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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 (inverse cumulative distribution function) Q(.), the scaled returns st = rt /vtt – 1, where vtt – 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 (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.

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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 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 . 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. A useful the transformed scaled returns zt. For example, the transformed scaled re- τ τ practical tool is the L-moment ratio diagram, which plots the 4– 6 values turns of the Hang Seng and S&P stock indexes had correlation in the range for several families of distributions together with the sample L-moment ra- 0.2–0.4 for most of the four-year period June 1994–May 1998. These 961 tios of fourth and sixth order for one or more sets of data. For a family of data points are plotted in figure 3, and show little deviation from meta- τ τ distributions indexed by a single shape parameter, the 4– 6 values lie on Gaussianity. The elliptical contours that should be exceeded with nominal a smooth curve on the diagram. If the sample data points lie close to the frequencies 5% and 1% have actual outlier frequencies of 6.2% and 1.3% curve for one of these families, there is reason to believe that a distribu- respectively. Similar results are observed with other pairs of series. tion from that family may give a good fit to the data. We are not limited to assuming that the dependence structure on the Figure 1 is an example of an L-moment ratio diagram. It shows the pop- normal-transformed scale is that of a multivariate . Thus ulation curves for three families of symmetric distributions and the sample we avoid the pitfalls of using correlation and elliptical distributions de- L-moments of scaled returns of several forex rates, stock indexes and in- scribed by Embrechts, McNeil & Strauman (1999). However, the evidence terest rates. The sample L-moments were calculated from five years of data in our data for more complex dependence structures, such as greater cor- (June 1993–May 1998), about 1,250 points. The data points cluster around relation between extreme market movements, is quite weak (surprisingly the Student’s t line, indicating that this distribution is most likely to give a weak, compared with our prior opinions) and the meta-Gaussian model good fit to the data. All the data points are far from the normal distribu- gives an adequate fit to the data. tion’s L-moment ratios, showing that the normal distribution is not a real- More thorough and accurate tests of the meta-Gaussian model could istic model for this kind of data. The data for the dollar/baht exchange rate be devised. However, the most important test is the adequacy of the en- have much higher L-moment ratios than the other data sets, reflecting the tire VAR calculation procedure, which is described below (“Back-testing”). fatter tail of this distribution arising from the large number of extreme jumps in the series during the period 1993–98. Scenario generation The goodness of fit of a distribution identified by inspection of an L-mo- The use of the model to generate scenarios of future market variable val- ment ratio diagram can be confirmed by standard methods. Quantile-quan- ues is straightforward. It involves two stages: estimation of the current tile plots of data versus a normal probability scale (equivalent to a plot on model parameters, based on analysis of recent data, and simulation of ran- normal probability paper) are often useful for financial data. Figure 2 shows dom values from the model and its current parameters. such a plot, comparing the Nikkei stock index scaled returns with several  Estimation steps. τ distributions that have the same 4 values as the data, ie, 0.212. The Stu- 1. For each series (say, i = 1, ... , K): (i) dent’s t distribution, which would be chosen by looking at the L-moment 1.1. Calculate the logarithmic returns r t. (i) ratio diagram in figure 1, appears to give a reasonable fit to the data. 1.2. Calculate current and past volatilities v tt – 1. (i) (i) (i) We recommend that at least 1,000 data points should be used when cal- 1.3. Calculate rescaled returns s t = r t /vtt – 1. culating sample L-moment ratios for plotting on a diagram such as figure 1.4. Fit distribution to scaled returns. Let Fi be the CDF of the fitted distri- 1. Smaller samples do not permit adequate discrimination between the bution. (Our current model fits a Pearson-type VII distribution by the types of distributions typical of financial data. (By the way, do not try this method of L-moments, ie, by equating the sample and population L-mo- approach with ordinary moments: fourth- and sixth-order moments are in- ments of orders one, two and four. Because the shape parameter of the finite for all the stable distributions and most of the t distributions plotted distribution – which controls the fatness of the tail, analogously to the de-

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3. Scatter plot of transformed scaled returns of Hang Seng and S&P stock 4. Summary of backtesting results indexes: Jun 1994–May 1998

