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.
Prepayment Risk and Mortgage Pass-Throughs
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From the course by Columbia University
Financial Engineering and Risk Management Part I
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From the lesson
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.
Meet the Instructors
Martin HaughCo-Director, Center for Financial Engineering
Industrial Engineering & Operations Research
Garud IyengarProfessor
Industrial Engineering and Operations Research Department
In this module we're going to discuss Prepayment risk and Mortgage
pass-throughs. Prepayment risk is a particular type of
risk that mortgage backed securities are exposed to.
Prepayment risk refers to the ability of homeowners to prepay their mortgages.
So mortgage holders or homeowners often have a option to prepay their mortgage
early. And that creates prepayment risk for
investors in mortgage-backed-securities. We're also going to discuss Mortgage
pass-throughs securities. A Mortgage pass-throughs is the simplest
example of a mortgage-backed security, and so we're going to spend some time as
well on this module, in this module, discussing pass-throughs.
Many mortgage holders in the US are allowed to prepay the mortgage principal
earlier than scheduled. Payments made in excess of the scheduled
payments are called prepayments. Now, there are many possible reasons for
prepayments. Number one, homeowners must prepay the
entire mortgage when they sell their home, and there are many reasons why
somebody might need to sell their home, maybe they're moving for a new job, maybe
they're getting divorced, and so on. Homer, homeowners can also refinance
their mortgage at a better interest rate. So for example say a homeowner took out a
mortgage when the interest rates were very high on mortgages, and maybe five or
six years later, they'd see that interest rates are much lower.
Well the ability to prepay allows the homeowner to pre-pay their mortgage and
then take out a new mortgage at a much lower level of interest.
A third reason is the homeowner may simply default on their mortgage
payments. Maybe they're behind on their payments,
they can't catch up and they smiply default.
In this case, if the mortgage is insured then the insurer will pre-pay the
mortgage as well. And finally, for another example, the
home might be destroyed by floods or fire or by some other catastrophe and
insurance proceeds in this case will pre-pay the mortgage.
So there are many possible reasons for pre-payments.
The second reason we've highlighted here is a very valuable reason.
It's the prepayment option that mortgage holders have.
In particular they can benefit when interest rates go lower.
And so this is a prepayment option and like options it has a positive value to
the owner of the option, in this case the US homeowner typically.
So prepayment modeling is therefore an important feature of pricing
mortgage-backed securities. And the value of some mortgage-backed
securities is extremely dependent on prepayment behavior.
Again, we'll come to some examples of mortgage-backed securities in later
modules. For now, we're going to consider the
simplest type of mortgage-backed security, and that is the Mortgage
pass-through. Now before I go on I should mention that
pretty much everything I'm going to be saying here is very US centric.
The mortgage markets are the markets from mortgage bank securities originated in
the US in the 1980s. And that's where most of the the modeling
and the if you like action in the mortgage market is taking place.
So I'm going to be focusing mainly on the US but that's fine.
Because pretty much everything I say will apply in some form or another to mortgage
markets generally. Also the bigger picture that I want to
get at here is the general idea of securitization and how mortgage-backed
securities and indeed asset-backed securities can be created.
How are Mortgage pass-throughs constructed.
Well in practice mortgages are often sold on to third parties who can then pool
these mortgages together to created mortgage-backed securites.
In the US, the third parties are either government sponsored agencies such as
Ginnie Mae, Freddie Mac or Fannie Mae, or other non-agency third parties such as
commercial banks. Mortgage-backed securities that are
issued by the government-sponsored agencies are guaranteed against default.
It is not true of agency mortgage-backed securities.
What do we mean by default here? Well what I mean by default here is just
that the homeowner might be default on their mortgage payments.
They just may not make their mortgage payments and, and therefore the mortgage
can go into default. Well, in that situation, if the
mortgage-backed security was issued by Fannie Mae or Freddie Mac, for example,
then the agency will step in and guarantee the payments.
This is not true in general of non-agency mortgage-backed securities.
So the modeling of mortgag- backed securities therefore depends on whether
they are agency or non agency mortgage-backed securities.
The simplest type of MBS is the pass through MBS, where a group of mortgages
are pooled together. Investors in this MBS receive monthly
payments representing the interest and the principal payments of the underlying
mortgages. So, here's a diagram explaining how this
works. We saw this before in the last module.
