19.2.1 The Distribution of Returns

Starting with the notations, R will stand for the (logarithmic) return of the asset, position, or portfolio computed over a given time interval, and f_R and F_R, respectively, the density probability and cumulative distribution functions of the random variable R. The support of the density function is noted as [l, u], the lower and upper bounds, l and u, being possibly equal to infinity (it is the case for the Gaussian distribution). Let R₁, R₂,…, R_n be n returns observed at n time intervals of frequency f.

19.2.2 Extremes Defined as Minimal and Maximal Returns

Extremes can be defined as the minimum and the maximum of the n random variables R₁, R₂,…, R_n. We note Y_n the highest return (the maximum) and Z_n the lowest return (the minimum) observed over n trading time intervals.⁶

The extreme value theorem (EVT) is interested in the statistical behavior of the minimum and maximum of random variables. It is analogous to the central limit theorem (CLT), which is interested in the statistical behavior of the sum of random variables.⁷ Both theorems consider the asymptotic behavior of the variables in order to get results that are independent of the initial distribution. In the EVT framework, extremes will have to be selected from a very long time interval, whereas in the CLT framework the sum is computed over a very long time interval. In order to get nondegenerated limiting distributions, the variables of interest have to be standardized first. This is illustrated below in the EVT case.

If the variables R₁, R₂,…, R_n are statistically independent and drawn from the same distribution (hypothesis of the random walk for stock market prices), then the exact distribution of the maximum Y_n is simply given by

19.1

The distribution of extremes depends mainly on the properties of F_R for large values of r. Indeed, for small values of r, the influence of F_R(r) decreases rapidly with n. Hence, the most important information about the extremes is contained in the tails of the distribution of R. From Formula (19.1), it can be concluded that the limiting distribution of Y_n is null for r less than the upper bound u and equal to 1 for r greater than u. It is a degenerate distribution.

As explained in Longin (1996), the exact formula of the extremes and the limiting distribution are not, however, especially interesting. In practice, the distribution of the parent variable is not precisely known and, therefore, if this distribution is not known, neither is the exact distribution of the extremes. For this reason, the asymptotic behavior of the maximum Y_n is studied. Tiago de Oliveira (1973) argues, “As, in general, we deal with sufficiently large samples, it is natural and in general sufficient for practical uses to find limiting distributions for the maximum or the minimum conveniently reduced and use them.” To find a limiting distribution of interest, the random variable Y_n is transformed such that the limiting distribution of the new variable is a nondegenerate one. The simplest transformation is the standardization operation. The variate Y_n is adjusted with a scaling parameter α_n (assumed to be positive) and a location parameter β_n. In the remainder of the paper, the existence of a sequence of such coefficients (α_n > 0, β_n) is assumed. Extreme value theory specifies the possible nondegenerate limit distributions of extreme returns as the variable n tends to infinity.⁸ In statistical terms, a limit cumulative distribution function denoted by satisfies the following condition: . Gnedenko (1943) showed that the extreme value distribution (EVD) is the only nondegenerate distribution that approximates the distribution of extreme returns . The limit distribution function is given by

19.2

The parameter ξ, called the tail index, gives a precise characterization of the tail of the distribution of returns. Distributions with a power-declining tail (fat-tailed distributions) correspond to the case ξ > 0, distributions with an exponentially declining tail (thin-tailed distributions) to the case ξ = 0, and distributions with no tail (finite distributions) to the case ξ < 0.⁹ The EVD is called a Fréchet distribution, a Gumbel distribution, and a Weibull distribution. The Gumbel distribution can be regarded as a transitional limiting form between the Fréchet and the Weibull distribution.

EVT gives an interesting result: whatever the distribution of the parent variable R, the limiting distribution of the extremes always has the same form. The distribution of the extremes for two different parent processes is differentiated by the values of the standardizing coefficients α_n and β_n and the tail index ξ.

