Introduction to Mortgage Mathematics and Mortgage-Backed Securities

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Del curso dictado por Columbia University

Financial Engineering and Risk Management Part I

1393 calificaciones

Columbia University

Financial Engineering and Risk Management Part I

1393 calificaciones

Financial Engineering is a multidisciplinary field drawing from finance and economics, mathematics, statistics, engineering and computational methods. The emphasis of FE & RM Part I will be on the use of simple stochastic models to price derivative securities in various asset classes including equities, fixed income, credit and mortgage-backed securities. We will also consider the role that some of these asset classes played during the financial crisis. A notable feature of this course will be an interview module with Emanuel Derman, the renowned ``quant'' and best-selling author of "My Life as a Quant". We hope that students who complete the course will begin to understand the "rocket science" behind financial engineering but perhaps more importantly, we hope they will also understand the limitations of this theory in practice and why financial models should always be treated with a healthy degree of skepticism. The follow-on course FE & RM Part II will continue to develop derivatives pricing models but it will also focus on asset allocation and portfolio optimization as well as other applications of financial engineering such as real options, commodity and energy derivatives and algorithmic trading.

De la lección

Introduction to Mortgage Mathematics and Mortgage-Backed Securities

Basic mortgage mathematics; mechanics of mortgage-backed securities (MBS) including pass-throughs, principal-only and interest-only securities, and CMOs; pricing of MBS; MBS and the financial crisis.

Introduction to Mortgage Mathematics and Mortgage-Backed Securities20:36

Conoce a los instructores

Martin Haugh
Co-Director, Center for Financial Engineering
Industrial Engineering & Operations Research
Garud Iyengar
Professor
Industrial Engineering and Operations Research Department

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

mortgage backed securities.

We're going to look at more 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.

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

And some of the mechanics behind the basic or

more 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 receivables, 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 to a different home owner.

What we do is, we combine these 10,000 mortgages into one big pool of mortgages.

So these 10 000 mortgages formed the collateral for

the mortgage back 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 are right now with seen example of this later on, but

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

We combine these 10 000 mortgages into accrual 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 labelled Tranche A to Tranche E, and the payments,

the mechanism,

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

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

mortgages.

So this is the process of securitization, now you may 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,

anyone of this individuals mortgages might be risky by itself.

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

So anyone mortgage by itself might be to risk 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 willing to bear that risk.

We will look at some examples of the 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 played 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.

And 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 in 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'll mention 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.

So if some of you have it or a few local libraries 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've 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 equal 0, t equals 1 and so on, up until t equals 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 equals 360 when the mortgage has finally been paid off.

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

So if you'd 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

interests 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

out until the end of the mortgage then clearly.

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

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

So this is an important fact and

we're going to analyse 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 times 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 principle alone will still be Mk-1.

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

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

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

And that's true for k= 0, 1, 2 up to 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 1?

Well what we can do is we can iterate it.

For example, we know that M1 is therefore equal to

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

We can now use this with k=2.

So M2 = 1+c times M1, and

M1 is 1+c times M0-B,

and we have a -B out here.

So therefore, this is equal

to 1 + c squared M0 minus

the sum of 1 + c times.

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

get to the power of j, with 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'll get to follow

the expression to the 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 our standard formula for

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

Now, we are not done yet remember that Mn = 0 after M repayment period so

mortgage has been paid off.

So if we take k equal to n, so take,

k = n in equation 2.

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?

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

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

And we know n, the number 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 would 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.

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

So we can put B in for this, 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.

Suppose now that we wish to compute the present value of the mortgage assuming

a deterministic world.

And by the way, what do we mean by deterministic world?

Well what I mean is that there's no uncertainty in the world.

We're using a very stylized situation here.

There is no randomness, there is no possibility of default so

prepayments though of course, these are very important features in practise.

In that situation, we can actually compute the present value of a level mortgage.

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

We obtain that the fair mortgage value is and

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

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

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

Now just to keep in mind or you can think of r as 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 a risk free interest rate.

R in general will 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.

So in this case,

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

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

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

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

And so not so surprisingly we get F0 = M0, and that is 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 home owners must charge a larger rate of interest z,

to account for the possibility of default, prepayment,

servicing fees on the mortgage.

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

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

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

the mortgage from the bank's perspective.

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

So in some sense, you can think of F0-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 the level payment mortgages

where you could do similar kinds of analysis with other types of mortgages

including adjustable rate mortgages or ARMs or so on.

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

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

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

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

So Ik is equal to c times Mk-1,

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

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

So 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 = B-cMk-1 of the remaining principle.

So Ik is the interest that is paid at period k and

Pk is the principal that is paid at period k.

And of course note that Ik + Pk = B, which must be

the case because we pay B dollars in every period.

So 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.

So in 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.

And 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.

And 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 on mortgage backed securities,

there are three worksheets.

The first worksheet is called single mortgage cash flows, and

it simply shows you how 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 and if I change it to 360 for

example we'll see that the spreadsheet adjusts appropriately.

So we see that we got 360 months appearing.

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

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

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

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 and

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

So this is d.

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

So 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.

So 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.

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

into a monthly interest payment and a scheduled principal payment.

So that's what these quantities are here.

Note that the monthly interest and

the scheduled principal always sum up to the monthly payment.

So in any cell here,

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

So monthly interest plus scheduled principal repayment

is equal to the total monthly payment.

And then of course in order to get the ending mortgage balance,

we simply subtract the scheduled principal repayment from the monthly payment.

And of course after 18 periods the outstanding mortgage balance is

zero at the end.

So 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 180 for

a 15-year mortgage, or a 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.

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

the interest payment's 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 pert of

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

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

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

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

So here, let's make it say at

$200,000 mortgage.

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

But, most of that payment is going to pay interest, 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 order of 25% is going to pay principal.

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

because it's a three 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 this time period.

So, a much smaller interest amount is due and

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

And this observation was 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 principal.

That will 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.

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