15 CoVar 4 Normal Historical 10 t(260) t(130) 2 t(65)

5 0

2

–2 Observed frequency (%) Pointwise 90% 1 confidence region 0.5 Transformed scaled returns: S&P 500 –4 0.5 1 2 5 10 –4 –2 0 2 4 Nominal frequency (%) Transformed scaled returns: Hang Seng For an accurately calibrated VAR calculation method, the nominal and observed The ellipses contain 95% and 99% of the probability mass of a bivariate frequencies of extreme events should be equal. (A square-root transformation Gaussian distribution with correlation 0.35 is used for the frequency axes, because this is an approximate variance- stabilising transformation for small binomial frequencies. This means that the confidence intervals for different nominal frequencies become a band of approximately constant height and that the plotted points have approximately grees of freedom of a Student’s t distribution – is difficult to estimate ac- equal sampling variability curately, we use five years of past data in this estimation step.) (i) Φ–1 1.5. Transform the recent scaled returns to Gaussianity, via z t = (Fi(st)). 2. Calculate the covariance matrix R of the transformed scaled returns. plays the number of days that profit and loss exceeded VAR for each of  Simulation steps. the different methods. The “lower CL” and “upper CL” columns give the 1. Generate a multivariate Gaussian random vector [z1, ... , zK] with mean endpoints of confidence intervals with an approximate level of 90%. They zero and covariance matrix R. are calculated from the binomial distribution B(n, p) with n = 1,838 and 2. Transform its elements to have the required marginal distribution (Pear- p equal to the nominal VAR level. Because of the discrete character of the –1 Φ son-type VII), si = Fi ( (zi)). distribution, the actual coverage of the interval is not exactly 90% and is (i) 3. Form the rescaled returns ri = sivt + 1t. given in the fourth column of the table. (i) (i) 4. Convert to the original variables, pt + 1 = p t exp(ri). The simulation steps The covariance, normal and historical methods, which represent the can be repeated to generate as many scenarios as required. current state of the art, tend to give too many outliers of the calculated VAR limits. This is particularly true of the covariance and normal methods Back-testing at the smaller outlier probabilities, and reflects these methods’ inaccurate We tested the new model on six large portfolios of the bank, mainly in- modelling of the tail of the distribution of returns. Figure 4 shows the re- terest rate dependent. For comparison with the new model, we used the sults for the different methods with the confidence bands. It clearly demon- normal method (ie, Monte Carlo with lognormal distribution), the histori- strates the superiority of the t-based methods: their plotted points all lie cal and the covariance methods. The dependence of the portfolios on the within the confidence band, throughout the frequency range 1–10%. market parameters was nearly linear.  Test procedure. The one-day VAR measure at 1%, 2.5%, 5% and 10% Conclusion levels was calculated for each day of a period of approximately one year We have described a fairly simple Monte Carlo simulation method that al- ending in July 1999. The estimation of model parameters and the scenario lows for non-Gaussianity in logarithmic returns and dependence between generation was done once a month. The new method (the “t method”) these non-Gaussian quantities. In particular, L-moment methods identify  was run with 260, 130 and 65 observations used for volatility-based 2 scal- the Pearson-type VII distribution as a suitable model for many market vari- ing (estimation step 1.3) and estimation of the covariance matrix (estima- ables. The resulting VAR calculation procedure gives accurate calibration tion step 2). The number of observations used for parameter estimation in of the frequencies of rare events under realistic market conditions. Our the normal method and for simulation in the historical method was 260. back-testing used linear portfolios that were not very complex. However,  Test results. We compared the one-day VAR estimate with the profit we believe that good market modelling improves VAR estimation in both and loss resulting from the bank’s position at the end of the day and the simple and complex portfolios. market move for the following day. Over all six portfolios, 1,838 compar- Our approach can also be used to generate scenarios for more than one isons were made. The results are summarised in table A. The table dis- day ahead, although in this case it may be necessary to estimate serial de-

A. Observed number of profit and losses in excess of VAR

VAR level Lower CL Upper CL Coverage CoVar Normal Historical t 260 t 130 t 65 1% 12 25 0.901 48 47 29 16 20 21 2.5% 36 57 0.899 68 70 76 51 47 57 5% 77 107 0.903 107 108 130 96 88 97 10% 163 205 0.906 168 165 213 172 164 175

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pendence in the levels or the volatilities of daily returns. Our method is conceptually straightforward and could be combined REFERENCES with other models for various components of a VAR model. For example, other volatility estimates (eg, based on Garch models) could be used, and Basle Committee on Banking Supervision, 1996 other marginal distributions could be fitted to the scaled returns (eg, the Capital accord to incorporate market risks “spike” model mentioned above for short-term interest rates). One could Embrechts P, L de Haan and X Huang, 1999 Modelling multivariate extremes also use models inspired by extreme-value theory that directly estimate the Preprint, ETH Zurich tail weight by some procedure such as Hill’s estimator (1975). However, Embrechts P, A McNeil and D Strauman, 1999 the explicit use of extreme-value theory, eg, by fitting a generalised ex- Correlation: pitfalls and alternatives treme-value distribution to the maximum return in each 100-day block of Risk May, pages 69–71 data, is harder to justify. Since several years of data are available, L-mo- Hill B, 1975 A simple general approach to inference about the tail of a distribution ment methods enable a distribution to be identified from the data and there Annals of Statistics 3, pages 1,165–1,174 is no need to adopt a priori the distribution predicted by extreme-value Hosking J, 1990 theory. Moreover, as mentioned in Embrechts, de Haan & Huang (1999), L-moments: analysis and estimation of distributions using linear combinations of extreme-value theory alone only allows for the treatment of fairly low-di- order statistics Journal of the Royal Statistical Society, series B, 52, pages 105–124 mensional problems, while typical portfolios involve hundreds of variables. Hosking J, 1998 Finally, we note that our procedure, because it models dependence on the L-moments Gaussian scale, can easily incorporate existing models of Gaussian de- In Encyclopedia of Statistical Sciences, update volume 2, edited by S Kotz, C pendence, such as a discrete binomial approximation based on principal Read and D Banks, pages 357–362. New York, Wiley I Hosking J, 1999 components (Jamshidian & Zhu, 1997). L-moments and their applications in the analysis of financial data Research Report RC21466, IBM Research Division, Yorktown Heights, New York Jonathan Hosking is a research staff member at IBM, TJ Watson Jamshidian F and Y Zhu, 1997 Research Center, Yorktown Heights, New York. Gabriel Bonti is in Scenario simulation: theory and methodology Finance and Stochastics 1, pages 43–67 the group risk controlling department at Deutsche Bank in Frank- furt. Dirk Siegel is practice leader for financial markets consult- Jorion P, 1997 Value at risk: the new benchmark for controlling market risk ing at IBM in Frankfurt. The authors would like to thank Bernhard New York, McGraw-Hill Grudzinski at IBM and Richard Shepherd at Deutsche Bank, who Kelly K and R Krzysztofowicz, 1997 did the substantial programming work leading to the code that is A bivariate meta-gaussian density for use in hydrology the basis of the results presented in this paper Stochastic Hydrology and Hydraulics 11, pages 17–31 Comments on this article can be posted on the technical discussion forum on the Risk Web site at http://riskpublications.com/risk

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