INTRODUCTION

Extreme asset price movements appear to be more pronounced over time and have major consequences for an economy's financial stability and monetary policies. This article investigates the extreme behaviour of equity market returns and quantifies the probabilities of these losses. Taking 14 major equity markets, the study examines similarities and divergences in the tail returns in equity markets from around the world. By their very nature, the estimation of extreme returns is highly dependent on accurate modelling of rare events, and the article models extreme equity returns using extreme value theory.

The article provides predictions of the frequency and severity of extreme returns for a comprehensive set of American, European and Asian markets. Using daily returns the study is able to incorporate the effects of major financial crises such as the 1987 crash, the Asian crises and more recent excessive price movements by illustrating and analysing the extent of market movements across all markets. Previous studies have examined tail behaviour under a number of different headings including portfolio allocation,1 risk management,2 methodological issues3 and for different assets such as currencies,4 whereas this study aims to comprehensively examine tail returns across the main global markets and identify similarities and divergences in the recent decades.

Extreme price movements are found during periods of manias and crashes,5 adequately describing the scenario of the 30 per cent fall in US equities over a week during the 1987 crash.6 This article assumes that an extreme return occurs if market movements exceed some predetermined threshold value on either side of a probability distribution of equity returns. Specifically, the article calculates ex post unconditional tail probabilities for global equity markets separately and uses these to determine the frequency of occurrence of large price movements.7 These measures are underpinned by an analysis of the unconditional distributions of American, Asian and European equity markets.

The remainder of the article proceeds as follows. In the next section, the estimation procedures are presented with a brief synopsis of the theoretical underpinnings of extreme value theory. The section after that provides a description of the markets indexes chosen for analysis and their time-varying dynamics. The penultimate section presents the empirics detailing unconditional extreme value estimates. Finally, a summary of the article and some conclusions are given in the last section.

THEORY AND ESTIMATION METHODS

We begin by providing a short synopsis of the salient features of extreme value theory as it applies to modelling extreme financial returns (for comprehensive details8). The theoretical framework distinguishes three types of unconditional asymptotic distributions that model tail realisations, the Gumbel, the Weibull and, the one of concern to this study, the fat-tailed Fréchet distribution. The fat-tailed property has been documented for the extreme returns of many financial time series, such as index returns,9 single equities,10 foreign exchange11 and derivatives.2 The property indicates the propensity for financial time series to exhibit upside and downside returns of very large magnitude relative to the normal distribution for given probability levels. The fat-tailed property causes a relatively slow decay for convergence towards the limit, vis-à-vis the normal distribution.

Begin by assuming that a random variable, such as financial returns, is independent and identically distributed (iid) and belonging to the true unknown cumulative probability density function F(r).12 Taking the full distribution, returns are defined as the equity index's first difference of daily logarithmic price, r t =ln(p t )−ln(pt−1), measuring daily price movements. To examine extreme tail returns only, let M n be the maxima of n random variables placed in ascending order such that M n =max{R1, R2, …, R n }, and the (tail) probability that the maximum value exceeds a certain price change, r,

This represents the tail probability.

Although the exact distribution is allowed to be unknown, asymptotically it behaves like a fat-tailed distribution. The Fréchet extreme value distribution unifies fat-tailed distributions to have tail equivalence and allows for unbounded moments. Asymptotically, it allows the tail to vary with −α, which follows a power law. The consequence of the power law is that the fat-tailed returns decline at a slow rate in comparison with other distributional shapes. These alternative distributions can be divided into three separate groups depending on the value of the tail index α. A commonly assumed class of distributions used for financial returns includes the set of thin-tailed densities, and most notably among these, the normal or lognormal distributions. This classification of densities includes the normal and exponential distributions, and these belong to the Gumbel distribution, having a characteristic of tails decaying exponentially. In contrast, the classification of a Weibull distribution (α<0) includes the uniform example where the tail is bounded by having a finite right end point and is a short-tailed distribution. Of primary concern to the analysis of fat-tailed distributions is the Fréchet classification, and examples of this type generated here are the Cauchy, Student-t and sum-stable distributions.

By unifying fat-tailed distributions by having tail equivalence, it implies that distributions such as Student-t and sum-stable distributions exhibit identical limiting tail behaviour. The tail index, α, measures the degree of tail thickness and the number of bounded moments. For example, a tail index of two implies that the first two moments, the mean and variance, exist, whereas financial studies have cited a value between 2 and 4, suggesting that not all the first four moments of the price changes are always finite.13 The tail index has also been used to distinguish between different distributions with for instance, α interpreted as representing the degrees of freedom of a Student-t distribution and equals the characteristic exponent of the sum-stable distribution for α<2.

