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Quantitative Risk Management / Portfolio Credit Risk and Credit Derivatives

Monte Carlo for Portfolio Credit Risk: Importance Sampling for Rare Events

The Basel IRB formula gives the 99.9% VaR analytically — but only in the large-portfolio limit and for the restricted class of homogeneous-group Gaussian threshold models. For realistic heterogeneous portfolios — different exposures, LGDs, PDs, factor loadings, and non-Gaussian copulas — the loss distribution must be computed by simulation. The naive approach fails: at α=99.9%\alpha = 99.9\%, only 1 in 1,000 simulation paths will produce a loss above the VaR threshold. With 100,000 paths you get roughly 100 exceedances — giving the 99.9% quantile an estimation standard error of ∼10%\sim 10\% even with a large simulation budget.
Importance sampling (IS) solves this by changing the sampling measure to oversample bad scenarios, then reweighting the results. For a Bernoulli mixture model, IS has a clean two-level structure:
  1. Outer IS (factor level): shift the distribution of the common factor Ψ\Psi to concentrate draws in adverse macro states (the "tail region" where many defaults occur simultaneously). This is exponential tilting of a normal distribution — adding a mean shift μ\mu to Ψ\Psi.
  2. Inner IS (conditional default level): given each factor draw ψ\psi, inflate each firm's conditional default probability via exponential tilting of the individual Bernoulli variables, targeting the event {L≥c}\{L \ge c\}.

Combining both levels (Algorithm 11.25) achieves dramatic variance reduction — Figure 11.3 in QRM shows that 10,000 paths with full IS beats 10,000,000 paths of naive MC for a 20-default probability in a 100-firm portfolio.

Learning objectives. By the end of this note you should be able to:
  • explain why naive Monte Carlo is inefficient for rare-event credit tail estimation;
  • define the IS estimator and its likelihood ratio r(x)=f(x)/g(x)r(x) = f(x)/g(x);
  • state the exponential tilting construction and explain how it shifts the mean of a distribution;
  • describe Algorithm 11.24 (inner IS for conditional defaults) and Algorithm 11.25 (full two-level IS);
  • explain how to choose the IS mean μ\mu for the factor distribution by solving an optimisation problem.

In this lesson

    1. The rare-event problem
    1. Importance sampling
    1. Exponential tilting
    1. Inner IS: tilting the conditional default probabilities
    1. Outer IS: tilting the factor distribution
    1. Implementation notes

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