The reduced form approach to modeling credit risky portfolios has a number of advantages over the older Merton or copula approach to the simulation of a credit risky portfolio.   In what follows, “reduced form” refers to the process of estimating default probabilities with a broad array of explanatory variables, including macro-economic factors, instead of using the Merton structural model approach.  In the Merton approach, the probability of default is derived from a highly stylized model of the capital structure of the company where there is only one random variable: the value of the assets of the firm.  In the reduced form approach, analysts either imply default probabilities from the observable values of traded securities or estimate them from a historical data base of defaults and the relevant explanatory variables.  The reduced form approach and its accuracy relative to the Merton structural model are explained at length by Bharath and Shumway (2008), Campbell et al (2008), and Jarrow, Mesler and van Deventer (2004).  Each of these studies employs logistic regression to estimate default probabilities from a historical default data base. Advocates of the reduced form approach argue that it allows default probability estimation with much higher accuracy than the Merton approach because any potential explanatory variable can be employed. Merton default probabilities, by contrast, are constrained in accuracy because the only explanatory variables that are consistent with the theory are the value of company assets, asset volatility, asset correlation with the assets of all companies, interest rates, the liabilities of the firm, and the difference between the expected return of the assets of all companies and the risk free interest rate.  Campbell et al (2008) argue powerfully that default model fitting with an unconstrained selection of inputs is obviously more accurate than model fitting where the selection of inputs is highly constrained.

There are two types of inputs to a reduced form model.  The first type of inputs is a set of macro-economic factors that affect some or all risky counterparties.  The second type of input is counterparty specific.  In the case of a publicly listed counterparty, these inputs would be financial ratios or inputs related to the counterparty’s stock price. Given these inputs, reduced form portfolio modeling consists of the following steps:

1. The analyst selects some or all of the explanatory variables to model as random variables
2. Probability distributions of these random inputs are selected, as is the correlation between random variables
3. Default probabilities for each counterparty are simulated forward on a multi-period basis
4. Credit spreads for each counterparty are derived from the simulated default probabilities and other random variables, typically macro economic variables, that drive the supply and demand for credit
5. Each transaction in the portfolio is valued at each time step in the multi-period simulation, and related cash flows and financial accruals are generated.
6. The value of the portfolio is the sum of the values of individual transactions
7. The value of tranches of the portfolio (whether the tranching is by maturity like a mortgage-backed security or by seniority from a credit loss perspective like a collateralized debt obligation) is then derived from the “waterfalls” of cash flows to each tranche, as described by the indenture of the tranched transaction.

The key principles of portfolio modeling in a reduced form context are given by Jarrow, Lando and Yu (2005).  They argue that, because of diversification, “risk neutral” default probabilities relevant to valuation will be indistinguishable from empirical default probabilities that are generated from the reduced form estimation process above, typically with logistic regression.  This argument is undergoing extensive testing in the rich data environment of the current credit crisis.

The reduced form approach to portfolio management is completely consistent with the modern risk neutral valuation of derivative securities.  Portfolio management using the Merton or copula approach, by contrast, relies on a number of very strong simplifying assumptions, as explained by Jarrow, Li, Mesler and van Deventer (2007).  The predominant assumptions used by market participants are dramatically different from the reduced form approach.  First, Merton simulation normally focuses on the distribution of losses rather than valuation per se, whereas the reduced form approach produces both outputs.  Second, the Merton simulation commonly assumes a single period analysis, not a multi-period analysis.[1]  Thirdly, the Merton/copula approach typically assumes that the correlation in the returns on company asset values for companies A and B is the same value for all pairs of companies.  This in turn implies that there is only one common random “macro” variable in the portfolio simulation, instead of the multiple random variables allowed under the reduced form approach.  Fourth, the Merton or copula approach normally assumes that default probabilities are not random, although they can be allowed to drift over time.  The reduced form approach, by contrast, allows for default probabilities to be either random or non-random.  Reduced form model advocates argue that only random default probability simulation correctly captures the rise and fall of default rates due to the business cycle.

In recent papers, Jarrow, Li, Mesler and van Deventer (2007) and Jarrow and van Deventer (2008) compare the implications of portfolio modeling with the Merton/copula approach versus reduced form approach.  Their results are consistent with CDO prices observed in the 2007-2008 credit crisis.  The authors find that the reduced form approach can be parameterized flexibly to be as consistent with market prices as the analyst desires.  By contrast, this is often not possible using the Merton or copula approach because there is only one input variable (correlation) at the analyst’s disposal.  Bloomberg.com reported that market CDO prices for super senior tranches in March, 2008 were not consistent with Merton/copula simulations even if the correlation were allowed to rise to 100%.  The reduced form approach, by contrast, has enough input flexibility that market prices can be matched very precisely, even if the same default probabilities are used as in the Merton/copula approach.  For that reason, the reduced form approach to credit portfolio management is increasingly regarded as the best practice approach to portfolio simulation.
 

References
S. Agca and S. Islam, 2007, “Can CDO Equity be Short on Correlation,” working paper, George Washington University.

S. Bharath and T. Shumway, May 2008, “Forecasting Default with the Merton Distance-to-Default Model,” Review of Financial Studies.

John Y. Campbell, Jens Hilscher, and Jan Szilagyi, December 2008, “In Search of Distress Risk,” Journal of Finance.

R. Jarrow and Donald R. van Deventer, 2008, “Synthetic CDO Equity: Short or Long Correlation Risk?” Journal of Fixed Income.

R. Jarrow, Li Li, Mark Mesler, and Donald R. van Deventer, 2007, “CDO Valuation: Fact or Fiction,” The Definitive Guide to CDOs, edited by Gunter Meissner, Risk Publications.

R. Jarrow, Mark Mesler, and Donald R. van Deventer, January 2006, Kamakura Default Probabilities Technical Report, Kamakura Risk Information Services, Version 4.1, Kamakura Corporation memorandum.  

R. Jarrow, D. Lando, F. Yu, 2005, “Default Risk and Diversification:  Theory and Empirical Applications,” Mathematical Finance, 15 (1), 1-26.

R. Jarrow and D. van Deventer, 2005, “Estimating Default Correlations Using a Reduced Form Models,” Risk Magazine, (January).

Donald R. Van Deventer, Kenji Imai, and Mark Mesler, 2004, Advanced Financial Risk Management: An Integrated Approach to Credit Risk and Interest Rate Risk Management, John Wiley & Sons.
 

[1] The single period nature of commercially available copula or Merton simulation models was discussed by Michel Araten, Managing Director, JPMorganChase, in a presentation summarizing the findings of a vendor comparison conducted by the International Association of Credit Portfolio Managers, ICBI Risk Management Conference, Geneva, December 2006.