Modelling extremal stock returns in a stable Paretian environment

Modelling Extremal Stock Returns

Table  of  NA

   

   

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1.  INTRODUCTION THE EMPIRICAL DISTRIBUTION OF STOCK RETURNS  1

1.1  STOCK RETURNS VOLATILITY AND ASSET ALLOCATION  1

1.2  STYLISED FACTS OF STOCK RETURN DISTRIBUTIONS  4

2.  MODELS FOR STOCK RETURN DISTRIBUTIONS  8

2.1  AN OVERVIEW OF FULL PARAMETRIC RETURN DISTRIBUTION MODELS  8

2.2  BASIC APPROACHES TO INFERENCE ABOUT EXTREMAL RETURNS  14

3.  TECHNICAL BACKGROUND FOR TAIL INFERENCE  18

3.1  EXTREME VALUE THEORY EVT  18

3.2  THE STABLE PARETIAN MODEL  25

4.  ESTIMATION OF THE STABLE PARETIAN INDEX  31

   

4.1  DESIRABLE PROPERTIES OF AN ESTIMATOR AND EVALUATION CRITERIA  31

4.2  ESTIMATION METHODOLOGIES AND PREREQUISITES  33

4.3  TAIL ESTIMATORS  37

4.3.1  Intuition behind tail estimation  37

4.3.2  Where does the tail start  38

4.3.3  Hill estimator  41

4.3.4  Modifications of the Hill estimator  44

4.3.5  Generalisations of the Hill estimator  49

4.3.6  Pickands estimator  54

4.3.7  Generalisations of the Pickands estimator  56

4.3.8  De Haan Resnick estimator  57

4.3.9  De Haan Pereira estimator  59

4.4  ESTIMATION VIA THE PEAKS OVER THRESHOLD POT METHOD  60

4.4.1  ML techniques  60

4.4.2  Method of probability weighted moments PWM  61

4.4.3  Falk Marohn estimator  61

4.4.4  Choice of the threshold level  62

4.5  CHARACTERISTIC FUNCTION TECHNIQUES  63

4.5.1  Method of Moments estimators MME  64

4.5.2  Minimum Distance estimators MDE  65

4.5.3  Regression type estimators  65

4.6  MAXIMUM LIKELIHOOD ESTIMATORS MLE  66

4.6.1  Algorithms for computation of the stable Paretian PDF  66

4.6.2  The ML estimation procedure  68

4.7  QUANTILE BASED ESTIMATORS  69

4.7.1  Quantile estimator by Fama and Roll  69

4.7.2  McCulloch estimator  70

4.8  OTHER APPROACHES  71

4.9  WHICH ESTIMATOR IS THE BEST ONE  72

   

Modelling Extremal Stock Returns

5.  AN EMPIRICAL COMPARISON OF ESTIMATORS  74

5.1  SIMULATION STUDY OF TAIL ESTIMATORS  74

5.1.1  Performance with Pareto data and small sample properties  75

5.1.2  Performance with Fréchet data  85

5.1.3  Performance with stable data  89

5.2  CONSIDERATION OF MODIFIED TAIL ESTIMATORS  95

5.3  APPLICATION TO STOCK RETURN DATA  103

5.3.1  Daily stock returns  106

5.3.2  Weekly stock returns  111

6.  CONCLUSION SUMMARY  116

APPENDIX  TO MAIN TEXT  118

LIST  OF REFERENCES  NA

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Modelling Extremal Stock Returns

List  of  NA

   

Table  Bias and standard deviation of tail estimators for different sample fractions with  NA

(  data replications  76

   

   

   

   

Table  Bias and standard deviation of tail estimators for different sample fractions with  NA

(  data replications  77

   

   

   

   

Table  Asymptotic behaviour of estimator bias for different rules of selecting the sample fraction  NA

0.22)  vs rule with Pareto data  NA

   

   

   

k   NA

   

   

   

   

centage  rule  NA

2,000).   81

   

   

Table  Asymptotic behaviour of tail estimator standard deviation for different rules of  NA

   

the  sample fraction with Pareto data  NA

   

   

   

   

   

   

50,  max  NA

2,000).   82

Table  Asymptotic behaviour of tail estimator mean squared error MSE for different rules of  NA

   

the  sample fraction with Pareto data  NA

   

   

   

   

   

   

50,  max  NA

2,000).   83

Table  Bias and standard deviation of tail estimators for different sample fractions with  NA

(  data replications  85

   

   

   

   

Table  Bias and standard deviation of tail estimators for different sample fractions with  NA

(  data replications  86

   

   

   

   

Table  Bias of tail estimators for different sample fractions with stable Paretian  NA

   

(  replication  89

   

Table  Bias of tail estimators for different sample fractions with stable Paretian  NA

   

(  replication  90

   

Table  Bias of tail estimators for different sample fractions with stable Paretian  NA

   

(  replication  91

   

Table  Bias and standard deviation of original tail estimators and two modifications for  NA

sample  fractions with Pareto data replications  96

   

   

   

   

   

Table  Bias and standard deviation of original tail estimators and two modifications for  NA

sample  fractions with Fréchet data replications  98

   

   

   

   

   

Table  Bias of original tail estimators and two modifications for different sample fractions with  NA

Paretian  data replication  100

   

   

   

