Extreme Value Approach Research Articles

Decay factor optimisation in time weighted simulation — Evaluating VaR performance

Abstract We propose an optimisation approach for determining the optimal decay factor in time weighted (BRW) simulation. The backtesting of the BRW simulation, which involves different decay factors, together with a broad range of competing VaR models, has been performed on a sample of seven stock indexes and two commodities: gold and WTI oil. The results obtained show that the BRW simulation with an optimised decay factor relative to the Lopez (1998) size-adjusted function is among the best performing VaR models, second only to the conditional extreme value approach ( McNeil & Frey, 2000 ). The optimised decay factors are sufficiently stable over time, giving economic justification to the optimisation because they do not change over longer time periods. Unlike most of the VaR models tested, in the large majority of cases, the optimised BRW model passes the Basel II criteria but yields significantly lower VaR forecasts than the extreme value approaches, thus resulting in a lower idle capital, i.e. lower costs.

Modeling old-age mortality risk for the populations of Australia and New Zealand: An extreme value approach

Old-age mortality for populations of developed countries has been improving rapidly since the 1950s. This phenomenon, which is often referred to as ‘rectangularization’ of mortality, implies an increased survival at advanced ages. With this increase comes different challenges to actuaries, economists and policy planners. A reliable estimate of old-age mortality would definitely help them develop various demographic and financial projections. Unfortunately, data quality issues have made the modeling of old-age mortality difficult and we need a method that can extrapolate a survival distribution to extreme ages without requiring accurate mortality data for the centenarian population. In this paper, we focus on a method called the threshold life table which systematically integrates extreme value theory to the parametric modeling of mortality. We apply the threshold life table to model the most recent period (static) mortality rates for the populations of Australia and New Zealand. We observe a good fit to the raw data for both populations. We then extend the model to predict the highest attained age, which is commonly referred to as ‘omega’ or ω in the actuarial literature, for the populations of Australia and New Zealand. On the basis of the threshold life table, the central estimates of ω for Australia and New Zealand are 112.20 and 109.43, respectively. Our estimates of ω are reasonably consistent with the validated supercentenarian in these countries.

Longevity bond premiums: The extreme value approach and risk cubic pricing

The purpose of this study is to analyze the securitization of longevity risk with an emphasis on longevity risk modeling and longevity bond premium pricing. Various longevity derivatives have been proposed, and the capital market has experienced one unsuccessful attempt by the European Investment Bank (EIB) in 2004. After carefully analyzing the pros and cons of previous securitizations, we present our proposed longevity bonds, whose payoffs are structured as a series of put option spreads. We utilize a random walk model with drift to fit small variations of mortality improvements and employ extreme value theory to model rare longevity events. Our method is a new approach in longevity risk securitization, which has the advantage of both capturing mortality improvements within sample and extrapolating rare, out-of- sample longevity events. We demonstrate that the risk cubic model developed for pricing catastrophe bonds can be applied to mortality and longevity bond pricing and use the model to calculate risk premiums for longevity bonds.

Risk of catastrophic terrorism: an extreme value approach

AbstractThis paper models the stochastic behavior of large‐scale terrorism using extreme value methods. We utilize a unique dataset composed of roughly 26,000 observations. These data provide a rich description of domestic and international terrorism between 1968 and 2006. Currently, a credible worst‐case scenario would involve losses of about 5000 to 10,000 lives. Also, the return time for events of such magnitude is shortening every year. Today, the primary threat is from conventional weapons, rather than from chemical, biological and/or radionuclear weapons. However, pronounced tails in the distribution of these incidents suggest that this threat cannot be dismissed. Copyright © 2009 John Wiley & Sons, Ltd.

Decay Factor Optimisation in Time Weighted Simulation - Evaluating VAR Performance

We propose an optimisation approach to determine the optimal decay factor in time weighted (BRW) simulation. The backtesting of the BRW simulation involving different decay factors together with a broad range of competing VaR models has been performed on a sample of seven stock indexes and two commodities - gold and WTI oil. The obtained results show that the BRW simulation with optimised decay factor in regard to Lopez (1998) size-adjusted function is among the best performing VaR models, second only to the conditional extreme value approach (McNeil, Frey, 2000). The optimised decay factors are sufficiently stable over time, giving economic justification to the optimisation since they do not change for longer time periods. Unlike most of the tested VaR models, in a large majority of cases, the optimised BRW model passes the Basel 2 criteria but yields significantly lower VaR forecasts if compared to the extreme value approaches, resulting thus in lower idle capital i.e. lower costs.

