CHAPTER 18
ANALYSIS OF RESIDENTIAL
MORTGAGE-BACKED SECURITIES
CHAPTER SUMMARY
There are two approaches to the analysis of residential mortgage-backed securities including
pass-throughs, collateralized mortgage obligations (CMOs), and stripped mortgage-backed
securities. They are: (1) the static cash flow yield methodology, and (2) the MonteCarlo
STATIC CASH FLOW YIELD METHODOLOGY
The static cash flow yield methodology begins with the computation of the cash flow yield
measure used for pass-throughs and based on some prepayment assumption.
Vector Analysis
Limitations of the Cash Flow Yield
The same shortcomings found in the yield to maturity approach are also present in application of
the cash flow yield measure: (i) the projected cash flows are assumed to be reinvested at the cash
Yield Spread to Treasuries
The yield for a RMBS will depend on the actual prepayment experience of the mortgages in the
pool. Nevertheless, the convention in all fixed-income markets is to measure the yield on
Static Spread
The static spread is the yield spread in a static scenario (i.e., no volatility of interest rates) of the
bond over the entire theoretical Treasury spot rate curve, not a single point on the Treasury yield
curve.
Effective Duration
Modified duration is a measure of the sensitivity of a bond’s price to interest-rate changes,
assuming that the expected cash flow does not change with interest rates. Modified duration is
consequently not an appropriate measure for mortgage-backed securities, because prepayments
To illustrate the effective duration calculation, consider tranche data: P_ = 102.1875; P+ =
98.4063; P0 (initial price) = 100.2813; Δy= 0.0025. Substituting into the duration formula yields
modified duration (with P_ = 102.1875) =
( )
0
2y
PP
P
−+
−
=
)0025.0)(2813.100(2
4063.981875.102 −
= 7.54.
To further illustrate the effective duration calculation, consider tranche data: P_ = 101.9063 (at
200 PSA; basis points decrease); P+= 98.3438 (at 150 PSA; basis point increase); P0 = 100.2813;
Δy= 0.0025. Substituting into the duration formula yields
effective duration (P_ = 101.9063) =
( )
0
2y
PP
P
−+
−
=
)0025.0)(2813.100(2
3438.989063.101 −
= 7.11.
Effective Convexity
To illustrate the convexity formula, consider the above tranche data. The standard convexity is
approximated as follows:
( )
0
2
0
2
+−
+−
P P P
Py
=
( )
( )( )
2
98 4063 102 1875 2100 2813
100 2813 0 0025
+−. . .
..
= 24.930.
( )
2
0
Py
( )( )
2
100 2813 0 0025
..
The standard convexity indicates positive convexity, whereas the effective convexity indicates
they have negative convexity. The difference is even more dramatic for bonds not trading near
par.
Prepayment Sensitivity Measure
The value of a RMBS will depend on prepayments. To assess prepayment sensitivity, market
participants have used the following measure: the basis point change in the price of an RMBS for
a 1% increase in prepayments. Specifically, we have:
prepayment sensitivity = (Ps– P0)100
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wherePs= price (per $100 par value) assuming a 1% increase in prepayment speed and P0 =
initial price (per $100 par value) at assumed prepayment speed.
Notice that a security that is adversely affected by an increase in prepayment speeds will have
a negative prepayment sensitivity while a security that benefits from an increase in prepayment
speed will have a positive prepayment sensitivity.
MONTE CARLO SIMULATION METHODOLOGY
For some fixed-income securities and derivative instruments, the periodic cash flows are path
dependent. This means that the cash flows received in one period are determined not only by the
current and future interest-rate levels but also by the path that interest rates took to get to the
current level.
