Modeling Frailty Correlated Defaults with Multivariate Latent Factors

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Introduction

Goal: assess risk of a portfolio with loans to a number of firms.

Use firm-level data and aggregate up.

Need good model on the firm-level and on the aggregate level.

Findings

Additional variables are needed in hazard model in Duffie, Saita, and Wang (2007) and Duffie et al. (2009).

Similar to Lando and Nielsen (2010) finding.

Assumption of linearity on the log-hazard scale is too strong.

Evidence of additional time-varying latent factors.

Similar to Lando et al. (2013) and Jensen, Lando, and Medhat (2017) but our model can directly be used for forecasting.

Data set

Use US rated firms as is common.

Data set

Market data from CRSP and financial statement variables from Compustat.

Default events from Moodys Default Risk Service Database.

Mainly missed interest payment and chapter 11 bankruptcy.

Include rated firms with an SIC code outside 6000-6999 range and less than 9000.

Sample is from January 1980 to October 2016.

Monthly default rate

Blue line is a generalized additive model. Gray areas are recession periods from the National Bureau of Economic Research.

Base models

Start with hazard models as in Duffie, Saita, and Wang (2007) and Duffie et al. (2009).

Base models

We model the instantaneous hazard rate of firm \(i\) at time \(t\)

\[ \lambda_i(t) = \lim_{h\rightarrow 0^+}\frac{P\left(T_i \leq t + h\mid T_i \geq t\right)}{h} \]

as a piecewise constant function of firm variables \(\vec x_{ik}\) and macro variables \(\vec m_k\)

\[ \lambda_i(t) = \lambda_{ik}= \exp\left(\alpha+\vec\beta^\top\vec x_{ik} + \vec\gamma^\top\vec m_k\right), \quad k - 1 < t \leq k \]

Base models

Similar model in Duffie, Saita, and Wang (2007).

Firm variables: distance-to-default and past one year excess return.

Macro variables: market return and one year T-bill rate.

Base models

Conditional independence assumption may be violated.

Duffie et al. (2009) suggest to use a so-called frailty variable

\[ \begin{aligned} \lambda_{ik} &= \exp\left(\alpha+\vec\beta^\top\vec x_{ik} + \vec\gamma^\top\vec m_k +A_k\right) \\ A_k &= \theta A_{k-1}+\epsilon_k & \epsilon_k\sim N\left(0,\sigma^2\right) \end{aligned} \]

Additions

Add additional covariates and non-linear effects to model in Duffie, Saita, and Wang (2007).

Additions

  • Working capital to size.
  • Operating income / size.
  • Market value / total liabilities.
  • Net income / size.
  • Total liabilities / size.
  • Current ratio.
  • Log relative market size.
  • Idiosyncratic volatility.

All have been used previously.

E.g., see Shumway (2001) and Chava and Jarrow (2004). Size is defined as 50 pct. total assets and 50 pct. market value.

Add splines

Add non-linear effect to the idiosyncratic volatility and net income / size.

Estimates

The figures in the parentheses are \(Z\)-scores except for the splines which are likelihood-ratio test statistics.

Estimates

Large difference in log-likelihood.

Similar to evidence by Lando and Nielsen (2010) and Bharath and Shumway (2008).

Splines

Partial effect on log-hazard scale. Lines are 95 pct. confidence intervals. Effect is similar to Christoffersen, Matin, and Mølgaard (2018).

Splines

Partial effect on log-hazard scale. Lines are 95 pct. confidence intervals.

Add frailty

First random intercept and then a random slope.

Additional frailty variable

Add frailty as in Duffie et al. (2009)

\[ \begin{aligned} \lambda_{ik} &= \exp\left(\alpha+\vec\beta^\top\vec x_{ik} + \vec\gamma^\top\vec m_k + A_k\right) \\ A_k &= \theta A_{k-1}+\epsilon_k & \epsilon_k\sim N\left(0,\sigma^2\right) \end{aligned} \]

Capture excess clustering defaults.

All hazard are multiplied \(\exp A_k\).