More interestingly, the same limiting distribution is obtained if the i.i.d. hypothesis is relaxed. Berman (1964) has shown that the same result stands if the variables are correlated and if the series of the squared correlation coefficients is finite. A common model is a discrete mixture of Gaussian distributions. In this particular case, the Gumbel distribution is still the limiting distribution of the extremes (see Leadbetter et al., 1983). De Haan et al. (1989) show that, if the returns followed an ARCH(1) process, the variable Y_n would have a limiting Fréchet distribution. Following their research, I detail below the relationship between the parameters of the ARCH process and those of the distribution of the extremes. Recall that an ARCH(1) process is given by two equations:

19.3a

19.3b

The realized return R_t observed at time t is decomposed into an expected part noted as computed one period before at time t − 1, and an unexpected part noted as ϵ_t known at time t only. The expected variance h_t varies over time and is conditioned upon the past value of the innovation ϵ_t−1. The ARCH models reflect quite well the time-varying behavior of volatility and especially the clustering of extremes. After a big shock (i.e., a large value for ϵ_t−1), one expects a high level of variance and then more big shocks in the future. The coefficient a₁ reflects the persistence of volatility (or the correlation of absolute returns). A high value of a₁ implies a high level of persistence, many clusters of extremes, and finally a fat-tailed unconditional distribution of returns. The tail index ξ is related to the degree of persistence a₁ by the following formula:

19.4

Assuming a conditional Gaussian distribution for the innovation ϵ, Eq. (19.4) becomes

19.5

where Γ is the gamma function and π the constant pi. For a given value of the parameter a₁, a unique value of ξ is obtained by solving Eq. (19.5). For example, for a₁ equal to 0.5, the tail index ξ is equal to 0.42.

These results show that the assumption of independence is less important for extreme values than would seem at first sight. Let us note that the extremes are (asymptotically) drawn from an unconditional distribution, even if the parent variable is drawn from a conditional distribution.

19.2.3 Extremes Defined as Return Exceedances

Extremes can also be defined in terms of exceedances with reference to a threshold denoted by θ. For example, positive θ-exceedances correspond to all observations of R greater than the threshold θ. As results for negative exceedances can be deduced from those for positive exceedances by consideration of symmetry, I focus on the case (R > θ), which defines the right tail of the distribution of returns. The probability that a return R is higher than θ, denoted by probability p_θ, is linked to the threshold θ and the distribution of returns F_R by the relation: p = 1 − F_R(θ).

As explained in Longin and Solnik (2001), the cumulative distribution of θ-exceedances, denoted by and equal to (F_R(x) − F_R(θ))/(1 − F_R(θ)) for x > θ, is exactly known if the distribution of returns F_R is known. However, in most financial applications, the distribution of returns is not precisely known and, therefore, neither is the exact distribution of return exceedances. For empirical purposes, the asymptotic behavior of return exceedances needs to be studied. Extreme value theory addresses this issue by determining the possible nondegenerate limit distributions of exceedances as the threshold θ tends to the upper point u of the distribution. In statistical terms, a limit cumulative distribution function denoted by satisfies the following condition: . Balkema and De Haan (1974) and Pickands (1975) show that the generalized Pareto distribution (GPD), , is the only nondegenerate distribution that approximates the distribution of return exceedances . The limit distribution function is given for x > θ by

19.6

where σ, the dispersion parameter, depends on the threshold θ and the distribution of returns F_R, and ξ, the tail index, is intrinsic to the distribution of returns F_R.

19.3.1 The Parametric Approach

The parametric approach assumes that minimal returns and maximal returns selected over a given period are exactly drawn from the EVD given by Formula (19.2) or, alternatively, that negative and positive return exceedances under or above a given threshold are exactly drawn from the distribution given by Formula (19.6). With either definition of extremes, the asymptotic distribution contains three parameters: ξ, α_n, and β_n, or extremes defined as minimal or maximal returns selected from a period containing n returns, or alternatively, ξ, σ_θ, and p_θ for extremes defined as negative or positive return exceedances under or above a given threshold θ. Under the assumption that the limit distribution holds, the ML method gives unbiased and asymptotically normal estimators (see Tiago de Oliveira (1973) for the system of equations). The system of nonlinear equations can be solved numerically using the Newton–Raphson iterative method. Note that the regression method (see Gumbel (1958)) gives biased estimates of the parameters but may be used to get initial values for the ML algorithm.

In practice, EVDs can be estimated with different values of the number of returns contained in the selection period n (for minimal and maximal returns) or, alternatively, with different values of the threshold θ (for negative and positive return exceedances). A goodness-of-fit test such as a Sherman test can then be carried out in order to choose the most relevant values from a statistical point of view.

