Predicting the Loss Given Default Distribution with the Zero-Inflated Censored Beta-Mixture Regression that Allows Probability Masses and Bimodality

Abstract

We propose a new procedure to predict the loss given default (LGD) distribution. Studies find empirical evidence that LGD values have a high concentration at the endpoint 0. Thus, we first use a logistic regression to determine the probability that the LGD value of a defaulted debt equals zero. Further, studies find empirical evidence that positive LGD values have a low concentration at the endpoint 1 and a bimodal distribution on the interval (0,1). Therefore, we use a right-tailed censored beta-mixture regression to model the distribution of positive LGD data. To implement the proposed procedure, we collect 5554 defaulted debts from Moody’s Default and Recovery Database and apply an expectation–maximization algorithm to estimate the LGD distribution. Using each of the k-fold cross-validation technique and the expanding rolling window approach, our empirical results confirm that the new procedure has better and more robust out-of-sample performance than its alternatives because it yields more accurate predictions of the LGD distribution.

Appendices

A sketch of the proof

To decompose ℓ_ZCBR(δ_{0, 1}, ρ₁, α_{1, 1}, β_{1, 1}, α_{2, 1}, β_{2, 1}, η₁), we first replace its components p_{ZCBR, 0}(x_i) and p_{ZCBR, 1}(x_i) with δ₀(x_i) and η{1 − δ₀(x_i)}[1 − Ω{(1 + ρ)⁻¹; α₁(x_i), β₁(x_i)}], respectively, and rearrange the result as:

$$ {\displaystyle \begin{array}{c}{\ell}_{ZCBR}\left({\delta}_{0,1},{\rho}_1,{\alpha}_{1,1},{\beta}_{1,1},{\alpha}_{2,1},{\beta}_{2,1},{\eta}_1\right)\\ {}={\sum}_{i=1}^n\ln \left(\begin{array}{c}{\delta}_0{\left({x}_i\right)}^{I\left({y}_i=0\right)}{\left[\left\{1-{\delta}_0\left({x}_i\right)\right\}{f}_{ZCBR,1}\left({y}_i|{x}_i\right)\right]}^{I\left\{{y}_i\in \left(0,1\right)\right\}}\times \\ {}{\eta}^{I\left({y}_i=1\right)}{\left\{1-{\delta}_0\left({x}_i\right)\right\}}^{I\left({y}_i=1\right)}{\left[1-\varOmega \left\{\frac{1}{1+\rho };{\alpha}_1\left({x}_i\right),{\beta}_1\left({x}_i\right)\right\}\right]}^{I\left({y}_i=1\right)}\end{array}\right)\end{array}}={L}_1+{L}_2, $$

where

$$ {\displaystyle \begin{array}{c}{L}_1={\sum}_{i=1}^n\ln \left[{\delta}_0{\left({x}_i\right)}^{I\left({y}_i=0\right)}{\left\{1-{\delta}_0\left({x}_i\right)\right\}}^{I\left\{{y}_i\in \left(0,1\right)\right\}}{\left\{1-{\delta}_0\left({x}_i\right)\right\}}^{I\left({y}_i=1\right)}\right],\\ {}{L}_2={\sum}_{i=1}^n\ln \left({f}_{ZCBR,1}{\left({y}_i|{x}_i\right)}^{I\left\{{y}_i\in \left(0,1\right)\right\}}{\eta}^{I\left({y}_i=1\right)}{\left[1-\varOmega \Big\{\frac{1}{1+\rho };{\alpha}_1\left({x}_i\right),{\beta}_1\left({x}_i\right)\Big\}\right]}^{I\left({y}_i=1\right)}\right).\end{array}} $$