Firms are conditionally more frail in some years.

Estimation

Use Monte Carlo expectation maximization algorithm.

Use dynamichazard package (Christoffersen 2018) in R (R Core Team 2018).

dynamichazard contains an implement of the particle smoother suggested by Fearnhead, Wyncoll, and Tawn (2010).

Estimates

Weak evidence

Twice the difference in log-likelihood is 3.4.

Duffie et al. (2009) find 22.6. Difference between \(\mathcal M_1\) with and without frailty is 14.5.

Similar to Lando and Nielsen (2010) finding.

Multivariate frailty

Extend to random slope

\[ \begin{aligned} \lambda_{ik} &= \exp\left(\alpha+\vec\beta^\top\vec x_{ik} + \vec\gamma^\top\vec m_k +A_k + B_k z_{ik}\right) \\ \begin{pmatrix} A_k \\ B_k \end{pmatrix} &= \begin{pmatrix} \theta_1 & 0 \\ 0 & \theta_2\end{pmatrix} \begin{pmatrix} A_{k - 1} \\ B_{k - 1}\end{pmatrix} +\vec \epsilon_k \\ \epsilon_k&\sim N\left(\vec 0, Q\right) \end{aligned} \]

Multivariate frailty

Add random slope to log relative market value.

Two firms \(i\) and \(j\) differ only by log relative market value with \(x_{jkl} = x_{ikl} + \Delta x\). Then the relative hazards is

\[ \frac{\lambda_{jk}}{\lambda_{ik}} = \exp\left((\beta_l + B_k)\Delta x\right) \]

Index \(l\) corresponds to the log relative market value.

Evidence from the literature

Lando et al. (2013) find a time-varying effect of log pledgeable assets.

Filipe, Grammatikos, and Michala (2016) find a non-constant log total assets coefficient in their robustness check.

Jensen, Lando, and Medhat (2017) find that small firms may be less or more cyclical than large firms

depending on which other variables are in the model.

Azizpour, Giesecke, and Schwenkler (2018) find that periods of large amount of defaulted debt is followed by higher aggregate default levels.

Estimates

Estimates

Estimates

Predicted random effects

Predicted values of \(A_k\), left, and \(B_k\), right. Dashed lines are 68.3% point-wise prediction intervals.

Out-of-sample

Goal: model both firm-level and aggregate level accurately.

Options

Use monthly frailty model and make models for covariates.

Estimation error or poor model for covariates can yield bad performance.

Duan, Sun, and Wang (2012) find inferior performance.

Setup

Estimate yearly models using data up to October and make one year ahead forecast.

Concordance index

Fraction of correctly ordered event times.

Equivalent to area under the receiver operating characteristic curve if we only have one discrete period.

0.5 is random chance and 1 is a perfect ordering.

Concordance index

Open diamonds: model as in Duffie, Saita, and Wang (2007), open triangles: model with splines and the interaction, closed triangles: same as open triangles with a random intercept, and closed diamonds: same as before with a random log relative market value slope.

Concordance index

Sometimes more than 5 pct. higher fraction of correctly ordered event times.

Predicted hazard rates

Look at predicted distribution of the average hazard rate.

Predicted hazard rates

Bars are 90 pct. prediction intervals. Crosses are realized hazard rates.

Conclusions

Additional variables are needed in hazard model in Duffie, Saita, and Wang (2007) and Duffie et al. (2009).

Similar to Lando and Nielsen (2010) finding.

Assumption of linearity on the log-hazard scale is too strong.

Evidence of additional time-varying latent factors.

Similar to Lando et al. (2013) and Jensen, Lando, and Medhat (2017) but our model can directly be used for forecasting.

Next steps

Performance on quarter or half-year forecasts.

Non-listed and/or non-rated firms could yield better estimate of frailty variable.

Out-of-sample test with monthly hazard models.

Thank you!

Slides are on rpubs.com/boennecd/PhD-wipt-19.

Paper is at ssrn.com/abstract=3339981.

