Corporate Default Models

dummy slide

Why?

\[ \definecolor{gray}{RGB}{192,192,192} \def\vect#1{\boldsymbol #1} \def\bigO#1{\mathcal{O}(#1)} \def\Cond#1#2{\left(#1 \mid #2\right)} \def\diff{{\mathop{}\!\mathrm{d}}} \]

Motivation

Want to model the loss distribution for a corporate debt portfolio.

E.g. a bank that provides loans to firms.

Important for both regulators and holders of the portfolio.

Will focus on whether the lenders default.

A component in bottom-up models.

Motivation: Concrete

The portfolio consists of \(n_t\) loans to firms \(R_t = \{i_{1t}, i_{2t}, \dots, i_{n_{t}t}\}\).

The exposure to each firm in interval \(t\) is \(E_{it}\in(0,\infty)\).

E.g. the current amount outstanding.

The default indicator \(Y_{it}\in \{0,1\}\) is 1 if firm \(i\) defaults in interval \(t\).

If the firm defaults then \(G_{it}\in [0,1]\) fraction of \(E_{it}\) is lost.

That is, \(G_{it}E_{it}\) is lost.

Motivation: Concrete

The loss in interval \(t\) is

\[L_t = \sum_{i \in R_t}E_{it}G_{it}Y_{it}\]

\(E_{it}\in(0,\infty)\): exposure, \(G_{it}\in [0,1]\) loss-given-default, and \(Y_{it}\in \{0,1\}\) is the default indicator.

Motivation: Concrete

The loss in interval \(t\) is

\[L_t = \sum_{i \in R_t}\color{gray}{E_{it}G_{it}}Y_{it}\]

\(E_{it}\in(0,\infty)\): exposure, \(G_{it}\in [0,1]\) loss-given-default, and \(Y_{it}\in \{0,1\}\) is the default indicator.

Focus on default indicators.

Need accurate joint model of \(Y_{it}\) for \(i\in R_t\).

Particularly to model the tail of \(L_t\).

Contributions

Typical Model

Given firm variables \(\vec x_{it}\) and macro variables \(\vec m_t\) one models the probability of default as

\[g(P(Y_{it} = 1 \mid \vec x_{it}, \vec m_t)) = \vec \beta^\top \vec x_{it} + \vec\gamma^\top\vec m_t \]

where \(g\) is a link function

e.g. logit function. See Shumway (2001), Chava and Jarrow (2004), Duffie, Saita, and Wang (2007), and Campbell and Szilagyi (2008).

I.e. firms are only correlated through observable firm variables and macro variables.

Add Random Effects

May observe excess clustering of defaults.

E.g. an omitted macro variable.

Thus, the model is extended to

\[\begin{aligned} g(P(Y_{it} = 1 \mid \vec x_{it}, \vec m_t, \alpha_t)) &= \vec \beta^\top \vec x_{it} + \vec\gamma^\top\vec m_t + \alpha_t \\ \alpha_t &= \theta \alpha_{t-1} + \epsilon_t \\ \epsilon_t & \sim N(0, \sigma^2) \end{aligned}\]

See Duffie et al. (2009), P. Chen and Wu (2014), and Qi, Zhang, and Zhao (2014). Koopman, Lucas, and Schwaab (2011), Koopman, Lucas, and Schwaab (2012), and Schwaab, Koopman, and Lucas (2017) use a similar model with group specific \(\alpha_t\)s or where groups load differently on \(\alpha_t\).

Contributions

Linearity and additivity are not obvious.

Though, we may expect a monotone partial effect.

\[g(P(Y_{it} = 1 \mid \vec x_{it}, \vec m_t,\alpha_t)) = \underbrace{\vec \beta^\top \vec f(\vec x_{it})}_? + \vec\gamma^\top\vec m_t + \alpha_t\]

Some suggestions that the assumptions are violated.

Min and Lee (2005), Berg (2007), Alfaro et al. (2008), Kim and Kang (2010), and Jones, Johnstone, and Wilson (2017).

Relax assumptions and focus on the firm-level and aggregate performance.

Contributions

Assumption of constant coefficients may be violated.

See Lando et al. (2013), Filipe, Grammatikos, and Michala (2016), and Jensen, Lando, and Medhat (2017).

\[g(P(Y_{it} = 1 \mid \vec x_{it}, \vec m_t, \alpha_t)) = \underbrace{\vec \beta^\top_t}_? \vec f(\vec x_{it}) + \vec\gamma^\top\vec m_t + \alpha_t\]

Relax assumptions with a model that can be directly used for forecasting.

Chapters

dynamichazard: Dynamic Hazard Models using State Space Models

Implementations of fast approximation methods to estimate discrete time survival models.

Can Machine Learning Models Capture Correlations in Corporate Distresses?

