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.
Collateralized Mortgage Obligations (CMOs)
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Columbia University 提供のコース
Financial Engineering and Risk Management Part I
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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.
コースの提供者
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 collateralized mortgage obligations or
CMOs. CMOs are a class of mortgage backed
securities that are created from underlying pools of mortgages or
underlying pass-through securities or indeed underlying mortgage backed
securities. We're going to see how tranches are
created. We're going to see an example of
sequential CMO where we've got four tranches, tranche A, tranche B, tranche C
and tranche D. And we'll see how these tranches are
created and how they've got different risk profiles that might appeal to
different investors. In this module we're going to discuss
briefly what are called collateralized mortgage obligations or CMOs.
CMOs are mortgage-backed securities that have been created by redirecting the
cash-flows from other mortgage securities.
They are created mainly to mitigate prepayment risk, and create securities
that are better suited to potential investors.
In practice, CMOs are often created from pass-through's, so we've seen already
what pass-through mortgage backed securities are, but they can also be
created from other mortgage backed securities, including for example,
principal loaning backed securities. Now there are many types of CMOs.
They include what are called sequential CMOs, CMOs with accrual bonds, CMOs with
floating rate and inverse floating rate tranches and planned amortization class
CMOs or PAC CMOs. We're only going to briefly discuss
sequential CMOs and I would point out here that the reason we're going through
this module is just to give ourselves an idea of how these securities are
constructed. To give ourselves another example of
securitization. The results of our worksheet, it's the
third worksheet in our XL workbook, which shows how a sequential CMO is created and
shows how the cashflows for the different tranches in a sequential CMO are created.
But I will emphasize here that this material, it's just there to show us how
these securities are constructed and to give us an example of these securities.
So, don't worry too much about the details, the details aren't particularly
important. The basic structure of a sequential CMO
is that there are several what are called tranches.
Now this word appears very often into the securitization world.
We're going to see it again when we discuss credit derivatives and what I
call CDOs. So the basic structure of a sequqnetial
CMO is that there are several tranches which are ordered in such a way that they
are retired sequentially. What do I mean by that?
Well, for example, the payment structure of a sequential CMO with tranches a, b,
c, and d might be as follows. Number one, periodic coupon interest is
disbursed to each tranche on the basis of the amount of principal outstanding in
the tranche in the beginning of the period.
Number two, all principal payments are disbursed to tranche A until it is paid
off entirely. After tranche A has been paid off, all
principal payments are dispersed to tranche B, until it is payed off
entirely. After tranche B has been paid off, all
principal payments are dispersed to tranche C until it is paid off entirely.
And then after tranche C has been paid off, all principal payments are dispersed
to tranche D, until it is paid off entirely.
So let's translate what we're seeing here, what we're seeing here is that
there's going to be four tranches, A, B, C, and D.
And when we look at the spreadsheet in a few minutes we'll see what we mean by
this. On each tranche the interest will be paid
to the investors in each tranche in every period.
But principal will not be paid to the holders of each tranche.
All of the principle payments will go towards paying tranche A investors.
And only when tranche A's principle has been paid off entirely will the principle
payments from the underlying mortgage pool be used to pay tranche B.
And then, only when all of the principle for tranche B has been paid off will the
principle payments go towards tranche C. Then only when tranche Cs principal has
been paid, will the principal be used to pay tranche D.
That's what we mean by retiring sequentially.
The advantages of doing this is that we're creating these four new securities,
tranche A, B, C and D in such a way that there's more certainty around when these
tranches will get their payments. Here's our earlier picture showing the
process of securitization. Again, you could imagine that we've got
10,000 mortgages forming a pool of mortgages.
These mortgages formed the collateral for our sequential CMO, so there's going to
be tranche A, tranche B, tranche C and tranche D.
These are all securities, so this is a security an investor can buy tranche a if
you like. This is a security, this is a security
and this is Is a security. So investors can come along and buy A, B,
C, or D. These are four different types of
securities. Their all constructed from the underlying
10,000 loans or 10,000 securities. But in order to get a better idea of
actually how this sequencial CMO works, its actually much easier to go to our
spread sheet and that's what we'll do. So here is our spreadsheet with our
sequential pay CMO. The first thing to keep in mind is that
this spreadsheet is constructed using the pass through spreadsheet we looked at
earlier. So the pass through spreadsheet we looked
at in an earlier module, this pass through mortgage, this, this pool of
mortgages, of $400 million worth of mortgages, for example.