Table  Overview of key figures of daily logarithmic equity and index  NA

(13  April April  103

Table  Overview of key figures of weekly logarithmic equity and index  NA

(13  April April Wednesday to Wednesday  105

Table  Accurate sample fraction for each of the applied tail estimators  105

Table  Tail index estimates for daily returns of stock price  NA

(13  April April  107

Table  Tail index estimates for daily returns of five blue chip  NA

(13  April April  109

Table  Tail index estimates for weekly returns of stock price  NA

(13  April April  112

Table  Tail index estimates for weekly returns of five blue chip  NA

(13  April April  114

   

Modelling Extremal Stock Returns

List  of  NA

   

Fig.  Histograms of Microsoft daily returns vs random numbers from normal  NA

with  same mean and std deviation illustrating fat tails and peakedness  6

Fig.  QQ plot of Dax daily and weekly logarithmic returns Jan Dec  7

   

   

D  and  NA

D   NA

with  Pareto data and Pareto  NA

   

   

   

Fig.  Fig  NA

   

mean  estimates standard deviation of replications  78

   

   

   

   

   

D   NA

D   NA

with  Pareto data and Pareto  NA

   

   

   

and  UPick and  NA

Fig.  Fig  NA

   

mean  estimates standard deviation of replications  79

   

   

   

   

   

D   NA

D   NA

   

for  rule vs percentage  NA

Fig.  Asymptotic behaviour of  NA

and  Peng and  NA

   

   

   

0.22  with Pareto data  NA

   

k   NA

   

   

   

   

   

   

   

50,  max  NA

2,000)   84

   

   

D  and  NA

D   NA

with  Fréchet data and Fréchet  NA

   

   

   

   

Fig.  Fig  NA

   

mean  estimates standard deviation of replications  87

   

   

   

   

   

D   NA

D   NA

with  Fréchet data and Fréchet  NA

   

   

   

   

and  dVGomes and  NA

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mean  estimates standard deviation of replications  88

   

   

   

   

   

D  and  NA

Dˆ  estimates with stable Paretian data stable Paretian  NA

   

   

   

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data,  and stable Paretian data replication  93

   

   

   

   

   

   

D   NA

D   NA

estimates  with stable data stable  NA

   

   

   

   

and  HillGomes and  NA

Fig.  Fig  NA

   

and  stable data replication  94

   

   

   

D   NA

D   NA

   

D   NA

   

   

   

   

   

   

   

   

   

D   NA

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Dc   NA

with  Pareto  NA

   

   

   

   

   

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/  dHR  NA

/  dHR  NA

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mean  estimates standard deviation of replications  97

   

   

   

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,  Peng  NA

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/  dVGomes  NA

,  Peng  NA

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mean  estimates standard deviation of replications  99

   

   

   

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stable  Paretian data replication  101

   

   

   

   

   

Fig.  Histograms of daily stock index returns Dax FTSE  NA

daily  equity returns Elf Aquitaine Microsoft logarithmic scale  104

Fig.  Histograms of daily equity returns Deutsche Bank Coca Cola  NA

logarithmic  scale  104

   

D  estimates of upper and lower tail index for daily  NA

Fig.   NA

of  Dax and FTSE stock price indices April April  108

   

   

D   NA

D   NA

estimates  of upper and lower tail index for daily  NA

Fig.   NA

of  Coca Cola Deutsche Bank Elf Aquitaine and Volkswagen  110

   

D   NA

estimates  of upper and lower tail index for weekly  NA

Fig.   NA

of  Dax and stock price indices April April  113

   

   

D   NA

Dˆ  estimates of upper and lower tail index for weekly  NA

/  dHR  NA

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of  Elf Aquitaine Deutsche Bank Coca Cola and Volkswagen  115

   

Modelling Extremal Stock Returns

1. Introduction: The Empirical Distribution of Stock Returns

1.1 Stock returns, volatility, and asset allocation

For a long time, it has been observed that when making investment decisions, individuals would not only look at their expected profit or rate of return, but also include the perceived risk inherent with the asset. Financial market theory has been capturing risk aversion of investors for over 50 years, based on the seminal work of MARKOWITZ (1952, pp. 77-79). It has become an indispensable element of financial models since then (SCHMID et al. (yns), p. 1).

The perception of asset risk is closely entwined with the probability of extremal returns. The likelihood of extremal events is reflected in the distribution of the random variable underlying the return-generating process, and especially in the shape of the tails: Where the probability of extremal returns is high, the tails of the distribution are rather ‘fat’ or ‘heavy’, whereas one speaks of ‘light’ tails when extremal returns occur very rarely. An important field of application for inference about the tail shape is the estimation of value at risk (VaR, for a definition, see HARRIS et al. 2001, p. 717), a concept for assessing the downside risk of portfolio values, which is closely related to the shape of the lower tail (DANIELSSON et al. 2000, p. 15). These findings are used to derive an optimal asset allocation. VaR calculation has traditionally been based on normally distributed security returns, yet it has been shown that results are dramatically different when the underlying model is non-normal (TOKAT et al. 2003, pp. 937-938; ORTOBELLI et al. (yns), pp. 1-2). Thus, given an observed risk aversion of investors, it is clear that the distribution of stock returns – and especially the shape of the tails – has far-reaching implications for risk assessment, portfolio management, and asset pricing (MITTNIK et al. 1999a, p. 236). Yet, albeit important, finance experts and statisticians still have considerable difficulties to understand extremal movements in stock prices (LONGIN 1996 , p. 383).

Basically, there are two approaches to shed some light on this question:

Tail inference based on full parametric assumptions. A natural way of gaining o insight about the likelihood of extreme price movements is to first establish a distributional model that fits well with empirical stock return distributions and then to estimate the parameter governing the tail behaviour (tail index).

