The asymptotic theory for the sample autocorrelations and extremes of a GARCH(1; 1) process is provided. Special attention is given to the case when the sum of the ARCH and GARCH parameters is close to one, i.e. when one is close to an infinite variance marginal distribution. This situation has been observed for various financial log--return series and led to the introduction of the IGARCH model. In such a situation the sample autocorrelations are unreliable estimators of their deterministic counterparts for the time series and its absolute values, and the sample autocorrelations of the squared time series have non--degenerate limit distributions. We discuss the consequences for a foreign exchange rate series. AMS 1991 Subject Classification: Primary: 62P20 Secondary: 90A20 60G55 60J10 62F10 62F12 62G30 62M10 Key Words and Phrases. GARCH, sample autocorrelations, stochastic recurrence equation, Pareto tail, extremes, extremal index, point processes, foreign exchange rates 1 Introduc...

The financial industry, including banking and insurance, is undergoing major changes. The (re)insurance industry is increasingly exposed to catastrophic losses for which the requested cover is only just available. An increasing complexity of financial instruments calls for sophisticated risk management tools. The securitization of risk and alternative risk transfer highlight the convergence of finance and insurance at the product level. Extreme value theory plays an important methodological role within risk management for insurance, reinsurance, and finance. 1.

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This paper develops efficient methods for computing portfolio value-at-risk (VAR) when the underlying risk factors have a heavy-tailed distribution. In modeling heavy tails, we focus on multivariate t distributions and some extensions thereof. We develop two methods for VAR calculation that exploit a quadratic approximation to the portfolio loss, such as the delta-gamma approximation. In the first method, we derive the characteristic function of the quadratic approximation and then use numerical transform inversion to approximate the portfolio loss distribution. Because the quadratic approximation may not always yield accurate VAR estimates, we also develop a low variance Monte Carlo method. This method uses the quadratic approximation to guide the selection of an effective importance sampling distribution that samples risk factors so that large losses occur more often. Variance is further reduced by combining the importance sampling with stratified sampling. Numerical results on a variety of test portfolios indicate that large variance reductions are typically obtained. Both methods developed in this paper overcome difficulties associated with VAR calculation with heavy-tailed risk factors. The Monte Carlo method also extends to the problem of estimating the conditional excess, sometimes known as the conditional VAR.

Given thegrowingneed formanaging financial risk, riskpredictionplays an increasing role inbanking and finance. In this studywecompare theout-of-sample performance of existing methods and some new models for predicting value-at-risk (VaR) in a univariate context. Usingmore than 30 years of the daily return data on theNASDAQ Composite Index, we find that most approaches perform inadequately, although several models are acceptable under current regulatory assessment rules for model adequacy. A hybrid method, combining a heavy-tailed generalized autoregressive conditionally heteroskedastic (GARCH) filter with an extreme value theory-based approach, performs best overall, closely followed by a variant on a filtered historical simulation, and a newmodel based on heteroskedastic mixture distributions. Condi-tional autoregressive VaR (CAViaR) models perform inadequately, though an exten-sion to a particular CAViaR model is shown to outperform the others.

This paper proposes a method for comparing and combining conditional quantile forecasts in an out-of-sample framework. We construct a Conditional Quantile Forecast Encompassing (CQFE) test as a Wald-type test of superior predictive ability. Rejection of CQFE provides a basis for combination of conditional quantile forecasts. Central features of our encompassing test are: (1) the use of the ‘tick ’ loss function; (2) a conditional, ratherthanunconditional approach to out-of-sample evaluation, and, (3) the derivation of our test in an environment with non-vanishing estimation uncertainty. Some of the advantages of our approach are that it allows the forecasts to be generated by using general estimation procedures and that it is applicable when the forecasts are based on both nested and non-nested models. The test is also relatively easy to implement using standard GMM techniques. An empirical application to Value-at-Risk evaluation illustrates the usefulness of our method.

We consider the estimation of quantiles in the tail of the marginal distribution of financial return series, using extreme value statistical methods based on the limiting distribution for block maxima of stationary time series. A simple methodology for quantification of worst case scenarios, such as ten or twenty year losses is proposed. We validate methods on a simulated series from an ARCH(1) process showing some of the features of real financial data, such as fat tails and clustered extreme values; we then analyse daily log returns on a share price.

Using one of the key property of copulas that they remain invariant under an arbitrary monotonous change of variable, we investigate the null hypothesis that the dependence between financial assets can be modeled by the Gaussian copula. We find that most pairs of currencies and pairs of major stocks are compatible with the Gaussian copula hypothesis, while this hypothesis can be rejected for the dependence between pairs of commodities (metals). Notwithstanding the apparent qualification of the Gaussian copula hypothesis for most of the currencies and the stocks, a non-Gaussian copula, such as the Student’s copula, cannot be rejected if it has sufficiently many “degrees of freedom”. As a consequence, it may be very dangerous to embrace blindly the Gaussian copula hypothesis, especially when the correlation coefficient between the pair of asset is too high as the tail dependence neglected by the Gaussian copula can be as large as, i.e., three out five extreme events which occur in unison are missed.

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 POT method 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

sive overview of financial risk management from the point of view of both Wall Street and the Ivory Tower. Most usefully, ABCD discuss a number of recent developments in the econometrics of time varying risk that hold vast promise for risk management applications: the dynamic conditional correlation model of Engle (2002) which permits large-scale, flexi-ble modeling of conditional covariance matrices; the use of high-frequency data to measure realized variances and covariances that has been developed largely by the authors; and the modeling of the full distribution of conditional returns. In this discussion I will just offer a couple of comments and extensions to ABCD’s very well organized survey. Unconditional vs Conditional Risk ABCD discuss extensively the pros and cons of both unconditional and conditional (dynamic) measures of risk. There is however an additional source of risk dynamics that is ignored in the paper and that, in fact, has not been studied much in the literature. Most financial assets are managed over time and it is therefore more important to study the risks of dynamic investment strategies rather than the risks of static portfolios. Especially for supervision and regulation purposes, it matters more to forecast the risk of a portfolio taking into account