# Introduction to Mortgage Arithmetic and Mortgage-Backed Securities – Half I – Introduction to Mortgage Arithmetic and Mortgage-Backed Securities

Within the subsequent sequence of modules, we will talk about mortgage arithmetic and mortgage backed securities. We’ll take a look at mortgage backed securities as a result of they are going to give us an instance of the method of securitization. That’s the course of by which new securities are created from swimming pools of underlying loans or mortgages. So, we will start, on this module, with fundamental mortgage arithmetic and an introduction to the mortgage markets. We known as it, in keeping with SIFMA, the Securities Business and Monetary Markets Affiliation, that within the third quarter of 2012, the entire excellent quantity of US bonds was $35.3 trillion. Now, should you take a look at this, you’ll be able to see that the mortgage market really accounted for 23.3% of this complete. So, the mortgage markets are subsequently large and they performed an enormous function within the monetary disaster of 2008 and 2009. And that is one of many causes we will discuss mortgages and mortgage backed securities over the subsequent few modules. And it is fascinating to grasp what they’re and the way they’re constructed and among the mechanics behind the fundamental or most traditional sorts of mortgage backed securities. Mortgage backed securities are a selected class of what are known as asset backed securities. These are belongings backed by underlying swimming pools of securities, comparable to mortgages, auto loans, bank card receivable,s pupil loans, and so forth. The method by which ABS, or mortgage backed securities, are created is commonly known as securitization. This is an image of how securitization works within the context of mortgages and mortgage backed securities. What we’ve listed here are 10,000 completely different mortgages, every of those mortgages corresponds with a special home-owner. What we do is we mix these 10,000 mortgages into one large pool of mortgages, so these 10,000 mortgages type the collateral for the mortgage backed securities that we’ll create. Right here, we’ve what are known as tranches, Tranche A, Tranche B, Tranche C, and Tranche D, and Tranche E. We can’t fear about these proper now, we’ll see an instance of this afterward, however mainly what we’re getting at is the next concept. We mix these 10,000 mortgages right into a pool of mortgages or a pool of loans. After which from this massive pool, we are able to assemble a sequence of various securities. Every of those securities are labeled Tranche A all the way down to Tranche E, and the funds, the mechanism, the danger traits of every of those securities are very completely different, regardless that they’re all constructed from the identical underlying pool of loans or mortgages. So, that is the method of securitization. Now, you may ask the next query, why trouble with securitization? So, why securitize? Nicely, a typical reply to this is that by securitizing, we’re enabling the sharing or spreading of danger. So, it’s, with the intention to share danger, should you like, any one in all these particular person mortgages is likely to be dangerous by itself. Possibly the proprietor of the house will default and never pay. So, anyone mortgage by itself is likely to be too dangerous for a small financial institution to carry, so as an alternative, what they’ll do is they’ll pull all of those mortgages collectively and then promote them off to traders who’re keen to reveal that danger. Not solely that, however by securitizing them and tranching them up like this, in order that the securities have various kinds of traits, completely different dangers profiles, you’ll be able to really share various kinds of dangers to various kinds of traders. And so, that is the primary concept behind securitization. The purpose is to share danger or promote it on to traders who’re keen to carry that danger. We are going to take a look at some examples of mortgage backed securities, however first, we should think about the arithmetic of the underlying mortgages. Now, there are numerous various kinds of mortgages, each right here within the US and in numerous elements of the world. We’ll think about simply level-payment mortgages. Degree-payment mortgages are mortgages the place a continuing settlement is paid each month till the top of the mortgage, in order that’s a level-payment mortgage. However there are different sorts of mortgages, for instance, adjustable-rate mortgages are mortgages the place the mortgage charge is reset periodically. And actually, these sorts of mortgages really play fairly an enormous function within the subprime disaster. In order I stated, we’re solely going to contemplate level-payment mortgages, however that is superb, it is essential to notice although that mortgage backed securities could also be constructed out of different mortgage sorts, as effectively. The development of mortgage backed securities, as I stated within the earlier slide, is an instance of securitization. And the identical concepts apply to asset backed securities, extra usually. And so, that is one of many objectives of those modules and mortgage backed securities. It is simply to indicate how the method of securitization may work. How one can mix swimming pools of loans, be they from mortgages or different markets, bank cards, or auto loans, for instance, how one can mix these swimming pools of loans and create new securities out of them. That is a really large a part of the monetary trade. And so, we will talk about that, however within the context of mortgage backed securities. Earlier than I am going on, I discussed {that a} very normal reference on mortgage