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Modelling Extremal Stock Returns

Letting  the tails speak for themselves In this case tail inference is made  NA

   

out  modelling the centre of the distribution Tail index estimation is based  NA

extremal  returns only This method is based on Extreme Value Theory  NA

This  paper is going to discuss both approaches the stable Paretian or  NA

   

distribution  serving as conceptual framework for the analysis Tail inference  NA

essentially  focused on estimation of the index which determines the shape of the  NA

   

Two  questions shall be  NA

Provided  that stock returns actually follow stable Paretian distribution what is  NA

   

best  estimator for the index  NA

   

Given  that deviations from the stable Paretian model have frequently  NA

   

observed,  how can we make inference about the tail shape and obtain an  NA

estimate  of even if the stable model does not hold  NA

   

It  shall be found later that the EVT based approach plays crucial role in  NA

suitable  estimators of the tail index under relaxed distributional  NA

Structure  of remainder of text The remainder of this paper is structured as  NA

In  section some stylised facts of empirical stock return distributions are  NA

Section  analyses how these empirical characteristics have been captured in  NA

return  distribution models following historical timeline The family of stable laws  NA

introduced,  along with theoretical and empirical findings on the  NA

Section  then shifts over to extremal returns short introduction to different ways  NA

making  inference about extremal returns is given focused on parametric models  NA

models  based on EVT which are most relevant  NA

Section  provides the technical framework for the estimation of the tail index  NA

section  gives necessary basics of EVT section introduces relevant  NA

background  on the stable Paretian  NA

Sections  and are the central parts of this paper Section aims to give  NA

prehensive  overview of estimators of the tail index based on the definition of  NA

properties  and estimation methodologies and prerequisites Common  NA

ators  are described and evaluated previous theoretical and practical evidence on  NA

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Modelling Extremal Stock Returns

performance  provided To conclude and summarise the theoretical part  NA

question  whether there is best estimator of the tail index is addressed in paragraph  NA

Section  provides results of empirical studies conducted In the simulation part  NA

focus  is on evaluation of estimator performance when sampling from known  NA

with  given tail index whilst paragraph contains applications of estimators to  NA

data.  Here the aim will be to give an indication of how fat tailed empirical stock  NA

distributions  actually are broken down into upper and lower tails as well as daily  NA

weekly  returns weekly  NA

Section  concludes by summarising the main findings and providing future  NA

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Modelling Extremal Stock Returns

1.2 Stylised facts of stock return distributions

In empirical comparisons, stock returns (especially daily stock returns) have consistently exhibited several characteristic features, which are known as ‘stylised facts’:

Dependence and volatility clustering. Financial time series are usually not inde- pendent, but exhibit stochastic dependence. Even though the linear dependence between returns of subsequent days is negligible, there is considerable dependence of squared returns (SCHMID et al. 2002, pp. 8-9).

However, when focusing on tail events exclusively, it can be observed that dependency decreases when thresholds are established (DANIELSSON et al. 2000, pp. 6-7). As for the correlation in the volatility of returns, conditional heteroskedasticity is a common feature, i.e. the volatility is not constant over time, but it varies (KRÄMER, pp. 14-15). Often a volatility clustering effect can be observed in all types of financial data, e.g. one day with an extremal stock return tends to be followed by another (EMBRECHTS et al. 1999, p. 413). In this case, the volatility is conditional on itself.

These findings often lead to the conclusion that it is easier to predict volatility than to predict prices or returns (SCHMID et al. 2002, p. 9).

But again, when focusing on extremal events, these appear to be almost “randomly scattered”

(DANIELSSON et al. 2000, p. 6), i.e. the clustering effect is much less prominent. In order to measure the degree of clustering of extreme values in financial time series, the

T 1 is of central importance. The extremal index measures the degree of short-range dependence exhibited by extreme values (ANCONA- NAVARRETE et al. 2000, p. 6). SMITH et al. (1994), WEISSMAN et al. (1998), and EMBRECHTS

et al. (1999, pp. 418-425) give a good overview of estimation techniques for the extremal index. The apparent violation of the iid assumption for stock returns can bring up severe

4

Modelling Extremal Stock Returns

Leptokurtosis. When plotting stock returns into a histogram and fitting a normal distribution (see Fig. 1-2), one would typically encounter two major fitting problems that can be summarised under the term ‘excessive kurtosis’ or ‘leptokurtosis’ vis-à-vis the normal distribution (see MILLS 1995, p. 323 or FRAHM 1998, p. 32).

Peakedness. In comparison to the Gaussian curve, the empirical distribution is o usually more peaked (cf. Fig. 1, rhs), i.e. the probability density in the very centre of the distribution is higher (KRÄMER (yns), pp. 10-11; MANDELBROT 1963, p. 395).

In order to measure the degree of peakedness, the following quantile-based estimator has been proposed (TREDE 1999, p. 17; SCHMID et al. (yns), p. 4): ˆ ˆ X

J where p

This estimator has been standardised, such that a positive value points towards a X ˆ represents the p -quantile of the empirical distribution function. distribution that is more peaked than the Gaussian. o Fat tails. The histogram of empirical returns would typically also deviate from the

normal distribution in that it is often thicker on the left and right tail (see Fig. 1, lhs). Financial data, including stock returns, typically exhibit fat tails, also called ‘heavy tails’, ‘long tails’, or ‘thick tails’ (KRÄMER (yns), p. 10). This fact has very important implications for the measurement and management of risk in portfolios (ORTOBELLI et al. (yns), pp. 2-4; TOKAT et al. 2003, p. 938). In a risk-averse environment, the amount of risky assets held by the investor depends on the probability of large deviations in asset prices (e.g. extremal stock returns). When the heaviness of tails is underestimated, this may well lead to an excessive

investment in risky assets, triggering sub-optimal asset allocation. Therefore, any model for tail inference that fails to reflect heavy tails is bound to show poor performance when applied in practice. Moreover, fat-tailedness may have important technical implications: As tails grow heavier, finite second and even first moments can fail to exist (BAMBERG et al. 2001,

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Modelling Extremal Stock Returns

(1963, p. 395) supports the infinite-variance hypothesis, other authors (e.g. SHIRYAEV 1999, p. 335) have questioned its validity.

