Extreme Value Theory for Risk Managers

We provide an overview of the role of extreme value theory (EVT) in risk management (RM), as a method for modelling and measuring extreme risks. We concentrate on the peaks-over-threshold (POT) model and emphasize the generality of this approach. Wherever the tail of a loss distribution is of interest, whether for market, credit, operational or insurance risks, the POT method provides a simple tool for estimating measures of tail risk. In particular we show how the POTmethod may be embedded in a stochastic volatility framework to deliver useful estimates of Value-at-Risk (VaR) and expected shortfall, a coherent alternative to the VaR, for market risks. Further topics of interest, including multivariate extremes, models for stress losses and software for EVT, are also discussed. 1 A General Introduction to Extreme Risk Extreme event risk is present in all areas of risk management. Whether we are concerned with market, credit, operational or insurance risk, one of the greatest challenges to the risk manager is to implement risk management models which allow for rare but damaging events, and permit the measurement of their consequences. This paper may be motivated by any number of concrete risk management problems. In market risk, we might be concerned with the day to day determination of the Value-atRisk (VaR) for the losses we incur on a trading book due to adverse market movements. In credit or operational risk management our goal might be the determination of the risk capital we require as a cushion against irregular losses from credit downgradings and defaults or unforeseen operational problems. Alongside these nancial risks, it is also worth considering insurance risks; the insurance world has considerable experience in the management of extreme risk and many methods which we might now recognize as belonging to extreme value theory have a long Alexander McNeil is Swiss Re Research Fellow at ETH Z urich and gratefully acknowledges the nancial support of Swiss Re. 1 history of use by actuaries. In insurance a typical problem might be pricing or building reserves for products which o er protection against catastrophic losses, such as excess-ofloss (XL) reinsurance treaties concluded with primary insurers. Whatever the type of risk we are considering our approach to its management will be similar in this paper. We will attempt to model it in such away that the possibility of an extreme outcome is addressed. Using our model we will attempt to measure the risk with a measurement which provides information about the extreme outcome. In these activities extreme value theory (EVT) will provide the tools we require. 1.1 Modelling Extreme Risks The standard mathematical approach to modelling risks uses the language of probability theory. Risks are random variables, mapping unforeseen future states of the world into values representing pro ts and losses. These risks may be considered individually, or seen as part of a stochastic process where present risks depend on previous risks. The potential values of a risk have a probability distribution which we will never observe exactly although past losses due to similar risks, where available, may provide partial information about that distribution. Extreme events occur when a risk takes values from the tail of its distribution. We develop a model for a risk by selecting a particular probability distribution. We may have estimated this distribution through statistical analysis of empirical data. In this case EVT is a tool which attempts to provide us with the best possible estimate of the tail area of the distribution. However, even in the absence of useful historical data, EVT provides guidance on the kind of distribution we should select so that extreme risks are handled conservatively. 1.2 Measuring Extreme Risks For our purposes, measuring a risk means summarising its distribution with a number known as a risk measure. At the simplest level, we might calculate the mean or variance of a risk. These measure aspects of the risk but do not provide much information about the extreme risk. In this paper we will concentrate on two measures which attempt to describe the tail of a loss distribution VaR and expected shortfall. We shall adopt the convention that a loss is a positive number and a pro t is a negative number. EVT is most naturally developed as a theory of large losses, rather than a theory of small pro ts. VaR is a high quantile of the distribution of losses, typically the 95th or 99th percentile. It provides a kind of upper bound for a loss that is only exceeded on a small proportion of occasions. It is sometimes referred to as a con dence level, although this is a misnomer which is at odds with standard statistical usage. In recent papers Artzner, Delbaen, Eber & Heath (1997) have criticized VaR as a measure of risk on two grounds. First they show that VaR is not necessarily subadditive so that, in their terminology, VaR is not a coherent risk measure. There are cases where a portfolio can be split into sub-portfolios such that the sum of the VaR corresponding to the sub-portfolios is smaller than the VaR of the total portfolio. This may cause problems if the risk-management system of a nancial institution is based on VaR-limits for individual books. Moreover, VaR tells us nothing about the potential size of the loss that exceeds it. Artzner et al. propose the use of expected shortfall or tail conditional expectation instead 2 of VaR. The tail conditional expectation is the expected size of a loss that exceeds VaR and is coherent according to their de nition. 