backed securities is the textbook Bond Markets, Evaluation, and Methods by Frank Fabozzi, however I ought to advise you it’s a particularly costly guide, and so I would not advocate that any of you really exit and buy it. Nonetheless, if a few of you will have it or in case your native library or faculty library has it, you may want to have a look, if you wish to be taught extra concerning the mechanics of mortgages and the way they work. So, as I stated, we will think about a typical level-payment mortgage. We’ll assume, for instance, that perhaps there are 360 intervals within the mortgage. So, that is t = 0, t = 1, and so forth, up till t = 360. So, this really would correspond to a 30 yr mortgage, as a result of there are 12 months in a yr, and so there could be 12 occasions 30 equals 360 intervals within the mortgage. So, we’ve an preliminary mortgage principal of M0. We’ll assume equal periodic funds of dimension B {dollars} are made in every interval. So, we will pay B {dollars} on the finish of each interval, and so forth, till t = 360, when the mortgage has lastly been paid off. We’ll assume that the coupon charge is C per interval. So should you like, that is simply the rate of interest due every interval on the mortgage. However we will use the time period coupon charge for this. There are a complete of n reimbursement intervals. So on this instance, I’ve drawn up right here, n is the same as 360. After which after the n funds, the mortgage principal and curiosity have all been paid. The mortgage is then stated to be absolutely amortizing. Which means that every settlement, B, pays each curiosity and among the principal. In any case, if we make the identical settlement B in each interval, and till the top of the mortgage, then clearly, every settlement B is paying each among the curiosity you owe the excellent principal, however it is also paying down among the excellent principal. So, this is a vital issue and we will analyze this over the subsequent couple of slides. We’ll use the next notation, we will let Mk denote the mortgage principal remaining after the kth interval. In that case, we are able to say that Mk = (1 + c)Mk- 1- B. Now, the place does this come from? Nicely, it comes from the next reality. The coupon charge is c per interval. So should you take care of interval k- 1, the excellent precept is Mk- 1. Nicely, within the subsequent interval, the excellent principal alone will nonetheless be Mk- 1, however additionally, you will owe a further c occasions Mk- 1 of curiosity. So subsequently, the excellent principal shall be (1 + c)Mk -1. However do not forget, additionally, you will have paid B {dollars} on the finish of that interval. So subsequently, the entire excellent precept, after the case interval, shall be (1 + c)Mk -1- B, and that’s true for ok = 0, 1, 2, up far as n, the entire variety of reimbursement intervals. However, be mindful, we stated that the mortgage ends after n intervals, when the whole mortgage has been paid off. So, that means, that Mn = 0. And this is essential, so this final couple of strains right here on the slide are essential. What can we are saying, or how can we use this expression right here in a single? Nicely, what we are able to do is we are able to iterate it. For instance, we all know that M1 is subsequently equal to (1 + c)M0, the preliminary mortgage principal, minus B, we are able to now use this with ok = 2. So M2 = (1+c)M1, and M1 is (1+c )M0- B, and we’ve a- B out right here, so subsequently, this is the same as (1+C)²M0- Σ(1+c) occasions, effectively let me put the minus right here, and it is a B right here, from C, to the facility of J, with J = 1, or J = 0, up so far as 1. So now, we might go on to M3 and repeat the identical calculation to get M3, by way of M0, B and C, and so forth. So that may go away us, basically, for ok, we’ll get the next expression, which is that Mk = 1 + C to the facility of ok, occasions M0- B occasions the sum from p = 0 to k- 1, occasions 1 + C to the facility of p. We will simplify this, that is only a easy geometric summation right here. And so, we are able to simply use a typical format for the sum of the geometric sequence to get this expression down in two. Now, we’re not executed but. Keep in mind that MN is the same as zero, after n reimbursement intervals, the mortgage has been paid off. So if we take ok equal to n, so take ok = n in equation two, and use the truth that Mn = 0, we’ll discover that B is the same as the next expression down right here. So, this provides us B, and that is very fascinating. Why is it very fascinating? Nicely, it tells us that if we’ve a level-payment mortgage, and we all know the preliminary loan quantity, or the preliminary principal M0, and we all know n, tnumber of time intervals, and we all know the coupon charge, effectively, we are able to compute what the proper worth of B is, so that B {dollars} paid in each interval will repay the mortgage after n intervals. By the way in which, that is very associated to the arithmetic of annuities, which you noticed again within the first week of this course. Shifting on, we are able to substitute our expression for B, again in up right here, so we are able to put B in for this expression right here, so we are able to substitute this expression right here in for B up in equation two, and we are able to get this expression down right here. And that is very good as a result of it tells us the worth of the excellent mortgage precept, after ok intervals, on the left-hand facet, that is the same as an expression on the right-hand facet, which solely depends upon M0, the preliminary mortgage principal, the coupon charge, C, the variety of time intervals in the whole mortgage, n, and the present interval, ok.