12 - 1
CHAPTER 12: SECURITIZATION AND
MORTGAGE-BACKED SECURITIES (MBSs)

SECURITIZATION
Securitization can be summarized as follows.
1. Financial assets (e.g. mortgage loans, accounts
receivable) accumulate on the balance sheet.
2. Assets with comparable maturity and risk structures are
“pooled” into a special purpose vehicle controlled by a
trustee.
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3. The trustee issues new securities to investors. The cash
flows from the new securities are just the periodic
repayments due from the original borrowers.
4. Often the new security includes a credit enhancer, i.e., a
AAA financial institution (e.g., the U.S. Federal
government, in some cases) extends a form of guarantee
against defaults on the loans.
5. A service manager manages the cash flows.

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Motivations for Securitization
Securitization provides a “secondary” market for mortgages.
This provides “liquidity” for mortgages, i.e., it allows the
holders of the primary securities (the mortgages) to
liquefy their securities should they wish to do so. This
should make mortgage financing more attractive to
lenders, and should result in a reduction in the margins
charged on mortgage finance (i.e., the “liquidity
premium” charged).

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MORTGAGES
Fixed-Rate Mortgages (FRMs)
For mortgages, interest rates are quoted as monthly
compounded rates. If R is the rate per year (compounded
monthly) then r = R
12
is the rate used for monthly
payments.
To find the loan payments, we use the formula for the present
value of an annuity. Let N = original term of the loan in
months, and let x = the monthly PMT.
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PV = 1 (1 ) NPMT r
r
    = 1 (1 )
Nx r
r
    .
Here, PV = F0, the face value of the loan taken (i.e., the
principal outstanding at month 0). Solving for x = PMT:
x = PMT = 0
[1 (1 ) ]N
F r
r  
.
Now, let Fn = the principal outstanding after the nth monthly
payment. There are N  n payments remaining. Fn is just
the PV of the remaining N  n payments:
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Fn = ( )1 (1 ) N n
x r
r
     .
Each month, your payment can be decomposed into two
terms: the interest payment
rFn  1 =
R
12
Fn  1.
and the principal payment
Fn  1  Fn.
We want to show that
interest payment + principal payment = x.
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Using our formula for Fn, the interest payment is
r Fn  1 = x[1  (1 + r)(N  n + 1)] = x[1 
1
1 1( )  r N n
]
and the principal payment each month is
Fn  1  Fn =
x
r
[1  (1 + r)(N  n + 1)  1 + (1 + r)(N  n)]
= x
r r N n( )1 1  
[1 + (1 + r)] = x
r N n( )1 1  
.
We have shown that
Interest payments + Principal payments = x
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each month, i.e., rFn 1 + (Fn  1 – Fn) = x. This leads to a
recursive formula for Fn:
Fn = (1 + r)Fn  1  x.
Example 1: Suppose you have a 30 year $500,000 mortgage
with a rate of 4.5% per year (i.e., 4.5% per year
compounded monthly.) Then r = 0.045
12
= 0.00375, and
N = 1230 = 360. Your monthly payments, x, are
x = F0
[1 (1 ) ]N
r
r  
= 500,000 360
0.00375
[1 (1.00375) ]
= 2,533.43.
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After 20 years, (n = 240 months, N  n = 120 months) your
principal outstanding is
F240 =
x
r
[1  (1 + r)(N  n)] =
5
2 3
0
5
.
3
0
, 3 .
037
4 [1  (1.00375)120]
= 244,448.95.
Your principal payment that month would be
1)1(  nNr
x = 121(1.
2
003
4
75
3. 3
)
,53 =1,610.71,
and your interest payment is
2,533.43  1.610.71 = 922.72.
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(Working recursively, in month 241, your interest payment
would be 244,448.950.00375 = 916.68, etc.)
Alternatively, the interest payment is
rF239 = x[1 – (1 + r)121]
= 2,533.43[1 – (1.00375)121] = 922.72.
(Note that at this stage we have paid a total of
2,533.43×240 = $608,023.20.)