Valuation modeling for CMOs is similar to valuation modeling for pass-throughs, although the
difficulties are amplified because the issuer has sliced and diced both the prepayment risk and
the interest-rate risk into smaller pieces called tranches. The sensitivity of the pass-throughs
Using Simulation to Generate Interest-Rate Paths and Cash Flows
The typical model that Wall Street firms and commercial vendors use to generate random
interest-rate paths takes as inputs today’s term structure of interest rates and a volatility
assumption. The term structure of interest rates is the theoretical spot rate (or zero-coupon) curve
implied by today’s Treasury securities. The volatility assumption determines the dispersion of
monthly interest rates are used to discount the projected cash flows in the scenario. The mortgage
refinancing rate is needed to determine the cash flow because it represents the opportunity cost
the mortgagor is facing at that time.
Calculating the Present Value for a Scenario Interest-Rate Path
Given the cash flow on an interest-rate path, its present value can be calculated. The discount
rate for determining the present value is the simulated spot rate for each month on the
12
T
where zT(n) = simulated spot rate for month T on path n, and fj(n) = simulated future one-month
rate for month j on path n.
The present value of the cash flow for month T on interest-rate path n discounted at the simulated
spot rate for month T plus some spread is
Determining the Theoretical Value
The present value of a given interest-rate path can be thought of as the theoretical value of
a pass-through assuming that the path was actually realized. The theoretical value of the
pass-through can be determined by calculating the average of the theoretical value of all the
interest-rate paths.
dispersion of the path values then the investor is warned about the potential variability of the
model’s value.
Simulated Average Life
Option-Adjusted Spread
The option-adjusted spread is a measure of the yield spread that can be used to convert dollar
differences between value and price. It represents a spread over the issuer’s spot rate curve or
Option Cost
The implied cost of the option embedded in any RMBS can be obtained by calculating the
difference between the OAS at the assumed volatility of interest rates and the static spread:
Effective Duration and Convexity
Effective duration and effective convexity can be calculated using the Monte Carlo method as
follows. First, the bond’s OAS is found using the current term structure of interest rates. Next,
Selecting the Number of Interest-Rate Paths
Let’s now address the question of the number of scenario paths or repetitions, N, needed to value
a RMBS. A typical OAS run will be done for 512 to 1,024 interest-rate paths.
The number of interest-rate paths determines how “good” the estimate is, not relative to the truth
but relative to the model. The more paths, the more average spread tends to settle down. It is
a statistical sampling problem.
Limitations of the Option-Adjusted Spread Measures
The OAS is a product of the valuation model. The valuation model may be poorly constructed
because it fails to capture the true factors that affect the value of particular securities. In Monte
Carlo simulation the interest-rate paths must be adjusted so that on-the-run Treasuries are valued
Illustration
We can use a plain vanilla deal (i.e., relatively simple derivative instrument with standard
features) to show how CMOs can be analyzed using the Monte Carlo simulation method. Using
TOTAL RETURN ANALYSIS
Neither the static cash flow methodology nor the Monte Carlo simulation methodology will tell
a money manager whether investment objectives can be satisfied. The performance evaluation of
Horizon Price for CMO Tranches
The most difficult part of estimating total return is projecting the price at the horizon date. In the
case of a CMO tranche the price depends on the characteristics of the tranche and the spread to
Option-Adjusted SpreadTotal Return
The total return and OAS frameworks can be combined to determine the projected price at the
horizon date. At the end of the investment horizon, it is necessary to specify how the OAS is
expected to change. The horizon price can be “backed out” of the Monte Carlo simulation model.
KEY POINTS
• There are two methodologies commonly used to analyze all RMBS (agency and nonagency):
cash flow yield methodology and Monte Carlo simulation methodology.
• The cash flow yield is the interest rate that will make the present value of the projected cash
flow from an RMBS equal to its market price. The cash flow yield assumes that (1) all the
cash flows can be reinvested at a rate equal to the cash flow yield, (2) the RMBS is held to the
• A methodology used to analyze path-dependent cash flow securities is the Monte Carlo
simulation. This methodology involves randomly generating many scenarios of future
interest-rate paths, where the interest-rate paths are generated based on some volatility
assumption for interest rates. The random paths of interest rates should be generated from an
arbitrage-free model of the future term structure of interest rates. The Monte Carlo simulation
methodology applied to RMBS involves randomly generating a set of cash flows based on
simulated future mortgage refinancing rates.
