Smooth Extremal Models in Finance and Insurance

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ABSTRACT

This article describes smooth nonstationary generalized additive modeling for sample extremes, in which spline smoothers are incorporated into models for exceedances over high thresholds. We summarize the smoothing methodology as a new tool for practical extreme value exploration in finance and insurance.

INTRODUCTION

Extreme value theory (EVT) has developed very rapidly over the past two decades both methodologically and with respect to applications. Whereas (nonlife) actuaries have, at least implicitly, used EVT techniques for a long time, mainly through the emergence of quantitative risk management, EVT has entered the finance stage more recently as a useful toolkit for describing nonstandard (more precisely nonnormal) price fluctuations. Econometricians for a long time were well aware of the so-called stylized facts of market data, which clearly showed that normal distribution-based models (i.e., Brownian motion technology) are only a first step into the direction of finding more realistic models. What was perhaps not so clear were the next steps, i.e., how to use this "heavy-tailed" reality in pricing, hedging, portfolio management, risk management and even banking, and insurance regulation. For the latter, despite the well-supported evidence for nonnormality, such tools like mean-variance optimization, value-at-risk (VaR), Sharpe-ratio, etc., play a very dominant role. EVT offers a pair of glasses through which to look at these types of questions more realistically. Embrechts, Kluppelberg, and Mikosch (1997) detail the mathematical theory of EVT and discuss its applications to financial and insurance risk management. Various updating material is to be found at http://www.math.ethz.ch/finance and http://www.risklab.ch. In Embrechts (2000), various papers highlight the current state-of-the-art on EVT modeling in Integrated Risk Management (also see Reiss and Thomas, 2001, and Coles, 2001, for very readable discussions).

The traditional approach to EVT is based on extreme value limit distributions. Here, a model for extreme losses, say, is based on the possible parametric form of the limit distributions of maxima over independent and identically distributed (i.i.d.) (or weakly dependent) data (see, for instance, Embrechts, Kluppelberg, and Mikosch, 1997, p. 121). For a complete discussion concerning possible dependencies on the data we refer to Coles (2001, Chapter 5). Another model is based on a so-called point process characterization. The resulting Peaks Over Threshold (POT) method appears more flexible and it considers exceedances over a threshold u. For a pictorial presentation of the POT method (see Figure 1). In Figure 1, [Z.sub.1], ..., [Z.sub.q] denote the ground up losses (say), u a (typically high) threshold, n the number of exceedances by [Z.sub.1], ..., [Z.sub.q] of the level u, and [W.sub.1], ..., [W.sub.n] the corresponding excesses (loss exceeding u minus u). The level u may, for instance, correspond to the attachment point or the lower level of an excess-of-loss reinsurance treaty; within finance, u could stand for a VaR number when managing market risk or a stress loss value within credit risk. A further example would correspond to larger operational losses [W.sub.1], ..., [W.sub.n] above a threshold u. For the latter, see Cruz (2002, Chapter 4). Mathematical theory (see Leadbetter, 1991) supports the condition of a possibly inhomogeneous Poisson process with intensity [lambda] for the number of exceedances combined with independent excesses W over the threshold. Given u, the excesses are treated as a random sample from the generalized Pareto distribution (GPD), with scale parameter [sigma] and shape parameter [kappa] (see (1) in "The Description of the Methodology" for the basic definitions).

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An additional advantage of the threshold method over the method of annual maxima is that, since each exceedance is associated with a specific event, it is possible to let the scale and shape parameters depend on covariates. …