Example 2 (read on own): Suppose you have a 30 year
$200,000 mortgage with a rate of 9% (i.e., 9% per year
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compounded monthly.) Then r = 0 09
12
. = 0.0075, and N =
1230 = 360. Your monthly payments, x, are
x = F0
[1 (1 ) ]N
r
r  
= 200,000
])0075.1(1[
0075.0
360
= 1,609.25.
After 20 years, (n = 240 months, N  n = 120 months) your
principal outstanding is
F240 =
x
r
[1  (1 + r)(N  n)] = 1,609.25
0.0075
[1  (1.0075)120]
= 127,036.54.
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Your principal payment that month would be
1)1(  nNr
x = 121
1,609.25
(1.0075)
= 651.58,
and your interest payment is
1,609.25  651.58 = 957.66.
(Working recursively, in month 241, your interest payment
would be 127,036.540.0075 = 952.77, etc.)
Alternatively, the interest payment is
rF239 = x[1 – (1 + r)121]
= 1609.25[1 – (1.0075)121] = 957.66.
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(Note that at this stage we have paid a total of
1609.25×240 = $386,220.00.)

Adjustable-Rate Mortgages (ARMs)
ARMs allow interest payments to be reset at periodic
intervals. The interest rates are determined by an index,
such as the one-year Treasury rate or the Cost-of-Funds
Index (COFI). We saw floating-rate debt in Chapter 16.
In this chapter we only discuss Fixed-Rate Mortgages
(FRMs).
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PREPAYMENTS
NOTE: the prepayment option results in “timing” risk for the
investors in the mortgages.
Factors Affecting Prepayments
Seasonality Factor. Families tend to move in the summer
time, so prepayments are more likely to occur then.
Refinancing Incentive. If interest rates drop, there is an
incentive for the homeowner to prepay the loan and
refinance it at the lower interest rate.
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Age of the Mortgage. Early in the loan, the interest payments
are high, so a change in interest rates gives more of an
incentive to prepay the loan in years 2 to 8. Late in the
loan, say after 25 years, there may be an incentive to pay
it off in order to secure the property’s title.
Family Circumstances. Job switching, divorce, etc. may
cause a family to move. Loss of a job or disability may
result in default.
NOTE 1: if the mortgage is assumable, then when one family
moves, the next family that moves in can assume the
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mortgages. The investors are unaffected. If, though, the
mortgage is not assumable, it has to be repaid in full,
which results in prepayments.
NOTE 2: if the mortgage is insured, then in case of default,
the insurer pays the investor the remaining principal on
the loan; i.e., default, in this case, has the same effect on
the investor as prepayment.
Housing Prices. If housing prices drop significantly, it may
be in the interests of the home-owner to default on the
mortgage and buy another house at a cheaper price.
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Mortgage Status (Premium Burnout). A drop in interest rates
tends to lead to more prepayments, but it has been found
that after some prepayments, things tend to stabilize; i.e.,
a number of mortgages remain outstanding. This is
called premium burnout.

Measures of the Probability of Prepayment
NOTE: for an individual homeowner, prepayments are (often)
“all-or-nothing”; that is, they either pay their entire
principal outstanding (and move, change banks, or
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whatever) or they continue with their constant monthly
payments. Investors in mortgage-backed securities,
however, only care about average (aggregate)
prepayments. So, we’ll model prepayments as if a small
amount is prepaid each month (since only a small number
of homeowners prepay their total principal outstanding in
a given month).
There are three measures of prepayments that we’ll look at.
1. Constant Monthly Mortality. In this model, we assume
that there is a constant probability, called the single
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monthly mortality rate, or SMM, that the mortgage will
be retired in a given month. The probability that the
mortgage will survive for one month is 1 – SMM; the
probability it will survive for one year is (1  SMM)12.
Usually annual probabilities are estimated first. We call the
annual prepayment rate the conditional prepayments rate
(CPR) = the probability of prepayment (i.e., of retiring
the loan) in a given year.
Note that the probability of the mortgage surviving for one
year is (1  SMM)12 = (1  CPR); so,
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SMM = 1  (1  CPR)
1
12 .
We need the SMM to understand our monthly cash-flows.
Example: If the CPR is 12%, then SMM = 1.0596%.
If SMM = 1.0596%, then the probability that the mortgage
survives for 30 months is
(1  SMM)30 = (1  0.010596)30 = (0.989404)30 = 0.72645.
The probability of the mortgage surviving for 30 months and
then being retired in month 31 is
(1  SMM)30SMM = (0.72645)(0.010596) = 0.00770.
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In general, the probability of the mortgage surviving for n
months and then be retired in month n + 1 is
(1  SMM)nSMM.