• The effective duration and effective convexity are calculated using the Monte Carlo
simulation methodology by holding the OAS constant and shifting the term structure up and
down.
• Total return is the correct measure for assessing the potential performance of CMO tranches
over a specified investment horizon.
ANSWERS TO QUESTIONS FOR CHAPTER 18
(Questions are in bold print followed by answers.)
1. Suppose you are told that the cash flow yield of a pass-through security is 11% and that
you are seeking to invest in a security with a yield greater than 10.7%. Answer the below
questions.
(a) What additional information would you need to know before you might invest in this
pass-through security?
To determine your chances of actually getting a 10.7% return, you would want to know the types
of securities backing the pass-through, the safety of the cash flows, and the expected volatility of
the cash flows.
If applicable, you would want to know the prepayment rate for a particular tranche formed from
the pass-through security as well as any stipulated guarantees in terms of interest and principal
payments. One should note that the greater the discount assumed to be paid for a tranche, the
more a tranche will benefit from faster prepayments. The converse is true for a tranche for which
a premium is paid. The faster the prepayments, the lower the cash flow yields.
(b) What are the limitations of the cash flow yield for assessing the potential return from
investing in a RMBS?
The limitations for the cash flow yield in a RMBS are like the yield to maturity for a bond where
it is assumed that the coupon payments can be reinvested at a rate equal to the yield to maturity
and the bond is held to maturity. These shortcomings are equally present in application of the
cash flow yield measure: (i) the projected cash flows are assumed to be reinvested at the cash
2. Using the cash flow yield methodology, a spread is calculated over a comparable Treasury
security. How is a comparable Treasury determined?
The repayment of principal over time makes it inappropriate to compare the yield of a RMBS to
a Treasury of a stated maturity. Instead, market participants have used two measures: Macaulay
duration and average life.
3. What is vector analysis?
One practice that market participants use to overcome the drawback of the PSA benchmark is to
assume that the PSA speed can change over time. This technique to deal with this drawback is
4. In the calculation of effective duration and effective convexity, why is a prepayment model
needed?
5. The following excerpt is taken from an article titled “Fidelity Eyes $250 Million Move
into Premium PACs and I-Os” that appeared in the January 27, 1992, issue of BondWeek,
pp. 1 and 21:
“Three Fidelity investment mortgage funds are considering investing this quarter a
total of $250 million in premium planned amortization classes of collateralized mortgage
obligations and some interest-only strips, said Jim Wolfson, portfolio manager … Wolfson
… will look mainly at PACs backed by 9-10% Federal Home Loan Mortgage Corp. and
Answer the below questions.
(a) Why would premium PACs and interest-only strips offer higher yields if the market
expects that prepayments will accelerate or are highly uncertain?
Prepayments will be expected to accelerate if interest rates are expected to decline or if there is
a greater possibility of decline due to general uncertainty as to which way rates will change. For
such a situation, investors would expect to deal with greater reinvestment rate risk. To
compensate for this risk, investors would have to be given higher rates of return for investing in
PAC tranches can reduce prepayment risk in a manner desired by an investor’s preference.
However, despite the redistribution of prepayment risk with sequential-pay and accrual
collateralized mortgage obligations (CMOs), there is still considerable prepayment risk. That is,
there is still considerable average life variability for a given tranche. This problem is mitigated
In early 1987, stripped MBS began to be issued where all the interest is allocated to one class
(the IO class) and the entire principal to the other class (the PO class). The IO class receives no
principal payments. IOs and POs are referred to as mortgage strips. The PO security is
purchased at a substantial discount from par value. The yield an investor will realize depends on
(b) What does Wolfson mean when he says: “You get paid in yield to take on negative
convexity”?
