And so, we're going to discuss that, but in the context of mortgage backed securities. Before I go on, I mentioned that a very standard reference on mortgage backed securities is the textbook Bond Markets, Analysis, and Strategies by Frank Fabozzi, but I should advise you it is an extremely expensive book, and so I wouldn't recommend that any of you actually go out and purchase it. Still, if some of you have it or if your local library or college library has it, you might want to take a look, if you want to learn more about the mechanics of mortgages and how they work. So, as I said, we're going to consider a standard level-payment mortgage. We're going to assume, for example, that maybe there are 360 periods in the mortgage. So, this is t = 0, t = 1, and so on, up until t = 360. So, this actually would correspond to a 30 year mortgage, because there are 12 months in a year, and so there would be 12 times 30 equals 360 periods in the mortgage. So, we have an initial mortgage principal of M0. We're going to assume equal periodic payments of size B dollars are made in each period. So, we're going to pay B dollars at the end of every period, and so on, until t = 360, when the mortgage has finally been paid off. We'll assume that the coupon rate is C per period. So if you like, this is just the interest rate due each period on the mortgage. But we're going to use the term coupon rate for this. There are a total of n repayment periods. So in this example, I've drawn up here, n is equal to 360. And then after the n payments, the mortgage principal and interest have all been paid. The mortgage is then said to be fully amortizing. This means that each payment, B, pays both interest and some of the principal. After all, if we make the same payment B in every period, and until the end of the mortgage, then clearly, each payment B is paying both some of the interest you owe the outstanding principal, but it's also paying down some of the outstanding principal. So, this is an important factor and we're going to analyze this over the next couple of slides. We're going to use the following notation, we're going to let Mk denote the mortgage principal remaining after the kth period. In that case, we can say that Mk = (1 + c)Mk- 1- B. Now, where does this come from? Well, it comes from the following fact. The coupon rate is c per period. So if you look after period k- 1, the outstanding principle is Mk- 1. Well, in the next period, the outstanding principal alone will still be Mk- 1, but you will also owe an additional c times Mk- 1 of interest. So therefore, the outstanding principal will be (1 + c)Mk -1. But don't forget, you will also have paid B dollars at the end of that period. So therefore, the total outstanding principle, after the case period, will be (1 + c)Mk -1- B, and that is true for k = 0, 1, 2, up far as n, the total number of repayment periods. But, keep in mind, we said that the mortgage ends after n periods, when the entire mortgage has been paid off. So, that implies, that Mn = 0. And this is very important, so this last couple of lines here on the slide are very important. What can we say, or how can we use this expression here in one? Well, what we can do is we can iterate it. For example, we know that M1 is therefore equal to (1 + c)M0, the initial mortgage principal, minus B, we can now use this with k = 2. So M2 = (1+c)M1, and M1 is (1+c )M0- B, and we have a- B out here, so therefore, this is equal to (1+C)²M0- Σ(1+c) times, well let me put the minus here, and it's a B here, from C, to the power of J, with J = 1, or J = 0, up as far as 1. So now, we could go on to M3 and repeat the same calculation to get M3, in terms of M0, B and C, and so on. So that will leave us, in general, for k, we will get the following expression, which is that Mk = 1 + C to the power of k, times M0- B times the sum from p = 0 to k- 1, times 1 + C to the power of p. We can simplify this, this is just a simple geometric summation here. And so, we can just use a standard format for the sum of the geometric series to get this expression down in two. Now, we're not done yet. Remember that MN is equal to zero, after n repayment periods, the mortgage has been paid off. So if we take k equal to n, so take k = n in equation two, and use the fact that Mn = 0, we will find that B is equal to the following expression down here. So, this gives us B, and this is very interesting. Why is it very interesting? Well, it tells us that if we have a level-payment mortgage, and we know the initial loan amount, or the initial principal M0, and we know n, tnumber of time periods, and we know the coupon rate, well, we can compute what the correct value of B is, so that B dollars paid in every period will pay off the mortgage after n periods. By the way, this is very related to the mathematics of annuities, which you saw back in the first week of this course. Moving on, we can substitute our expression for B, back in up here, so we can put B in for this expression here, so we can substitute this expression here in for B up in equation two, and we can get this expression down here. And this is very nice because it tells us the value of the outstanding mortgage principle, after k periods, on the left-hand side, that is equal to an expression on the right-hand side, which only depends on M0, the initial mortgage principal, the coupon rate, C, the number of time periods in the entire mortgage, n, and the current period, k.