Lecture slides mortgage_models_mooc_masterslides

Financial Engineering & Risk Management
Introduction to Mortgage Mathematics and Mortgage Backed
Securities
M. Haugh G. Iyengar
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
bonds was $35.3 trillion:
Government $10.7 30.4%
Municipal $3.7 10.5%
Mortgage $8.2 23.3%
Corporate $8.6 24.3%
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)
by 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
Xk−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 fair
mortgage value as
F0 =
Xn
k=1
B
(1 + r)k
= c(1 + c)nM0
(1 + c)n − 1 ×
(1 + r)n − 1
r(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 break 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. Iyengar
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
behavior.
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 or
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
remaining 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 expressed as a percentage of the current
outstanding principal level in the underlying mortgage pool.
Definition. The single-month mortality rate (SMM) is the CPR converted to a
monthly rate assuming monthly compounding.
The CPR and SMM are therefore related by
SMM = 1 − (1 − CPR)1/12
CPR = 1 − (1 − SMM)12.
6

Prepayment Conventions
In practice, the CPR is stochastic and depends on the mortgage pool and other
economic variables.
However, market participants often use a deterministic prepayment schedule as
a mechanism to quote MBS yields and so-called option-adjusted spreads etc.
The standard benchmark is the Public Securities Association (PSA) benchmark.
The PSA benchmark assumes the following for 30 year mortgages:
CPR =

6% × (t/30), if t 30
6%, if t 30.
where t is the number of months since the mortgage pool originated.
Slower or faster prepayment rates are then given as some percentage or multiple
of PSA.
7

The Average Life of an MBS
Given a particular prepayment assumption the average life of an MBS is defined
as
Average Life =
XT
k=1
kPk
12 × TP (6)
– where Pk is the principal (scheduled and projected prepayment) paid at
time k
– TP is the total principal amount
– T is the total number of months
– and we divide by 12 so that average life is measured in years.
It is immediate that the average life decreases as the PSA speed increases.
8

Mortgage Yields
In practice the price of a given MBS security is observed in the market place
and from this a corresponding yield-to-maturity can be determined.
This yield is the interest rate that will make the present value of the
expected cash-flows equal to the market price.
The expected cash-flows are determined based on some underlying
prepayment assumption such as 100 PSA, 300 PSA etc.
– 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 called a bond-equivalent yield.
Yields are clearly very limited when it comes to evaluating an MBS and
indeed fixed-income securities in general.
Indeed the option-adjusted-spread (OAS) is the market standard for quoting
yields on MBS and indeed other fixed income securities with embedded
options.
9

Prepayment Risks for Mortgage Pass-Throughs
An investor in an MBS 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, a pass-through investor is 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
– this is contraction risk.
When interest rates increase, prepayments tend to decrease and so there is less
prepaid principal that can be invested at the higher rates
– this is extension risk.
10

Principal-Only and Interest-Only MBS
M. Haugh G. Iyengar
Columbia University

Principal-Only and Interest-Only MBS
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.
Now recall our earlier expression for Mk:
Mk = (1 + c)kM0 − B
(1 + c)k − 1
c

. (7)
Using (7), we therefore obtain
Pk = B − c

(1 + c)k−1M0 − B
(1 + c)k−1 − 1
c

= (B − cM0) (1 + c)k−1.
3

Principal-Only MBS
The present value, V0, of the principal payment stream is therefore given by
V0 = (B − cM0) (1 + r)n − (1 + c)n
(r − c)(1 + r)n (8)
where, as before, r, is the per-period risk-free interest rate.
Can use l’Hôpital’s rule to check that
lim
c!r
V0 = n(B − rM0)
1 + r .
Now recall our earlier expression:
B = c(1 + c)nM0
(1 + c)n − 1 . (9)
If we use (9) to substitute for B in (8) then we obtain
V0 = cM0
(1 + c)n − 1 ×
(1 + r)n − (1 + c)n
(r − c)(1 + r)n . (10)
4

