Sequential Design and Spatial Modeling for Portfolio Tail Risk Measurement

We consider calculation of capital requirements when the underlying economic scenarios are determined by simulatable risk factors. In the respective nested simulation framework, the goal is to estimate portfolio tail risk, quantified via value-at-risk (VaR) or tail value-at-risk (TVaR), of portfolio losses in a given collection of future economic scenarios represented by factor levels at the risk horizon. Traditionally, evaluating portfolio losses in an outer scenario is done by computing a conditional expectation via inner-level Monte Carlo simulations and is computationally expensive. We introduce several inter-related machine learning techniques to speed up this computation, in particular by properly accounting for the simulation noise. Our main workhorse is an advanced Gaussian process (GP) regression approach that uses nonparametric spatial modeling to efficiently learn the relationship between the stochastic factors defining scenarios and corresponding portfolio values. Leveraging this emulator, we develop sequential algorithms that adaptively allocate inner simulation budgets to target the quantile region. The GP framework also yields better uncertainty quantification for the resulting VaR/TVaR estimators which reduce bias and variance compared to existing methods. We illustrate the proposed strategies with two case-studies in two and six dimensions.