Traditional asset and liability management (ALM) has ignored mortgage defaults and focused on interest rate-driven and mortgage age-driven prepayment. The events of the last two years have made it more obvious that prepayment and default are intimately linked and that home prices are a critical driver of both probabilities. This post explains how mortgage prepayment and default are modeled on an integrated basis using multinomial logit.

Traditional asset and liability management has modeled the impact of interest rates on market values, cash flows and financial accounting net income on a multi-period basis. The events of the last two years have made it clear that the sole focus on a single macro-economic factor, interest rates, has been insufficiently robust to prevent the effective failures of institutions like Lehman Brothers, Washington Mutual, New Century, IndyMac, Countrywide, FNMA, and FHLMC. The more modern approach to risk management is a fully integrated approach to credit risk, market risk and ALM much like the U.S. government mandated macro factor stress tests of February-April 2009. For more on this, see our April 28, 2009 blog post “Improving on the Fed’s Supervisory Capital Assessment Program, Step by Step”:

In the context of traditional interest rate risk management, default was almost always ignored by most practitioners for two reasons:

- Credit risk was in a “separate risk silo” managed by others
- Most silo-focused interest rate risk assessment software was incapable of modeling default even if the analyst had wanted to do so

Both of these constraints, one political and one technical, have now been removed at institutions at the forefront of best practice risk management. One of the tasks triggered by this new approach to integrated risk is that the traditional assumptions of ALM have to be modernized, one by one. In no area is this more true than in mortgage prepayment and default analysis.

The techniques traditionally used to mortgage prepayment analysis can be listed briefly as follows:

- Constant prepayment rate, either expressed as a conditional prepayment rate (annualized basis, “CPR”) or as a single monthly “mortality” rate. This assumption ignores the impact of interest rates and home prices on prepayment. It is also derived from behavior of mortgage pools rather than individual loan data.
- Prepayment rate as a linear function of mortgage age, interest rates and mortgage attributes. This approach also ignores home prices as a driving factor and is derived from pooled behavior rather than individual loan behavior
- Prepayment tables where the prepayment rate is driven by a table of N explanatory factors, typically the same factors as the linear prepayment function. Again, these tables are typically derived from pools of mortgages and the related prepayment rate on the pool.
- Rational prepayment models, based on the “all or nothing” valuation of the mortgage on an American option basis. Again, default and home prices are ignored.
- Rational prepayment subject to transaction costs, which recognizes the true transaction costs of refinancing and other implicit costs that cause the prepayment on mortgages to be less frequent than the fully rational approach would predict. For more on this approach, see Chapter 31 of
*Advanced Financial Risk Management*(van Deventer, Imai and Mesler, John Wiley & Sons, 2004)

In all of these approaches, the amount of prepayment that occurs, given market conditions, is never random. Given the inputs to each of these techniques, the prepayment amount is known with certainty.

More modern approaches differ in three key dimensions:

- Prepayment is recognized as being random, just like default, although the probability of prepayment can be estimated just like a default probability
- Prepayment and default are being analyzed at the individual loan level. The fact that those who analyzed mortgage collateral at the loan level fared much better than investors who did not (i.e. asset backed CDO investors and the rating agencies) during the credit crisis has not been lost on senior management.
- Default and prepayment are increasingly being modeled as mutually exclusive events that are modeled together, not separately.

The first step forward in recognizing home prices and other loan-specific inputs in the prepay/don’t prepay decision and the default/no default outcome was the use of logistic regression. Logistic regression is commonly used to model events that have only binary 0/1 outcomes. The technique has been widely applied in medical science, in mortality rate modeling in insurance, and in default modeling like the corporate d*efault models of Bharath and Shumway (Review of Financial Studies, 2008), Chava and Jarrow (*Journal of Banking and Finance, 2004), and Campbell et all (*Journal of Finance*, 2008). Many analysts have applied this technique to mortgage prepayment and default, estimating separate logistic regressions for these two events. The impact of common risk factors like home prices, interest rates, and mortgage age can be explicitly taken into account on a loan by loan basis. Conceptually, even factors like the DNA of the borrower could be taken into account, as we explained in our blog post of July 14, 2009, “Implications of DNA for Retail Default Risk and Consumer Credit Scoring”:

This logistic regression based approach has many extremely attractive attributes:

- Default and prepayment are recognized as random events, not events that are predicted with certainty
- Previously ignored factors like home prices can be incorporated into both prepayment and default modeling on a consistent basis Loss given default can be recognized as related to default probability by modeling home prices forward and using those home prices as inputs to the default and prepayment probability functions. In a default scenario, the simulated value of the home allows one to derive loss given default after applying appropriate liquidation costs and time lags.

Emerging best practice is to use multinomial logit to deal with two practical problems that arise when independent logistic regressions are used:

- With independent logistic regressions, the modeler is uncertain about which probability to apply first in a given period and a given scenario: the simulation of prepay/don’t prepay or the simulation of default/don’t default
- Similarly, it’s possible that the probabilities of prepayment and default could add up to more than 100%

In order to deal with these issues, a number of advanced techniques can be employed. One of the most popular techniques is multinomial logit, which allows N potential outcomes instead of the binary 0/1 outcomes assumed in normal logistic regression. Here are some potential specifications of outcomes from a multinomial logistic regression:

Model A outcomes:

- Default
- Prepayment
- Neither of the above

Model B outcomes:

- Default
- Prepayment for interest rate reasons, remaining in the same house
- Prepayment because of relocation
- None of the above

Model C outcomes:

- Default for purely non-medical financial reasons
- Default triggered by medical expenses
- Prepayment because of divorce
- Prepayment because of interest rate reasons
- Prepayment because of job relocation
- Prepayment because of voluntary relocation
- None of the above

These multiple outcomes offer the analyst and risk manager a much more precise assessment of the impact of interest rates and home prices on the value of a pool of mortgage loans. The multinomial logit coefficients, given the explanatory variables chosen by the analyst, are derived either using joint maximum likelihood estimation or by estimating a logistic regression for each outcome and making appropriate adjustments.

For those institutions that place a priority on correctly understanding risk adjusted shareholder value creation and business strategy, the logistic and multinomial logistic approaches are an essential step forward. For more on these techniques in practical use, please address comments and questions to info@kamakuraco.com.

Donald R. van Deventer

Kamakura Corporation

Honolulu

August 24, 2009