0:03

In the next series of modules, we're going to discuss mortgage mathematics and

Â mortgage backed securities.

Â We're going to look at more mortgage backed securities because they will give

Â us an example of the process of securitization.

Â That is the process by which new securities are created from

Â pools of underlying loans or mortgages.

Â So we're going to begin in this module with basic mortgage mathematics and

Â an introduction to the mortgage markets.

Â We called it according to SIFMA, the Securities Industry and

Â Financial Markets Association, that in the third quarter of 2012,

Â the total outstanding amount of US bonds was $35.3 trillion.

Â Now if you look at this you can see that the mortgage market actually accounted for

Â 23.3% of this total.

Â So the mortgage markets are therefore huge, and

Â they played a big role in the financial crisis of 2008 and 2009.

Â And that's one of the reasons we're going to talk about mortgages and

Â mortgage backed securities over the next few modules.

Â It is interesting to understand what they are and how they are constructed.

Â And some of the mechanics behind the basic or

Â more standard types of mortgage backed securities.

Â 1:50

Here, we have what are called Tranches, Tranche A, Tranche B, Tranche C,

Â and Tranche D, and Tranche E.

Â We won't worry about these are right now with seen example of this later on, but

Â basically what we're getting at is the following idea.

Â We combine these 10 000 mortgages into accrual of mortgages or a pool of loans.

Â And then from this large pool,

Â we can construct a series of different securities.

Â Each of these securities are labelled Tranche A to Tranche E, and the payments,

Â the mechanism,

Â the risk characteristics of each of these securities are very different.

Â Even though they're all built from the same underlying pool of loans or

Â mortgages.

Â So this is the process of securitization, now you may ask the following question.

Â Why bother with securitization?

Â So why securitize?

Â Well a standard answer to this, Is that by

Â securitizing we are enabling the sharing or spreading of risk.

Â So it is in order to share risk, if you like,

Â anyone of this individuals mortgages might be risky by itself.

Â Maybe the owner of the home will default and not pay.

Â So anyone mortgage by itself might be to risk for a small bank to hold.

Â So instead what they can do is they can pull all of these mortgages together.

Â And then sell them off to investors who willing to bear that risk.

Â 3:12

We will look at some examples of the mortgage backed securities, but

Â first we must consider the mathematics of the underlying mortgages.

Â Now there are many different types of mortgages, both here in the US and

Â in different parts of the world.

Â We're going to consider just level payment mortgages.

Â Level payment mortgages are mortgages where a constant payment is paid

Â every month until the end of the mortgage.

Â So that's a level payment mortgage, but there are other types of mortgages.

Â For example, adjustable rate mortgages are mortgages where the mortgage rate is

Â reset periodically.

Â And in fact these kinds of mortgages actually played quite a big role

Â in the subprime crisis.

Â 3:50

So as I said, we're only going to consider level payment mortgages, but that's fine.

Â It's important to note though, that mortgage-backed securities may be

Â constructed out of other mortgage types as well.

Â And the construction of mortgage-backed securities,

Â as I said in the previous slide is an example of securitization.

Â And the same ideas apply to asset-backed securities more generally.

Â And so that's one of the goals of these modules in mortgage-backed securities.

Â It's just to show how the process of securitization might work.

Â How you can combine pools of loans, be they from mortgages or other markets,

Â credit cards or auto loans, for example.

Â How you can combine these pools of loans and create new securities out of them.

Â That's a very big part of the financial industry and so

Â we're going to discuss that, but in the context of mortgage-backed securities.

Â Before I go on, I'll mention that a very standard reference on mortgage-backed

Â securities is the textbook Bond Markets, Analysis and Strategies by Frank Fabozzi.

Â But I should advise you it is an extremely expensive book and so

Â I wouldn't recommend that any of you actually go out and purchase it.

Â So if some of you have it or a few local libraries or college library has it,

Â you might want to take a look.

Â If you want to learn more about the mechanics of mortgages and how they work.

Â So as I've said, we're going to consider a standard level payment mortgage.

Â We're going to assume for

Â example, that maybe there are 360 periods in the mortgage.

Â So this is t equal 0, t equals 1 and so on, up until t equals 360.

Â So this actually would correspond to a 30 year mortgage,

Â because there are 12 months in a year.

Â And so there would be 12 times 30 equals 360 periods in the mortgage.

Â 5:56

We'll assume that the coupon rate is c per period.

Â So if you'd like, this is just the interest rate due each period

Â on the mortgage, but we're going to use the term coupon rate for this.

Â There are a total of n repayment periods.

Â So in this example I've drawn up here, n is equal to 360.

Â And then after the n payments, the mortgage principal and

Â interests have all been paid.

Â The mortgage is then said to be fully amortizing.

Â 6:20

This means that each payment B pays both interest and some of the principal.

Â After all, if we make the same payment B in every period

Â out until the end of the mortgage then clearly.