There is no unanimous definition of when a distribution can actually be called ‘fat- tailed’. BAMBERG et al. (2001, pp. 4-5) give an overview of different definitions, but at this point, it is sufficient to introduce another quantile-based estimator (TREDE 1999, p. 18; SCHMID et al. (yns), p. 4):

ˆ ˆ X

6% 0%

11% 17%

Fig. 1: Histograms of Microsoft daily returns (blue) vs. random numbers from normal distribution (grey) with same mean and std. deviation; illustrating fat tails (lhs) and peakedness (rhs). Skewness. Besides excessive kurtosis, it has sometimes been suggested that empirical stock return distributions show considerable skewness, such that inflexible symmetric models (such as the normal distribution) are inappropriate (SHIRYAEV 1999, p. 331).

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Modelling Extremal Stock Returns

A quantile-based measure of skewness has been suggested by SCHMID et al. (yns, p. 4):

ˆ ˆ ˆ ˆ

) ( ) ( X X X X

J

For symmetric distributions, 2

ˆ

J 0.

In this paper, skewness will implicitly be accounted for when discussing inference about

separately, such that potential differences would suggest asymmetry.

the tail index. The shape parameters for the upper and lower tail will be derived

Daily returns vs. lower-frequency returns. In contrast to daily returns, two of the above

stylised facts are less clear-cut when moving to weekly or monthly returns:

Volatility clustering. When sampling at weekly or monthly intervals, the volatility

o

clustering effect is very small. Thus, the deviation from an iid process is not

equally significant in this case (PAOLELLA 2001 , p. 1096).

o

Leptokurtosis. When moving to lower-frequency data, the goodness-of-fit with the

normal distribution tends to improve considerably (SCHMID et al. 2002, p. 8).

normal distribution provides a much better fit in the tail region (rhs), whereas in case of

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Modelling Extremal Stock Returns

2. Models for Stock Return Distributions

2.1 An overview of full parametric return distribution models

The following section describes common models to characterise the distribution of

logarithmic stock returns

n observations of the asset price n based on P P

P

, 1

0

. The log-returns can be

sampled at varying intervals (e.g. daily, weekly, or monthly). According to the additivity

i t regarded as the sum of daily returns of the corresponding week, this reasoning appears to be sensible, provided that the iid assumption is fulfilled. The hypothesis that stock returns follow a normal distribution used to be prevalent in statistics in the first half of the 20 th century, based on early work by Bachelier in 1900, who claimed that movements in stock prices are independent and normal (Brownian motion).

obtained by adding up higher frequency returns – a favourable feature: ¦ s i t s t

property of log-returns (see SCHMID et al. 2002, pp. 4-5), low frequency returns may be t

X

X X , , As this paper is concerned with the unconditional distribution of stock returns rather than conditional models or time series, the notation shall be simplified to n

1 X X

,...,

P

S n x x f Obviously, the assumption that stock returns follow this model is based on the Central Limit Theorem (CLT), stating that the sum of independent and identically distributed

V random variables

X

X

X

,...,

, 2

1

approximately follows a normal distribution for large n

(BOMSDORF et al. 1999, p. 44). As for instance weekly (logarithmic) returns can be

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Modelling Extremal Stock Returns

Scholes option pricing model, in which equity prices are assumed to follow a geometric Brownian motion (HARRIS et al. 2001, p. 738). Moreover, the normal distribution is also the standard model underlying the calculation of VaR (KLÜPPELBERG 2002, p. 1).

Whilst this assumption could be empirically sustained for weekly returns (FAMA 1963, p. 420), it has since then been rejected on various occasions for daily and higher-frequency returns, e.g. via non-parametric inspection (PAOLELLA 2001, p. 1095; KLÜPPELBERG 2002, pp. 8-9). HARRIS et al. (2001, pp. 725-736) adopt a simulation approach to demonstrate that the Gaussian hypothesis is not a suitable working assumption.

The central problem is that the normal model, appealing though it may be in theory, is obviously not capable of capturing any of the stylised facts described in section 1.2 – neither dependency nor excessive kurtosis nor skewness.

As a result of the normality assumption being found insufficient for daily returns, researchers have proposed various other models to better reflect the stylised facts (LINDEN 2001, pp. 159-160). This has led to two conflicting schools, the one of which relying on a theoretically sound model (e.g. stable distributions, normal mixtures), the other working with distributions picked according to their empirical goodness-of-fit.

Stable distributions. The earliest evidence of non-normality of daily and higher- frequency returns was provided in the early 1960s by MANDELBROT (1963, p. 395) and FAMA (1963, p. 421), see also HARRIS et al. (2001, p. 739).

Mandelbrot and Fama proposed the class of stable distributions for modelling stock returns, which had first been studied by Lévy in 1924, but since then found little attention. Within this class, the stable Paretian distribution was found capable of more accurately reflecting the stylised facts of empirical distributions. Even though there are other sub- classes of stable distributions, such as max-stable, min-stable, and multiplication-stable distributions (MITTNIK et al. 1993, pp. 270-290 provide a good overview), those have been less extensively studied.