19.3.2 The Nonparametric Approach

The previous methods assume that the extremes are drawn exactly from the EVD. Estimators for the tail index ξ, which do not assume that the observations of extremes follow exactly the EVD, have been developed by Pickands (1975) and Hill (1975). These estimators are based on order statistics of the parent variable R.

Pickands's estimator for the right tail is given by

19.7

where is the series of returns ranked in an increasing order, and q is an integer depending on the total number of returns N contained in the database. Pickands's estimator is consistent if q increases at a suitably rapid pace with N (see Dekkers and De Haan (1989)). Pickands's statistic is asymptotically normally distributed with mean ξ and variance . Pickands's estiamtor is the most general estimator because it can be used for all types of distributions.

Hill's estimator for the right tail is given by

19.8

Hill's estimator can be used in the case of the Fréchet distribution only (ξ > 0). In this situation, Hill's estimator is a consistent and the most efficient estimator. Consistency is still obtained under weak dependence in the parent variable R. Hill's statistic is asymptotically normally distributed with mean ξ and variance ξ².

In practice, as the database contains a finite number of return observations, the number of extreme returns q used for the estimation of the model is finite. As largely discussed in the extreme value theory literature, the choice of its value is a critical issue (see Danielsson et al., 2001; Huisman et al., 2001 for a discussion). On one hand, choosing a high value for q leads to few observations of extreme returns and implies inefficient parameter estimates with large standard errors. On the other hand, choosing a low value for q leads to many observations of extreme returns but induces biased parameter estimates, as observations not belonging to the tails are included in the estimation process. To optimize this tradeoff between bias and inefficiency, I use a Monte Carlo simulation method inspired by Jansen and De Vries (1991). Return time series are simulated from a known distribution for which the tail index can be computed. For each time series, the tail index value is estimated with a different number of extreme returns. The choice of the optimal value is based on the mean-squared error (MSE) criterion, which allows one to take into account the tradeoff between bias and inefficiency. The procedure is detailed in the appendix.

19.4.1 Distributions of Returns

Several distributions for stock returns have been proposed in the financial literature. Most of the empirical works in finance assume that continuously compounded rates of return on common stock or on a portfolio are normally distributed with a constant variance. The Gaussian distribution is consistent with the log-normal diffusion model made popular by the Black–Scholes–Merton option pricing formula. Moreover, most of the statistical tests lie on the hypothesis of normality. Unfortunately, there is now strong evidence that the distribution of the stock returns departs from normality. High kurtosis usually found in the data implies that the distribution is leptokurtic. The empirical distribution is fat-tailed; there are more extreme observations than predicted by the normal model. This is of great importance because the tails of the density function partly determine the level of the volatility. And volatility is certainly a most important variable in finance.

I review below the alternative models to the Gaussian distribution, and show how these models can be discriminated using the tail index.¹¹

Mandelbrot (1963) first suggested that the variance of certain speculative price returns could not exist. Studying cotton prices, he concluded that the stable Pareto distributions fitted the data better than the Gaussian distribution. Fama (1965) extended this approach to stock market prices.

If stock returns usually present fat tails, this does not imply that the variance is infinite. The mixture of Gaussian distributions and the unconditional Student-t distributions presents an excess of kurtosis but still possesses finite variance. Such models were proposed for stock prices by Press (1967) and Praetz (1972). A mixed distribution models the heterogeneity of the random phenomenon. The returns are drawn from different Gaussian distributions. Such a model has been used to take into account extreme price movements, such as stock market crashes, that do not fit in a model with a single distribution. Such events are assumed to be drawn from a distribution with a negative mean and high variance. Anomalies in the stock market like the “day effect” can also motivate this model.

The volatility varies in fact much more over time. Mandelbrot (1963) first found a “clustering effect” in volatility and pointed out that large changes in prices tend to be followed by large changes of either sign, and, similarly, that small changes tend to be followed by small changes of either sign. The ARCH process proposed by Engle (1982) models this feature and tends to fit quite well the behavior of volatility.

