TK 6413 / TK 5413 : ISLAMIC RISK MANAGEMENT

1. TK 6413 / TK 5413 : ISLAMIC RISK MANAGEMENT TOPIC 9: EXTREME VALUE THEORY IN RISK MANAGEMENT

2. (I) INTRODUCTION • A risk manager is often concerned with the distribution of the losses that are of low frequency and of high severity. Such types of losses lie in the upper tail of the loss distribution. The conventional approaches for estimating VaR do not therefore always account for fat tails of the distribution of returns. The distribution of returns are assumed to follow a unique distribution for the entire range of values. • The term extreme value theory (EVT) is used to describe the science of estimating the tails of a distribution. EVT acts complementarily and separates the distribution of returns that belongs to the tails. Thus, EVT attempts to find the optimal point beyond which all values belong to the tail. After finding the cut-off point that separates the body distribution from the tail distribution, the latter has to be modeled.

3. (II) APPROACHES TO ESTIMATE EXTREME VALUE LOSSES • In order to model the tail distribution, two approaches can be used namely; • The block maxima (BM) approach; • The peaks-over-threshold (POT) approach. • The Block Maxima Model: • Consider time series of loss data divided into independent blocks (one block equals one year) of the same size. The block maxima model focuses on the distribution of the largest events taken from each block. Loss Amount X Time O

4. For very large loss observations X, the limiting distribution of such normalized maxima is the generalized extreme value (GEV) distribution: • F(x)= • Where and • x refers to the maxima, and is the location parameter (often assumed to be 0), is the scale parameter, and is the shape parameter.

5. Peak Over Threshold Model • In the peak over threshold (POT) model, the focus of the statistical analysis is put on observations that lie above a certain high threshold. Loss Amount X Time O

6. The POT approach consists of two subcategories, semi-parametric and parametric approaches: • The semi-parametric models, which are constructed around the Hill estimator; • The parametric models, which are based on the Generalized Pareto Distribution (GPD); • The Hill estimator is considered the most appropriate procedure in order to provide the simplest solution for the class of problems faced. Hill estimators are calculated using the following steps: • The independency and identity of risk factors’ distribution is assured; • The time series are sorted according to descending (ascending) order • The following equation is applied in order to estimate the tail index:

7. Where, ∞ = , index m denotes the mthelement that • acts as the cut-off point between mean and extreme values; the determination of m is a rather vague subject, but in order to simplify its estimation two alternatives are proposed: • m as a result of the square root of the number of observations; or • m as a result of the 10th percentile of the number of observations. • Generalized Pareto Distribution (GPD) • Let X be a random variable representing losses with cumulative distribution function F. Let u be a certain high threshold. Then Fu is the distribution of losses above this threshold and is called conditional excess distribution function.

8. F(x) Fu(x) 1 1 x x-u For a sufficiently large threshold u, the conditional excess distribution function Fu of such extreme observations is summarized by a generalized Pareto distribution (GPD). The cumulative distribution function of GPD is as follows: F(x) = where o u

9. From above, mean (x) = Variance (x)=

10. (III) ADVANTAGES AND LIMITATIONS OF EXTREME VALUE THEORY • Advantages • EVT provides theoretical properties of the limiting distribution of extreme events. Hence, it provides direct treatment of the events of low frequency and high severity, both near the boundary of the range of observed data, and beyond. • POT model can be used with catastrophic losses that lie beyond a high threshold such as VaR. • Both theoretical and computational tools are available. The tail of loss distribution has a functional form determined a priori. • Non parametric estimators of the shape parameter – Hill estimator – posses nice asymptotic properties.

11. Pitfalls: • In the POT model, interpretations and estimations are based on a small sample of observations. This may lead to biased estimators that are very sensitive to the choice of the high threshold • The choice the high threshold in the POT model that determine extreme losses is based primarily on the visual examination of the mean excess plot. More rigorous techniques must be developed. • The analysis based on EVT relies on the distributional properties of extreme losses and puts little weight on the properties of low and medium scale observations.

12. (IV) EMPIRICAL STUDIES WITH OPERATIONAL LOSS DATA • Cruz (2002) applied the block maxima model to fraud data for a period of 5 years (1992 – 1996); he also tested using legal loss data for a business unit; he found the generalized extreme value (GEV) distribution to be a superior fit for the data utilized. • Moscadelli (2004) explores the data collected by the Risk Management Group (RMG) of the Basel Committee in June 2002’s Operational Risk Loss Data Collection Exercise (LDCE); Moscadelli applies the POT model to estimate operational risk capital charge; he concludes that the POT model is rarely violated in practice compared to the other models used.