2. FHA Experience. The Federal Home Administration has
estimates of
xt = the probability that the mortgage will survive to the end
of year t.
Let pt = the probability that the mortgage is retired during
year t. Then
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pt = Prob(survive to end of year t  1)
 Prob(survive to end of year t)
= xt1  xt,
so that xt = xt1 – pt.
One can also show that the conditional probability of
surviving through the end of year t, given that it has
survived until end of year t  1 is (call this conditional
probability yt):
yt =
)1yearofenduntilsurvive(
)yearofenduntilsurvive(
tProb
tProb
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= x
x
t
t1
= 1  p
x
t
t1
.
[In general, ( and )( | )
( )
P A BP A B
P B
 .]
Example: Suppose a mortgage has a probability of 0.60 of
surviving for 15 years and a probability of 0.54 of
surviving for 16 years. (Then, x15 = 0.60, and x16 = 0.54.)
The probability that the mortgage is retired during the 16th
year is p16 = 0.60  0.54 = 0.06.
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Given that the mortgage has survived the first 15 years, the
(conditional) probability that it will be retired in the 16th
year is 1  y16 = 1 
054
0 60
.
.
= 1  0.90 = 0.10.
To see this, note that
y16 = (survive 16 yrs. survive15)Prob
=
)15yearofenduntilsurvive(
)16yearofenduntilsurvive(
Prob
Prob
=
15
16
x
x
= 054
0 60
.
.
= 0.90;
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( 16 yrs. survive15)Prob retire = 1 – y16 = 1 
15
16
x
x

=
60.0
06.0 = 0.10.

3. The PSA Experience (Public Securities Association)
The idea here is that the probability of prepayment per month
increases (linearly) for the first 30 months, and then
levels off. As a benchmark, they assume that it levels off
at 6% per year, i.e., the CPR = probability of prepayment
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per year is = 6% per year after the first 30 months. This
is called 100% PSA. If you use a higher level rate, say
9% it’s called 150% PSA.







time (months)
CPR
6%
30
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In general (for 100% PSA)
Months 1 to 30: CPR = 6% n
30
per year,
where n is the number of months since the loan began, and
Months > 30: CPR = 6% per year.
The monthly probabilities (or rates) of prepayment, SMM,
are again just given by SMM = 1  (1  CPR)
1
12 .

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Mortgage Cash Flows with Prepayments
We now use these probabilities of prepayment to calculate
expected cashflows. Recall that for an individual
homeowner, it’s (usually) “all-or-nothing”; i.e., either
there is no pre-payment, or they prepay the total principal
outstanding. Investors in MBS’s, however, care about
aggregate payments. When many similar mortgage loans
are combined in the same pool, there is a “law of large
numbers” effect at work and prepayments can be
predicted “on average”. Here’s how it works.
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The CPR is given by
CPR1 = 6%
1
30
= 0.2%, CPR2 = 6%
2
30
= 0.4%,
CPR3 = 0.6%, CPR4 = 0.8%, CPR5 = 1.0%,…,
CPR30 = 6% = CPR31, etc.
But, these are annual rates; the corresponding monthly rates are
SMM1 = 1  (1  CPR1)
1
12 = 1  (1  0.002)
1
12 = 0.0001668.
SMM2 = 1  (1  0.004)
1
12 = 0.0003339; SMM3 = 0.0005014;
SMM4 = 0.0006691, SMM5 = 0.0008372 (per month).
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You can check that SMM30 = 0.005143 per month = SMM31,
etc.
Example: Now, suppose you invest $200,000 in a 30 year
MBS with a rate of 8% (i.e., 8% per year compounded
monthly.)
Your monthly payments, x, are given by
x = 0
[1 (1 ) ]N
F r
r 

 
,
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where r = 0 08
12
. , N = 360 months, and F0 = 200,000.
Check that x = 1,467.53. This would be the payment per
month if there were no prepayment.
Now, for each month, we need to calculate 1. the interest
payment, 2. the principal payment and 3. the expected
prepayments.