The stable Paretian model allows for modelling heavy tails as well as skewness (GHOSE et al. 1995, pp. 227-228), yet nesting the normal distribution as a special case (MITTNIK et al. 1999c, pp. 275-276). It is therefore more realistic and more flexible than the normal distribution (MITTNIK et al. 1993, pp. 268-269).

There has been considerable empirical evidence in favour of stable laws (e.g. DOSTOGLOU et al. (1999), pp. 57-58), and it has generally been regarded as a visible improvement vis-

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Modelling Extremal Stock Returns

à-vis the traditional normality assumption, being “more adherent to the reality of the market” (ORTOBELLI et al. (yns), p. 33).

The stable model is not specific to equity returns, but has been successfully applied to many sorts of asset prices, e.g. bonds and options (see for instance DOSTOGLOU et al. 1999, pp. 58-60).

Besides an improved practical performance, the class of stable distributions also exhibits attractive theoretical properties:

Generalised CLT. MANDELBROT (1963, pp. 399-401) and FAMA (1963, pp. 424-425) o show that stable variables are the only possible limits of sums of iid random variables. Economically, this means that a return over a certain period can be interpreted as the sum of many small asymptotically Paretian price changes (FAMA 1963, p. 426; also TOKAT et al. 2003, p. 939). This finding relies upon a generalised version of the CLT (GCLT) by Feller (1966) which does not rule out the normal distribution. Therefore, the stable family is compatible with many financial models originally built on the normality assumption, e.g. the portfolio theory (FIELITZ et al. 1983, p. 28).

Domains of attraction (DA). The stable model is applicable to sums of iid RVs o even if the stable model does not hold exactly, which is a highly desirable property, given that realisations of stock returns cannot be assumed to follow an ideal theoretical distribution model (MITTNIK et al. 1993, p. 265). Any distribution in the domain of attraction of a stable law will exhibit similar properties, allowing investors to base their decisions upon the idealised stable model. What is more, it can usually be checked whether or not the DA condition is met by just looking at the tails of a distribution (MITTNIK et al. 1993, p. 265).

Stability under aggregation. Moreover, stable Paretian distributions are invariant o under addition, which is an important property for financial arbitrage theory (HOLS et al. 1991, p. 295). More generally, one formulates that stable Paretian distributions belong to their own domain of attraction, i.e. this class is robust with respect to n- fold convolution and scaling (MITTNIK et al. 1993, p. 269). It is stable with respect to cross-portfolio as well as temporal aggregation. A necessary consequence is that the tail index D

(also called ‘characteristic exponent’) stays the same regardless of the sampling interval (TOKAT et al. 2003, p. 944; DEO 2002, p. 258).

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So, all in all, stable distributions are found to be a flexible, theoretically sound model for describing stock returns.

Yet on the other hand, several inconsistencies with the stable distribution have been identified in empirical case studies – which have triggered a search for alternative models:

Finite variance.

Stable Paretian distributions only possess finite moments of order o

d D 2. This implies an infinite variance except for the case of the normal distribution ( D 2). This property of stock returns has been questioned more than once. JANSEN et al. (1991, p. 24) derive the conclusion that, based on

higher moments. This contradicts the stable Paretian notion. empirical research, stock returns exhibit finite first and second moments, but infinite o Volatility clustering not reflected. The stable Paretian model is based upon an iid process. Even though MITTNIK et al. (1993, p. 266) state that, due to the stability property, iid assumptions may be relaxed, this turns out to be an issue when looking at volatility clusters: Generally, the model is theoretically not capable of displaying the volatility clustering effect described in section 1.2 (GROENENDIJK et al. 1995, pp. 253-254). This is even more important when considering that conditional volatility alone can be an explanatory factor for fat tails. Following GHOSE et al. (1995, p. 225), the phenomenon of heavy tails can be ascribed to volatility clustering, which leads them to conclude that ARCH-type models (with stable Paretian innovations) are more suitable for modelling financial data (also see PAOLELLA 2001, p. 1108-1110). This drawback can adversely affect inference, e.g. by limiting the applicability of testing procedures (PAOLELLA 2001, p. 1098).

against the use of stable Paretian models in finance. According to Mittnik et al. (2000, p. 390), this is the most serious argument standing o Stability property violated.

DEO (2002, p. 258) points out that the theoretically attractive property of invariance of the characteristic exponent vis-à-vis the sampling interval is inconsistent with empirical evidence. In many studies, it is found that when the sampling interval would be expanded (e.g. when shifting from daily to weekly returns), the estimate of D

would increase (MITTNIK et al. 1993, p. 269) – a violation of the temporal aggregation criterion. This critique also fits with the observation that the normal distribution ( D 2) tends to provide a better fit for

11

Modelling Extremal Stock Returns

stable Paretian distribution is probably not a suitable model for stock returns. HSU et al. (1974, pp. 110-113) suggest an alternative model with non-stationary parameters.