19.4.2 Test Based on Extreme Value Theory

An extreme value investigation allows one to discriminate among these non-nested models. Although all processes of returns lead to the same form of distribution of extreme returns, the values of the parameters of the distribution of extremes are in general different for two different processes. Especially, the value of the tail index ξ allows the discrimination of these processes. A tail index value equal to 0 implies a Gumbel distribution obtained for thin-tailed distributions of returns. A negative value for the tail index implies a Weibull distribution obtained for distributions of returns with finite tails. A positive value for the tail index implies a Fréchet distribution obtained for fat-tailed distributions of returns. More precisely, a value of ξ greater than 0.5 is consistent with a stable Pareto distribution. The Cauchy distribution corresponds to the special case ξ = 1. A value of ξ less than 0.5 is consistent with the ARCH process or Student's distribution. An interesting feature of the tail index is that it is related to the highest existing moment of the distribution. The tail index ξ and the highest existing moment denoted by k are simply related by k = 1/ξ (for ξ positive). When ξ is equal to 0, then all moments are defined (k = +∞). This is the case of the Gaussian distribution and the mixture of Gaussian distributions. For the stable Pareto distribution, k is less than 2 (the variance is not defined) and equal to the characteristic exponent. For the Student-t distributions, k is more than 2 and equal to the number of degrees of freedom. Table 19.1 summarizes these results.

Note: This table gives the type of extreme value distribution, the tail index value, and the highest existing moment for different models of returns commonly used in financial modeling. The tail index ξ and the highest existing moment k are related by: k = 1/ξ. The last two columns indicate the constraints on the coefficients ξ and k imposed by each model.

The tail index provides us with a straightforward test. Two particular unconditional distributions are considered below: the thin-tailed Gaussian distribution and the fat-tailed stable Pareto distribution.

19.4.2.1 The Gaussian distribution

As the Gaussian distribution for returns implies a Gumbel distribution for extreme returns, the tail index can be used for testing normality. The null hypothesis is stated as

If the tail index ξ is significantly different from 0, then the asymptotic distribution of extreme returns is not a Gumbel distribution. As a consequence, the Gaussian distribution for returns can be rejected. Alternatively, if the tail index ξ is not different from 0, then the asymptotic distribution is the Gumbel distribution. Such a result is not inconsistent with the normal model.

19.4.2.2 The stable Pareto distribution

As the Pareto distribution for returns implies a Fréchet distribution for extreme returns (with a constraint on the tail index value greater than 0.5), the tail index can also be used for testing the Pareto model. The null hypothesis is stated as

If the tail index ξ is significantly less than 0.5, then the asymptotic distribution of extreme returns is not a Fréchet one with a high tail index value. As a consequence, the stable Pareto distribution for returns can be rejected. Alternatively, if the tail index ξ is not significantly less than 0.5, then the asymptotic distribution is the Fréchet distribution with high tail index value. Such a result is not inconsistent with the stable Pareto model.

19.5.1 Data

I use logarithmic daily percentage returns of the S&P 500 index based on closing prices. Data are obtained from Yahoo Finance. The database covers the period January 1954–December 2015 and contains 16,606 observations of daily returns (Figures 19.1 and 19.2).

The daily returns have a slightly positive mean (0.029%) and a high standard deviation (0.946). The values of the skewness (−1.016) and the excess kurtosis (27.290) suggest departure from the Gaussian distribution. The first-order autocorrelation (generally attributed to a nontrading effect) is small (0.029) but significantly positive. Little serial correlation is found at higher lags. For the second moment, I find a strong positive serial correlation: 0.114 at lag 1. The correlation decreases slowly and remains significant even with a lag of 20 days (0.056), which suggests a strong persistence in volatility.