Focus on non-linear effects with tall and narrow sample of limited liability companies.

Modeling Frailty Correlated Defaults with Multivariate Latent Factors

Focus on time-varying effects with short and wide sample of US public companies.

Chapters

Particle Methods in the dynamichazard Package

Covers the particle based methods in the dynamichazard package which are used in chapter 3.

Chapter 1 and 4
dynamichazard: Dynamic Hazard Models using State Space Models

Particle Methods in the dynamichazard Package

R Packages

One of 6½ open source R packages I have authored or co-authored: dynamichazard, pre, mssm, rollRegres, parglm, and DtD.

The Oumuamua package is almost finished.

Covers implementations of fast approximation methods.

Covers the implemented particle based methods.

Alternatives

Interested in

\[P(Y_{it} = 1 \mid \vec x_{it}, Y_{i1} = \cdots = Y_{it-1} = 0) = h_t(\vec x_{it})\]

Diagnosis (Time zero)
End of followup

Can use non-parametric method.

First cross-section (Time zero)
Now
Period of interest

Use parametric model for \(h_t\).

State-Space Model

\[\begin{aligned} g(P(Y_{it} = 1 \mid \vec x_{it}, \vec m_t, \vec u_{it}, \vec\alpha_t)) &= \vec \beta^\top \vec x_{it} + \vec\gamma^\top\vec m_t + \vec\alpha_t^\top \vec u_{it} \\ \vec\alpha_t &= F \vec\alpha_{t-1} + \vec\epsilon_t \\ \vec\epsilon_t & \sim N(\vec 0, \Sigma) \end{aligned}\]

Yields intractable integral.

Approximations

Contains implementations of extended Kalman filters and an unscented Kalman filter.

The former is suggested by Fahrmeir (1992) and Fahrmeir (1994) and for the latter see Julierm and Uhlmann (1997) and Wan and Merwe (2000).

The approximations are commonly used in engineering.

The implementations are in C++, very fast, and support computation in parallel.

Monte Carlo Methods

The package also support

  • some of the particle filters suggested by Pitt and Shephard (1999) and Lin et al. (2005).
  • the particle smoothers suggested by Briers, Doucet, and Maskell (2009) and Fearnhead, Wyncoll, and Tawn (2010).
  • approximations methods suggested by Cappe and Moulines (2005) and Poyiadjis, Doucet, and Singh (2011).
Covered in chapter 4.

The mssm contains a new implementation of some of the above methods.

Chapter 2
Can Machine Learning Models Capture Correlations in Corporate Distresses?

Rastin Matin

Danmarks Nationalbank

Pia Mølgaard

Danmarks Nationalbank

Motivation

Recent papers suggest that “machine learning” models perform better on the firm-level.

Suggests that previous model may be misspecified.

Maybe some of the evidence of random effects are due to invalid assumptions of partial associations.

E.g. due to co-movements in covariates.

Findings

Find better firm-level performance with more complex models.

Smaller difference in firm-level performance.

Suggest a model to easily account non-linear effects and with time-varying effects.

Sample

Danish limited liability companies.

Tall sample: 198 929 firms.

Narrow sample: 14 years with annual time periods.

Limits ability to model changes through time.

Models

Generalized linear model.

Assumptions: linearity, additivity, and conditional independence.

Generalized additive model.

Assumptions: conditional independence.

Gradient boosted tree model.

Assumptions: conditional independence.

Generalized linear mixed model with splines.

Assumptions: conditional independence given the random effect.

Example of Partial Effects

Estimated partial effect in the generalized additive model of log of the size and net profit to a size measure of the firm.

Out-of-Sample

Firm-Level Performance (AUC)

◆: Generalized linear model, ▲: Generalized additive model, and ●: Gradient boosted tree model.

Aggregate Performance (Distress Rate)

◆: Generalized linear model, ▲: Generalized additive model, ●: Gradient boosted tree model, ■: realized rate, and ○: Generalized linear mixed model. Bars are 90 pct. prediction intervals.

Chapter 3
Modeling Frailty Correlated Defaults with Multivariate Latent Factors

Rastin Matin

Danmarks Nationalbank

Motivation

Add variables to model in Duffie, Saita, and Wang (2007) and Duffie et al. (2009).

Due to evidence provided by Lando and Nielsen (2010).

Relax linearity and additivity.

Add additional random effects in a model that can directly be used for forecasting.

Due to evidence provided by e.g. Lando et al. (2013).

Findings

Non-linear effect for

the idiosyncratic stock volatility of the firm, the net income over total assets, and log market value over total liabilities. Two of them are related to distance-to-default.

Weaker evidence of random baseline in the model

with additional variables and non-linear effects.

Time-varying relative market size effect.

Sample

US public firms.