Given the asymptotic relationship of the random variable to the fat-tailed distribution, non-parametric tail estimation takes place giving separate upside and downside tail probabilities. The tail probability estimator is obtained from taking a second-order expansion of Fn(r) as r → ∝, avoiding all higher-order terms in the expansion r, and rearranging to incorporate sample estimates. The tail probabilities focus on extreme price movements only and determine the probability of various price movements, p.

The statistical features of the tail probability estimator are equivalent to those of the tail estimator that assumes that returns belong to the fat-tailed Fréchet distribution. The widely used Hill14 moment estimator is used to determine tail quantiles and probabilities. The Hill estimator represents a maximum likelihood estimator of m order statistics:

To compare these estimators, the tail stability test of Loretan and Phillips15 is applied to examine whether the tail probabilities vary according to the index chosen assuming that the underlying data is iid with Fréchet-type tail behaviour:

for γx(γy) are used to denote different equity indexes x and y. Here the tail stability test determines the extent to which indexes x and y deviate from each other (V(γx−γy)). Furthermore, this statistic is rearranged to examine whether probabilities across the tails of the distribution are distinguishable. Thus, tail behaviour can be examined for constancy across indexes and across the upside and downside of any return distribution. If both hypotheses of constancy hold, it suggests that tail returns are first similar across indexes, and second similar across the distribution of an index itself.

Notwithstanding that the main analysis examines the unconditional distribution of returns, an introductory discussion of the conditional distribution of index returns gives us an understanding of the relationship between the magnitude of returns and their occurrence during periods of high and low levels of volatility. A description of the time-varying dynamics is provided from fitting a GARCH (1, 1) model to the returns series.15 This allows for modelling of serial dependency that exists in financial returns and provides a description of the conditional environment:

Here, volatility is time varying and modelled adaptively on past squared values of the disturbance term and past values of the conditional volatility process.

DATA DESCRIPTION

Daily logarithmic returns from a broad spectrum of equity market indexes representing major markets from around the world are analysed for the period between 1 January 1985 and 31 December 2000 (given the recent financial crises, we utilise an earlier period that could be considered normal market conditions). The data include three US, five European and six Asian equity market indexes allowing for an extensive investigation of extreme returns for major financial markets across a wide geographical spread.

Characteristics of the returns series are provided by the summary statistics in Table 1. Mean values indicate on average positive returns, and standard deviation values suggest daily volatility around 1 per cent, although in general the Asian series exhibit higher values pointing to their high level of inherent risk. These indexes also indicate the largest interquartile range with the largest individual return occurring for the Hong Kong index, with a daily loss in excess of 40 per cent. Clearly, all series are non-normal, given the results for the Jarque–Bera test statistic. This lack of normality is reflected in the excess skewness and, more importantly for this study, the excess kurtosis that is evident for all series analysed. This implies that any attempt to model these returns using a normal distribution would clearly underestimate the tail densities, and thus fails to adequately predict the likelihood of extreme events.

Table 1 Summary statistics for daily returns series
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To provide a description of the conditional environment and time-varying volatility, an AR (1) – GARCH (1, 1) model is fitted to the daily returns series. Conditional volatility is obtained allowing for an examination of the pattern and magnitude of return fluctuations in periods of tranquillity and turbulence. The model's parameters and associated probability values using Bollerslev and Wooldridge's16 robust standard errors are presented in Table 2. In addition, the commonly noted fat-tailed characteristic of financial returns is accounted for by modelling the error terms with Student-t distributions. Generally, the conditional volatility models are similar in their attributes, indicating that past return volatility impacts on current volatility as is typical of a GARCH-type process. Generally, the autoregressive term is significant in the conditional mean equation and both ARCH and GARCH effects are documented in the conditional volatility equation.

Table 2 GARCH (1, 1) model for daily returns series
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The post-fitting diagnostics suggest that the models appear well specified. The standardised residual series are white noise in all cases except for the United Kingdom, Singapore and Indonesian indexes as can be seen from the Ljung–Box test results. Furthermore, the serial correlation associated with financial returns series is removed after fitting the GARCH model, as the standardised residuals for the squares generally satisfy the null of no fourth-order linear dependence of the Ljung–Box Q2(12) tests. The only exception is the Hong Kong index. Turning to the tail modelling, we now examine the estimates of extreme value theory that concentrate on tail values only, thereby minimising event risk and provide estimates based on the unconditional features of the equity index returns.