Apart from inconsistencies between theoretical properties and empirical findings, there are some practical obstacles that impair the usefulness of the stable Paretian model in practice:

Slow rate of convergence. In practice, the rate of convergence towards a stable o law may be very slow, i.e. very high sample sizes are needed for the sum of iid RVs to converge towards a stable distribution. This reduces the attractiveness of working under DA conditions in practice (MITTNIK et al. 1993, p. 270). No closed-form density function. Stable distributions are difficult to handle o (MITTNIK et al. 1999c, p. 276). There is no closed-form expression for the PDF except for some special cases, which makes practical application more difficult. For example, ML estimation of the tail index requires a time-consuming approximation of the PDF prior to the estimation procedure (see section 4.6.1). Infinite variance property. Besides being called into question per-se, the infinite o variance property also triggers practical problems. The non-existence of a second moment leads to difficulties in inferential procedures, e.g. computing confidence intervals or testing (JAMMALAMADAKA 2003, p. 4; SHIRYAEV 1999, p. 335).

It is for those reasons that the stable Paretian model is not without controversy in empirical finance (GHOSE et al. 1995, p. 226).

Nevertheless, in this paper, this model shall be recurred to when estimating the tail index D

, yet bearing in mind its limitations. It shall also be investigated in detail under what conditions one can estimate the heaviness of the tail without requiring that the above model must hold exactly.

A more extensive technical description of stable laws in general and the stable Paretian model in particular is provided in section 3.2.

Other return distribution models. After the stable model had been ‘en vogue’ for some time, disappointing results of empirical studies and the complexity of the family made some financial economists search for alternative distribution models (LINDEN 2001, p. 159).

12

Modelling Extremal Stock Returns

Various proposals have been made, some based on normative reasoning, others driven by empirical fit (HARRIS et al. 2001, pp. 715-716). A brief overview is provided.

Mixtures of distributions. Mixtures of distributions are a widespread approach towards capturing the stylised facts of stock return distributions. Various mixture models have been proposed, the most notable of which is the mixture of normal distributions. MCLACHLAN et al. (2000) provide a good introduction to finite mixture models. The economic intuition behind mixtures is that stock returns are essentially driven by various independent influences (e.g. different groups of investors or types of information), which are captured in the components of the mixture model.

Normal mixtures. Given the complexity of D

-stable distributions, TREDE (1999, pp. o 19-20) suggests that mixtures of normal distributions are an attractive alternative, capable of displaying leptokurtosis and also skewness. Technically, a normal mixture is the weighted sum of l iid normally distributed RVs with different variance and possibly different mean (for a formal definition, see MCLACHLAN et al. 2000, pp. 6-7). The attractiveness of normal mixtures vis-à-vis D

-stable models is based on the fact that normal mixtures do not require the stability conjecture to hold (FIELITZ et al. 1983, pp. 34-35). Moreover, they are more easily applicable in 3

practice due to an algorithm that easily computes the ML estimates of the 1 ˜ l relevant parameters (EM algorithm, see FLURY 1997, pp. 656-659). It remains an open question whether or not normal mixtures outperform non-normal stable laws (SCHMID et al. (yns), p. 7; FIELITZ et al. 1983, pp. 34-35), yet there is some evidence

o that they may constitute a pragmatic alternative. Other mixtures. Other mixtures proposed include the Laplace mixture distribution (LINDEN 2001, pp. 160-162) and mixtures of non-normal stable distributions (FIELITZ

et al. 1983, pp. 32-34). o Other distributions. Alternative models suggested include the following distributions: Student- t distribution (e.g. JOHNSON et al. 1994, Vol. 2, pp. 362-374; SHIRYAEV 1999, p. 334). Some authors, such as BLATTBERG et al. (1974, pp. 263-277), have o

argued that the t

-distribution is more suitable for daily returns than the stable. o Log-normal distribution (e.g. HARTUNG 1998).

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Modelling Extremal Stock Returns

2.2 Basic approaches to inference about extremal returns

Whilst section 2.1 was focused on characterising models capturing the stylised facts of the whole (unconditional) return distribution, this section describes different approaches towards describing the focal point of interest – the tail of the distribution. Basically, there are five possibilities:

Use past empirical extremal realisations only to predict the likelihood of extremal o events (and thus tail thickness) in the future (non-parametric historical simulation). Have a normative model for the entire distribution (cf. 2.1) and derive tail thickness o by fitting the whole empirical distribution (homoskedastic parametric models). Account for stochastic volatility, all else being equal.

o Let the tails speak for themselves, i.e. concentrate on deriving a normative model o for the tail of the distribution, which is fitted using tail observations exclusively (models based on EVT).

Combine the aforementioned methods.

o

Non-parametric historical simulation (HS). Historical simulation methods rely on forecasting the distribution of future returns with just the help of an empirical distribution function (MCNEIL et al. 2000, p. 273). They can be easily implemented. Plus, few assumptions are required.

However, this approach is not suitable as a stand-alone method to draw conclusions about extremal returns:

Extremal observations – by their nature – are very rare, such that massive amounts o of data are needed or else the estimator will suffer a high variance. Extrapolation beyond empirical observations is not feasible, yet this is the crucial o point in tail estimation (DANIELSSON et al. 2000, pp. 11-12).

MILLS (1995, pp. 324-331) delivers an example of how HS can be applied to stock return distributions. Yet in this paper, the approach is not pursued any further.

14

Modelling Extremal Stock Returns

Parametric homoskedastic models. All models introduced in section 2.1 contain full parametric assumptions that can be used to describe extremal returns as well. For instance, assuming that unconditional stock returns follow a stable Paretian distribution, the model parameters can be estimated from a set of empirical data, e.g. using the ML method. The estimate obtained for the parameter D

describes the heavi- ness of the tails and is therefore a measure of the risk of extremal returns. In contrast to HS, this method allows for inference beyond the range of observations, yet there are two crucial drawbacks:

Neither of the above-mentioned unconditional models incorporates stochastic o volatility, even though this is a characteristic feature of stock returns. This means that obtained tail estimates cannot be based on the current volatility underlying short-term market fluctuations (MCNEIL et al. 2000, p. 274). This approach is therefore not suitable for gaining insight about the likelihood of short-term extremal losses, conditional upon current market volatility, over the next few days (p. 272). Under full parametric assumptions, central observations serve to draw inference o about the tails. This can lead to estimation biases (SHIRYAEV 1999, p. 336), left alone it is complicated if one is just interested in tail behaviour.

