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Quiz Instructions: Introduction to Mortgage Mathematics and
Mortgage-Backed Securities
In each of the following questions monthly mortgage coupon rates should be
calculated by simply dividing the annual rate by 12. You should also assume that all of the
securities pay monthly. You should also divide annual interest rates by 12 to get the
co
esponding monthly rate and assume monthly compounding when computing present
values.
1.
(Level-Payment Mortgages) Compute the monthly payment on a 30-year level payment
mortgage assuming an annual mortgage rate of 5% and an initial mortgage principal of
$400,000.
Submission Guideline: Give your answer rounded to two decimal places. For example, if
you compute the answer to be $73.2367, submit 73.24.
2,
(Mortgage Pass-Throughs) Consider a $400 million pass-through MBS that has just been
created (so the 'seasoning' of the pass-through is equal to 0). The underlying pool of
mortgages each has a maturity of 20 years and an annual mortgage coupon rate of 6%. The
pass-through rate of the mortgage pool is 5%. Assuming a prepayment multiplier of 100 PSA
what is the total amount of interest paid to the pass-through investors?
Submission Guideline: Give your answer in millions rounded to two decimal places. For
example, if you compute the answer to be $123,456,789,12, submit 123.46.
3.
(Mortgage-Pass Throughs) Refe
ing to the same mortgage pass-through of the previous
question, what is the total amount of the prepayments?
Submission Guideline: Give your answer in millions rounded to two decimal places. For
example, if you compute the answer to be $123,456,789,12, submit 123.46.
4.
(Mortgage-Pass Throughs) Refe
ing to the same mortgage pass-through of the previous
question, what is the total amount of the prepayments if the rate of prepayments increases to
200 PSA?
Submission Guideline: Give your answer in millions rounded to two decimal places. For
example, if you compute the answer to be $123,456,789,12, submit 123.46.
5.
(Principal-Only MBS and Interest-Only MBS) Suppose we construct principal-only (PO) and
interest-only (IO) mortgage-backed securities (MBS) using the mortgage pass-through of the
previous questions. Assume a prepayment multiplier of 100 PSA. What is the present value
of the PO MBS if we use an annual risk-free rate of 4.5% to value the cash-flows?
Submission Guideline: Give your answer in millions rounded to two decimal places. For
example, if you compute the answer to be $123,456,789,12, submit 123.46.
6.
(Principal-Only MBS and Interest-Only MBS) Refe
ing to the previous question, what is the
value of the IO MBS?
Submission Guideline: Give your answer in millions rounded to two decimal places. For
example, if you compute the answer to be $123,456,789,12, submit 123.46.
7.
(Principal-Only MBS and Interest-Only MBS) Refe
ing to the previous question, what is
the average life of the IO MBS?
Submission Guideline: Give your answer in years rounded to two decimal places. For
example, if you compute the answer to be XXXXXXXXXXyears, submit 12.12.
8.
(Principal-Only MBS and Interest-Only MBS) Suppose now that you purchased the IO MBS
of the previous question and that the price you paid was the same price that you calculated
in the previous question. The risk-free interest rate suddenly changes from 4.5% to 3.5%.
Everything else stays the same. How much money have you made or lost on your
investment?
Submission Guideline: Give your answer in millions rounded to two decimal places. For
example, if you compute the answer to be $123,456,789,12, submit 123.46.
9.
(Principal-Only MBS and Interest-Only MBS) Refe
ing to the previous question, suppose the
isk-free interest rate suddenly changes from 4.5% to 3.5% and that the pre-payment
multiplier changes from 100 PSA to 150 PSA. How much money have you made or lost on
your investment in the IO MBS?
Submission Guideline: Give your answer in millions rounded to two decimal places. For
example, if you compute the answer to be $123,456,789,12, submit 123.46.
Financial Engineering & Risk Management - Introduction to Mortgage Mathematics and Mortgage Backed Securities
Financial Engineering & Risk Management
Introduction to Mortgage Mathematics and Mortgage Backed
Securities
M. Haugh G. Iyenga
Department of Industrial Engineering and Operations Research
Columbia University
Mortgage-Backed-Securities Markets
Recall that according to SIFMA, in Q3 2012 the total outstanding amount of US
onds was $35.3 trillion:
Government $ XXXXXXXXXX%
Municipal $ XXXXXXXXXX%
Mortgage $ XXXXXXXXXX%
Corporate $ XXXXXXXXXX%
Agency $2.4 6.7%
Asset-backed $1.7 4.8%
Total $35.3 tr 100%
– the mortgage market accounted for 23.3% of this total!
The mortgage markets are therefore huge
– and played a big role in the financial crisis of 2008 / 2009.
MBS are a particular class of asset-backed securities (ABS)
– assets backed by underlying pools of securities such as mortgages,
auto-loans, credit-card receivables, student loans etc.
– the process by which ABS are created is often called securitization.
2
MBS Markets
We will look at some examples of MBS but first must consider the mathematics
of the underlying mortgages.
