For long, it was suspected that without access to INTEX or TREPP, it would be next to impossible to model securitized assets. It has been argued that these assets are fungible and therefore not easy to model on a transaction basis in any risk management solution, and the standard modelling techniques outlined take into consideration the links that INTEX or TREPP offer on cashflow pools and waterfalls. The conventional modeling approach to the valuation of securitized assets has taken on a simplistic hue, in that a link to a provider of standard asset-backed security information is enough to provide a valuation framework that would be largely accepted, but this comes at a distinct cost:

- The underlying information is taken at face value, and so are the variables governing the various aspects of cashflow generation
- There is no transparency on the underlying information
- For retail securitization, often-times there is no access to the underlying asset pool
- Prepayment speeds are based on a homogenous matrix and cannot be stressed
- Default is not a function that is explicitly captured

In spite of these shortcomings, the industry standard for modeling and valuation of any asset backed security is largely defined thus:

- Load the CUSIP, and tie it to INTEX/TREPP/SIMILAR
- On process, load all the relevant information relating to the transaction from the analytics provider
- Apply a logical discount factor to the cashflows generated
- Value the instrument

This approach essentially implies that the cashflows that are generated through the analytics provider is stand-alone, and do not behave in a dynamic manner, as these should, influenced in whole or in part through a combination of market factors, macro factors, and idiosyncratic customer behavior.

This discussion paper outlines a contrarian approach to modeling and valuing asset backed securities, and it is a *fait accompli* that the approach of linking an analytics provider’s data to the CUSIP is well established within Kamakura Risk Manager. The KRM solution can model securitization derivative instruments, such as REMIC, CMO, ABS, and CDO securitization tranche instruments. Tranched credit derivatives such as synthetic CDOs, tranched credit guarantees, and tranched credit default swaps are also considered securitization tranche instruments. Securitization tranche instruments are derivative instruments promising future cash flows derived from allocation of cash flows from an underlying asset pool and a deal tranche and cash flow waterfall structure. Normally, each asset in the underlying pool is a defaultable financial instrument with a specific issuer or counterparty. In KRM, many of these instruments are modeled using third-party deal databases and cash flow generators, e.g. those provided by Intex Solutions. KRM models the outstanding balances and the security identifier for these instruments, and the other attributes, deal structure, and underlying asset pool are obtained from the deal database.

KRM models the detailed contractual and behavioral characteristics of each financial instrument in the asset pool underlying a securitization tranche instrument. The contractual characteristics are obtained from the third-party deal database, while the behavioral characteristic are provided by prepayment, default, and recovery models KRM associates with each instrument in the underlying pool. KRM associates a specific prepayment, default, and recovery model for each instrument in the pool based upon the descriptive characteristics of the instrument, e.g. a particular type of mortgage loan. In general, the prepayment, default, and recovery model associated with instruments in the underlying pool are functions of underlying risk factor values as well as dynamic attributes of the associated instruments (e.g. remaining term), and prepayment, default, and recovery intensities and rates depend upon future values of these underlying risk factors.

The KRM Analytical Engine can calculate the arbitrage-free economic values of securitization tranche instruments like CMO or CDO tranches. The economic value of these securitization tranche instruments is calculated as the expected present value of the remaining future cash flows of the instrument at the valuation time under a risk-neutral distribution of the risk factor values affecting the future cash flows using a specific valuation yield curve. The extended Monte Carlo pricing method uses a stochastic process model to simulate changes in a valuation risk factor vector for the securitization tranche instrument to produce a collection of risk-neutral scenario sample paths over the remaining term of the instrument. The valuation risk factor vector consists of component risk factors that collectively provide the future values for the asset value, prepayment, default, and recovery models associated with all of the instruments in the underlying asset pool.

**Commercial asset-backed securities**

The alternative approach to modeling commercial asset-backed securities is to model the underlying assets directly, as opposed to generating a synthetic cashflow pool through the interaction with an analytics service provider.

Modeling commercial asset-backed securities therefore is a simple exercise of modeling the individual underlying assets, whilst keeping the same CUSIP across each transaction. This ensures that when any reporting is done, it can be at the CUSIP-level as opposed to the underlying transactions

This approach takes away the ambiguity relating to the cashflows as each individual transaction represents an element of the tranche that is sought to be modelled. This essentially means that each of these elements can be modelled independently for each of the following cashflow influencing traits:

- Prepayments
- Defaults
- Recoveries
- Change in interest rates

A critical element of accuracy is thus established when the CUSIP is now reviewed as a single instrument with different transaction elements, and each of these elements can have their own prepayment assumptions, probabilities of default, recovery rates and lags, and any other cashflow influencing feature. This leads to a greater degree of accuracy and greater control over modeling variables, which in turn provides results that are realistic, accurate, and reliable.