Principal-Only MBS
In the case r = c (10) reduces to
V0 = rnM0
(1 + r) [(1 + r)n − 1] . (11)
It is clear that the earlier mortgage payments comprise of interest payments
rather than principal payments
– only later in the mortgage is this relationship reversed.
Indeed this property is reflected in the fact that
lim
n!1
V0 = 0.
5

Interest-Only MBS
We can also compute the present value, W0 say, of the interest payment
stream
– again assuming there are no mortgage prepayments.
To do this we could compute the sum
W0 =
Xn
k=1
Ik
(1 + r)k
– but much easier to recognize that the sum of the principal-only and
interest-only streams must equal the total value of the mortgage, F0.
Now recall our earlier expression for F0:
F0 = c(1 + c)nM0
(1 + c)n − 1 ×
(1 + r)n − 1
r(1 + r)n . (12)
6

Interest-Only MBS
Since W0 = F0 − V0 we can use (12) and (10) to obtain
W0 = cM0
[(1 + c)n − 1](1 + r)n

(1 + c)n (1 + r)n − 1
r −
(1 + r)n − (1 + c)n
r − c

.
Moreover when r ! c it is easy to check that this reduces to
W0 = M0 −
rnM0
(1 + r) [(1 + r)n − 1]
as expected from (11) and since F0 = M0 when r = c.
7

Risks of Principal-Only and Interest-Only MBS
M. Haugh G. Iyengar
Columbia University

Duration of the Principal-Only MBS
Again we assume no prepayments and consider the durations of the PO and IO
cash-flow streams.
The duration of a cash-flow is a weighted average of the times at which each
component of the cash-flow is received
– a standard measure of the risk of a cash-flow.
It should be clear that the principal stream has a longer duration than the
interest stream.
If we let DP denote the duration of the principal stream, then it is given by
DP = 1
12V0
Xn
k=1
k Pk
(1 + r)k (13)
where we divide by 12 in (13) to convert the duration into annual rather than
monthly time units.
3

Duration of the Interest-Only MBS
Similarly, we can compute the duration, DI , of the interest-only stream as
DI = 1
12W0
Xn
k=1
k Ik
(1 + r)k
= 1
12W0
Xn
k=1
k (B − Pk)
(1 + r)k
= 1
12W0
Xn
k=1
k B
(1 + r)k −
V0
W0
DP. (14)
4

Principal-Only and Interest-Only MBS in Practice
To this point we have assumed that prepayments do not occur.
But this is not realistic: in practice pass-throughs do experience prepayments and
the PO and IO cash-flows must reflect these prepayments correctly.
But this is straightforward: the interest payment in period k is simply, as before,
Ik := cMk−1
where Mk−1 is the mortgage balance at the end of period k − 1.
Mk must now be calculated iteratively on a path-by-path basis as
Mk = Mk−1 − ScheduledPrincipalPaymentk − Prepaymentk,
for k = 1, . . . , n and where ScheduledPrincipalPaymentk is now the scheduled
principal payment (adjusted for earlier prepayments) in period k.
5

The Risks of PO and IO MBS
The risk profiles of principal-only and interest-only securities are very different
from one another.
The principal-only investor would like prepayments to increase.
The interest-only investor wants prepayments to decrease
– after all the IO investor only earns interest at time k on the mortgage
balance remaining at time k.
In fact the IO security is that rare fixed income security whose price tends to
follow the general level of interest rates:
– when interest rates fall the value of the IO security tends to decrease
– and when interest rates increase the expected cash-flow increases due to
fewer prepayments but the discount factor decreases
– the net effect can be a rise or fall in value of the IO security.
Question: What happens to the value of a PO security when interest rates (i)
increase and (ii) decrease?
6

Collateralized Mortgage Obligations (CMOs)
M. Haugh G. Iyengar
Columbia University