Â Each payment B is paying both some of the interest due on the outstanding principle,

Â but it's also paying down some of the outstanding principle.

Â So this is an important fact and

Â we're going to analyse this over the next couple of slides.

Â We're going to use the following notation,

Â we're going to let Mk denote the mortgage principal remaining after the kth period.

Â In that case, we can say that Mk = 1+c times Mk-1-B.

Â Now where does this come from?

Â Well it comes from the following fact.

Â The coupon rate is c per period.

Â So if you look after period k-1, the outstanding principle is Mk-1.

Â Well in the next period, the outstanding principle alone will still be Mk-1.

Â But you'll also own additional c times Mk-1 of interest.

Â So therefore, the outstanding principal will be 1 + c times Mk1-1.

Â But don't forget, you will also have paid b dollars at the end of that period.

Â So therefore the total outstanding principle

Â after the kth period will be 1+c times Mk-1-B.

Â And that's true for k= 0, 1, 2 up to far as n,

Â the total number of repayment periods.

Â But keep in mind, we said that the mortgage ends after n periods,

Â when the entire mortgage has been paid off.

Â So that implies that Mn = 0.

Â And this is very important, so

Â this last couple of lines here on the slide are very important.

Â What can we say or how can we use this expression here in 1?

Â Well what we can do is we can iterate it.

Â For example, we know that M1 is therefore equal to

Â 1 + C times M0 the initial mortgage principal minus B.

Â We can now use this with k=2.

Â So M2 = 1+c times M1, and

Â M1 is 1+c times M0-B,

Â and we have a -B out here.

Â So therefore, this is equal

Â to 1 + c squared M0 minus

Â the sum of 1 + c times.

Â Well let me put the minus here and it's a B here,

Â get to the power of j, with j = 0 up as far as 1.

Â So now we could go on to M3 and repeat the same

Â calculation to get M3 in terms of M0 B and C and so on.

Â So that will leave us in general for k, we'll get to follow

Â the expression to the Mk = 1 + c to the power of k times M0-B

Â times the sum from p = 0 to k-1 times 1 + c to the power of P.

Â We can simplify this, this is just a simple geometric summation here.

Â And so we can just use our standard formula for

Â the sum of a geometric series to get this expression down in two.

Â 9:50

k = n in equation 2.

Â And use the fact that Mn = 0,

Â we will find that B is equal to the following expression down here.

Â So this gives us B, And this is very interesting.

Â Why?

Â Well it tells us that if we have a level payment mortgage and

Â we know the initial loan amount or the initial principle M0.

Â And we know n, the number of time periods, and we know the coupon rate.

Â Well we can compute what the correct value of B is, so

Â that b dollars paid in every period would pay off the mortgage after n periods.

Â By the way, this is very related to the mathematics of annuities,

Â which you saw back in the first week of this course.

Â 10:34

Anyway moving on, we can substitute our expression for B back in up here.

Â So we can put B in for this, and we can get this expression down here.

Â And this is very nice, because it tells us the value of the outstanding

Â mortgage principle after k periods on the left hand side.

Â That is equal to an expression on the right hand side, which only depends on M0.

Â The initial mortgage principal, the coupon rate c,

Â the number of time periods in the entire mortgage n and the current period k.

Â 11:37

If we assume, the risk free interest rate of r per period.

Â We obtain that the fair mortgage value is and

Â we're going to use F0 to denote the fair mortgage value.

Â Then the fair mortgage value is equal to the sum of the Bs divided by

Â 1 + r to the power of k from k = 1 to the power of n.

Â Now just to keep in mind or you can think of r as the borrowing rate for the banks.

Â The banks that write the mortgages or that lend the money out to the homeowners.

Â Presumably they can borrow at r and for all intents and

Â purposes here, we can imagine r to be a risk free interest rate.

Â R in general will certainly not be equal to c.

Â C is the coupon rate or if you like,

Â the interest rate that the homeowner must pay on their mortgage.

Â R is the interest rate that the banks use to discount their payments.

Â So in this case,

Â we're going to get F0 equals the sum of the b over 1 + r to the power of the k.

Â Again this is a geometric series, we can easily calculate this term and

Â then we can substitute in for B using our expression on the previous slide.

Â If we do that we'll get this expression here, expression number five.

Â Note that if r = c then actually F0 = M0 because if r = c, this term would

Â cancel out with this term, and this term could cancel out with this term.

Â And so not so surprisingly we get F0 = M0, and that is exactly as we would expect.

Â 13:01

In general however, r is less than c, and

Â that is because the banks who write the mortgages.

Â Or lend the money out to the home owners must charge a larger rate of interest z,

Â to account for the possibility of default, prepayment,

Â servicing fees on the mortgage.

Â They must make some profit, they must also account for payment uncertainty and so on.

Â So in general r will be less than c, and the difference here between r and

Â c accounts for the difference between F0, the fair value for

Â the mortgage from the bank's perspective.

Â And M0 the amount of money that the home owner is being lent in the first place.