We've got 10,000 mortgages, of course the number doesn't have to be 10,000, it
could be 5,000 or 20,000 mortgages. We've got mortgage number one, mortgage
number two, up to mortgage number 10,000. All of these mortgages are pulled
together into one mortgage pull and from that mortgage pull we can create new
securities. These new securities are mortgage-backed
securities and maybe they've got names like Tranche A, Tranche B down to Tranche
D and Tranche E and so on. For now let's not worry about these
tranches. In fact, in the case of the Mortgage
pass-through security we don't have any tranches.
But we will some examples later on where you can have tranches.
So these 10,000 mortgages formed the collateral for the mortgage-backed
security. and so when we create these new
securities out of the underlying pool of 10,000 mortgages, this process is often
called securitization. And as mentioned earlier, the economic
reasons for securitization is that the desire of people to spread risk.
Most people most agents most investors do not like risks.
They need to be compensated for holding risky securities.
So, one could view these 10,000 mortgages, individually, as being very
risky. If you hold any one of these mortgages,
there will be a substantial chance that the homeowner will default, for example,
or pre-pay, and so there will be risk associated with these mortgages.
But by pulling them together, creating new securities, we can sell these new
securities on to investors who want the particular type of risk.
And so that is the economic or financial motivation behind securitization.
The pass-through coupon rate, and we're referring now to our Mortgage
pass-through is strictly less than the average coupon rate of the underlying
mortgages. Now this is due to fees associated with
servicing the mortgages. Somebody has to collect the mortgage
payments every month and do the book work and bookkeeping to make sure that the
homeowners are up to date on their payments, and so on.
We're going to assume that our mortgage-backed securities are agency
issued, and are therefore default free. Now, I'm going to give you a couple of
definitions here. You'll find when we discuss
mortgage-backed securities that there are many definitions.
The weighted average coupon rate, WAC, is a weighted average of the coupon rates in
the mortgage pool with weights equal to the mortgage amounts still outstanding.
Similarly, the weighted average maturity is a weighted average of the remaining
months to maturity of each mortgage in the mortgage pool with weights equal to
the mortgage amounts still outstanding. Now there's no need to worry about the
specific details of these definitions. I just want to mention that there are
many definitions associated with mortgage-backed securities.
We also need to discuss some important prepayment conventions, that are often
used by market participants, when quoting yields, and prices, of mortgage backed
securities. But first we need some definitions.
One definition, is the following, the conditional prepayment rate, the CPR, is
the annual rate at which a given mortgage pool prepays.
It is expressed as a percentage of the current outstanding principal level in
the underlying pool, to this is the so called conditional prepayment rate.
I very related definition is the single-month mortality rate.
The SM and the SMM is the CPR converted to a monthly rate assuming monthly
compounding. And so therefore the SMM and CPR are
related as follows. Now the CPR is expressed as an annual
rate, and so it is probably the rate that makes most sense to us.
We like to think in terms of annual rates.
We think of interest rates expressed as annual rates, and so likewise we might
think of a conditional prepayment rate as an annual rate.
However, when modeling the payments of a mortage-backed security we typically need
the coresponding monthly rate and that's because payments in mortgage-backed
securities typically take place monthly and so we wont be able to convert our CPR
to SMM, and that's what we do here. We'll also see how the SMM and CPR are
used in the spreadsheet that is associated with these modules to
calculate the various cash flows underlying the Mortgage pass-through and,
indeed, other mortgage-backed securities. In practice, of course, the CPR is
stochastic, it is random, and it depends on the mortgage pool and other economic
variables. We saw, for example, earlier that
prepayments tend to go up when interest rates go down.
And that's because it is more attractive to homeowner's to prepay their entire
mortgage and then refinance at a better rate which is possible when interest
rates have gone down. So the prepayment rate for a given
mortgage pool will certainly depend on the economic variables.
That are present at any given time. That having been said, market
participants, that is, people who work with mortgage-backed securities in the
marketplace. They often use a deterministic
pre-payment schedule as a mechanism to quote mortgage-backed security yields,
and so-called option adjusted spreads. Now we're not going to discuss option
adjusted spreads it all in this course that's fine so you don't have to worry
about what they are. The standard benchmark is what is called
the Public Securities Association benchmark or the PSA benchmark.
The PSA assumes the CPR is equal to 6% times t over 30 if t is less than or
equal to 30 and I should mention here that C is now measured in months.
Here and then after 30 months, CPR is equal to 6%.
So the assumption here is that the conditional prepayment rate of a mortgage
is 60% if it's more than 30 months old. If it's less than 30 months old, the
conditional prepayment rate is 6% times t over 30.