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Month 1: Interest payment = 200,0000 08
12
. = 1,333.33.
(Scheduled) Principal pmt = x  Int. pmt
= 1,467.53  1,333.33 = 134.20.
(Expected) Prepayments at month 1 are computed by
applying SMM to the remaining principal:
Expected Prepayment
= Probability of prepaymentamount of prepayment
= SMM1Principal outstanding
= 0.0001668(200,000  134.20) = 33.34.
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(NOTE 1: these are expected prepayments, which should hold
on average. NOTE 2: if the SMM1 = 0.0001668, then this
means that out of 10,000 mortgages, roughly 2 ≈
10,000SMM1 homeowners prepay their total principal
outstanding. The remaining homeowners keep making
their constant monthly payments.)
Total Cash Flow: 1,467.53 + 33.34 = 1,500.87.
Month 2: The principal outstanding is now
F0 – Scheduled Princ. pmt – Expected Princ. Pre-pmt
= 200,000  134.20  33.34 = 199,832.46
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and the loan has 359 months remaining. The new
mortgage payment is
x = 199,832.46
])1(1[ 359 r
r = 1,467.28 < 1,467.53.
Again, for an individual homeowner, x remains constant.
These calculations are taken from the MBS investor’s
point of view and only hold on average.
That breaks down into
Interest payment = 199,832.460 08
12
. = 1,332.22.
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(Scheduled ) Principal pmt = 1,467.28  1,332.22
= 135.07.
(Expected) Prepayment = SMM2(199,832.46  135.07)
= 0.0003339(199,832.46  135.07) = 66.69.
Total Cash Flow = 1,467.28 + 66.69 = 1,533.97.
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Month 3:
Principal Outstanding = 199,832.46  135.07  66.69 =
199,630.71. Check that
Mortgage Payment = 1466.79
Interest Payment = 1330.87
Scheduled Principal Pmt = 135.92
Expected Principal Prepayment = 100.02.
Total Cash Flow = 1466.79 + 100.02 = 1566.81.
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ASIDE: for those who are interested.
MORTGAGE-BACKED SECURITIES (MBS)
Cash Flows and Market Conventions
Amount of Cash Flow. Generally, due to fees, the investor
gets 50 basis points less than the coupon of the loan
portfolio.
Timing of Cash flow. Payments to investors are made on the
15th of the month.
Market Conventions.
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GNMAs are quoted in 1
32
much like T-bonds.
The pool factor, pf(t) is
pf(t) =
B
P
t ,
where Bt is the par value remaining at date t and P is the
original par value.
Example: Consider a $100 million par value of GNMA
issued some time ago. The pool factor is now 0.8, and
the quoted price is “958” = 95 8
32
= 95.25. An investor
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owns 15 million original par of this GNMA. The market
value is
Investor’s Par value remaining = 15  pf(t) = 15  0.8 = 12.
Quoted Market Value = 12  0.9525 = 11.43 million.
Accrued Interest is given by
ait =
SD M
30
 c  1
12
 Bt,
where SD is the settlement date, M is the first day of the
month within which t falls, Bt is the principal balance,
and c is the annual coupon rate.
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Example: (continued) Suppose that, in the previous
example, c = 10% per year, and SD  M = 14 days. Then
ait =
14
30
 0.10  1
12
 12 million, = 46,666.67.
So, the cash price would be 11.43 million + 46,666.67.
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VALUATION FRAMEWORK
At any given time, a homeowner has several options: do not
refinance, prepay the mortgage, default on the loan.
The two main variables affecting these decisions are
 Interest rates
 House price(s)
If interest rates drop sufficiently, the homeowner has an
incentive to: Prepay the mortgage? Default? Do not
refinance?

12 - 42
For the investor, this means that the value of the MBS will
tend to have an upper? lower? bound.

This is called negative convexity or compression to par.
Here’s the graph:
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What is par in this case?
Bond

MBS

Par

Price

Yield

12 - 44

If housing prices drop significantly, the homeowner has an
incentive to: Prepay the mortgage? Default? Do not
refinance?

MORTGAGE DERIVATIVES
Strips
There are two types of mortgage strips.
Interest-Only (IO) Strip. In this case, the interest payments
are stripped from the mortgage and sold separately from
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the principal payments. If total prepayment is made,
interest payments cease, so the IO strip becomes
worthless. This means that as interest rates decrease, the
probability of prepayment increases, and so the value of
the IO strip decreases. (Usually if r , then the price .)
If r causes the price to , what can we say about an IO
strip’s duration?