I now give some statistics about the extremes. Let us first consider the definition of extremes as the minimum and maximum returns selected over a given time period. Considering yearly extremes, I get 66 observations for each type of extreme over the period January 1950–December 2015. The top 20 yearly largest daily market falls and market rises are reported in Table 19.2. Both types of extreme are widely spread. For the largest declines, the minimum value (−22.90%) is obtained in October 1987, the second minimum values (−9.47%) during the subprime crisis in 2008, and the third minimum value (−7.11%) during the Asian crisis in 1997. The lowest yearly minimum daily returns (−1.33%) is observed in 1972. For the largest rises, the maximum value (+810.96%) is observed in October 2008 a few days after the market crash. Let us then consider the definition of extremes as negative and positive return exceedances under or above a given threshold. The top 20 largest daily market falls and market rises are reported in Table 19.3. As expected, the two definitions of extremes lead to similar sets of extreme observations. However, due to some clustering effect, extreme returns tend to appear around the same time.¹² This effect is especially severe for the stock market crash of October 1987 and the recent crisis. Among the top 20 largest daily market falls, the stock market crash of October 1987 appears twice: October 19 (−22.90%) and October 26 (−8.64%); the subprime crisis appears eight times: September 21 (−9.21%), October 7 (−5.91%), October 9 (−7.92%), October 15 (−9.47%), October 22 (−6.30%), November 19 (−6.31%), November 20 (−6.95%), and December 1 (−9.35%). The same remark applies to top 20 largest daily market rises. The period of extreme volatility following the stock market crash of October 19, 1987, contains three top positive return exceedances: October 21 (+8.71%), October 20 (+5.20%), and October 29 (+4.81%); the subprime crisis appears eight times: September 30 (+5.28%), October 13 (+10.96%), October 20 (+4.66%), October 28 (+10.25%), November 13 (+6.69%), November 21 (+6.13%), November 24 (+6.27%), and December 26 (+5.01%).

Note: This table gives the 20 lowest yearly minimum daily returns and the 20 highest yearly maximum daily returns in the S&P 500 index over the period January 1954–December 2015. Yearly extreme returns are selected over nonoverlapping years (containing 260 trading days on average).

Note: This table gives the 20 lowest negative daily returns and the 20 highest positive daily returns in the S&P 500 index over the period January 1954–December 2015.

19.5.2 Tail Index Estimates

The approaches described in Section 19.4 are now used to estimate the tail index. The empirical results are reported in Table 19.4 for parametric estimates using minimum and maximum returns, in Table 19.5 for parametric estimates using negative and positive return exceedances, and in Table 19.6 for nonparametric estimates.

Note: This table gives the tail index estimates using minimum and maximum returns observed over a given time period. Minimum and maximum returns are selected over different periods: from 1 month to 1 year. The number of minimum or maximum returns used in the estimation process is given below in parentheses in the first column. The parameters of the distributions of minimum and maximum returns are estimated by the maximum likelihood method (only the tail index estimates are reported). Asymptotic standard errors are given below in parentheses. The result of Sherman's goodness-of-fit test is given in brackets with the p-value (probability of exceeding the test-value) given next in curly brackets. The 5% confidence level at which the null hypothesis of adequacy (of the estimated asymptotic distribution of extreme returns to the empirical distribution of observed extreme returns) can be rejected is equal to 1.645.

Note: This table gives the tail index estimates using negative and positive return exceedances below or above a given threshold. Return exceedances are selected with fixed threshold values: ±1%, ±2%, ±3%, and ±4% (percentage point below or above the mean of returns). In the last row, return exceedances are selected with optimal threshold values: −2.60% for the left tail and +3.27% for the right tail (see Appendix for the description for obtaining optimal threshold). In the first column, the number of return exceedances used in the estimation process is given below in parentheses for both negative and positive return exceedances. The parameters of the distributions of negative and positive return exceedances are estimated by the maximum likelihood method (only the tail index estimates are reported). Asymptotic standard errors are given below in parentheses. The result of Sherman's goodness-of-fit test is given in brackets with the p-value (probability of exceeding the test value) given next in curly brackets. The 5% confidence level at which the null hypothesis of adequacy (of the estimated asymptotic distribution of extreme returns to the empirical distribution of observed extreme returns) can be rejected is equal to 1.645.

Note: This table gives the tail index estimates based on nonparametric methods developed by Pickands (1975) and Hill (1975). For each method, the optimum number of tail observations is computed by simulation (see Appendix and Table A1). The optimum number is given in parentheses for both the left and right tails below the method name in the first column. Asymptotic standard errors of the tail index estimates are given below in parentheses.