Large Portfolio Risk Management with Dynamic Copulas

Modeling the dynamic high-dimensional multivariate distribution is very useful for active risk management and optimal portfolio allocation; however, available dynamic models are not easily applied for high-dimensional problems due to the curse of dimensionality. In the light of the recent development of multivariate GARCH techniques for a large number of underlying securities, I extend the framework of the Dynamic Conditional Correlation/Equicorrelation (DCC/DECO) (Engle, 2002 and Engle and Kelly, 2008) and an extreme value approach (McNeil and Frey, 2000) into a series of Dynamic Conditional Elliptical Copulas. By constructing portfolios of 89 stocks from CDX-listed …rms between 1995 and 2005, I examine Value at Risk (VaR) and Expected Shortfall (ES) by Monte Carlo simulation for passive portfolios and dynamic optimal portfolios through Mean-Variance and ES criteria. I …nd: (1) Modeling the marginal distribution is important for the dynamic high-dimensional multivariate models. (2) Neglecting the dynamic dependence in the copula causes over-aggressive risk management. (3) The DCC/DECO Gaussian copula and t-copula work very well for both VaR and ES. The DCC copula is necessary for value-weighted portfolios. For equally-weighted portfolios, the DECO copula performs about as well as the DCC copula. (4) Grouped t-copula and t-copula with dynamic degrees of freedom further match the fat tail. (5) Correctly modeling dependence structure makes an improvement in portfolio optimization. The optimal portfolio by ES does a good job against the tail risk in both DECO and DCC copulas. (6) Since portfolio optimization induces statistical error maximization, the assumption of multivariate t innovations with exogenously given degrees of freedom provides a ‡exible and applicable method for optimal portfolio risk management.

A New Class of Models for Bivariate Joint Tails

SummaryA fundamental issue in applied multivariate extreme value analysis is modelling dependence within joint tail regions. The primary focus of this work is to extend the classical pseudopolar treatment of multivariate extremes to develop an asymptotically motivated representation of extremal dependence that also encompasses asymptotic independence. Starting with the usual mild bivariate regular variation assumptions that underpin the coefficient of tail dependence as a measure of extremal dependence, our main result is a characterization of the limiting structure of the joint survivor function in terms of an essentially arbitrary non-negative measure that must satisfy some mild constraints. We then construct parametric models from this new class and study in detail one example that accommodates asymptotic dependence, asymptotic independence and asymmetry within a straightforward parsimonious parameterization. We provide a fast simulation algorithm for this example and detail likelihood-based inference including tests for asymptotic dependence and symmetry which are useful for submodel selection. We illustrate this model by application to both simulated and real data. In contrast with the classical multivariate extreme value approach, which concentrates on the limiting distribution of normalized componentwise maxima, our framework focuses directly on the structure of the limiting joint survivor function and provides significant extensions of both the theoretical and the practical tools that are available for joint tail modelling.

Point and confidence interval estimates for a global maximum via extreme value theory

The aim of this paper is to provide some practical aspects of point and interval estimates of the global maximum of a function using extreme value theory. Consider a real-valued function f:D→ℝ defined on a bounded interval D such that f is either not known analytically or is known analytically but has rather a complicated analytic form. We assume that f possesses a global maximum attained, say, at u*∈D with maximal value x*=max u f(u)≐f(u*). The problem of seeking the optimum of a function which is more or less unknown to the observer has resulted in the development of a large variety of search techniques. In this paper we use the extreme-value approach as appears in Dekkers et al. [A moment estimator for the index of an extreme-value distribution, Ann. Statist. 17 (1989), pp. 1833–1855] and de Haan [Estimation of the minimum of a function using order statistics, J. Amer. Statist. Assoc. 76 (1981), pp. 467–469]. We impose some Lipschitz conditions on the functions being investigated and through repeated simulation-based samplings, we provide various practical interpretations of the parameters involved as well as point and interval estimates for x*.

