Importance Sampling and Quantile Estimation for Concentration Credit Risk
Efficient algorithms for assessing concentration in credit portfolios
Q.T. van Hattem (TU Delft - Electrical Engineering, Mathematics and Computer Science)
LE Meester – Mentor (TU Delft - Applied Probability)
A. Papapantoleon – Graduation committee member (TU Delft - Applied Probability)
Joris Bierkens – Graduation committee member (TU Delft - Statistics)
R. Vedder – Mentor (Triodos Bank)
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Abstract
This thesis investigates efficient Monte Carlo methods for estimating the 99.9% Value-at-Risk of
concentrated credit portfolios modelled through a normal copula framework. Crude Monte Carlo
simulation is inefficient when estimating extreme loss levels. To address this inefficiency, variance
reduction techniques are applied, including importance sampling and its adaptive variant. This thesis
provides empirical evidence that these techniques make extreme quantile estimation computationally
feasible in a realistic portfolio credit risk setting.
Multiple methods from the literature are reviewed to approach an appropriate importance sampling
proposal. Bernoulli tilting conditional on the common factors is an effective method when obligors
are weakly correlated. However, when large losses are mainly driven by common factors in the credit
portfolio, shifting their means turns out to be more effective. This has to be done with care, since an
improper shift may increase the variance. To this end, a deterministic method that approaches the mode
of the zero-variance distribution is discussed. Furthermore, adaptive importance sampling is studied,
which aims to minimize the variance directly by iteratively updating the shift based on past samples.
This thesis introduces a method that uses the generalized Poisson–Binomial distribution, together with
an inverse Fourier transform, to approximate the zero-variance distribution nearly exactly for simple
portfolio cases.
A deterministic mean-shifted importance sampling (M-IS) and an adaptive importance sampling
(AIS) approach are applied on stylized portfolios. These portfolios are constructed such that they mimic
real-world conditions. The M-IS method achieves an estimated variance reduction factor of 175 on a
large portfolio with 20 000 obligors. Performance on the actual Triodos Bank portfolio delivered similar
performance, but detailed results are withheld for reasons of confidentiality. For a moderately sized
portfolio consisting of 1 000 obligors, M-IS is no more effective than crude Monte Carlo because of the
higher impact of idiosyncratic concentrations. In this case, AIS finds a better balance between the
various kinds of concentrations. With conservative parameters, AIS achieves an estimated variance
reduction factor of 5 on the moderately sized portfolio. Bernoulli tilting could further increase the
variance reduction in these portfolio scenarios. Moreover, as the number of samples increases, AIS further
reduces variance. Finally, this thesis investigates how to construct asymptotic confidence intervals for
the extreme quantile estimates of these stylized portfolios and demonstrates via experiments that they
exhibit asymptotic normal behaviour despite the discreteness of the loss distribution.