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Student-t Factor Models and Copula-Based Credit Risk Estimation

A recent study by Kang and Shahabuddin [2005] proposed a novel approach to estimating Value-at-Risk (VaR) for student-t factor models. The method involves sampling V, then Z1,…,Zm, given V, and exponentially twisting V by a parameter that is the solution of a constrained optimization problem.

Variance Reduction Techniques

The authors suggest using stratified sampling to minimize computational costs, as the original approach requires multiple numerical optimization procedures per sample. This methodology was later extended to general single-factor models by Bassamboo et al. [2008].

Another method for applying variance reduction to Student-t factor models is presented in Chan and Kroese [2010]. This approach estimates VaR by calculating the expectations of truncated gamma random variables.

Worked Example: Gaussian Factor Model

To illustrate these methods, a worked example was conducted using a Gaussian factor model. The portfolio consisted of 1000 assets with factor loadings drawn uniformly from (0,1/√m). Barriers were set at ρi = Φ−1(1−Pi), where Pi = 0.01∗(1+sin(16πi/n)).

Three methods were used to estimate VaR and Expected Shortfall (ES): the CMC approach, Glasserman and Li’s method, and Cross-Entropy (CE). The results show that CE and Glasserman and Li’s estimators outperform the CMC estimator as α increases. However, running times are implementation-specific and vary significantly between methods.

Copula Models

In credit risk modeling, copulas have become a popular tool for expressing dependency structures. A copula is a multivariate distribution function with uniform marginals that describes the dependence structure between random variables U1,…,Un. These can be transformed into random variables X1,…,Xn with arbitrary distributions F1,…,Fn.

Three types of copulas are commonly used in credit risk modeling: Gaussian, Student-t, and Archimedean. The Gaussian copula is popularized by Li [2000], while the other two exhibit tail dependence. These models can be applied to estimate VaR and ES for credit portfolios.

Conclusion

In conclusion, this article highlights recent advancements in estimating Value-at-Risk (VaR) and Expected Shortfall (ES) for student-t factor models using copulas. The methods proposed by Kang and Shahabuddin [2005] and Bassamboo et al. [2008] provide a framework for variance reduction in credit risk modeling. Further research is needed to explore the application of these methodologies in real-world scenarios.

Table 2: Estimated VaR and ES for a Gaussian Factor Model

Note: Running times are implementation-specific and not included in this article.