Expected Shortfall (ES) in several variants has been proposed as remedy for the deficiencies of Value-at-Risk (VaR) which in general is not a coherent risk measure. In fact, most definitions of ES lead to the same results when applied to continuous loss distributions. Differences may appear when the underlying loss distributions have discontinuities. In this case even the coherence property of ES can get lost unless one took care of the details in its definition. We compare some of the definitions of Expected Shortfall, pointing out that there is one which is robust in the sense of yielding a coherent risk measure regardless of the underlying distributions. Moreover, this Expected Shortfall can be estimated effectively even in cases where the usual estimators for VaR fail.

We discuss the coherence properties of Expected Shortfall (ES) asafinancial risk measure. This statistic arises in a natural way from the estimation of the “average of the 100p% worst losses ” in a sample of returns to a portfolio. Here p is some fixed confidence level. We also compare several alternative representations of ES which turn out to be more appropriate for certain purposes. Key words: Expected Shortfall; Risk measure; worst conditional expectation; tail conditional expectation; value-at-risk (VaR); conditional value-at-risk (CVaR); coherence; subadditivity. 1 A four years impasse Risk professionals have been looking for a coherent alternative to Value at Risk (VaR) for four years. Since the appearance, in 1997, of Thinking Coherently by Artzner et al [3] followed by Coherent Measures of Risk [4], it was clear to risk practitioners and researchers that the gap between market practice and theoretical progress had suddenly widened enormously. These papers in fact faced for the first time the problem of defining in a clearcut way what properties a statistic of a portfolio should have in order to be considered a sensible risk measure. The answer to this question was given through a complete characterization of such properties via an axiomatic formulation of the concept of coherent risk measure. With this result, risk management became all of a sudden a science in itself with its own rules correctly definedinadeductiveframework. Surprisingly enough, however, VaR, the risk measure adopted as best practice by essentially all banks and regulators, happened to fail the exam for being admitted in this science. VaR is not a coherent risk measure because it simply doesn’t fulfill one of the axioms of coherence.

The viability of an insurance company depends critically on the size and frequency of large claims. An accurate modelling of the distribution of large claims contributes to correct pricing’s and reserving’s decisions while maintaining, at an acceptable level, the unexpected fluctuations in the results through reinsurance. We provide a model for large losses and we extrapolate through a simulation a scenario based on separate estimation of loss frequency and loss severity according to extreme value theory, with particular reference to generalized Pareto approximations. We present an application to the fire claims of an insurance company. One conclusion is that the distribution of fire claims is long-tailed and an accurate exploratory data analysis is done to detect heavy tailed behavior and stability of parameters and statistics across different thresholds. We simulate the impact of a quota share and an excess of loss reinsurance structure on the distribution of total loss and on economic risk capital. We provide also a tool to price and investigate how different reinsurance programs can affect economic risk capital and explain the rationale of the choice of the optimal reinsurance programmes to smooth economic results.

OF MASTER'S THESIS Department of Engineering Physics and Mathematics PO Box 2200, FIN-020105 HUT, FINLAND Author: Samppa Nylund Department: Department of Engineering Physics and Mathematics Major subject: Systems Analysis and Operations Research Minor subject: Business Finance and Financial Economics English title: Value-at-Risk Analysis for Heavy-Tailed Financial Returns Finnish title: Positiivisesti huipukkaiden osaketuottojen Value-at-Risk-analyysi Number of pages: 77 Chair: Mat-2 Applied Mathematics Supervisor: Ahti Salo Instructor: Ahti Salo Abstract: Market risk refers to the risks of financial losses arising from adverse movements in market prices and rates. Sources of market risk include equity prices, interest rates, foreign exchange rates and commodity prices. The interest in managing market risk has grown during the last two decades among financial institutions and other market participants. A market risk measure called Value-atRisk (VaR) has been adopted as the most important measure of market risk by most financial institutions. VaR based risk management is also required by many regulatory bodies. VaR measures the maximum expected loss in monetary terms at a given confidence level over a given forecast period. There are several alternative techniques to estimate VaR measures. The RiskMetrics system, introduced by investment bank J.P. Morgan in 1994, has become de facto standard within the industry. This system is based on the assumption that the returns of financial instruments follow the normal distribution. Yet the results of many empirical studies suggest that at high confidence levels (at confidence levels over 95%) extreme losses are more frequent than predicted by the RiskMetrics technique. This is be...