Code is at bit.ly/github-US_PD_data and bit.ly/github-US_PD_models.

Extra slides

Slides with coefficient estimates in annual models, more predicted random effects, and marginal effect of log relative market value.

Estimates -- annual models

Predicted random effect in random intercept model

Predicted random effects in annual models

Marginal effect of log relative market value

Generalized additive model with tensor product spline on time and the log relative market value only. The \(z\)-axis is the log hazard.

References

Azizpour, S, K. Giesecke, and G. Schwenkler. 2018. “Exploring the Sources of Default Clustering.” Journal of Financial Economics 129 (1): 154–83. doi:https://doi.org/10.1016/j.jfineco.2018.04.008.

Bharath, Sreedhar T., and Tyler Shumway. 2008. “Forecasting Default with the Merton Distance to Default Model.” The Review of Financial Studies 21 (3): 1339–69. doi:10.1093/rfs/hhn044.

Chava, Sudheer, and Robert A. Jarrow. 2004. “Bankruptcy Prediction with Industry Effects *.” Review of Finance 8 (4): 537–69. doi:10.1093/rof/8.4.537.

Christoffersen, Benjamin. 2018. Dynamichazard: Dynamic Hazard Models Using State Space Models. https://github.com/boennecd/dynamichazard.

Christoffersen, Benjamin, Rastin Matin, and Pia Mølgaard. 2018. “Can Machine Learning Models Capture Correlations in Corporate Distresses?”

Duan, Jin-Chuan, Jie Sun, and Tao Wang. 2012. “Multiperiod Corporate Default Prediction—A Forward Intensity Approach.” Journal of Econometrics 170 (1): 191–209. doi:https://doi.org/10.1016/j.jeconom.2012.05.002.

Duffie, Darrell, Andreas Eckner, Guillaume Horel, and Leandro Saita. 2009. “Frailty Correlated Default.” The Journal of Finance 64 (5). Blackwell Publishing Inc: 2089–2123. doi:10.1111/j.1540-6261.2009.01495.x.

Duffie, Darrell, Leandro Saita, and Ke Wang. 2007. “Multi-Period Corporate Default Prediction with Stochastic Covariates.” Journal of Financial Economics 83 (3): 635–65. doi:https://doi.org/10.1016/j.jfineco.2005.10.011.

Fearnhead, Paul, David Wyncoll, and Jonathan Tawn. 2010. “A Sequential Smoothing Algorithm with Linear Computational Cost.” Biometrika 97 (2). [Oxford University Press, Biometrika Trust]: 447–64. http://www.jstor.org/stable/25734097.

Filipe, Sara Ferreira, Theoharry Grammatikos, and Dimitra Michala. 2016. “Forecasting Distress in European Sme Portfolios.” Journal of Banking & Finance 64: 112–35. doi:https://doi.org/10.1016/j.jbankfin.2015.12.007.

Jensen, Thais, David Lando, and Mamdouh Medhat. 2017. “Cyclicality and Firm-Size in Private Firm Defaults.” International Journal of Central Banking 13 (4): 97–145.

Lando, David, and Mads Stenbo Nielsen. 2010. “Correlation in Corporate Defaults: Contagion or Conditional Independence?” Journal of Financial Intermediation 19 (3): 355–72. doi:https://doi.org/10.1016/j.jfi.2010.03.002.

Lando, David, Mamdouh Medhat, Mads Stenbo Nielsen, and Søren Feodor Nielsen. 2013. “Additive Intensity Regression Models in Corporate Default Analysis.” Journal of Financial Econometrics 11 (3): 443–85. doi:10.1093/jjfinec/nbs018.

R Core Team. 2018. R: A Language and Environment for Statistical Computing. Vienna, Austria: R Foundation for Statistical Computing. https://www.R-project.org/.

Shumway, Tyler. 2001. “Forecasting Bankruptcy More Accurately: A Simple Hazard Model.” The Journal of Business 74 (1). The University of Chicago Press: 101–24. http://www.jstor.org/stable/10.1086/209665.