François Longin

Is the correlation in international equity returns constant: 1960–1990?

We study the correlation of monthly excess returns for seven major countries over the period 1960-1990. We find that the international covariance and correlation matrices are unstable over time. A multivariate GARCH(1,1) modelling with constant conditional correlation helps capture some of the evolution in the conditional covariance structure. We include information variables in the mean and variance equations. The volatility of markets changed somewhat over the period 1960-1990 and the proposed GARCH modelling allows to capture this evolution in variances. However tests of specific deviations lead to a rejection of the hypothesis of a constant conditional correlation. An explicit modelling of the conditional correlation indicates an increase of the international correlation between markets over the past thirty years. We also find that the correlation rises in periods when the conditional volatility of markets is large. There is some preliminary evidence that economic variables such as the dividend yield and interest rates contain information about future volatility and correlation that is not contained in past returns alone. However, further theoretical work is required to provide a satisfactory model. The effects are not of great magnitude but are often statistically significant.

The Asymptotic Distribution of Extreme Stock Market Returns

This article presents a study of extreme stock market price movements. According to extreme value theory, the form of the distribution of extreme returns is precisely known and independent of the process generating returns. Using data for an index of the most traded stocks on the New York Stock Exchange for the period 1885-1990, the author shows empirically that the extreme returns obey a Frechet distribution. Copyright 1996 by University of Chicago Press.

From value at risk to stress testing: The extreme value approach

This article presents an application of extreme value theory to compute the value at risk of a market position. In statistics, extremes of a random process refer to the lowest observation (the minimum) and to the highest observation (the maximum) over a given time-period. Extreme value theory gives some interesting results about the distribution of extreme returns. In particular, the limiting distribution of extreme returns observed over a long time-period is largely independent of the distribution of returns itself. In financial markets, extreme price movements correspond to market corrections during ordinary periods, and also to stock market crashes, bond market collapses or foreign exchange crises during extraordinary periods. An approach based on extreme values to compute the VaR thus covers market conditions ranging from the usual environment considered by the existing VaR methods to the financial crises which are the focus of stress testing. Univariate extreme value theory is used to compute the VaR of a fully-aggregated position while multivariate extreme value theory is used to compute the VaR of a position decomposed on risk factors.

Optimal margin level in futures markets: Extreme price movements

Along with price limits and capital requirements, the margin mechanism ensures the integrity of futures markets. Margin committees and brokers in futures markets face a trade‐off when setting the margin level. A high level protects brokers against insolvent customers and thus reinforces market integrity, but it also increases the cost supported by investors and in the end makes the market less attractive. This article develops a new method for setting the margin level in futures markets. It is based on “extreme value theory,” which gives interesting results on the distribution of extreme values of a random process. This extreme value distribution is used to compute the margin level for a given probability value of margin violation desired by margin committees or brokers. Extreme movements are central to the margin‐setting problem, because only a large price variation may cause brokers to incur losses. An empirical study using prices of silver futures contracts traded on COMEX is also presented. The comparison of the extreme value method with a method based on normality shows that using normality leads to dramatic underestimates of the margin level.

Beyond the VaR

“Value at Risk has become a well-known and widely used approach to evaluating risk exposure, even though what VaR measures, in an important sense, is really the value not at risk. The 5% VaR level, for example, gives a lower bound on portfolio value that holds 95% of the time. But the VaR does not tell how large the loss is likely to be on the 5% of the occasions that the VaR level is penetratedÑonly that it will be worse than the 5% VaR value. To address this problem, Longin discusses a related concept, the expected value of the loss conditional on its being greater than the VaR, known as BVaR. For a lognormal distribution, there is a simple relationship between VaR and BVaR. But actual return distributions typically differ considerably from the lognormal, particularly in the extreme tails that are the focus of the VaR calculation. BVaR involves both the VaR and the shape of the entire tail of the distribution function. It is also better than VaR in taking account of the presence of securities with non-linear payoffs in the portfolio. Longin presents an analysis with historical data to show how VaR and BVaR calculations would differ for linear and non-linear positions based on the S&P 500 stock index and its options, under different assumptions about the returns distribution (e.g., lognormal, extreme value).”

Minimal returns and the breakdown of the price-volume relation

Abstract We look at stock-market prices and transaction volume on the day of minimal (the minimum for that year) daily returns, from 1885 to 1990. We find that large (in absolute terms) minimal returns show little correlation with transaction volume.

Value at Risk: Une nouvelle approche fondée sur les valeurs extrêmes

This paper presents extreme value theory and its application to the computation of the value at risk of a position. This statistical theory allows quantification of the behavior of extreme moveme nts in prices and rates such that a new measure for catastrophe or bankruptcy risk can be defined. Empirically, it is shown that the Frechet distribution models this type of movement well. Extreme movements are associated with both little tremors like market adjustments or corrections during ordinary periods, and also earthquake-like stock market crashes, bond market collapses or foreign exchange crises observed during extraordinary periods. The approach based on extreme values then covers market conditions ranging from the usual environemnt considered by the existing VaR methods to the financial crises which are the focus of stress testing. The approach based on extreme values is then applied to a position in the French stock market using extreme value theory which characterizes the limit distribution of extreme returns. This method is then compared to different methods of the traditional approach which describe the statistical behavior of all returns (the historic distribution, the normal distribution and conditional processes like the GARCH process or the exponential weighted moving average used in RiskMetrics(tm) used to describe the variance). These empirical results allow to evaluate the French regulation on market risks.

Asymmetric Exceedance-Time Model: An Optimal Threshold Approach Based on Extreme Value Theory

We propose an innovative asymmetric exceedance-time model with optimal thresholds. This model examines the impact of extreme downside and upside shocks and determines how the duration between past and present extreme shocks affects the dependent variable. We use extreme value theory (peak-over-threshold method) to model extremes, proposing a procedure for the automatic computation of optimal thresholds, at the point where the fitting of the extreme value distribution is maximized. We apply our model to S&P 500 equity index and GBP/USD exchange rate data and show that a strong statistical relation coincides with optimal threshold levels.