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Accounting Consumer
Behavior Economics Finance
Management
Human values in organizations
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 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
|
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, 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) 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:
Monte
Carlo simulation and Binomial Tree. The first
is more flexible, because offers potentially infinite number
of combinations, but of course, more complicated because of
number of path to run on. The third lattice approach also
exists: intermediate form of those two.[6] These models are
also useful because CMO (and other fixed income securities)
are interest rate path-dependent, in other words, cash flow,
received on one period is determined not only by the current
interest rate level, but also by the path that interest rates
took to get the current level. The prepayment rates are also path-dependent
because today’s prepayment rate depends on whether there have been prior opportunities to refinance
since the underlying mortgages were originated (prepayment
burnout). In addition, CMO valuation adds another level of
path dependence, because subordinate tranche’s cash flow depends
on collateral and senior cash flow.
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 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 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.
|