Negative convexity has the same impact on the price performance of a RMBS as it does on the
performance of a callable bond. When interest rates decline, a bond with an embedded call
(c) What measure is Wolfson using to assess the risks associated with prepayments?
Because Wolfson is looking at securities that can experience a decline when prepayment
increases, Wolfson wants a measure that captures this. Thus, Wolfson appears to be cognizant of
the prepayment sensitivity measure. More details on this measure are supplied below.
The value of a RMBS will depend on prepayments. To assess prepayment sensitivity, market
participants have used the prepayment sensitivity measure that determines the basis point change
6. In an article titled “CUNA Mutual Looks for Noncallable Corporates” that appeared in
the November 4, 1991, issue of BondWeek, p. 6, Joe Goglia, a portfolio manager for CUNA
Mutual Insurance Group, stated that he invests in “planned amortization class tranches,
which have less exposure to prepayment risk and are more positively convex than other
mortgage-backeds.” Is this true?
As seen below there are a lot of factors to consider before we assume that a PAC tranche will
absolutely have less exposure to prepayment risk and will be more positively convex that other
mortgage-backed securities.
The creation of a mortgage-backed security cannot make prepayment risk disappear. This is true
for both a pass-through and a CMO. Thus the reduction in prepayment risk (both extension risk
Generally, the market will take the greater convexity bonds into account in pricing them. How
much should the market want investors to pay up for convexity? If investors expect that market
7. What is a path-dependent cash flow security?
8. Why is a pass-through security a path-dependent cash flow security?
In the case of mortgage pass-through securities, prepayments are path dependent because this
month’s prepayment rate depends on whether there have been prior opportunities to refinance
9. Give two reasons why a CMO tranche is a path-dependent cash flow security.
Pools of pass-throughs are used as collateral for the creation of collateralized mortgage
10. Explain how, given the cash flow on the simulated interest-rate paths, the theoretical
value of a RMBS is determined.
Given the cash flow on an interest-rate path, its present value can be calculated. The discount
rate for determining the present value is the simulated spot rate for each month on the
interest-rate path plus an appropriate spread. The spot rate on a path can be determined from the
for month T on interest-rate path n discounted at the simulated spot rate for month T plus some
spread is:
PV[CT(n)] =
1
()
1 ( )
T/T
T
Cn
z n K++
1 2 360
where PV[path(n)] is the present value of interest-rate path n.
The present value of a given interest-rate path can be thought of as the theoretical value of
a pass-through if that path was actually realized. The theoretical value of the pass-through can be
determined by calculating the average of the theoretical value of all the interest-rate paths. That
is, the theoretical value is equal to
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411
distribution rules of the deal.
11. Explain how, given the cash flow on the simulated interest-rate paths, the average life of
a RMBS is determined.
Given the cash flow on the simulated interest-rate paths, the average life can be determined for
each path. The average life reported in a Monte Carlo analysis is the average of the average lives
12. Suppose that a support bond is being analyzed using the Monte Carlo simulation
methodology. The theoretical value using 1,500 interest-rate paths is 85. The range for the
path present values is a low of 45 and a high of 110. The standard deviation is 20 points.
How much confidence would you place on the theoretical value of 85?
13. In a well-protected PAC structure, what would you expect the distribution of the path
present values and average lives to be compared to a support bond from the same CMO
structure?
PAC tranches are structured to meet the designs of the investor who prefers a reduction is risk.
The greater predictability of the cash flow for PAC bonds occurs because there is a principal
repayment schedule that must be satisfied. PAC bondholders have priority over all other classes
in the CMO issue in receiving principal payments from the underlying collateral. The greater
14. Suppose that the following values for a RMBS are correct for each assumption:
PSA Assumption
Value of Security
191
113.20
192
112.90
200
112.40
202
111.70
210
110.70
Assuming that the value of the security in the market is 112.40 based on 220 PSA. What is
the prepayment sensitivity of this security?