So basically the CPR grows linearly, for 30 months and then is flat.
So this is time t in months, this is 6%, and this is the CPR.
And then slower or faster prepayment rates are given as some percentage or
multiple of PSA. So, for example 2 times CPA refers to
mortgage pool that repays at twice this rate.
A 50% CPR refers to a mortgage pool where the condition of prepayment is half of
this CPR. Giving a particular prepayment assumption
the average life of a mortgage-backed security is given as the following.
Sells equal to the sum from k equals1 to capital T, k times Pk divided 12 times
TP. Where Pk's the principle scheduled and
projected prepayment paid at time k, TP's the total principle amount, capital T is
the total number of months. And, we also divide by 12, so that
average life is measured in years. It should be clear that the average life
decreases as the PSA speed increases. And that makes sense, because if the PSA
speed increases, well then you're going to get more of your principle payments
earlier in the life of the mortgage. So the Pk's for k small, will actually be
larger when the PSA speed increases. And so average life gives you a measure
of the average life, or how long you have to wait to get the payments associated
with the mortgage or mortgage. Implicit in the calculations of the
average life is some assumption regarding the speed of prepayments.
In practice the price of a given mortage-backed security is observed in
the marketplace and from this a corresponding yield to maturity can be
determined. This yield is the interest rate that will
make the present values of the expected cash flows equal to the market price.
So if you recall what a yield to maturity is in the case of a fixed income bond
where the cash flows are fixed, well, it'll be some quantity that satisfies the
following. So it will be n, i equals 1, ci over 1
plus lambda to the power of i. Now if lambda, if the payments are made
semi-annually and we're compunding semi-annually then I should divide lambda
by 2 here and multiply the i by 2. But basically lambda is the quantity that
makes this equation correct. Where P0 is the value of the fixed income
security in the marketplace and Ci is the cashflow that you receive at time I when
you own this fixed income security. The problem with mortgages is, and
mortgage-backed securities, is that the Ci's are uncertain.
You don't know what they're going to be and that is because of prepayments.
So when you are calculating a lambda or yield maturity for mortgage-backed
security, you have to fix the ci's, and the way to fix the ci's is to make some
assumption about prepayments. So that's what market participants do.
So the expected cash flows are determined, or based on some underlying
prepayment assumptions such as 1 PSA, 300 PSA, etcetera.
So any quoted yield must be with respect to some prepayment assumptions.
When the yield is quoted as an annual rate based on semi-annual compounding, it
is often called a bond-equivalent yield. Yields actually are very limited when it
comes to evaluating a mortgage-backed security.
And indeed, fixed income securities in general, that don't take account the term
structure of interest rates. They don't take the prepayment option
into account. so yields are very limited, and indeed
the, the market place typically uses option adjusted spreads as the market
standard for quoting yields in mortgage-backed securities, and indeed
other fixed income securities with embedded options.
But, as I mentioned before, we're not going to be discussing option-adjusted
spread so there's no need to worry about that in this course.
An investor in a mortgage-back, backed security pass-through is, of course,
exposed to interest rate risk in that the present value of any fixed set of
cash-flows decreases as interest rates increase.
However pass-through investors also exposed to prepayment risk, in particular
contraction risk and extension risk. When interest rates decline prepayments
tend to increase and the additional prepaid principal can only be invested at
lower interest rates. So this is contraction risk.
So if you think about it. When interest rates fall if you own a
home and therefore have a mortgage, it is in your interest to prepay that mortgage
early and then re-finance the mortgage at a much lower level of interest.
If you do this then the people who have invested in these mortgage-backed
securities. They're going to get payments sooner than
they expected, and these payments can only be invested at the lower interest
rates. So this is called contraction risk.
The opposite also provides a risk, which is called extension risk.
In this case, when interest rates increase, the prepayments tend to
decrease, and that makes sense. After all, why would I want to prepay a
mortgage early when I'm going to have to re-finance it at higher interest rates.
So therefore, interest rates increase implies prepayments tend to decrease, and
therefore I'm getting less prepaid principals than I expected.
And so I'm not going to invest that principal, that prepaid principal at the
higher interest rates that might now be prevailing.
This is called extension risk. So, to summarize, in addition to the
regular interest rate risk that fixed income securities have Mortgage-backed
securities and Mortgage pass-throughs are also exposed to prepayment risk, in
particular contraction risk and extension risk.
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