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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

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 additi