**Retail asset-backed securities**

These assets, that are typically mortgage-backed securities, student loans, or credit cards, are more difficult to model simply because of the volumes involved. If a creative solution is preferred, a similar approach to that outlined for commercial asset-backed securities may be adopted, but since the underlying information is not readily available, a set of synthetic transactions can be created to represent the CUSIP, and the modeling approach can then parallel the approach elucidated above. However, if such synthesis is not to the user’s palate, a Monte Carlo approach can be incorporated to model these securities.

The KRM Analytical Engine applies each risk-neutral scenario sample path for the valuation risk factor vector to the asset value, prepayment, default, and recovery models associated with each instrument in the underlying asset pool to simulate future cash flows for the instrument on each sample path. The resulting simulated cash flows for each instrument in the asset pool on each sample path incorporate the effects of prepayments, defaults, and recoveries on the sample path as well as the scheduled contractual cash flows for the instrument. The KRM Analytical Engine uses the resulting cash flow sample paths for all of the instruments in the asset pool to generate aggregate cash flow sample paths for the pool corresponding to the scenario sample paths for the valuation risk factor vector.

The KRM Analytical Engine applies the deal tranche and cash flow waterfall structure for the securitization tranche instrument to each resulting aggregate cash flow sample path for the asset pool to generate the corresponding remaining tranche cash flows for the sample path. The KRM Analytical Engine calculates the arbitrage-free economic value of the securitization tranche instrument as the expected present value of the remaining tranche cash flows of the instrument at the valuation time across the valuation risk factor vector scenario sample paths using a specific valuation yield curve. The Monte Carlo pricing methods for securitization tranche instruments with asset value, prepayment, default, and recovery models for instruments in underlying asset pools assure that the credit-adjusted economic value of each securitization tranche instrument is based on consistent simultaneous simulation of all of the risk factors underlying the asset pricing, prepayment default intensity, and value recovery models for the instruments in underlying asset pools.

Due to the magnitude of the number of risk factors required to determine the credit-adjusted economic value of defaultable instruments with dynamic prepayment, default and recovery models, analytical pricing methods are generally not available, and numerical methods must be used to calculate the economic values of these contracts. Theoretically, lattice pricing methods could be applied in some limited cases where the number of component risk factors in the valuation risk factor vector is small, but this involves a multidimensional lattice structure that can become quite extensive for even relatively small numbers of component risk factors. Consequently, Monte Carlo valuation methods generally are applied to determine the credit-adjusted economic value of defaultable instruments or instruments with underlying asset pools consisting of prepayable and/or defaultable instruments.

Application of Monte Carlo pricing methods to instruments with dynamic prepayment, default, and recovery models requires definition of the valuation risk factor vector for each instrument and specification of an associated vector stochastic process model for the valuation risk factor vector. This model describes the interdependent simultaneous evolution of the valuation risk factor vector under the pricing probability measure, i.e. it must use risk-neutral probabilities when simulating changes in individual component risk factors in the valuation risk factor vector and applying those changes to generate valuation risk factor vector scenario sample paths. The Monte Carlo pricing methods for instruments with dynamic prepayment, default, and recovery models apply Monte Carlo sampling techniques to the stochastic process model for the (extended) valuation risk factor vector to simulate change in vector values and generate scenario sample paths for the vector.

Monte Carlo pricing methods for financial instruments with dynamic prepayment models use the prepayment intensity related components of the valuation risk factor vector scenario sample paths generated by the simulation to evaluate the prepayment intensity functions for the issuer of the instrument and to simulate potential prepayment events for the issuer. The sample paths are applied to dynamically vary the prepayment intensities, which are integrated over time to determine random prepayment times for the issuer. When the issuer has a prepayment time prior to the earlier of the expiration date or the default date of the instrument, a prepayment event occurs, the principal balance of the instrument is distributed to the holder, and the contractual terms of the instrument are terminated.

