1. Introduction.
A
collateralized mortgage obligation is a security backed by a pool of mortgages
and structured to transfer
prepayment or interest rate risk from one group of security holders to another.
Their major difference from mortgage pass-through securities is the
mechanism by which interest and principal are paid to security holders. Payments
on a CMO are broken into four component categories, “tranches,” which receive
payments usually in sequential[1] order: all principal
payments go to the A tranche until
the A tranche principal is completely
returned, then the next tranche begin to receive principal.
Exhibit 1. Structure of CMO
|
Mortgage 1 |
|
Mortgage 2 |
|
Mortgage N |
Mortgage pool |
|
A Tranche |
|
B Tranche |
|
C Tranche |
|
Other tranches |
Legend
Mortgage pool: collection of mortgage loans assembled by an originator: Government National Mortgage Association (GNMA), Federal Home Loan Mortgage Corporation (FHLMC), or Federal National Mortgage Association (FNMA).
Tranche: class of bonds in a CMO offering (up to 50 or more) which shares the same characteristics. There are following types: Planned Amortization Class (PAC), Targeted Amortization Class (TAC), Companion[2], Z-Tranches (Accretion Bonds or Accrual Bonds). May be designed also as Principal-Only (PO) Securities or Interest-Only (IO) Securities.
Investors have choice among different transhes and, therefore, should be willing to pay different prices for securities of different expected maturities. In this respect pricing of CMO similar to pricing of long- and short-term bonds, and the market yield for a CMO can be expressed as a weighted average of the yields for transhes.
2. Factors, affecting value
of CMO
However, because CMO are prepayable, one cannot measure returns or values easily. The keys here are the collateral, transhes’ classes, original interest rates, prepayment assumptions, current interest and prepayment rates.
2.1. Interest rates.
Market interest rates affect CMOs in two major ways. First, as with any bond, when interest rates rise, the market price or value of most types of outstanding CMO tranches drops in proportion to the time remaining to the estimated maturity, because investors miss the opportunity to earn a higher rate (“extension risk” of investing in CMO). Conversely, when rates fall, prices of outstanding CMOs generally rise, creating the opportunity for capital appreciation if the CMO is sold prior to the time when the principal is fully repaid. However, if principal is repaid earlier than was expected, investors would have to reinvest it at lower interest rates (“call risk”).
The spread between long- term and short-term interest rates also plays an important role in the pricing of the CMO: when it increases (decreases), the sum of the prices of the CMO's tranches is more (less) likely to exceed the price of the underlying security. [1]
2.2. Average life and prepayment assumptions.
The term “average life” is more often used for discussing mortgage securities, than their stated maturity date. The average life is the average time that each principal dollar in the pool is expected to be outstanding, based on certain assumptions about prepayment speeds. If prepayment speeds are faster than expected, the average life of the CMO will be shorter than the original estimate; if prepayment speeds are slower, the CMO's average life will be extended. Prepayment estimates based usually on the Standard Prepayment Model of The Bond Market Association, which contains historic prepayment rates for each particular type of mortgage loan under various economic conditions from various geographic areas, and assumes that new mortgage loans are less likely to be prepaid than somewhat older ones. Projected and historical prepayment rates are expressed as “percentage of PSA”. 100% PSA means prepayment rate (CPR)[3] 6% a year after 30 months, for 30 year mortgages. Annual prepayment rate is estimated as PSA*6%*t/30, where t is time, in formula, less or equal 30. 50 % PSA means one-half the CPR of the PSA benchmark. Both CPR and PSA are used as benchmarks for prepayment rates, or speeds.
For example if the collateral is a pool of GNMA 12% loans, the CMO may be priced to reflect 100% PSA. If interest rates drop to 8%, prepayments may speed up to, perhaps, 300% PSA. The CMO matures more quickly than had been expected and Tranche A in this case would be completely paid off by month 31, instead of 64th.
2.3. The underlying collateral.
The following factors should be considered [7]: mortgage type, coupon and maturity, cash flow pattern of mortgages, geographic distribution, due-on-sale provisions, prepayment history. This information is needed to forecast prepayments under various scenarios.
