# CAIMS Prize Award

Department of Computer Science, University of Toronto

Supervisor: K. Jackson

Thesis title: Computation of value-at-risk: The fast convolution method, dimension reduction and perturbation theory

Abstract:

Value-at-risk is a measure of market risk for a portfolio. Market risk is the chance that the portfolio declines in value due to changes in market variables. This thesis is about the computation of value-at-risk for portfolios with derivatives and for models for returns that have a distribution with fat tails.

We introduce a new Fourier algorithm, the fast convolution method, for computing value-at-risk. The fast convolution method is different from other Fourier methods in that it does not require that the characteristic function of the portfolio returns be known explicitly. Our new method can therefore be used with more general return models. In the thesis we present experiments with three return models: the normal model, the asymmetric T model and a model using the non-parametric Parzen density estimator. We also discuss how the fast convolution method can be extended to compute the value-at-risk gradient, present a proof of convergence and illustrate the performance of the method with examples.

We develop and compare two methods for dimension reduction in the computation of value-at-risk. The goal of dimension reduction is to reduce computation time by finding a small model that captures the main dynamics of the original model. We compare the two methods for an example problem and conclude that the method based on mean square error is superior. Finally, we present an optimization example that illustrates that dimension reduction may reduce the time to compute value-at-risk while maintaining good accuracy.

We develop a perturbation theory for value-at-risk with respect to changes in the return model. By considering variational properties, we derive a first-order error bound and find the condition number of value-at-risk. We argue that the sensitivity observed in empirical studies is an inherent limitation of value-at-risk.