Principal-Only (PO) Strip. PO strips receive all of the
principal payments from the mortgage and none of the
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interest. Here, if r , then probability of prepayment ;
prepayment means the holder receives par sooner, which
is good news for them; the value of the PO strip goes up.
One way to see this is to think of the MBS as a portfolio
of the IO strip and the PO strip, and then use the formula
for the duration of a portfolio:
(1 )MBS IO POD xD x D   .
If DMBS > 0 and DIO < 0, then we must have DPO > 0.
Note that the duration of an IO strip is not always negative; it
may be positive if there is little chance of re-financing
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occurring. But, it is possible for its duration to become
negative.
COLLATERALIZED MORTGAGE OBLIGATIONS
(CMOs)
A CMO is similar to a CDO in a few respects. Like a CDO,
a CMO begins with a portfolio of assets. The difference
is that, with a CMO the tranches are based on mortgage
repayments rather than defaults.
A typical CMO has four tranches. Each tranche receives
coupon payments, but the first tranche initially receives
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all the prepayments until the tranche is retired. The
second tranche (which has been receiving coupon
payments all along) begins receiving prepayments after
the first tranche is retired. The second tranche then
continues to receive coupon payments and prepayments
until it is retired, and so on. (The fourth tranche is
sometimes called the Z tranche.)
This procedure reduces the timing risk for investors.
Note that CMOs still earn a risk premium.

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A VALUATION MODEL (Monte Carlo)
The procedure used to value MBSs involves the following
steps.
1. Specify an interest rate process. (See Options#3.)
2. Specify a model of prepayments. This may involve such
factors as interest rates (prepayment incentive) the age of
the mortgage, housing prices, and premium burnout.
3. Run a Monte Carlo simulation to simulate interest-rate
paths and (random) prepayment times. For the i th
simulation, the monthly interest rates will be r1(i),
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r2(i), ... , rN(i). (Usually, N = 360, i.e., the mortgage is a
30-year mortgage.) Using these one-period rates we
compute the zero rate for n periods in path i, denoted
zn(i), by
   0 0,1 ( ) exp ( ) ( ) ( )nn n nz i r i t r i t B i       ,
where 112t  , and 0, ( )nB i is the money-market account,
as discussed in Options#3.
4. For each month along each path, identify the total cash
flows, denoted C1(i), C2(i), ... , CN(i). If prepayment
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occurs in month n, say, then at that date you’ll get one
more interest payment plus the principal outstanding; all
cash flows in later months will be equal to zero, i.e., Cj(i)
= 0 for all j > n.
5. Next, we calculate the present value of each cash flow
using the randomly generated interest rates:
P(i) = C i
z i
1
11
( )
( )
+ C i
z i
2
2
21
( )
[ ( )]
+ ... + C i
z i
N
N
N
( )
[ ( )]1
.
Note that if we use 360 steps, then we need 360 randomly
generated standard normal random variables here.
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6. The calculation in Step 5 above just represents one
discounted payoff from one sample path. Repeat this
process for a number of paths (usually thousands) and for
each path, the price is determined. Call the number of
paths simulated M.
Again, if each path uses 360 steps, and we use 1000
paths, we are generating 360,000 random variables.
7. Compute the average of all the discounted payoffs.
(Variance reduction procedures can also be applied.) The
price given by the model, Pmodel is then
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Pmodel =
1
1M
P i
i
M
( )

 .
As usual, this average is our estimate of the (risk-neutral)
discounted expectation, which is the price. We can now
compare our model value, Pmodel, to the market value, V,
of the security. Call the difference (i.e., the error)
  e = Pmodel  V.
If e > 0, (i.e., Pmodel > V) then the model price is greater than
the market value, so we believe the security is cheap
(under-priced).
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On the other hand, we can set Pmodel = V by increasing the
discount factor, i.e. we select a z > 0 and add z to each of
the zero rates (in each of the simulations) z1(i), z2(i), ...,
z360(i), until
V = Pmodel
= 










M
i ziz
iC
ziz
iC
M 1 360360
360
1
1
])(1[
)(
)(1
)(1 
This factor z is referred to as the option-adjusted spread
(OAS). A positive OAS indicates that the security is
12 - 55
cheap (i.e., under-priced); a negative OAS indicates that
the security is rich (i.e., overpriced).