Then, through a straightforward calculation, the quantities L₁ and L₂ become:

$$ {\displaystyle \begin{array}{c}{L}_1={\sum}_{i=1}^n\ln \left[{\delta}_0{\left({x}_i\right)}^{I\left({y}_i=0\right)}{\left\{1-{\delta}_0\left({x}_i\right)\right\}}^{I\left({y}_i>0\right)}\right]\\ {}={\sum}_{i=1}^n\left[I\left({y}_i=0\right)\ln \left\{{\delta}_0\left({x}_i\right)\right\}+\left\{1-I\left({y}_i=0\right)\right\}\ln \left\{1-{\delta}_0\left({x}_i\right)\right\}\right]\\ {}={\sum}_{i=1,{y}_i=0}^n\ln \left[{\delta}_0\left({x}_i\right)/\left\{1-{\delta}_0\left({x}_i\right)\right\}\right]+{\sum}_{i=1}^n\ln \left\{1-{\delta}_0\left({x}_i\right)\right\}\equiv {\ell}_{ZCBR,1}\left({\delta}_{0,1}\right),\end{array}} $$

The proof for the decomposition of ℓ_ZCBR(δ_{0, 1}, ρ₁, α_{1, 1}, β_{1, 1}, α_{2, 1}, β_{2, 1}, η₁) is complete.

An EM algorithm

To describe the EM algorithm for finding the maximizer of ℓ_{ZCBR, 2}(ρ₁, α_{1, 1}, β_{1, 1}, α_{2, 1}, β_{2, 1}, η₁), we set the following notation. Let

$$ {\displaystyle \begin{array}{l}{p}_1\left(x,{\theta}_1\right)=1-\varOmega \left\{\frac{1}{1+\rho };{\alpha}_1(x),{\beta}_1(x)\right\},\\ {}{f}_{M,1}\left(y|x,{\theta}_1\right)={p}_1{\left(x,{\theta}_1\right)}^{I\left(y=1\right)}{\left[\frac{1}{1+\rho}\times \omega \left\{\frac{y}{1+\rho };{\alpha}_1(x),{\beta}_1(x)\right\}\right]}^{I\left\{y\in \left(0,1\right)\right\}},\\ {}{f}_{M,2}\left(y|x,{\theta}_2\right)=\omega \left\{y;{\alpha}_2(x),{\beta}_2(x)\right\},\end{array}} $$

where ρ = ln{1 + exp(ρ₁)}, θ₁ = (ρ₁, α_{1, 1}, β_{1, 1}), and θ₂ = (α_{2, 1}, β_{2, 1}). Using these results, the conditional mixture distribution of the positive LGD data is:

$$ {f}_M\left(y|x,\theta \right)=\eta {f}_{M,1}\left(y|x,{\theta}_1\right)+\left(1-\eta \right){f}_{M,2}\left(y|x,{\theta}_2\right), $$

where y ∈ (0, 1], $\eta =\frac{\exp \left({\eta}_1\right)}{1+\exp \left({\eta}_1\right)},$ and θ = (θ₁, θ₂, η₁) = (ρ₁, α_{1, 1}, β_{1, 1}, α_{2, 1}, β_{2, 1}, η₁).

To apply the EM algorithm to maximize ${\ell}_{ZCBR,2}\left(\theta \right)={\sum}_{i=1,{y}_i>0}^n\ln \left\{{f}_M\left({y}_i|{x}_i,\theta \right)\right\},$ we introduce a set of dummy latent data z_i that are indicator variables linking observations y_i > 0 to the mixture components f_{M, 1}(y_i| x_i, θ₁) and f_{M, 2}(y_i| x_i, θ₂). Thus, we write the log-likelihood function of the observed and latent data as:

$$ {\ell}_{ZCBR,2}^{\ast}\left(\theta, {z}_1,\cdots, {z}_n\right)={\sum}_{i=1,{y}_i>0}^n\ln \left\{{f}_M^{\ast}\left({y}_i,{z}_i|{x}_i,\theta \right)\right\}\equiv {\ell}_1\left({\theta}_1\right)+{\ell}_2\left({\theta}_2\right)+{\ell}_3\left({\eta}_1\right), $$