Â So in some sense, you can think of F0-M0,

Â as being the amount of money that the bank is earning from the mortgage.

Â But that money must be used to handle these effects here.

Â The possibility of default, prepayment, servicing fees and so

Â on as I mentioned already.

Â 14:07

I also want to mention the fact that we can also decompose the payment B that is

Â paid in every period into an interest component under principal component.

Â And this is very easy to do, since we know Mk-1,

Â we can compute the interest, let's cal it Ik in every period.

Â So Ik is equal to c times Mk-1,

Â afterall Mk-1 is the outstanding principle at the end of period k-1.

Â So c times Mk-1 is the interest that is due on that principle in time period k.

Â So this is the interest that is paid in time period k.

Â Therefore that means we can interpret the kth payment which is B,

Â as paying Pk = B-cMk-1 of the remaining principle.

Â 15:23

So in any time period k as I said, we can easily break down the payment B into

Â scheduled principal payment and a scheduled interest payment, Ik.

Â And we're actually going to use this observation later to create principal only

Â and interest only mortgage backed securities.

Â These are an interesting class of mortgage backed securities.

Â And we will see that when we actually use a pool of mortgages

Â to create these new securities.

Â We're actually creating new securities that have very different risk profiles,

Â but we will return to that in a later module.

Â 15:51

In the spreadsheet that goes with these modules on mortgage backed securities,

Â there are three worksheets.

Â The first worksheet is called single mortgage cash flows, and

Â it simply shows you how a single level payment mortgage works.

Â So that we can see everything here on the same on the one sheet on the one screen

Â I've assumed that there's just 18 periods in the mortgage.

Â In reality there might be 240, or 360 and if I change it to 360 for

Â example we'll see that the spreadsheet adjusts appropriately.

Â So we see that we got 360 months appearing.

Â But just so that we can see anything see everything in the same screenshot,

Â I'm going to assume that this is just aiding periods.

Â We start off with a mortgage loan of $20000, the mortgage rate is 5%.

Â Now this is an annual rate, this is not c, this is an annual rate but

Â we can easily convert it into a monthly rate and

Â what's what we do here when we calculate the monthly payments.

Â So this is d.

Â This monthly payment in cell C3 is our B from the slides.

Â So we see how to calculate B using C2 which is the monthly rate.

Â As well of course C1 which is M0, the initial mortgage loan.

Â 17:07

So what we have here is we have the 18 months.

Â We see the beginning monthly balance.

Â This is the outstanding principle on the mortgage at the beginning of each month.

Â We see the monthly payment is $1155.61.

Â It's the same monthly payment in every period.

Â And as we saw in the final slide, we can break this payment down

Â into a monthly interest payment and a scheduled principal payment.

Â So that's what these quantities are here.

Â Note that the monthly interest and

Â the scheduled principal always sum up to the monthly payment.

Â So in any cell here,

Â you'll see that these numbers always sum up to the corresponding cell in column E.

Â 18:03

And of course after 18 periods the outstanding mortgage balance is

Â zero at the end.

Â So you can play with the spreadsheet if you like.

Â As I said, in practice, you typically have a much longer term of the loan.

Â Instead of 18 periods, you might have 240 for a 20-year mortgage, or 180 for

Â a 15-year mortgage, or a 360 for a 30-year mortgage.

Â Another interesting observation to make.

Â Is that in the earlier part of the mortgage, the monthly interest

Â payments are larger than they are in the latter part of the mortgage.

Â So that up here, you see in the earlier months of the mortgage,

Â the interest payment's around $80, $75, and so on.

Â But they're much smaller later in the mortgage.

Â On the other hand, the principal repayments in the earlier pert of

Â the mortgage are smaller in this case around $1,075,

Â $1,080 versus later in the mortgage when they're $1,141, $1,146, $1,150.

Â The impact isn't so obvious here but if I switch to say a 360

Â period mortgage you will see that this effect is much greater.

Â So here, let's make it say at

Â $200,000 mortgage.

Â So here, what you'll notice is the monthly payment is $1073.

Â But, most of that payment is going to pay interest, look at that.

Â Of that $1073, in the earlier part of the mortgage life,

Â most of that is going to pay interest and

Â only a smaller fraction of it maybe on the order of 25% is going to pay principal.

Â However, if I scroll down towards the end of the mortgage which is a long way down

Â because it's a three 360 months, then, you will see that now

Â only a small part of the monthly payment is going to pay monthly interest and

Â that's because the outstanding principle is much smaller in this time period.

Â So, a much smaller interest amount is due and

Â a much larger fraction of the monthly payment is going to pay the principal.

Â And this observation was important.

Â It's worthwhile knowing for anybody who's thinking about taking out a level

Â payment mortgage, that in the earlier part of the mortgage life,

Â most of their monthly payments will actually be going towards paying interest,

Â and only a small fraction would be going towards paying principal.

Â That will be reversed towards the end of the life of the mortgage.

Â