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Technical Article Risk Technical article Risk is the world's leading financial risk management magazine. Risk’s Cutting Edge articles are a showcase for the latest thinking and research into derivatives tools and techniques, with comprehensive coverage of the most recent advances in areas such as option pricing and hedging, market risk, credit risk, swaps and Monte Carlo methods. www.risk.net/technical Copyright statement © Risk Waters Group Ltd, 2003. All rights This PDF is made available for personal use reserved. No parts of this article may be only. Under the following terms you are reproduced, stored in or introduced into any granted permission to view online, print out, retrieval system, or transmitted, in any form or store on your computer a single copy, or by any means, electronic, mechanical, provided this is intended for the benefit of photocopying, recording or otherwise the individual user. Should you wish to use without the prior written permission of the multiple copies of this article, please contact copyright owners. [email protected] HOSKING.QXD 03/05/00 13:36 Page 1 Value-at-risk Beyond the lognormal Although value-at-risk has suffered a fair amount of criticism in recent years, it remains a central platform of any market risk management system. In the first of two VAR-themed articles, Jonathan Hosking, Gabriel Bonti and Dirk Siegel propose a new method for incorporating non-lognormality into Monte Carlo VAR simulation ost risk managers recognise that market returns with location and scale parameters in addition to the degrees of freedom) are not lognormally distributed but exhibit so- gives a satisfactory approximation for most data sets. called fat tails. This well-established phenomenon Short-term interest rates, whose daily returns contain many exact zero is important when measuring value-at-risk, as the values, may require a different model, perhaps a fat-tailed distribution with quantiles in VAR are directly related to the tails of a “spike” of non-zero probability mass at zero. This is a subject of contin- the distributions of market parameters. The more uing research. Mextreme a quantile required, the more sensitive is the result to the correct Meta-Gaussian joint distributions. The probability integral transform modelling of the tails of the distribution. permits any continuous distribution to be transformed into any other. We Two of the most widely used approaches to calculating VAR, however, use it to transform scaled returns so that their distribution is Gaussian. The Φ–1 are based on the (log) normality assumption. They are the covariance transformation is zt = (F(st)), where F is the cumulative distribution method of JP Morgan’s RiskMetrics and the Monte Carlo simulation ap- function (CDF) of the distribution of scaled returns and Φ–1 is the inverse proach. The historical simulation technique does not suffer from this draw- CDF of the standard Gaussian distribution. back but can be criticised for other reasons (Jorion, 1997, pages 195–196). A simple model for the joint dependence of market variables is that As part of a research project between Deutsche Bank and IBM, we have after this transformation of the scaled returns to Gaussianity, the joint dis- devised a method for calculating VAR that overcomes the problems asso- tribution of the transformed variables is multivariate Gaussian. The joint ciated with the assumption of normality. Our basic approach uses Monte distribution of the original variables has been termed the “meta-Gaussian” Carlo simulation, which has the desirable property of being able to gen- distribution (Kelly & Krzysztofowicz, 1997). Tests, which we describe erate large scenario sets, thereby improving the statistical properties of the below, of this meta-Gaussian model for scaled returns indicate that it gives VAR estimator. We generate the scenarios by modifying the classical Monte an adequate fit to the data. Carlo approach to take into account the fat-tailed distributions of market Time-varying dependence. In the meta-Gaussian distribution, depen- variables. The method uses new statistical techniques for identifying and dence between the variables is measured by the correlation matrix of the estimating fat-tailed distributions, and includes a model of statistical de- transformed Gaussian variables. These correlations vary substantially over pendence between quantities that are not normally distributed. time, so they are estimated from relatively short stretches of data. Analo- In this paper, we describe the rationale behind our approach, give de- gously to our volatility estimates, we estimate the correlation from the last tails of the mathematical framework and report encouraging results ob- 260 trading days’ values of each pair of transformed series of scaled returns. tained by applying the method to several