To assess prepayment sensitivity, market participants have used the following measure. This
measure determines the basis point change in the price of an RMBS for a 1% increase in
prepayments. We have: prepayment sensitivity = (Ps – P0)100
15. An analysis of a CMO structure using the Monte Carlo method indicated the following,
assuming 12% volatility:
OAS
(basis points)
Static Spread
(basis points)
Collateral
95
130
Tranche
PAC I A
50
80
PAC I B
55
85
PAC I C
80
100
PAC II
95
125
Support
75
275
(a) Calculate the option cost for each tranche.
The implied cost of the option embedded in any RMBS can be obtained by calculating the
difference between the OAS at the assumed volatility of interest rates and the static spread. We
use the below formula:
between the spread that would be earned in a static interest-rate environment (the static spread)
and the spread after adjusting for the homeowner’s option.
Below we compute the option cost for each tranche.
In general, a tranche’s option cost is more stable than its OAS in the face of market movements.
This interesting feature is useful in reducing the computational costs of calculating the OAS as
(b) Which tranche is clearly too rich?
The support tranche is rich relative to Treasuries. We might add that a typical OAS run will be
done for 512 to 1,024 interest-rate scenario paths to value a RMBS. The scenarios generated
using the simulation method look very realistic and, furthermore, reproduce today’s Treasury
(c) What would happen to the static spread for each tranche if a 20% volatility is assumed?
At the higher level of assumed interest-rate volatility of 20%, the static spread would fall because
(d) What would happen to the OAS for each tranche if a 20% volatility is assumed?
16. Why would the option-adjusted spread vary across dealer firms?
interest-rate paths will make the average present value of the paths equal to the observed market
price (plus accrued interest). The spread among dealer firms will differ to the extent interest-rate
paths and their volatility assumptions differ.
17. Explain how the number of interest-rate paths used in the Monte Carlo simulation
methodology is determined.
The number of interest-rate paths used in the Monte Carlo simulation methodology is determined
by the number of sample paths necessary to get a good statistical sample to obtain a price
estimate within a tick. More details are given below.
What is the number of scenario paths or repetitions, N, needed to value a RMBS? A typical OAS
18. Explain why you agree or disagree with the following statement: “When the Monte
Carlo simulation methodology is used to value a RMBS, a PSA assumption is employed for all
interest-rate paths.”
As seen in the illustration and Exhibit 18-9, the collateral value of a CMO is not always
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415
However, the value of the tranches is influenced so that there is a need to employ an accurate
PSA assumption to gage how the price will change when prepayment assumptions change.
19. What assumption is made about the OAS in calculating the effective duration and
effective convexity of a RMBS?
20. What are the limitations of the option-adjusted spread measure?
Although the OAS measure is much more useful than the static cash flow yield measure, it still
suffers from major pitfalls. These limitations apply not only to the OAS for RMBS but also the
OAS produced from a binomial model. First, the OAS is a product of the valuation model. The
valuation model may be poorly constructed because it fails to capture the true factors that affect
21. What assumptions are required to assess the potential total return of a RMBS?
The measure that should be used to assess the performance of a security or a portfolio over some
investment horizon is the total return. The total dollars received from investing in a RMBS
consist of (i) the projected cash flow from the projected interest payments and the projected
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416
second step requires assumption of a reinvestment rate. Finally, either of the methodologies
described in this chapter—cash flow yield or Monte Carlo simulation —can be used to calculate
the price at the end of the investment horizon under a particular set of assumptions. Either
approach requires assumption of the prepayment rate and the Treasury rates (i.e., the yield curve)
at the end of the investment horizon. The cash flow yield methodology uses an assumed spread
to a comparable Treasury to determine the required cash flow yield, which is then used to
compute the projected price. The Monte Carlo simulation methodology requires an assumed
OAS at the investment horizon. From this assumption, the OAS methodology can produce the
horizon price.
22. What are the complications of assessing the potential total return of a CMO tranched
using the total return framework?
The most difficult part of estimating total return is projecting the price at the horizon date. In the
case of a CMO tranche the price depends on the characteristics of the tranche and the spread to