There are many different types of mortgages including:
1. level-payment mortgages
2. adjustable-rate mortgages (ARMs)
3. balloon mortgages
4. and others.
We will only consider level-payment mortgages
– but MBS may be constructed out of other mortgage types as well.
The construction of MBS is an example of securitization
– the same ideas apply to asset-backed securities more generally.
A standard reference on mortgage-backed securities is Bond Markets, Analysis and Strategies (Pearson)
y F.J Fabozzi. But it is very expensive!
4
Basic Mortgage Mathematics for Level-Payment Mortgages
We consider a standard level-payment mortgage:
Initial mortgage principal is M0 := M .
We assume equal periodic payments of size B dollars.
The coupon rate is c per period.
There are a total of n repayment periods.
After the n payments, the mortgage principal and interest 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.
If Mk denotes the mortgage principal remaining after the kth period then
Mk = (1 + c)Mk−1 − B for k = 0, 1, 2, . . . , n (1)
with Mn = 0.
5
Basic Mortgage Mathematics for Level-Payment Mortgages
Can iterate (1) to obtain
Mk = (1 + c)kM0 − B
k−1∑
p=0
(1 + c)p
= (1 + c)kM0 − B
[
(1 + c)k − 1
c
]
. (2)
But Mn = 0 and so we obtain
B = c(1 + c)
nM0
(1 + c)n − 1 . (3)
Can now substitute (3) back into (2) and obtain
Mk = M0
(1 + c)n − (1 + c)k
(1 + c)n − 1 . (4)
6
The Present Value of a Level-Payment Mortgage
Suppose now that we wish to compute the present value of the mortgage
assuming a deterministic world
- with no possibility of defaults or prepayments.
Then assuming a risk-free interest rate of r per period, we obtain that the fai
mortgage value as
F0 =
n∑
k=1
B
(1 + r)k
= c(1 + c)
nM0
(1 + c)n − 1 ×
(1 + r)n − 1
(1 + r)n . (5)
Note that if r = c then (5) immediately implies that F0 = M0
– as expected!
In general, however, r < c, to account for the possibility of default, prepayment,
servicing fees, profits, payment uncertainty etc.
7
Scheduled Principal and Interest Payments
Since we know Mk−1 we can compute the interest
Ik := cMk−1
on Mk−1 that would be due in the next period, i.e. period k.
This also means we can interpret the kth payment as paying
Pk : = B − cMk−1
of the remaining principal, Mk−1.
So in any time period, k, we can easily
eak down the payment B into a
scheduled principal payment, Pk , and a scheduled interest payment, Ik
– we will use this observation later to create principal-only and interest-only
MBS.
8
Financial Engineering & Risk Management
Prepayment Risk and Mortgage Pass-Throughs
M. Haugh G. Iyenga
Department of Industrial Engineering and Operations Research
Columbia University
Prepayment Risk
Many mortgage-holders in the US are allowed to pre-pay the mortgage principal
earlier than scheduled
– payments made in excess of the scheduled payments are called
prepayments.
There are many possible reasons for prepayments:
1. homeowners must prepay entire mortgage when they sell their home
2. homeowners can refinance their mortgage at a better interest rate
3. homeowners may default on their mortgage payments
– if mortgage is insured then insurer will prepay the mortgage
4. home may be destroyed by flooding, fire etc.
– again insurance proceeds will prepay the mortgage.
Prepayment modeling is therefore an important feature of pricing MBS
– and the value of some MBS is extremely dependent on prepayment
ehavior.
Will now consider the simplest type of MBS
– the mortgage pass-through.
2
Mortgage Pass-Throughs
In practice, mortgages are often sold on to third parties who can then pool
these mortgages together to create mortgage-backed securities (MBS).
In the US the third parties are either government sponsored agencies (GSAs)
such as Ginnie Mae, Freddie Mac or Fannie Mae, or other non-agency third
parties such as commercial banks.
MBS that are issued by the government-sponsored agencies are guaranteed
against default
– not true of non-agency MBS.
The modeling of MBS therefore depends on whether they are agency o
non-agency MBS.
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 principal payments of the underlying mortgages.
3
Mortgage Pass-Throughs
The pass-through coupon rate, however, is strictly less than than the
average coupon rate of the underlying mortgages
– due to fees associated with servicing the mortgages.
Will assume that our MBS are agency-issued and are therefore default-free.
Definition. The weighted average coupon rate (WAC) is a weighted average of
the coupon rates in the mortgage pool with weights equal to mortgage amounts
still outstanding.
Definition. The weighted average maturity (WAM) is a weighted average of the
emaining months to maturity of each mortgage in the mortgage pool with
weights equal to the mortgage amounts still outstanding.
5
Prepayment Conventions
There are important prepayment conventions that are often used by market
participants when quoting yields and prices of MBS.
– but first need some definitions.
Definition. The conditional prepayment rate (CPR) is the annual rate at which a
given mortgage pool prepays. It is