We analyze a bank that operates under the Basel credit and market risk requirements, and that maximizes its value through recapitalizations, dividends, and liquid asset investments. According to our model, the market risk requirement may postpone recapitalization and this way increase the bank’s default probability. We show that this is indeed the case if the expected return and volatility of the liquid asset portfolio are high, i.e., then the market risk requirement raises the default probability of the bank. In this sense the market risk requirement is inefficient.

Crisis events such as the 1987 stock market crash, the Asian Crisis and the bursting of the Dot-Com bubble have radically changed the view that extreme events in financial markets have negligible probability. This paper argues that the use of the Generalized Extreme Value (GEV) distribution to model the Risk Neutral Density (RND) function provides a flexible framework that captures the negative skewness and excess kurtosis of returns, and also delivers the market implied tail index of asset returns. We obtain an original analytical closed form solution for the Harrison and Pliska (1981) no arbitrage equilibrium price for the European option in the case of GEV asset returns. The GEV based option prices successfully remove the well known pricing bias of the Black-Scholes model. We explain how the implied tail index is efficacious at identifying the fat tailed behaviour of losses and hence the left skewness of the price RND functions, particularly around crisis events.

Abstract. From the practitioners ’ point of view, one of the most interesting questions that tail studies can answer is what are the extreme movements that can be expected in financial markets? Have we already seen the largest ones or are we going to experience even larger movements? Are there theoretical processes that can model the type of fat tails that come out of our empirical analysis? Answers to such questions are essential for sound risk management of financial exposures. It turns out that we can answer these questions within the framework of the extreme value theory. This paper provides a step-by-step guideline for extreme value analysis in the MATLAB environment with several examples.

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The viability of an insurance company depends critically on the size and frequency of large claims. An accurate modelling of the distribution of large claims contributes to correct pricing’s and reserving’s decisions while maintaining through reinsurance an acceptable level of the unexpected fluctuations in the results. We present an application to the fire claims of an insurance company providing a model for large losses that we evaluate through simulations based on both a traditional and a Peaks over Threshold’s approach. Under the first one we estimate separately loss frequency, according to Negative Binomial distibution studying a claims number development triangle, and loss severity, according to Generalized Pareto distribution. A Peaks over Threshold’s approach is then developped estimating jointly frequency and severity distribution and considering the time dependence of data. We calculate the economic risk capital as the difference between the expected loss, defined as the expected annual claims amount, and the 99.93th quantile of the total cost distribution corresponding to a Standard&Poor’s A rating; we then simulate the impact of a quota share and an excess of loss reinsurance structure on the distribution of total cost amount and on economic risk capital. We provide a tool to price alternative programs and investigate how they can affect economic risk capital and explain the rationale of the choice of the optimal reinsurance programmes to smooth economic results.

It is of great importance for those in charge of managing risk to understand how financial asset returns are distributed. Practitioners often assume for convenience that the distribution is normal. Since the 1960s, however, empirical evidence has led many to reject this assumption in favor of various heavy-tailed alternatives. In a heavy-tailed distribution the likelihood that one encounters significant deviations from the mean is much greater than in the case of the normal distribution. It is now commonly accepted that financial asset returns are, in fact, heavy-tailed. The goal of this survey is to examine how these heavy tails affect several aspects of financial portfolio theory and risk management. We describe some of the methods that one can use to deal with heavy tails and we illustrate them using the NASDAQ composite index.