3. Valuing CMO.
Because CMO’s value depends on risk of mortgage
prepayment, which in turn depend on interest rates, economic conditions, etc,
pricing CMO begins with creating model of cash flows. This model must satisfy
arbitrage free criteria: there are no arbitrage opportunities referring to
model prices at all points in time, or each cash flow is priced correctly by
the interest rates’ generation process . An arbitrage opportunity is defined
as: the ability to make zero net
investment, to have no probability of loss, to have a positive probability of
gain.
There are two commonly used
models to built cash flow in:
3.1. Binomial lattice method.
It’s a simplest way to model an evolution of interest rates, which
widely used for valuing bonds with embedded options; in case of CMO, it’s a
callable bond, but the call option may be executed by “issuer” at any time. In
addition, issuer has distributed the prepayment risk into transhes and sensitivity
of each tranche to prepayment risk and interest risk is unequal.
Building a binomial tree begins today (root of the tree) and its node
(vertical column of dots) contains the current time rate A, say 10%.
H4
H3
H2 HL4
H1 HL3
A HL2 HL4
L1 HL3
L2 HL4
L3
L4
Legend
L- the lowest one-year rates 1, 2, 3, 4
–years forward
H- the highest one-year rates 1, 2, 3, 4
–years forward
HL- the middle one year rates 2, 3, 4 –years forward
All other nods contain interest rates,
which are equally likely as one period elapses and the logarithm of one period
rate obeys a binomial distribution with p=0.5
The relationship between H and L is as
follows:
H=L*e,2C where e= 2.71828.., C- assumed volatility of the one year rate.
CMO valuation begins with assigning prices to the dots in the final
period’s node. These prices take into consideration mortgage prepayment speed
under different interest rates. Next
step is to determine prices in the nodes to the left, as an average between
prices in two preceding nodes, discounted by current node’s interest rate.
Example [4, p. 404]
Valuation CMO, class A, carrying 25% of the principal of a pool of
30-year mortgages with 12% interest. The prepayment speed for this type of CMO
is 12,41 per hundred.
|
Short rate lattice (1) |
Pool size lattice (2) |
||||||||||
|
10% |
1 |
||||||||||
|
9,5% |
11,5% |
0,95 |
0,98 |
||||||||
|
9% |
11% |
13% |
0,903 |
0,931 |
0,960 |
||||||
|
8,5% |
10,5% |
12,5 |
14,55 |
0,857 |
0,884 |
0,912 |
0,941 |
||||
1)
Spot lattice with
current short rate 10%
2)
Pool size lattice
under assumption that prepayment rate is
5% if short rates go down and 2% if they go up.
|
Principal tree (3) |
|||||||
|
25.000 |
|||||||
|
16.436 |
19.526 |
||||||
|
7.969 |
10.756 |
10.759 |
13.725 |
||||
|
0 |
2.041 |
2.026 |
4.704 |
2.029 |
4.707 |
4.703 |
7.551 |
3)
Calculation of
principal owed to class A. Initial is 25 ( 25% of total 100).
New principal ( for example 7.551)is the old one with interest 12%
(13.725 x 1.12), minus the total payment made by remaining pool (12.42 x 0.96)
plus interest payments for classes B & C (0.25 x 12% x 2) minus the new prepayment
amounts ( (0.96-0.941)(13.725 + 50 + 25 (1.12)3 )
|
Value tree (4) |
|||||||
|
25.758 |
|||||||
|
28.627 |
28.041 |
||||||
|
18860 |
18508 |
21969 |
21.665 |
||||
|
9.405 |
8.953 |
12.074 |
12.026 |
12.078 |
12.029 |
15.351 |
15.207 |
4)
Value of the class
using backward calculation. Value in
earlier node is equal to its cash flow plus the discounted expected value of
the successor node :
15.207= 7.551x1.12/1.145 + 12.41 x 0.96- 2 x 3 + (0.96-0.941)(13.725 +
50 + 25 (1.12)3
Final value 25.758 is normalized to the
base of 100: 4x 25.758= 103.032
A
binomial tree has several drawbacks when applied to CMO.
-
First, there is not so much freedom
for rates to change at each time step;
-
second, there are very few overall
rate choices early on, while there are 360 choices of
terminal
rates in a monthly tree.
