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Multiperiod Corporate Default Prediction– A Forward Intensity Approach
Jin-Chuan Duana,b Jie Suna Tao Wangb
aRisk Management Institute, National University of SingaporebDepartment of Finance, National University of Singapore
2010 NTU International Conference on FinanceTaipei, December 2010
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Research Question
Default/bankrutpcy prediction over different future periods
Term structures of default probabilitiesShort-term vs. long-term
A forward intensity approachReduced form model
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Outline
1 Literature Review
2 A forward intensity approach to multiperiod default prediction
3 Data and covariates
4 Empirical results
5 Conclusion
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Literature Review
Outline
1 Literature Review
2 A forward intensity approach to multiperiod default prediction
3 Data and covariates
4 Empirical results
5 Conclusion
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Literature Review
Literature Review
Discriminat analysisBeaver (1966, 1968), Altman (1968), etc.Model output: credit scores
Binary response models: logit/probit regressionsOhlson (1980), Zmijewski (1984), etc.Model output: default probability in the next one period
Campbell, et al (2008): multiple logit models
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Literature Review
Literature Review (Cont’d)
Recent development: duration analysisShumway (2001), Chava and Jarrow (2004), etc.
Duffie, et al (2007): intensity model with stochastic covariatesTwo Poisson processes
1 Default/bankruptcy2 Other exit: merger and acquisition, etc.
Instantaneous intensityInstantaneous rate of occurrenceFunctions of the covariates
Time-series dynamics of the covariates
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A forward intensity approach to multiperiod default prediction
Outline
1 Literature Review
2 A forward intensity approach to multiperiod default prediction
3 Data and covariates
4 Empirical results
5 Conclusion
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A forward intensity approach to multiperiod default prediction
Model Setup
Spot combined exit intensity: "average" rate of exitoccurrence
it (�) ≡ − ln(1 − Fit (�))
�= −
ln Et
[exp
(−∫ t+�
t (�is + �is)ds)]
�
Fit (�): the time-t conditional distribution function of the combinedexit time evaluated at t + � .�is: instantaneous intensity for default.�is: instantaneous intensity for other exit.
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A forward intensity approach to multiperiod default prediction
Model Setup (Cont’d)
Forward exit intensity: forward rate of exit occurrence
git (�) ≡F ′it (�)
1 − Fit (�)= it (�) + ′it (�)�
Forward default intensity: forward rate of default occurrence
fit (�) ≡ e it (�)� limΔt→∞
Pt (t + � < �Di = �Ci ≤ t + � + Δt)Δt
= e it (�)� limΔt→∞
Et
[∫ t+�+Δtt+� exp
(−∫ s
t (�iu + �iu)du)�isds
]Δt
�Di : default time of the i-th firm.�Ci : combined exit time of the i-th firm.The default probability over [t , t + � ] becomes∫ �
0e− it (s)sfit (s)ds
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A forward intensity approach to multiperiod default prediction
Model Setup (Cont’d)
Model fit (�) and git (�) directly as functions of state variablesavailable at time t and the horizon of interest, � .git (�) ≥ fit (�)
Xit = (xit ,1, xit ,2, ⋅ ⋅ ⋅ , xit ,k ): the set of the state variables
fit (�) = exp(�0(�) + �1(�)xit ,1 + �2(�)xit ,2 + ⋅ ⋅ ⋅�k (�)xit ,k
)git (�) = fit (�) + exp
(�0(�) + �1(�)xit ,1 + �2(�)xit ,2 + ⋅ ⋅ ⋅�k (�)xit ,k
)Discretize the model for empirical implementation
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A forward intensity approach to multiperiod default prediction
Model Setup (Cont’d)
Forward default probability
Cumulative default probability
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A forward intensity approach to multiperiod default prediction
Estimating the Forward Intensity Model
MLE
Decomposable
Non-sequential, parallelizable
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Data and covariates
Outline
1 Literature Review
2 A forward intensity approach to multiperiod default prediction
3 Data and covariates
4 Empirical results
5 Conclusion
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Data and covariates
Data
Sample period: 1991-2009.
Database:CompustatCRSPBloomberg
12,196 U.S. public companies (both industrial and financial),1,030,305 firm-month observations.
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Data and covariates
Covariates
3-month treasury rateTrailing 1-year S&P500 returnDistance to defaultCash and shor-term investments/Total assetsNet income/Total assetsRelative sizeMarket to book ratioIdiosyncratic volatility
Level and trend
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Empirical results
Outline
1 Literature Review
2 A forward intensity approach to multiperiod default prediction
3 Data and covariates
4 Empirical results
5 Conclusion
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Empirical results
Parameter Estimates
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Empirical results
Parameter Estimates (Cont’d)
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Empirical results
Aggregate Number of Defaults
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Empirical results
In-Sample Accuracy
1 month 3 months 6 months 12 months 24 months 36 months93.41% 91.42% 88.37% 82.70% 72.09% 64.77%
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Empirical results
Out-of-Sample (Over Time) Accuracy
1 month 3 months 6 months 12 months 24 months 36 months91.99% 90.21% 87.36% 83.42% 74.35% 68.53%
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Empirical results
Term Structures of Default Probabilities(Lehman Brothers)
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Empirical results
Term Structures of Default Probabilities (Cont’d)
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Conclusion
Outline
1 Literature Review
2 A forward intensity approach to multiperiod default prediction
3 Data and covariates
4 Empirical results
5 Conclusion
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Conclusion
Conclusion
A forward intensity approach for the prediction of corporatedefaults over different future periods is proposed.Maximum likelihood analysis is conducted on a large sample ofU.S. industrial and financial firms spanning the period 1991-2009.Several frequently used covariates are shown to be useful forprediction at both short and long horizons.The forward intensity model is amenable to aggregation, whichallows analysts to assess default behavior at the portfolio and/oreconomy level.
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