Bayesian confidence intervals for probability of default and asset correlation of portfolio credit risk

Abstract

We derive Bayesian confidence intervals for the probability of default (PD), asset correlation (Rho), and serial dependence (Theta) for low default portfolios (LDPs). The goal is to reduce the probability of underestimating credit risk in LDPs. We adopt a generalized method of moments with continuous updating to estimate prior distributions for PD and Rho from historical default data. The method is based on a Bayesian approach without expert opinions. A Markov chain Monte Carlo technique, namely, the Gibbs sampler, is also applied. The performance of the estimation results for LDPs validated by Monte Carlo simulations. Empirical studies on Standard & Poor’s historical default data are also conducted.

Appendix

In Sect. 3, we use the GMM method with the CU approach to estimate the prior parameters. The GMM method with CU estimators is based on the expectation values (15) and (16). To derive these expectation values, we introduce two lemmas.

Lemma 1

Andersen and Sidenius (2004). For any constants $a$ and $b$,

$$\begin{aligned} \int \limits _{-\infty }^{\infty } \Phi (a z + b) \phi (z) \text { d}z = \Phi \left( {b \over \sqrt{1+a^2}} \right) . \end{aligned}$$

Lemma 2

Let $y \sim \hbox {Binomial}(n, p)$, then, for $n \ge k$,

$$\begin{aligned} \mathsf{E}\left[ {y(y-1) \cdots (y-k+1) \over n(n-1) \cdots (n-k+1)} \right] = p^k, \quad k=1, 2, \ldots . \end{aligned}$$

From Lemmas 1 and 2, it can be shown that

$$\begin{aligned} \mathsf{E}\left[ \lambda _t(p, \rho ,z_t)|p, \rho \right]&= \int \limits _{-\infty }^{\infty } \Phi \left( {\Phi ^{-1}(p) - \sqrt{\rho } z_t \over \sqrt{1-\rho }} \right) \phi (z_t) \text { d}z_t \nonumber \\&= p, \nonumber \\ \mathsf{E}\left[ \lambda _t^k(p, \rho ,z_t)|p, \rho \right]&= \int \limits _{-\infty }^{\infty } \Phi ^k \left( {\Phi ^{-1}(p) - \sqrt{\rho } z_t \over \sqrt{1-\rho }} \right) \phi (z_t) \text { d}z_t,\quad k=2, 3, 4. \end{aligned}$$

Now, the expectation value (15) is

$$\begin{aligned} \mathsf{E}(\bar{y}_t)&= \mathsf{E}[\mathsf{E}(\bar{y}_t|z_t, p, \rho )]\nonumber \\&= \mathsf{E}[\lambda _t(p, \rho , z_t)]\nonumber \\&= \mathsf{E}[\mathsf{E}[\lambda _t(p, \rho , z_t)|p, \rho ]] \nonumber \\&= \mathsf{E}(p) \nonumber \\&= \int \limits _0^1 p\times \pi (p) \text { d}p \nonumber \\&= {a_p \over a_p+b_p}, \end{aligned}$$

and, for $k=2, 3, 4$, we obtain (16),

$$\begin{aligned}&\mathsf{E}\left[ \bar{y}_t \left( \bar{y}_t - {1 \over n_t} \right) \cdots \left( \bar{y}_t - {k-1 \over n_t} \right) \right] \nonumber \\&\quad = \mathsf{E}\left\{ \mathsf{E}\left[ \bar{y}_t \left( \bar{y}_t - {1 \over n_t} \right) \cdots \left( \bar{y}_t - {k-1 \over n_t} \right) \right] \Bigg |z_t, p, \rho \right\} \nonumber \\&\quad ={n_t(n_t-1) \cdots (n_t-k+1) \over n_t^k} \mathsf{E}\left[ \lambda _t^k( p, \rho , z_t) \right] \nonumber \\&\quad ={(n_t-1) \cdots (n_t-k+1) \over n_t^{k-1}} \mathsf{E}\left\{ \mathsf{E}\left[ \lambda _t^k(p, \rho , z_t)|p, \rho \right] \right\} \nonumber \\&\quad ={(n_t-1) \cdots (n_t-k+1) \over n_t^{k-1}} \int \limits _0^1 \int \limits _0^1\int \limits _{-\infty }^{\infty } \Phi ^k\left( \frac{\Phi ^{-1}(p)-\sqrt{\rho }z_{t}}{\sqrt{1-\rho }}\right) \nonumber \\&\qquad \times \pi (p) \times \pi (\rho )\times \phi (z_t)\text { d}z_t \text { d}p \text { d}\rho . \end{aligned}$$