Collateralized Mortgage Obligations (CMOs)
Collateralized mortgage obligations (CMOS) are mortgage-backed securities that
have been created by redirecting the cash-flows from other mortgage securities
– 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 but they can also be
created from other MBS including, for example, principal-only MBS.
There are many types of CMOs including
- sequential CMOs
- CMOs with accrual bonds
- CMOs with floating-rate and inverse-floating-rate tranches
- planned amortization class (PAC) CMOs.
We’ll briefly describe sequential CMOs.
2

Sequential CMOs
The basic structure of a sequential CMO is that there are several tranches which
are ordered in such a way that they are retired sequentially.
For example, the payment structure of a sequential CMO with tranches A, B, C
and D might be as follows:
Sequential CMO Payment Structure
1. Periodic coupon interest is disbursed to each tranche on the basis of the
amount of principal outstanding in the tranche at the beginning of the
period.
2. 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 disbursed to
tranche B until it is paid off entirely. After tranche B has been paid off all
principal payments are disbursed to tranche C until it is paid off entirely.
After tranche C has been paid off all principal payments are disbursed to
tranche D until it is paid off entirely.
3

Pricing Mortgage-Backed-Securities
M. Haugh G. Iyengar
Columbia University

Prepayment Modeling
In many respects, the prepayment model is the most important feature of
any residential MBS pricing engine.
Term-structure models are well understood in the financial engineering
community
– but this is not true of prepayment models.
The main problem is that there is relatively little publicly available
information concerning prepayments rates
– so very difficult to calibrate prepayment models.
One well known publicly available model is due to Richard and Roll (1989)
– they model the conditional prepayment rate (CPR) whose definition
we recall:
Definition. The CPR is the rate at which a given mortgage pool prepays. It is
expressed as a percentage of the current outstanding principal level in the
underlying mortgage pool.
2

A Particular Prepayment Model (Richard and Roll 1989)
The model of Richard and Roll assumes
CPRk = RIk × AGEk ×MMk × BMk
where
- RIk is the refinancing incentive
e.g. RIk = .28 + .14 tan−1 (−8.57 + 430 (WAC − rk(10))) (15)
where rk(10) is the prevailing 10-year spot rate at time k.
- AGEk is the seasoning multiplier, e.g. AGEk = min (1, t/30)
- MMk is the monthly multiplier.
- BMk is the burnout multiplier
e.g. BMk = .3 + .7 Mk−1
M0
where Mk is the remaining principal balance at time k.
3

Choosing a Term Structure Model
We also need to specify a term-structure model in order to fully specify the
mortgage pricing model.
The term structure model will be used to:
(i) discount all of the MBS cash-flows in the usual risk-neutral pricing framework
(ii) to compute the refinancing incentive according to (15), for example.
Whatever term-structure model is used, it is important that we are able to
compute the relevant interest rates analytically
– for example, r10(k) in the prepayment model of Richard and Roll.
Such a model would first need to be calibrated to the term structure of
interest rates in the market place as well as liquid interest rate derivatives.
The actual pricing of MBS then requires Monte-Carlo simulation
– very computationally intensive
– analytic prices not available.
4

The Financial Crisis
The so-called sub-prime mortgage market played an important role in the
financial crisis of 2008-2009.
Sub-prime mortgages are mortgages that are issued to home-owners with very
weak credit
– the true credit quality of the home-owners was often hidden
– the mortgages were often ARMs with so-called teaser rates
– very low initial mortgage rates intended to “tease” the home-owner into
accepting the mortgage.
The financial engineering aspect of the MBS-ABS market certainly played a role
in the crisis
– particularly when combined with the alphabet soup of CDO’s and
ABS-CDO’s
– these products are too complex and too difficult to model.
But there were many other causes including: moral hazard problem of mortgage
brokers, ratings agencies, bankers etc, inadequate regulation, inadequate risk
management and poor corporate governance.
5

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