1.3 Extreme Value Theory The approach to EVT in this paper follows most closely Embrechts, Kl uppelberg & Mikosch (1997); other recent texts on EVT include Reiss & Thomas (1997) and Beirlant, Teugels & Vynckier (1996). All of these texts emphasize applications of the theory in insurance and nance although much of the original impetus for the development of the methods came from hydrology. Broadly speaking, there are two principal kinds of model for extreme values. The oldest group of models are the block maximamodels; these are models for the largest observations collected from large samples of identically distributed observations. For example, if we record daily or hourly losses and pro ts from trading a particular instrument or group of instruments, the block maxima method provides a model which may be appropriate for the quarterly or annual maximum of such values. We see a possible role for this method in the de nition and analysis of stress losses (McNeil 1998) and will return to this subject in Section 4.1. A more modern group of models are the peaks-over-threshold (POT) models; these are models for all large observations which exceed a high threshold. The POT models are generally considered to be the most useful for practical applications, due to their more eÆcient use of the (often limited) data on extreme values. This paper will concentrate on such models. Within the POT class of models one may further distinguish two styles of analysis. There are the semi-parametric models built around the Hill estimator and its relatives (Beirlant et al. 1996, Danielsson, Hartmann & de Vries 1998) and the fully parametric models based on the generalized Pareto distribution or GPD (Embrechts, Resnick & Samorodnitsky 1998). There is little to pick and choose between these approaches both are theoretically justi ed and empirically useful when used correctly. We favour the latter style of analysis for reasons of simplicity both of exposition and implementation. One obtains simple parametric formulae for measures of extreme risk for which it is relatively easy to give estimates of statistical error using the techniques of maximum likelihood inference. The GPD will thus be the main tool we describe in this paper. It is simply another probability distribution but for purposes of risk management it should be considered as equally important as (if not more important than) the Normal distribution. The tails of the Normal distribution are too thin to address the extreme loss. We will not describe the Hill estimator approach in this paper; we refer the reader to the references above and also to Danielsson & de Vries (1997). 2 General Theory Let X1; X2; : : : be identically distributed random variables with unknown underlying distribution function F (x) = PfXi xg. (We work with distribution functions and not densities.) The interpretation of these random risks is left to the reader. They might be: Daily (negative) returns on nancial asset or portfolio { losses and pro ts Higher or lower frequency returns 3 Operational losses Catastrophic insurance claims Credit losses Moreover, they might represent risks which we can directly observe or they might also represent risks which we are forced to simulate in some Monte Carlo procedure, because of the impracticality of obtaining data. There are situations where, despite simulating from a known stochastic model, the complexity of the system is such that we do not know exactly what the loss distribution F is. We avoid assuming independence, which for certain of the above interpretations (particularly market returns) is well-known to be unrealistic. 2.1 Measures of Extreme Risk Mathematically we de ne our measures of extreme risk in terms of the loss distribution F . Let 1 > q 0:95 say. Value-at-Risk (VaR) is the qth quantile of the distribution F VaRq = F (q); where F 1 is the inverse of F , and expected shortfall is the expected loss size, given that VaR is exceeded ESq = E [X j X > VaRq] : Like F itself these are theoretical quantities which we will never know. Our goal in risk measurement is estimates d VaRq and c ESq of these measures and in this chapter we obtain two explicit formulae (6) and (10). The reader who wishes to avoid mathematical theory may skim through the remaining sections of this chapter, pausing only to observe that the formulae in question are simple. 2.2 Generali