real-world portfolios. Statistical analyses Model specifications L-moment methods to identify marginal distributions. L-moments Logarithmic daily returns. The model is based on logarithmic daily are measures of the salient features – location, scale and shape – of prob- returns rt = log(pt /pt – 1), where pt is the market variable (stock index, ability distributions or data samples. They were introduced by Hosking forex rate or interest rate) on trading day t. (1990) and are further described in Hosking (1998, 1999). For a distribu- Time-varying volatility. To allow for non-constant volatility, we model tion with quantile function (inverse cumulative distribution function) Q(.), the scaled returns st = rt /vtt – 1, where vtt – 1 is the volatility of trading the L-moment of order r is defined by: day t, as estimated using data up to day t – 1. This estimated volatility is 1 * λ=∫ PuQudu− () () used because, when simulating scenarios, we will be simulating data for rr10 * * tomorrow (day t + 1) but will have only data up to today (day t) from where Pr is a polynomial of degree r. Specifically, the Pr are shifted Le- which to calculate tomorrow’s volatility. gendre polynomials, which are orthogonal on the interval (0, 1) with con- λ λ We use a simple estimate of volatility, the second sample L-moment, 2, stant weight function. 1 is the mean of the distribution and 2 is a scale τ λ λ of the last 260 daily returns. The second L-moment is a dispersion mea- or dispersion measure. The L-moment ratios r = r / 2 are dimensionless sure analogous to the standard deviation, but is less affected by extreme measures of the shape of the distribution; they all take values between –1 τ data values than are ordinary moments and is therefore more suitable for and +1. Analogously to the ordinary moments, 3 is a measure of skew- τ use with data drawn from fat-tailed distributions (Hosking, 1990). We use ness and 4 is a measure of kurtosis (in a sense that can be formalised; 260 days of past data to satisfy regulatory requirements (Basle Committee see Hosking, 1990, page 109). on Banking Supervision, 1996, page 44). If the mean of the distribution exists, then so do all the higher-order L- Non-Gaussian marginal distributions. Although scaling the returns moments. This means that L-moments can describe fat-tailed distributions to allow for non-constant volatility can, under some theoretical models, whose variance or higher-order ordinary moments may be infinite. account for fat-tailedness in market variables, in practice it is not sufficient. From a sample of data, the L-moments of the distribution from which L-moment analysis, which we describe below, indicates that distribu- the sample was drawn can be estimated by “sample L-moments”, which tions of returns are approximately symmetric and have heavier tails than are linear combinations of the ordered data values. Corresponding sample the Gaussian. A Pearson-type VII distribution (a Student’s t distribution L-moment ratios can also be defined. 59 • RISK • MAY 2000 HOSKING.QXD 03/05/00 13:36 Page 2 Value-at-risk 2. Nikkei stock index: 1. L-moment ratio diagram Jun 1993–May 1998 0.3 15 Data Ger. Jap. US HK Thai Nikkei scaled daily returns Forex v. $ 6 τ Stock 10 Int 2-yr 0.2 Int 5-yr 5 Int 10-yr 0 0.1 Scaled returns –5 Distributions Distributions Stable –10 Stable α = 1.82 Sixth-order L-moment ratio, Student's t Normal Student's t 4.2 df Power exp Power exp β = 1.13 0.0 –15 0.1 0.2 0.3 0.4 –4 –2 0 2 4 τ Fourth-order L-moment ratio, 4 Normal variate By comparing the sample data points with the population L-moment curves for Normal probability plot of scaled returns of the Nikkei stock index, and of three different distributions, one can choose the best distribution to fit to the data distributions fitted to the data The scaled returns of market variables have little skewness. The sam- in figure 1, and the sample moments can be wildly unstable across differ- τ ple estimate of 3 is generally less than 0.05 in absolute value, and dis- ent five-year samples, because their values are strongly affected by the plays little consistency across different data series. We are therefore content most extreme observations in the sample.) to use symmetric distributions to fit the data. Tests of meta-Gaussianity. Testing the hypothesis of meta-Gaussiani- For symmetric distributions, all the odd-order L-moment ratios are zero ty is complicated by the time variability of correlations between market and the even-order L-moment ratios carry the information about the shape variables. For periods during which correlations are approximately con- τ τ of the distribution. In particular, the relation between 4 and 6 can be stant, standard tests and judgements of joint Gaussianity can be made for used to distinguish different families of symmetric distributions.
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