where

$$ {\displaystyle \begin{array}{c}{f}_M^{\ast}\left({y}_i,{z}_i|{x}_i,\theta \right)={\left\{\eta {f}_{M,1}\left({y}_i|{x}_i,{\theta}_1\right)\right\}}^{z_i}{\left\{\left(1-\eta \right){f}_{M,2}\left({y}_i|{x}_i,{\theta}_2\right)\right\}}^{\left(1-{z}_i\right)},\\ {}{\ell}_1\left({\theta}_1\right)={\sum}_{i=1,{y}_i>0}^n{z}_i\ln \left\{{f}_{M,1}\left({y}_i|{x}_i,{\theta}_1\right)\right\},\\ {}{\ell}_2\left({\theta}_2\right)={\sum}_{i=1,{y}_i\in \left(0,1\right)}^n\left(1-{z}_i\right)\ln \left\{{f}_{M,2}\left({y}_i|{x}_i,{\theta}_2\right)\right\},\\ {}{\ell}_3\left({\eta}_1\right)={\sum}_{i=1,{y}_i>0}^n\left\{{z}_i\ln \left(\eta \right)+\left(1-{z}_i\right)\ln \left(1-\eta \right)\right\}.\end{array}} $$

The EM algorithm for maximizing ${\ell}_{ZCBR,2}^{\ast}\left(\theta, {z}_1,\cdots, {z}_n\right)$ comprises the following two-step procedure. The first step of the algorithm involves making an expectation:

$$ E\left\{{\ell}_{ZCBR,2}^{\ast}\left(\theta, {z}_1,\cdots, {z}_n\right)|{\theta}^{(k)}\right\}\equiv {\tilde{\ell}}_1\left({\theta}_1\right)+{\tilde{\ell}}_2\left({\theta}_2\right)+{\tilde{\ell}}_3\left({\eta}_1\right). $$

Here ${\theta}^{(k)}=\left\{{\theta}_1^{(k)},{\theta}_2^{(k)},{\eta}_1^{(k)}\right\}$ and ${\tilde{\ell}}_1\left({\theta}_1\right),$ ${\tilde{\ell}}_2\left({\theta}_2\right),$ and ${\tilde{\ell}}_3\left({\eta}_1\right)$ are ℓ₁(θ₁), ℓ₂(θ₂), and ℓ₃(η₁) with z_i replaced by ${\tilde{z}}_i.$ The quantity ${\tilde{z}}_i$ has the formula:

$$ {\tilde{z}}_i=E\left\{Z|{y}_i,{x}_i,{\theta}^{(k)}\right\}=\frac{\eta^{(k)}{f}_{M,1}\left({y}_i|{x}_i,{\theta}_1^{(k)}\right)}{\eta^{(k)}{f}_{M,1}\left({y}_i|{x}_i,{\theta}_1^{(k)}\right)+\left\{1-{\eta}^{(k)}\right\}{f}_{M,2}\left({y}_i|{x}_i,{\theta}_2^{(k)}\right)}, $$

for each i = 1, ⋯, n, where ${\eta}^{(k)}=\frac{\exp \left\{{\eta}_1^{(k)}\right\}}{1+\exp \left\{{\eta}_1^{(k)}\right\}}.$ The second step of the algorithm involves maximizing $E\left\{{\ell}_{ZCBR,2}^{\ast}\left(\theta, {z}_1,\cdots, {z}_n\right)|{\theta}^{(k)}\right\}$ to obtain ${\theta}^{\left(k+1\right)}=\arg {\max}_{\theta }E\left\{{\ell}_{ZCBR,2}^{\ast}\left(\theta, {z}_1,\cdots, {z}_n\right)|{\theta}^{(k)}\right\}.$ It is separately performed by finding ${\theta}_1^{\left(k+1\right)}=\arg {\max}_{\theta_1}{\tilde{\ell}}_1\left({\theta}_1\right),$ ${\theta}_2^{\left(k+1\right)}=\arg {\max}_{\theta_2}{\tilde{\ell}}_2\left({\theta}_2\right),$ and ${\eta}_1^{\left(k+1\right)}=\arg {\max}_{\eta_1}{\tilde{\ell}}_3\left({\eta}_1\right).$¹⁷ Thus, ${\theta}^{\left(k+1\right)}=\left\{{\theta}_1^{\left(k+1\right)},{\theta}_2^{\left(k+1\right)},{\eta}_1^{\left(k+1\right)}\right\}.$ The two-step procedure continues until the convergence criterion ‖θ^(k + 1) − θ^(k)‖ < 0.005 is satisfied.¹⁸ The notation ‖ψ‖ denotes the Euclidean length of the given vector ψ.