Probabilistic Overview of Probabilities of Default for Low Default Portfolios by K. Pluto and D. Tasche

Abstract

This article gives a probabilistic overview of the widely used method of default probability estimation proposed by K. Pluto and D. Tasche. There are listed detailed assumptions and derivation of the inequality where the probability of default is involved under the influence of systematic factor. The author anticipates adding more clarity, especially for early career analysts or scholars, regarding the assumption of borrowers' independence, conditional independence and interaction between the probability distributions such as binomial, beta, normal and others. There is also shown the relation between the probability of default and the joint distribution of $\sqrt{\varrho}X-\sqrt{1-\varrho}Y$, where $X$, including but not limiting, is the standard normal, $Y$ admits, including but not limiting, the beta-normal distribution and $X,\,Y$ are independent.

Figures (7)

• Figure 1: The probability density function whose cumulative distribution function is $F_{\alpha,\,\beta,\,\varrho}(y)$, $y\in\mathbb{R}$.
• Figure 2: The cumulative distribution function $F_{\alpha,\,\beta,\,\varrho}(y)$, $y\in\mathbb{R}$.
• Figure 3: The probability density function whose cumulative distribution function is $\tilde{F}_{n-k,\,k+1,\,\varrho}(p)$, $p\in(0,\,1)$.
• Figure 4: The cumulative distribution function $\tilde{F}_{n-k,\,k+1,\,\varrho}(p)$, $p\in(0,\,1)$.
• Figure 5: The joint density $\varphi(x)bn_{\alpha,\,\beta,\,0,\,1}(y)$, $(x,\,y)\in\mathbb{R}^2$ and the line $\sqrt{\varrho}x-\sqrt{1-\varrho}y=\Phi^{-1}(p)$, when $\alpha=5$, $\beta=2$, $\varrho=1/2$ and $p=1/10$. The red colored volume corresponds to $\sqrt{\varrho}x-\sqrt{1-\varrho}y<\Phi^{-1}(p)$, while the blue one is $1-\gamma=0.869$. In other words, the inequality \ref{['main_ineq']} with $1-\gamma=0.869$, $k=1$, $n=6$ and $\varrho=1/2$ is satisfied when $p\in(0,\,1/10]$.

Theorems & Definitions (10)

• Proposition 1
• Proposition 2
• Proposition 3
• Corollary 4
• proof : Proof of Proposition \ref{['beta_prop']}
• proof : Proof of Proposition \ref{['main_prop']}
• proof : Proof of Proposition \ref{['prop:dist_eq']}
• proof : Proof of Corollary \ref{['corollary']}
• Example 5
• Example 6