Can be used to actually create a new set of securities and these new sets of
securities give us the sequential pay CMO.
So up here what we have is the following, we have a total of $400 million, that's
from the underlying pass through on the other worksheet.
We've split this $400 million into different notions, $194.5 million for
tranche A, $36 million for tranche B, $96.5 million for transche C and $73
million for tranche D. We're going to assume that the underlying
coupon rate for all four charges is 7.5% and of course, this is the same 7.5% that
we saw on the pass through. The underlying pass through had a pass
through rate of 7.5%. So, that gives us these four new
securities. So we've got four new securities, each of
them have different principle amounts but the sum of the four principal amounts
must be equal to 400 million dollars, the sum of the principal of the underlying
pass through security. We then have our months, again, just to
show us all the payments on one screen, I've assumed that we're only going to
have 17 payments, or 17 months. in reality, you can have many more months
than this, up to maybe 240 or 360, as I said earlier.
we start off with our balances for each if the tranche, so tranche A balance is
194.5 to begin with, tranche B is 36, which is what we have up here, tranche C
is 96.5, which is what we have here, and tranche D is 73.0.
Now, what we do is, we follow the instructions on the, on the and two
slides ago, which tells us how interest and principle are paid on each of these
trenches. So notice that the principle is
decreasing on trench eight, for the first nine months until there is no interest or
no principle remaining. It is only at that point that the
principle in trench B starts to be repaid.
Notice that the principle on trans B remains 36 on every period until the
principle on trans A has been repaid and only at that point does the principle on
trans B start being repaid. Likewise the principle on trans C remains
96.5 until such time as the principle on trans B has been repaid entirely, only
then is tranche C paid down. Likewise, on Tranche D it starts off with
a principal of 73 and it remains 73 until the principal on all the previous
Tranches has been paid down and only after that point is the principal on
Tranche D paid. Notice these principal payments by the
way Are the principle payments that are calculated from the underlying
passthrough on the previous worksheet. If you look at the formula, you'll see
that it refers to the passthrough worksheet.
Passthrough here is the worksheet here. So the principle payments are computed
from the underlying passthrough and they are here.
So, 22.533 That is the principal payment that appears over here.
So what we're doing here is constructing a mortgage-backed-security, or CMO, with
different tranches, so an investor could invest in tranche a or another investor
could invest in tranche b, and so on. And they're going to get the schedule of
cash flows. one thing to notice here as well is we've
actually calculated the average life of each tranche .
So you can see how the average life is calculated by looking at this formula.
Not surprisingly, the average life in years of tranche A, is less than tranche
B, is less than trance C, and is less than trance D.
So for trans A, it's 0.4 years, for trans B it's 0.8 years, trans C it's 1.04
years, trans D it's 1.33 years. And of course that's understood by the
fact that the principle in trans B gets payed, and so on.
So what we've succeeded in doing here, is taking the underlying pass through and
instead of just leaving it as a pass 3 mortgage backed security, we constructed
a more complicated mortgage backed security, where the underlying charges
have very dis, different risk profiles. Some investors might like a, a security
with a shorter average life, other investors might prefer a security with a
longer average life. And so they might prefer trans d to trans
a. And why are we doing all of this?
Well it goes back to that comment we made in an earlier module about
securitization. The economic, or big picture goal behind
securitization is the idea of spreading the risk.
An investor may not want to invest in the pass through security.
They may not like an average life of .76 years, however, that same investor may
like a security with an average life of 0.4 and so they may be willing to buy
this tranche, even though they would not buy the underlying pass-through.
Likewise, there may be another investor who does not like the pass-through
security, but they do like tranche D, with the longer average life.
So by constructing new securities with very different risk profiles out of the
same underlined pool of loans, you can actually potentially appeal to more
investors. And therefore share the risk among
investors who want to hold different types of risks.
This prepayment multiplier by the way is the same prepayment multiplier of the
underlying pass through. Remember, it's the underlying pass
through security which is actually creating the cash flows that feed into
the four trenches of the CMO. Finally, in practice you would actually
have many more periods here, it wouldn't just be 17 periods, it would be maybe 240
and if I change it 240 here, it will also change appropriately over here.
So now you can get a better sense. So in this case with a 240 period
Sequential CMO, the average life of tranche A is four years, the average life
of tranche C is 11.93 years, and so on. And the, these tranches will have there,
therefore have very different risk profiles.
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