Due to a
combination of the timing of principal payments and discounting effects, a tree
adds a lot of rate choices to a part of the valuation that is going to result
in very little increase in precision for most CMO [5]
3.2. Monte Carlo simulation
This model involves simulation of sufficiently large number[4] of potential interest paths in order to assess the security along these different paths, i.e. calculate the path’s present value. Generation of these random interest rates’ paths uses as input today’s term structure of interest rates and volatility assumption. The term structure is the theoretical spot rate curve implied by today’s Treasury securities. The simulations should be normalized so that the average simulated price of a zero-coupon bond equals today’s actual price.
Calculation of PV in each month is determined by discounting cash flow in that month by respective spot rate. Simulated spot rate for month t on path n is the product of simulated future rates’ factors on path n in the power of 1/t . Consequential steps in determining PV, or theoretical value of CMO, are shown in Exhibit 3.
Exhibit
3. Monte-Carlo simulation in valuing CMO.
Step 1. Simulated paths of 1-months future interest rate.
|
Interest rate path number |
|||||
|
Month |
1 |
2 |
3 |
.n… |
N |
|
1 |
F11 |
F12 |
F13 |
|
F1N |
|
2 |
F21 |
F22 |
F23 |
|
F2N |
|
3 |
F31 |
F32 |
F33 |
|
F3N |
|
4 |
F41 |
F42 |
F43 |
|
F4N |
|
.t... |
|
|
|
|
|
|
358 |
F358-1 |
F358-2 |
F358-3 |
|
F358-N |
|
359 |
F359-1 |
F359-2 |
F359-3 |
|
F359-N |
|
360 |
F360-1 |
F360-2 |
F360-3 |
|
F360-N |
Notation: F- 1-month future rate for month t on path n.
Step2 . Simulated paths of 1-months refinancing rate. Estimated on the basis of found relationship between short-term rates and refinancing rates.
|
Interest rate path number |
|||||
|
Month |
1 |
2 |
3 |
.n… |
N |
|
1 |
R11 |
R12 |
R13 |
|
R1N |
|
2 |
R21 |
R22 |
R23 |
|
R2N |
|
3 |
R31 |
R32 |
R33 |
|
R3N |
|
4 |
R41 |
R42 |
R43 |
|
R4N |
|
.t .... |
|
|
|
|
|
|
358 |
R358-1 |
R358-2 |
R358-3 |
|
R358-N |
|
359 |
R359-1 |
R359-2 |
R359-3 |
|
R359-N |
|
360 |
R360-1 |
R360-2 |
R360-3 |
|
R360-N |
Notation: R- mortgage refinancing rate, for month t on path n.
Step 3. Simulated cash flows on each of the interest rate paths on the base of prepayment model. This model may include prepayments at 80%, 120%…. .of the base case.
|
Interest rate path number |
|||||
|
Month |
1 |
2 |
3 |
.n… |
N |
|
1 |
C11 |
C12 |
C13 |
|
C1N |
|
2 |
C21 |
C22 |
C23 |
|
C2N |
|
3 |
C31 |
C32 |
C33 |
|
C3N |
|
4 |
C41 |
C42 |
C43 |
|
C4N |
|
.t .... |
|
|
|
|
|
|
358 |
C358-1 |
C358-2 |
C358-3 |
|
C358-N |
|
359 |
C359-1 |
C359-2 |
C359-3 |
|
C359-N |
|
360 |
C360-1 |
C360-2 |
C360-3 |
|
C360-N |
Notation: C- cash flow for month t on path n.
Step 4. Simulated paths of monthly spot rates.
|
Interest rate path number |
|||||
|
Month |
1 |
2 |
3 |
.n… |
N |
|
1 |
Z11 |
Z12 |
Z13 |
|
Z1N |
|
2 |
Z21 |
Z22 |
Z23 |
|
Z2N |
|
3 |
Z31 |
Z32 |
Z33 |
|
Z3N |
|
4 |
Z41 |
Z42 |
Z43 |
|
Z4N |
|
.t .... |
|
|
|
|
|
|
358 |
Z358-1 |
Z358-2 |
Z358-3 |
|
Z358-N |
|
359 |
Z359-1 |
Z359-2 |
Z359-3 |
|
Z359-N |
|
360 |
Z360-1 |
Z360-2 |
Z360-3 |
|
Z360-N |
|
ZT-N= ((1+F1N)(1+F2N)….(1+FT-N))1/t – 1 Example for month 4, path 2: Z42= ((1+F12)(1+F22)(1+F32)(1+F42))1/4-1 |
|||||
Notation: Z- spot rate for month t on path n.