Short sample: 3 020 firms.

Wide sample: 1980 to 2016 with monthly time intervals.

Estimates without Random Effects

The figures in the parentheses are Wald \(\chi^2\) statistics. \(\mathcal{M_1}\): model similar to Duffie, Saita, and Wang (2007), \(\mathcal{M_2}\): model with additional variables, and \(\mathcal{M_3}\): model with non-linear effects and an interaction.

Estimates without Random Effects

Large difference in the log-likelihood.

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

Adding Time-Varying Coefficients

\[\begin{aligned} \vec z_{it} &= (\vec x_{it}^\top, \vec m_t^\top, u_{it}, \alpha_t, b_t)^\top \\ g(P(Y_{it} = 1 \mid \vec z_{it})) &= \vec \beta^\top \vec x_{it} + \vec\gamma^\top\vec m_t + \alpha_t + b_tu_{it} \\ \begin{pmatrix}\alpha_t \\ b_t \end{pmatrix} &= \begin{pmatrix}\theta_1 & 0 \\ 0 & \theta_2 \end{pmatrix} \begin{pmatrix}\alpha_{t-1} \\ b_{t-1} \end{pmatrix} + \vec\epsilon_t \\ \vec\epsilon_t & \sim N\left(\vec 0, \begin{pmatrix} \sigma_1^2 & \rho\sigma_1\sigma_2 \\ \rho\sigma_1\sigma_2 & \sigma_2^2 \end{pmatrix}\right) \end{aligned}\]

Adding Time-Varying Coefficient

\[\begin{aligned} \color{gray}{\vec z_{it}} & \color{gray}= \color{gray}{(\vec x_{it}^\top, \vec m_t^\top, u_{it}, \alpha_t, b_t)^\top} \\ \color{gray}{g(P(Y_{it} = 1 \mid \vec z_{it}))} & \color{gray}= \color{gray}{\vec \beta^\top \vec x_{it} + \vec\gamma^\top\vec m_t} + \alpha_t + b_tu_{it} \\ \begin{pmatrix}\alpha_t \\ b_t \end{pmatrix} &= \begin{pmatrix}\theta_1 & 0 \\ 0 & \theta_2 \end{pmatrix} \begin{pmatrix}\alpha_{t-1} \\ b_{t-1} \end{pmatrix} + \vec\epsilon_t \\ \vec\epsilon_t & \sim N\left(\vec 0, \begin{pmatrix} \sigma_1^2 & \rho\sigma_1\sigma_2 \\ \rho\sigma_1\sigma_2 & \sigma_2^2 \end{pmatrix}\right) \end{aligned}\]

Estimates with Random Effects

The figures in the parentheses are Wald \(\chi^2\) statistics. \(\mathcal{M_4}\): model with non-linear effects, an interaction, and a random intercept and \(\mathcal{M_5}\): same as \(\mathcal{M_4}\) with a random relative market size slope.

Estimates with Random Effects

The figures in the parentheses are Wald \(\chi^2\) statistics. \(\mathcal{M_4}\): model with non-linear effects, an interaction, and a random intercept and \(\mathcal{M_5}\): same as \(\mathcal{M_4}\) with a random relative market size slope.

Estimates with Random Effects

The figures in the parentheses are Wald \(\chi^2\) statistics. \(\mathcal{M_4}\): model with non-linear effects, an interaction, and a random intercept and \(\mathcal{M_5}\): same as \(\mathcal{M_4}\) with a random relative market size slope.

Out-of-Sample

Firm-Level Performance (AUC)

Blue: lowest, black: highest. ◇: model as in Duffie, Saita, and Wang (2007), ▽: + covariates and non-linear effects, ▲: + random intercept, and ◆: + random size slope.

Firm-Level Performance (AUC)

Blue: lowest, black: highest. ◇: model as in Duffie, Saita, and Wang (2007), ▽: + covariates and non-linear effects, ▲: + random intercept, and ◆: + random size slope.

Aggregate Performance (Default Rate)

Bars: 90% prediction interval. ○: realized rate, ◇: model as in Duffie, Saita, and Wang (2007), ▽: + covariates and non-linear effects, ▲: + random intercept, and ◆: + random size slope.

Aggregate Performance (Default Rate)

Bars: 90% prediction interval. ○: realized rate, ◇: model as in Duffie, Saita, and Wang (2007), ▽: + covariates and non-linear effects, ▲: + random intercept, and ◆: + random size slope.

Partial Effect

Predicted partial effect in the final model.

Marginal Effect

Estimated marginal effect from a generalized additive model.

Thank You!

Slides are at rpubs.com/boennecd/PhD-defence.

Markdown is at github.com/boennecd/Talks.

Most of the code is at github.com/boennecd.

References on next slide.

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