Computing Portfolio Risk with Optimised Polynomial Expansions
Professor Rodney Wolff, The University of Queensland
11th Apr 2014 11:00 am - Rm 498 Merewether Bldg H04
The application of orthogonal polynomial expansions to estimation of probability density functions has considerable attraction in financial portfolio theory, particularly for accessing features of a portfolio's profit/loss distribution. This is because such expansions are given by the sum of known orthogonal polynomials multiplied by an associated weight function. When the weight function is the Normal density, in classical models for financial profit/loss, the Hermite system constitutes the orthogonal polynomials. Now in the case of estimators, orthogonal polynomials are simply linear combinations of moments. For low orders, such moments have substantive interpretation as concepts in finance, namely for tail shape. Hence, orthogonal polynomial expansion methods provide a transparent indication of how empirical moments can affect the distribution of portfolio profit/loss, and hence associated risk measures which are based on tail probability calculations. However, naive applications of expansion methods are flawed. The shape of the estimator's tail can undulate, under the influence of the constituent polynomials in the expansion, and can even exhibit regions of negative density. This paper presents techniques to redeem these flaws and to improve quality of risk estimation. We show that by targeting a smooth density which is sufficiently close to the target density, we can obtain expansion-based estimators which do not have the shortcomings of equivalent naive estimators. In particular, we apply optimisation and smoothing techniques which place greater weight on the tails than the body of the distribution. Numerical examples using both real and simulated data illustrate our approach. We further outline how our techniques can apply to a wide class of expansion methods, and indicate opportunities to extend to the multivariate case, where distributions of individual component risk factors in a portfolio can be accessed for the purpose of risk management.
* joint work with Kohei Marumo (Saitama University, Japan)