Step 5. Present value of cash flows for month T on interest rate path N discounted at the respective simulated spot rate plus some spread K.
PV(TN)= CTn / (1+ZTN + K)T
K-spread, or option adjusted spread, when added to all spot rates on all paths, makes the average present value of the paths equal the observed market price plus accrued interest.
Step 5. Present value of each path is the sum of PV’s for each month on that path.
Step 6. Theoretical value of CMO is the average of all PV’s, calculated in Step 5.
This value can not be used for pricing CMO’s separate transhes without information about distribution of path’s present values. For example, standard deviation for well protected PAC[5] bond should be small, while for support transhes must be large.
This process is also applied for calculation of CMO average life.
Conclusion: Monte-Carlo simulation gives good estimates of CMO price if one have empirical model of relationship between short-term rates and refinancing rates and prepayment model.
3.3. Multinomial lattice method.
In contrast
to a binary tree, at the first time step, there are four choices for where
rates may go; after that there are five possible transitions from each node. Many of these transitions end up at
the same rate, so there is some amount of "recombination" after the
initial spreading out of rate nodes.
Exhibit
4. Multinomial tree.
A
The level of rates depends on 1-month's rate volatility
and mean reversion parameter, which controls how much the mean of rate
movements (ratios) pull rates towards some long-term average when they get very
high or very low.
Conclusion: This
method combines simplicity of a binary tree with flexibility of Monte-Carlo and designed to price large
set of CMO using a small set of paths.
CMO’s
transhes.
The main goal of my project is to discuss CMO pricing, not classes; but because one of those classes was mentioned as an example, here’s the description of this only class.
Planned Amortization Class (PAC) Tranches.
PAC tranches are similar to a "sinking fund" bond: fixed principal payment schedule that directs cash-flow irregularities caused by faster- or slower- than-expected prepayments away to other companion thranshes in CMO structure. When prepayments are minimal, the PAC payments are met first and the companion may have to wait. When prepayments are heavy, the PAC pays only the scheduled amount, and the companion class absorbs the excess. " Type I PAC" tranches maintain their schedules over the widest range of actual prepayment speeds- say, from 100% to 300% PSA. " Type II" and " Type III PAC" tranches can also be created with lower priority for principal payments from the underlying loans than the primary or Type I tranches. They function as support tranches to higher-priority PAC tranches and maintain their schedules under increasingly narrower ranges of prepayments.
PAC tranches are now the most common type of CMO tranche, constituting over 50% of the new-issue market. Because they offer a high degree of investor cash-flow certainty, PAC tranches are usually offered at lower yields.
Conclusion
Valuation of CMOs is quite challenging, because payment stream of a mortgage is not fixed in advance. This timing uncertainty is somewhat alleviated by the averaging effect derived from a pool, but can not be entirely eliminated, because the prepayment pattern remains unpredictable.
Two major models used for valuing CMO: binomial lattice and Monte-Carlo. If we are trying binomial tree method, monthly payments would require very large trees. Other difficulty is that principal amounts are path dependent. For this reason, in practice, CMO valuation is based on simulation (Monte-Carlo) method. If we want to achieve low sampling error in valuation, tradeoffs between method’s flexibility, volatility structure and number of data (paths) needed must be investigated.
Sources
1. Yoon Dokko, Robert H. Edelstein, and Kenneth T. Rosen, “Collateralized Mortgage Obligations (CMOs): Mortgage Money Cheaper by the Slice?”
2. Yoon Dokko “Pricing Collateralized Mortgage Obligations”
3. Richard A.Brealey “Principles of Corporate Finance”
4. David Luenberg “Investment Science”
5.
Jacob Boudoukh
Pricing
Mortgage-Backed Securities in a Multifactor Interest
Rate Environment: A Multivariate Density
Estimation Approach.
6.
Eknatt Belbase A Lattice
Implementation of the Black-Karasinski Rate process.
7. Frank Fabozzi “The Handbook of Fixed Income Securities.”
8. Frank Fabozzi “Readings in Investment Management”
9. Frank Fabozzi “Valuation of Fixed Income Securities and Derivatives”.