Let us begin to analyze the results for each estimation method as the tail index value tends to vary according to the method used and also to the parameter used to implement a particular method (i.e., the length of the selection period, the threshold value, and the number of tail observations). Let us consider the left tail, for example. For the left tail, the tail index estimate varies between 0.226 and 0.509 for the parametric method using minimum returns observed a given period, from 0.185 to 0.466 for the parametric method using negative return exceedances under a given threshold, and from 0.212 to 0.328 for the nonparametric methods. As the parametric approach assumes that the asymptotic distribution holds for finite samples, it is important to check the goodness of fit of the distribution to empirical data. For minimum returns, the Sherman test (reported in Table 19.4) shows that it seems cautious to select the extremes over a period longer than a semester. Similarly, for negative return exceedances, the Sherman test (reported in Table 19.5) shows that it seems cautious to select extremes under a threshold value lower than −3%. Looking at the nonparametric approach, Pickands's estimate is positive, suggesting that Hill's estimator can be used as it is restrained to the case of a positive tail index. Under this assumption, Hill's estimator is more precise than Pickands's estimator: the standard error of Hill's estimate is almost four times lower than the one of Pickands's estimate (see Table 19.6).

The first result is about the sign of the tail index, which determines the type of EVD. All tail index estimates are positive, implying that the distribution of extreme returns is a Fréchet distribution consistent with fat-tailed distribution of returns.

The second result is about the relative asymmetry between the left tail and the right tail. The tail index estimates for the left tail are systematically higher than the one for the right tail. This statement can be formalized by testing the null hypothesis H₀: ξ^max = ξ^min. For the usual confidence level (say 5%), this hypothesis is sometimes rejected by the data, indicating that the left tail is heavier than the right tail.

19.5.3 Choice of a Distribution of Stock Market Returns

Two particular unconditional distributions are considered: the Gaussian distribution and the stable Pareto distribution by testing, respectively, the null hypotheses H₀: ξ = 0 and H₀: ξ > 0.5. Empirical results are reported in Table 19.7. Three confidence level are considered: 1%, 5%, and 10%. The lower the confidence level, the harder it is to reject the null hypothesis.

Note: This table gives the result of the choice of a particular distribution for returns based on the tail index. Two particular distributions are considered: the Gaussian distribution characterized with a tail index value equal to 0 (Panel A) and the stable Pareto distribution characterized with a tail index value higher than 0.5 (Panel B). For the Gaussian distribution, the null hypothesis is H₀: ξ = 0. For the stable Pareto distribution, the null hypothesis is H₀: ξ > 0.5. Three estimators are used: the parametric maximum likelihood (ML) estimator based on minimum and maximum returns and return exceedances and the Hill nonparametric estimator. Three confidence levels are considered: 1%, 5%, and 10%.

19.5.4 Highest Existing Moment

The tail index can be used to compute the highest defined moment of the distribution of returns. Technically, it corresponds to the highest integer k such that E(R^k) is finite. I proceed as follows: I consider a set of null hypotheses H₀(k) defined by ξ < 1/k. If the null hypothesis H₀(k) is rejected at a given confidence level, then the moment of order k is not defined at this level. The null hypothesis H₀(+∞) defined by ξ ≤ 0 serves as a limiting case. If the null hypothesis H₀(+∞) is rejected, then not all moments are defined.

Table 19.8 gives the empirical results concerning the highest existing moment by looking at each tail independently. Three confidence levels are considered: 1%, 5%, and 10%. The lower the confidence level, the easier it is to accept the existence of lower moments. As expected, the conclusion of the test depends on the method used for estimating the tail index. However, general results emerged. The first result is that the second moment (the variance) seems to be always defined, as the null hypothesis H₀(2) is never rejected. The fourth moment seems, however, not always defined. The second result is the relative asymmetry between the left tail and the right tail. The highest existing moment by considering the left tail is always lower than the highest existing moment by considering the right tail, suggesting that the left tail is heavier than the right tail. Moreover, by looking at the right tail, all moments seem defined in most of tests.

Note: This table gives the highest existing moment of the distribution of stock market returns by investigating the weight of extreme price movements. For a given level of confidence, equal to 1%, 5%, and 10%, the null hypotheses H₀(k) defined by: ξ > 1/k where k is equal to 1, 2, 3, … is studied. A t-test and its associated p-value are computed. The null hypothesis H₀(+∞): ξ < 0 serves as the limiting case. The highest integer k for which H₀(k) is not rejected at the given level is reported in the table. If the null hypothesis H₀(+∞) is not rejected, then all the moments are defined. Three estimators are used: the parametric maximum likelihood (ML) estimator based on minimum and maximum returns and return exceedances and the Hill nonparametric estimator. Three confidence levels are considered: 1%, 5%, and 10%.