This paper quantifies the credit risk loss distribution of the Spanish financial system by introducing a general Monte Carlo importance sampling (IS) approach. We start obtaining all the required information for the standard credit risk model. Then we quantify the loss distribution under the standard IS method and allocate the total risk over the different institutions in the Spanish financial system. We extend the current IS framework to deal with more general assumptions like random recoveries and market valuation. We also study the variability of the risk measures over the business cycle and the possible variability due to the model parameters uncertainty. Our results show that this approach can be very useful for banking supervisors from a macroprudential point of view and that the risk allocation can vary considerably depending on the valuation model under analysis. 

We propose a new method that extends the saddlepoint approximation to allocate credit risk. This method applies a Taylor expansion to the inverse Laplace transform around an arbitrary point to characterize the loss distribution of a portfolio. It is based on Hermite polynomials. From a computational point of view, our method is less demanding than other approximate methods. We also extend the current saddlepoint methods to deal with random recoveries and market valuation. Considering a portfolio that includes Spanish financial institutions, we show that these extensions can characterize the risk of the portfolio very well. The risk allocation method generates more accurate results than other approximate methods, with few calculations for default mode models and pure macroeconomy driven recoveries. We also find that modeling mixed idiosyncratic and macroeconomic random recoveries does not generate much greater risk than a pure macroeconomic random recoveries model.

Financial institutions and regulators usually measure credit risk only over a one-year time horizon. Hence, current statistical models can generate closed-form expressions for the one-year loss distribution. Losses over longer horizons are considered using scenario analysis or Monte Carlo simulation. This paper proposes a simple multi-period credit risk model and uses Taylor expansion approximations to estimate the multi-period loss distribution. In this paper we extend the currently available second-order Taylor expansion approximations to credit risk with a third-order term and we use this new approximation to obtain the loss distribution in the multi-period framework. Our results show that the approximation is more accurate under recessions or for portfolios with high probability of default. We also show that, in general, the effect of this third-order adjustment is quite small.

This paper proposes an approximate formula to measure the credit risk of portfolios under random recoveries. This formula is based on a Taylor expansion and enables having recoveries that are correlated with the default rates over the business cycle. We show how to calibrate the corresponding models and the accuracy of the approximation using defaulted corporate bonds data for the period 1982–2014. Our results show that the proposed formula can be used to approximate the loss distribution of a portfolio under random correlated recoveries in a very satisfactory way. Moreover, this kind of recovery models could be easily implemented under the Basel capital requirements regulation to improve the credit risk measurement. 

This paper explores the ability of the Machine Learning (ML) techniques to calibrate models that replicate the outputs of the Vasicek (1987) credit risk model. In the general case, estimating the loss distribution in this model requires computationally demanding Monte Carlo simulations while the ML approach only requires an initial calibration process. For different granular or concentrated portfolios, our results show that using just two variables (the confidence level and a Gaussian copula-based loss distribution estimate), the tree-based models provide fast and accurate estimates of the real loss distribution.