  Every "x"-adjustment in the so-called xVA financial risk management framework
relies on the computation of exposures. Considering thousands of Monte Carlo
paths and tens of simulation steps, a financial portfolio needs to be evaluated
numerous times during the lifetime of the underlying assets. This is the
bottleneck of every simulation of xVA. In this article, we explore numerical
techniques for improving the simulation of exposures. We aim to decimate the
number of portfolio evaluations, particularly for large portfolios involving
multiple, correlated risk factors. The usage of the Stochastic Collocation (SC)
method, together with Smolyak's sparse grid extension, allows for a significant
reduction in the number of portfolio evaluations, even when dealing with many
risk factors. The proposed model can be easily applied to any portfolio and
size. We report that for a realistic portfolio comprising linear and non-linear
derivatives, the expected reduction in the portfolio evaluations may exceed
6000 times, depending on the dimensionality and the required accuracy. We give
illustrative examples and examine the method with realistic multi-currency
portfolios consisting of interest rate swaps and swaptions.