Predicting tail-related risk measures: The consequences of using GARCH filters for non-GARCH data

Abstract We investigate the consequences for Value-at-Risk and expected shortfall purposes of using a GARCH filter on various mis-specified processes. In general, we find that the McNeil and Frey (McNeil, A.J. and R. Frey, 2000, Estimation of Tail-Related Risk Measures for Heteroscedastic Financial Time Series: An Extreme Value Approach, Journal of Empirical Finance 7, 271–300.) two step procedure has very good forecasting properties. Using an unconditional non-filtered tail estimate also appears to perform satisfactorily for expected shortfall measurements but less so for VaR computations. Methods assuming specific densities such as the Gaussian or Student- t may yield wrong predictions. Thus, the use of an adequacy test for filtered data to given densities appears relevant. The paper builds on recent techniques to obtain thresholds for extreme value computations. Statistical tests for the expected shortfall, based on the circular bootstrap, are developed.

The tail behavior of extreme stock returns in the Gulf emerging markets

PurposeIn this paper, the aim is to investigate the tail behavior of daily stock returns for three emerging stock in the Gulf region (Bahrain, Oman, and Saudi Arabia) over the period 1998‐2005. In addition, the aim is also to test whether the distributions are similar across these markets.Design/methodology/approachFollowing McNeil and Frey, Wanger and Marsh, and Bystrom, extreme value theory (EVT) methods are utilized to examine the asymptotic distribution of the tail for daily returns in the Gulf region. As a first step and to obtain independent and identically distributed residuals series, the returns are prefiltered with an ordinary time‐series model, taking into account the observed Gulf return dynamics. Then, the “Peaks‐Over‐Threshold” (POT) model is applied to estimate the tails of the innovational distribution.FindingsNot only is the heavy tail found to be a facial appearance in these markets, but also POT method of modelling extreme tail quantiles is more accurate than conventional methodologies (historical simulation and normal distribution models) in estimating the tail behavior of the Gulf markets returns. Across all return series, it is found that left and right tails behave very different across countries.Research limitations/implicationsThe results show that risk models that are able to exploit tail behavior could lead to more accurate risk estimates. Thus, participants in the Gulf equity markets can rely on EVT‐based risk model when assessing their risks.Originality/valueThe paper extends previous studies in two aspects. First, it extends the classical unconditional extreme value approach by first filtering the data by using AR‐FIAPARCH model to capture some of the dependencies in the stock returns, and thereafter applying ordinary extreme value techniques. Second, it provides a broad analysis of return dynamics of the Gulf markets.

A Generalized Extreme Value Approach to Financial Risk Measurement

This paper develops an unconditional and conditional extreme value approach to calculating value at risk (VaR), and shows that the maximum likely loss of financial institutions can be more accurately estimated using the statistical theory of extremes. The new approach is based on the distribution of extreme returns instead of the distribution of all returns and provides good predictions of catastrophic market risks. Both the in‐sample and out‐of‐sample performance results indicate that the Box–Cox generalized extreme value distribution introduced in the paper performs surprisingly well in capturing both the rate of occurrence and the extent of extreme events in financial markets. The new approach yields more precise VaR estimates than the normal and skewed t distributions.

Price Limit and Volatility in Taiwan Stock Exchange: Some Additional Evidence from the Extreme Value Approach

We reexamine the effects of price limits on stock volatility of Taiwan Stock Exchange using a new methodology based on the Extreme-Value technique. Consistent with the advocates of price limits, we find that stock market volatility is sharply moderated under more restrictive price limits.

An Extreme Value Approach to Estimating Interest-Rate Volatility: Pricing Implications for Interest-Rate Options

This paper proposes an extreme value approach to estimating interest-rate volatility, and shows that during the extreme movements of the U.S. Treasury market the volatility of interest-rate changes is underestimated by the standard approach that uses the thin-tailed normal distribution. The empirical results indicate that (1) the volatility of maximal and minimal changes in interest rates declines as time-to-maturity rises, yielding a downward-sloping volatility curve for the extremes; (2) the minimal changes are more volatile than the maximal changes for all data sets and for all asymptotic distributions used; (3) the minimal changes in Treasury yields have fatter tails than the maximal changes; and (4) for both the maxima and minima, the extreme changes in short-term rates have thicker tails than the extreme changes in long-term rates. This paper extends the standard option-pricing models with lognormal forward rates to accommodate significant kurtosis observed in the interest-rate data. This paper introduces a closed-form option-pricing model based on the generalized extreme value distribution that successfully removes the well-known pricing bias of the lognormal distribution.

Modelling Catastrophic Risk in International Equity Markets: An Extreme Value Approach

This letter uses the Block Maxima Extreme Value approach to quantify catastrophic risk in international equity markets. Risk measures are generated from a set threshold of the distribution of returns that avoids the pitfall of using absolute returns for markets exhibiting diverging levels of risk. From an application to leading markets, the letter finds that the Nikkei is more prone to catastrophic risk than the FTSE and Dow Jones Indexes.