Full parametric homoskedastic methods are applied in this paper in the course of tail index estimation. The effects triggered by volatility clustering are thus not explicitly accounted for.

Parametric models with stochastic volatility. Methods incorporating the current volatility background include ARCH and GARCH and related models (for an overview, see DANIELSSON et al. 2000, pp. 13-14). This approach is set to tackle the first weakness of the unconditional models described in the previous paragraph.

An appealing property of these models is that the implied unconditional return distributions exhibit fat tails, reflecting a crucial stylised fact (RACHEV et al. 2000, p. 4). Yet they do not belong to the DA of stable distributions, but the heavy-tailedness is caused by stochastic volatility effects (GHOSE et al. 1995, pp. 226-227).

While the basic GARCH model works with normal innovations (MCNEIL et al. 2000, p. 273), there are more advanced models with non-normal stable residuals. A good description of

15

Modelling Extremal Stock Returns

so-called ‘stable GARCH’ processes can be found in MITTNIK et al. (2000, pp. 392-404) and RACHEV et al. (2000, pp. 275-293).

Whilst standard GARCH models perform poorly in tail inference (DANIELSSON et al. 2000, p. 13), conditional volatility models with stable Paretian innovations are shown to satisfactorily reflect the stylised facts of return distributions, even when using smaller samples (MITTNIK et al. 2000, p. 411).

Since this paper is focused on the stable Paretian hypothesis, the conditional volatility approach will rather remain in the background. However, even when acknowledging that stochastic volatility approaches reflect the current state of the art in stock return models, Paretian tail inference stays important when it comes to innovations: “…if one wishes to interpret the error term as a random variable representing the sum of many external effects which cannot be realistically captured by the model, the stable Paretian is the only valid candidate” (PAOLELLA 2001, p. 1095).

Models based on Extreme Value Theory. An EVT-based approach avoids modelling the entire distribution. Instead, it strives to make accurate predictions about extremal events by modelling only the tails of the distribution, i.e. only the sub-sample of extremal observations that provides information about extremal events is included in the tail inference procedure (KLÜPPELBERG 2002, p. 3).

This approach shows several striking advantages:

The tail estimate cannot, in any case, be distorted by central observations o (DANIELSSON et al. 2002, p. 3). In a seminal paper, DUMOUCHEL (1983, pp. 1023- 1028) showed that the estimate of the tail index D

of a stable law can be severely disrupted by deviations from the ideal model. He therefore suggested to “let the tails speak for themselves” (p. 1025). Doing so, it is possible to work under asymptotic conditions, such that the whole procedure is more flexible and robust. It is not necessary to be strictly committed to a specific model as presented in section 2.1 (HOLS et al. 1991, p. 287).

Another important aspect is that, by using EVT, one avoids modelling the entire o distribution, which is inefficient and time-consuming (PAOLELLA 2001, p. 1096).

Notwithstanding these aspects, EVT models do not usually capture stochastic volatility (MCNEIL et al. 2000, p. 274).

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Modelling Extremal Stock Returns

The central problem coming along with tail-only methods is to determine the right number k d of upper order observations to be included in the sub-sample (EMBRECHTS et al. n 1999, p. 326). On the one hand, the number of observations included should not be too small to limit the standard deviation of the tail estimator; on the other hand, including too big a share of the overall sample would lead to an estimation bias since central observations get included. So in case of finite samples, one is confronted with a bias- variance trade-off (for details, refer to section 4.1).

Besides imposing an exact D

-stable model, tail inference based on EVT shall be the second focus of this paper.

Combinations. Several combinations of the above methods have been suggested:

(1, 3, 4). One notable example is provided in MCNEIL et al. (2000, p. 274). A o GARCH approach is combined with HS (for the central part of the distribution of the innovations) and EVT (serving the purpose of tail inference for residuals, which are closer to iid -ness than raw data). Capturing the stylised facts of fat tails and conditional volatility, this is a fairly sophisticated model.

(1, 4). A semi-parametric combination of HS with EVT-based tail inference can be o found in DANIELSSON et al. (2000, pp. 18-23).

(2, 3). Several combinations of conditional heteroskedasticity models (GARCH or o modified GARCH) with D

-stable innovations have already been mentioned. The same has been investigated for exponential or Student- t residuals (MITTNIK et al. 2000, p. 390).

These combinations will not be investigated any further in this paper.

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Modelling Extremal Stock Returns

3. Technical Background for Tail Inference

3.1 Extreme Value Theory (EVT)

This section provides an introduction to the necessary basics of EVT.

Most generally, EVT serves to draw inference about the likelihood of extremal events in

the absence of a closed distributional model for the sampling distribution (HALL et al. 1997,

p. 1311). It is especially useful when extrapolating beyond the range of available data

(“predicting the unpredictable”, EMBRECHTS et al. 1999, p. VII).

EVT has been applied in various fields, such as hydrology (REISS et al. 1997, pp. 233-

244), insurance (MCNEIL 1996, pp. 5-18), and environmental issues (REISS et al. 1997, pp.

257-264).

