It is basic and important to compute the portfolio VaR (PVaR) and the loss density function of a credit portfolio in risk management. To estimate these quantities, we focus on calculating the (joint) default probabilities. We model the market returns data by copula methods such as Gaussian copula and Student's t copula. Student's t copula provides a heavy-tailed distribution, so it is more commonly used in practice than normal distribution.

Basic Monte Carlo method doesn't work well for rare event simulation because its standard error is relatively large. We develop efficient importance sampling algorithms to accurately estimate the (joint) default probabilities. These algorithms are capable of increasing the possibilities of rare events and reducing the standard error of the estimators. We use a transformation method developed by Glasserman et al. (2002) to construct moment generating functions associated with proposed importance sampling algorithms. We compare the efficiency of our importance sampling with other (conditional) importance sampling algorithms.

For applications, we evaluate (1) the default leg premium of Basket Default Swap (2) the loss density function of a portfolio and (3) PVaR. The first application is useful for CDO pricing. The second and third applications are useful for risk management of a credit portfolio. In particular for the third application, we apply Gaussian copula and Student's t copula model to fit a company's stock price and its CDS price and estimate the PVaR. Some backtesting diagnostics are used to examine the performance of PVaR estimates.

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