EXTREME VALUE FINDINGS

We have already seen similarities and divergences across the full distribution of index returns and we now turn our attention to the extremes. To begin, the Hill tail estimates are calculated and compared across the distribution of each index. Next, extreme value quantiles for very low probabilities are discussed. Tests of stability in tail behaviour between the respective markets are then outlined. Finally, comparative tail probability estimates are scrutinised for the 14 major equity markets.

Hill tail estimates and associated quantiles for major equity markets are given in Table 3 and are now discussed. Alternative approaches in determining the optimal number of tail returns, m, analysed are available,17 and the approach adopted here follows the bootstrap procedure that minimises the asymptotic mean squared error of the Hill estimator. The initial starting value chosen for the procedure is n0.6. To overcome the possible biases in the number of tail returns chosen and the inferred tail probabilities, Hill estimates for a large range of tail returns are calculated and plotted as Hill plots. Inferences on extreme upside and downside returns imply that the constancy of the tail index value is paramount. Tail constancy is examined and confirmed for the equity indexes by the Hill plots, with an illustration given for the Hong Kong index in Figure 1. Here constant estimates are obtained for a large spectrum around the number of tail values chosen. The maximum number of values required in tail index estimation occurs for the US NASDAQ and US DOW JONES with m=155, although all estimation uses less than 5 per cent of each indexes returns (5 per cent=209 returns from a sample size of 4174 returns).

Table 3 Hill estimates for daily returns series and associated quantiles
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Figure 1
figure 1

Hill plot for upper tail returns of the Hong Kong index.Notes: The Hill estimates are reasonably constant, using between 70 and 200 tail values, and diverge considerably from using a very small number of tail returns.

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Turning now to the Hill values themselves, point estimates in Table 3 for tail index values are generally between 2 and 4 within small confidence intervals. An exception to this is the Indonesian index with an estimate of 1.81 based on the upper tail of its distribution.18 In general, the lower the tail index estimates, the fatter the density mass of the tail. This rule recognises the impact of the very large spikes occurring for some of the Asian series, especially the Indonesian series, and also the US NASDAQ. The implication is of a propensity for greater price movements for these series in accordance with the noted maximum and minimum values. In contrast, the European and American indexes have higher tail estimates indicating relatively thinner tails. An exception to this generalisation is the Asian Japanese index with a Hill estimate of 3.41 for the lower tail of the distribution. This only indicates that the Nikkei 225, although from an Asian market, represents a reasonably stable market and behaves similarly to the European and American indexes. Similar conclusions are made for the well-diversified old economy index, the S&P500, which has a relatively thinner tail with higher Hill estimates compared with the well-diversified new economy index, the NASDAQ. Surprisingly, the thinnest tail documented is for the Italian index, and one would not normally consider it as being the safest market.19

Using the Hill estimates, inferences are also made regarding the number of defined moments of an empirical distribution. Previous studies usually indicate a Hill estimate of between 2 and 4, suggesting that the first two moments, the mean and variance, are defined, but this is not necessarily so for higher moments such as kurtosis.13 Thus, for a second moment to exist, tail index values should not be less than 2, which is a hypothesis that is not rejected for any index (with the exception of Indonesia), thereby supporting the existence of a second moment with a critical value of 1.64. Turning to the fourth moment, kurtosis is defined if tail index estimates are greater than 4. This hypothesis, although never supported for the point estimates themselves, is generally rejected with the exception of incidences for UK, French, Dutch, Italian and Japanese markets, respectively. Overall the results are in line with previous studies where there is ambiguity whether a fourth moment is defined for financial series.

Extreme equity market returns quantiles are also presented in Table 3. These extremal equity index values provide evidence on the severity and timing of extreme financial returns. The quantiles represent estimated extremal returns based on various probabilities, for example, Q1/n occurring once over the sample period of 16 years and for Q1/2n occurring once over the sample period of 32 years. The low probability levels for in-sample quantiles occur 0.024 per cent of the time, whereas for out-of-sample quantiles occur 0.012 per cent of the time given the sample size (n=4174 returns) chosen for analysis. The evidence supports the view that (with the exception of Japan) the Asian markets exhibit a greater propensity for extreme returns. For instance, there is a 1 in 4714 chance that an upside return of 38.38 per cent would occur for the Indonesian index that exhibits the largest extreme returns whereas the corresponding loss for the UK index is 6 per cent. Also in terms of geographical location, the European market with the largest extreme values is the Amsterdam index from Holland. For the United States the NASDAQ provides an exception to the relatively stable American indexes and may be driven by the uncertainty associated with the technology sector.