As a consequence of the valuation requirements outlined above, the credit-adjusted economic value of a prepayable and/or defaultable instrument generally depends upon at least four risk factors for unilateral contracts, such as bonds, and at least six risk factors for bilateral contracts, such as derivative instruments. The four risk factors minimally required for unilateral contracts are: (i) one or more asset value or asset pricing risk factors describing the market and/or macroeconomic risk underlying the instrument, (ii) one or more prepayment intensity risk factors affecting prepayments of the instrument, (iii) one or more default intensity risk factors for the issuer, and (iv) one or more value recovery risk factors for the issuer. The six risk factors minimally required for bilateral contracts are: (i) one or more asset value or asset pricing risk factors describing the market and/or macroeconomic risk underlying the instrument, (ii) one or more prepayment intensity risk factors affecting prepayments of the instrument, (iii and iv) one or more default intensity risk factors for each counterparty, and (v and vi) one or more value recovery risk factors for each counterparty.

Depending upon the structure of the prepayment intensity, default intensity, value recovery, and asset value models associated with a financial instrument, some of these risk factors may be the same, and these common risk factors must be included a valuation risk factor vector for the instrument together with the remaining unique risk factors. Each of the component risk factors in the resulting valuation risk factor vector is mapped to the appropriate component of the asset value risk factor vector, the prepayment intensity risk factor vector, the default intensity risk factor vector(s), and the value recovery risk factor vector(s) using an index vector, and this mapping is used to provide values for the asset value, prepayment, default, and recovery models when simulating scenario sample paths for the valuation risk factor vector.

**Valuation Models**

The KRM Analytical Engine uses arbitrage-free valuation methods to determine the economic values of securitization tranches, such as investments in individual tranches of CMO, CDO, or ABS deals. These methods model the attributes of the securitization tranche, such as its outstanding balance, coupon rate, and stated maturity as well as the overall structure of the securitization deal to which the tranche belongs. This requires that the assets underlying the securitization tranche be individually modeled and that the reserve accounts, hedging instruments, cash flow waterfall, trigger mechanisms, and other material aspects of the securitization deal be modeled and analyzed.

The KRM Analytical Engine calculates the economic value of a securitization tranche as the expected present value with respect to the distribution of yield curve and other risk factor paths to the maturity of the instrument of the projected future cash flows allocated by the deal cash flow waterfall structure to the tranche. The present value of each future cash flow is determined by discount rates obtained from the valuation yield curves on a yield curve path for the currency denomination of the tranche at the tenors of the future cash flows. When performing a securitization tranche valuation, the KRM Analytical Engine generates a sample set of yield curve and other risk factor sample paths using Monte Carlo simulation techniques.

The projected future cash flows allocated to a securitization tranche for a given yield curve and risk factor sample path are determined by application of the cash flow waterfall for the securitization deal. This allocation is based on the aggregate cash flows from the underlying assets over a payment period and the balances in reserve accounts at the end of the period. The allocation may be modified by trigger mechanisms in the deal structure, and the aggregate cash flows may be modified by any hedging instruments incorporated in the deal structure. Normally, the aggregate cash flows from the underlying assets are the primary source of cash flows allocated to the securitization tranche.

The aggregate cash flow from the underlying assets during each period under a given yield curve and risk factor sample path is the sum of the cash flow amounts for each underlying asset during the period. The cash flow amount for a given underlying asset during a period depends upon the characteristics of the asset as well as the yield curve and risk factor sample path. For example, if the underlying asset is a fixed- or floating-rate instrument, such as a mortgage loan, the cash flow for the asset will be the interest and principal amounts paid during the period. These amounts can depend upon yield curve points and risk factor values that are incorporated in the yield curve and risk factor sample path.

Underlying assets for a securitization deal may have embedded options, such as call provisions or rate caps and floors. These options will be exercised when the value or their underlying assets is less than or greater than the option exercise price/rate on an option exercise date. The underlying asset value is described by a yield curve point or risk factor value for the yield curve and risk factor sample path during each payment period, which is provided by the cash flow analysis for the underlying asset. Exercise of embedded options in an underlying asset will result in modification of the cash flow projection for the asset, and it will modify the aggregate cash flows from the underlying asset pool during the exercise period.

**Conclusion**

As the reader can appreciate, the best approach to model securitized assets is to strip bare the underlying either in actual form or in a synthesized form to gain the most accuracy in generating valuation and loss results. This also is the purest form of modeling such complex securities. The analytics provided by agencies can only help up to a point, but if the user wishes to model the riskiness of these instruments holistically, an integrated approach is warranted and the Kamakura Risk Manager analytical engine provides this through its world class risk solutions suite of products.