Extreme values and fat tails of multifractal fluctuations

In this paper we discuss the problem of the estimation of extreme event occurrence probability for data drawn from some multifractal process. We also study the heavy (power-law) tail behavior of probability density function associated with such data. We show that because of strong correlations, the standard extreme value approach is not valid and classical tail exponent estimators should be interpreted cautiously. Extreme statistics associated with multifractal random processes turn out to be characterized by non-self-averaging properties. Our considerations rely upon some analogy between random multiplicative cascades and the physics of disordered systems and also on recent mathematical results about the so-called multifractal formalism. Applied to financial time series, our findings allow us to propose an unified framework that accounts for the observed multiscaling properties of return fluctuations, the volatility clustering phenomenon and the observed "inverse cubic law" of the return pdf tails.

Value‐at‐risk under extreme values: the relative performance in MENA emerging stock markets

Purpose – The paper aims to investigate the relative performance of the most popular value‐at‐risk (VaR) estimates with an emphasis on the extreme value theory (EVT) methodology for seven Middle East and North Africa (MENA) countries.Design/methodology/approach – The paper calculates tails distributions of return series by EVT. This allows computing VaR and comparing the results with Variance‐Covariance method, Historical simulation, and ARCH‐type process with normal distribution, Student‐t distribution and skewed Student‐t distribution. The paper assesses the performance of the models, which are used in VaR estimations, based on their empirical failure rates.Findings – The empirical results demonstrate that the return distributions of the MENA markets are characterized by fat tails which implies that VaR measures relies on the normal distribution will underestimate VaR. The results suggest that the extreme value approach, by modeling the tails of the return distributions, are more relevant to measure VaR...

Modelling catastrophic risk in international equity markets: an extreme value approach

This letter uses the Block Maxima Extreme Value approach to quantify catastrophic risk in international equity markets. Risk measures are generated from a set threshold of the distribution of returns that avoids the pitfall of using absolute returns for markets exhibiting diverging levels of risk. From an application to leading markets, the letter finds that the Nikkei is more prone to catastrophic risk than the FTSE and Dow Jones Indexes.

Price risk in the NYMEX energy complex: An extreme value approach

We estimate tail parameters and construct risk statistics for unconditional distributions of daily logarithmic price changes of the NYMEX energy complex and apply the conditional extreme value method proposed by A. J. McNeil and R. Frey (2000) for estimating VAR and related risk statistics from the tails of conditional distributions for these commodities. The unconditional distribution of spot market price declines is found to be fat tailed relative to the normal for all commodities examined. Backtesting of candidate conditional risk measurement methods indicates that the conditional extreme value method is significantly more accurate for measuring risk exposure due to price declines for 7 of the 10 price series examined. © 2005 Wiley Periodicals, Inc. Jrl Fut Mark 25:309–337, 2005

A comparison of extreme value theory approaches for determining value at risk

This paper compares a number of different extreme value models for determining the value at risk (VaR) of three LIFFE futures contracts. A semi-nonparametric approach is also proposed, where the tail events are modeled using the generalised Pareto distribution, and normal market conditions are captured by the empirical distribution function. The value at risk estimates from this approach are compared with those of standard nonparametric extreme value tail estimation approaches, with a small sample bias-corrected extreme value approach, and with those calculated from bootstrapping the unconditional density and bootstrapping from a GARCH(1,1) model. The results indicate that, for a holdout sample, the proposed semi-nonparametric extreme value approach yields superior results to other methods, but the small sample tail index technique is also accurate.

Does ignoring multidestination trips in the travel cost method cause a systematic bias?

The present paper demonstrates that treating multidestination trips (MDT) as single‐destination trips does not involve any systematic upward or downward bias in consumer surplus (CS) estimates because the direct negative effect of a price increase (treating MDT as a single‐destination trip) is offset by a shift in the estimated demand curve. Still, ignoring MDT can greatly underestimate or overestimate the CS. In addition, we demonstrate that there is a sound theoretical basis for using preference information for allocating travel costs between different sites included in the MDT package. A novel extreme value approach is proposed, which does not require any overly restrictive assumptions about consumer preferences. This approach is applied to the zonal travel cost model of the Bellenden Ker National Park, Australia. Parametric and non‐parametric estimation techniques are used for calculating CS estimates, and the effects of different MDT treatments and estimation methods are compared.