In addition to comparing extreme quantiles for different geographical areas, we can also determine the variation across upside and downside extreme returns. Some of the Asian contracts including Indonesia, Japan and Malaysia have larger extreme returns associated with the upside distribution than the downside. In contrast, the outcomes for the less risky European and American markets suggest that extreme positive returns are smaller than negative outcomes. For example, the UK index exhibits an extremal return of 6.00 per cent at the probability Q1/n from analysing the upper tail in comparison to 9.75 per cent for the lower tail. Comparison of upside and downside extreme returns is further applied to out-of-sample estimates, where similar conclusions are obtained for the tails of the distribution.

It is interesting to formally determine the extent to which tail behaviour deviates across markets and trading positions. Using the stability test discussed in Loretan and Phillips,13 estimates are presented in Table 4 determining the extent to which the Hill tail estimates of each index deviates from each other for each trading position. Overall, the vast majority of tail estimates for the indexes analysed are similar in magnitude, with very few statistically significant estimates. This is particularly pronounced in examining the downside tail statistics, with all but 13 cases of 91 having similar sized tail values for a critical value of 1.96. Thus, the fat-tailed behaviour associated with extreme financial returns is not just prone to affecting certain geographical markets, but impacts equity markets per se. Deviations that do occur tend to be from the relatively thin-tailed Japanese and the relatively fat-tailed Indonesian index. This implies that the extreme tail return behaviour is reasonably similar across all equity markets and is generally homogeneous across American, European and, to a lesser degree, Asian markets. Turning to comparing tail behaviour of the markets for the upside returns, similar conclusions are inferred from examining stability across indexes (although there is a greater degree of divergence with 35 significant test statistics).

Table 4 Tail stability tests for daily returns series
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An interesting extension for the Hill index values is to provide tail probability estimates, and these are presented in Table 5 and are now discussed. These provide information on the probability of these indexes incurring price movements that reach certain thresholds such as 10 per cent. The tail probabilities implicitly feed directly from the Hill index values. Tail probability estimates are provided for three thresholds and for extreme negative and positive returns on an annualised basis. The predetermined extreme losses chosen allow for a thorough investigation into the propensity for any of the market to experience daily losses of a very large magnitude. Overall, a clear distinction can be made from examining the likelihood of experiencing various extreme returns for the markets in the different geographical regions. The tail probability values for the Asian indexes dwarf their American and European counterparts. Exceptions are the relatively low values for the well-developed Japanese index and the relatively high values for the technology-dominated US NASDAQ.

Table 5 Tail probability estimates for daily returns series
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To illustrate, taking the Malaysian index the estimate of 0.3440 suggests a very high probability of occurrence of −10 per cent. These extreme returns would occur once every (k=1/p:1/0.3340) 3 years approximately, whereas the occurrence for the UK index is much less estimated at every (1/0.0589) 20 years approximately. These may appear to be rare events but two important points remain. First, the events occur with some frequency – for example, every 20 years might suggest two times in the average lifespan of a professional investor – and second, of more importance is the size of these rare events occurring at daily frequency, which have disastrous conclusions for a range of economic agents. The most risky European and Asian markets for large extreme returns are Holland and Indonesia, respectively, whereas the safest markets are the United Kingdom and Japan. For America, the NASDAQ index of equities represents the riskiest in terms of tail behaviour in contrast to the relatively safe S&P500. Similar probability findings hold across the Asian, American and European bourses at the different thresholds. Thus, these estimates represent a greater propensity for equity market crashes occurring in Asian markets with respect to other international markets.

SUMMARY AND CONCLUSIONS

This article examines the prediction of the frequency and severity of extreme market returns for a range of global equity indexes. The emphasis is on the statistical calculation of extreme price movements and their associated consequences using extreme value theory. This article comprehensively investigates the extreme behaviour of equity market returns and quantifies the probabilities of these losses. Taking 14 major world equity markets using American, European and Asian indexes, the study is able to ascertain similarities and divergences in the extreme tail returns from around the world – which is an issue not explored in previous studies.

The article reports a number of interesting findings. First, the analysis confirms the non-normality of equity market returns, and in particular the leptokurtosis indicative of fat-tailed distributions. Similarly, the returns series exhibit positive tail indexes, implying that the limiting extreme value distributions are characterised by a (fat-tailed) Fréchet distribution. Second, the extreme returns associated with lower tails are generally higher in absolute terms than those of the upper tails; this implies that large negative movements are more severe in magnitude than large positive movements. Third, with the exception of Japan, the extreme returns of the Asian market indexes are higher than their US and European counterparts, suggesting that the frequency and severity of extreme returns on these markets are greater than those on Western markets.