In the context of stock returns, one makes use of EVT to meet DuMouchel’s (1983)

paradigm of ‘letting the tails speak for themselves’, estimating the tail index from extreme

observations exclusively (DANIELSSON et al. 2002, p. 2).

The first paragraph of this section sets out the central result of EVT, the Fisher-Tippett

theorem. This is followed by a description how this result can be used to make tail

inference using upper order statistics. The third paragraph establishes a modified

approach, based on excesses over a high threshold.

Theorem by Fisher/Tippett and Extreme Value Distributions (EVD). The central

finding one can derive from EVT is that the form of the asymptotic distribution of extreme

returns is not dependent on the return-generating process (LONGIN 1996, p. 384): of iid RVs with CDF F . EVT is concerned Given is a stationary sequence X X X ,..., , 2

n 1

with the limiting behaviour of the order statistic

n n X X X 2 1 ,..., analogue to the well-known Central Limit Theorem (CLT, see BOMSDORF 2000, pp. 127- 128): ) 1 .

n X M : 1

,

max

The theory can be equivalently extended to a greater number of upper order statistics

(EMBRECHTS et al. 1999, pp. 196-204).

The central result of classical EVT is called the Fisher-Tippett theorem or Extremal Types

theorem (EMBRECHTS et al. 1999, p. 121; JANSEN et al. 1991, p. 19), an upper-order

18

Modelling Extremal Stock Returns

Theorem 3.A (Fisher-Tippett theorem).

If there exist suitable normalising constants

then F belongs to the maximum domain of attraction

(MDA) of one of three types of ˜

exp )

( x

x

x

actual distribution of the underlying RV X (DANIELSSON et al. 2002, p. 6). A sketch of the proof can be found in EMBRECHTS et al. (1999, p. 122) and FRAHM (1998, p. 26).

[

19

Modelling Extremal Stock Returns

Positive values of [

correspond to the Fréchet case, negative values correspond to the

Even though no detailed knowledge about F is required, it is essential to decide which limit law applies, or put differently, whether the extreme value index is positive or negative.

What about dependency? It should be noted that the above results apply to iid maxima in the first place. However, remedies have been developed for situations in which maxima exhibit a dependence structure (KLÜPPELBERG 2002, p. 8). Moreover, it has been shown by Berman in 1964 that the results do not change if variables are correlated, provided that the sum of squared correlation coefficients remains finite (LONGIN 1996, p. 387). Methodologically, in case of dependency, the normalising constant n a

has to be modified

T , the extremal index, whereas the tail index D

remains

D D T Given the frequent observation that larger values tend to occur in clusters, T is a measure

of the degree of clustering of extremes (ANCONA-NAVARRETE et al. 2000, p. 6). Various methods have been developed to estimate the extremal index, the simplest of which are the blocks and runs estimators (see WEISSMAN et al. 1998). A good overview is given by

SMITH et al. (1994). Since this problem can essentially be handled independently from tail index estimation in that most results can easily be transferred, estimation procedures for D can be discussed autonomously (HOLS et al. 1991, p. 290). Modelling heavy tails with upper order statistics. How can the stylised facts of

empirical stock return distributions be modelled in the context of EVT? Obviously, it has to be identified which limit law corresponds to the phenomenon of heavy tails. Put differently,

20

Modelling Extremal Stock Returns

dF

One says that the right tails

,

x

MDA

F

then

 \

lim

then the tail ) ( 1 x F

varies regularly at infinity and

MDA  x F D )

.

Regular variation at infinity has the following implication (JANSEN et al. 1991, p. 19):

Theorem 3.D. If the tail

21

Modelling Extremal Stock Returns

power and not all moments are finite. This holds true for stable Paretian laws, but also for Student- t distributions and ARCH-type processes (LONGIN 1996, p. 387; JANSEN et al. 1991, p. 20). All distributions that exhibit fat (upper) tails in the sense of the EVT are nested within the Fréchet case. On the other hand, normal distributions as well as normal mixtures lead to a limit law of Gumbel type.

As for the stable Paretian type, condition 2 is clearly in accordance with D

-stable distributions. Moreover, the tail index coincides with the index D characterising the shape

of the limiting distribution (KEARNS et al. 1997, p. 171; LONGIN 1996, p. 387). It can be shown analytically that, for 0

(FRAHM 1998, p. 38). As one might expect from the discussion of suitable return distributions, there is strong empirical support in favour of the Fréchet hypothesis. LONGIN (1996, pp. 394-400)

demonstrates empirically that the distributions of minima and maxima of a selection of the most actively traded American stocks follow a Fréchet distribution. In accordance with the assumption of a stable Paretian distribution for stock returns, it will henceforth be assumed that the type II limit law (Fréchet case) is applicable to extremal

stock returns. This is central to the construction of tail index estimators based on upper order statistics (section 4.3). Tail convergence and the Generalised Pareto distribution (GPD). Apart from analysing the weak convergence of maxima, there is another branch of EVT concerned with the distribution of excesses over a high threshold u . The so-called PoT (peaks over threshold)

method was first used by hydrologists in the 1970s, and has since then been extended to other fields, such as finance (EMBRECHTS et al. 1999, p. 366). The appeal of the PoT method is based upon the finding that for sufficiently high levels u , the conditional distribution of the RV X above u converges to a generalised Pareto distribution (GPD) – regardless of the shape of the underlying distribution of X (CAERS et al. 1999, pp. 191-192). In other words, if a high enough threshold is chosen, the data above this threshold will exhibit Generalised Pareto behaviour. This result was first

22