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Finance
1

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: 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[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