Universität zu Köln
Wirtschafts- und Sozialwissenschaftliche Fakultät

Diplomarbeit im Wahlpflichtfach Statistik
Vorgelegt in der Diplomprüfung im Studiengang Betriebswirtschaft

Modelling extremal stock returns in a stable Paretian environment

Köln 2003

vorgelegt von

Hendrik Kohleick

Table of Contents

List of Tables ... IV
List of Figures ... V

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 (1975) ... 41
4.3.4 Modifications of the Hill estimator ... 44
4.3.5 Generalisations of the Hill estimator ... 49
4.3.6 Pickands estimator (1975) ... 54
4.3.7 Generalisations of the Pickands estimator ... 56
4.3.8 De Haan-Resnick estimator (1980) ... 57
4.3.9 De Haan-Pereira estimator (1999) ... 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 (1999) ... 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 (1971) ... 69
4.7.2 McCulloch estimator (1986) ... 70
4.8 Other approaches ... 71
4.9 Which estimator is the best one? ... 72

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 ... VI
List of Abbreviations ... XI
List of Symbols ... XII

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 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).
• Letting the tails speak for themselves. In this case, tail inference is made with-out modelling the centre of the distribution. Tail index estimation is based on extremal returns only. This method is based on Extreme Value Theory (EVT).

This paper is going to discuss both approaches, the stable Paretian or -stable distribution serving as a conceptual framework for the analysis. Tail inference is essentially focused on estimation of the index , which determines the shape of the tails.
Two questions shall be answered:

• Provided that stock returns actually follow a stable Paretian distribution, what is the best estimator for the index ?
• Given that deviations from the stable Paretian model have frequently been observed, how can we make inference about the tail shape (and obtain an accurate estimate of ) even if the stable model does not hold exactly?

It shall be found later that the EVT-based approach plays a crucial role in identifying suitable estimators of the tail index under relaxed distributional assumptions.

Structure of remainder of text. The remainder of this paper is structured as follows:

In section 1.2, some stylised facts of empirical stock return distributions are described. Section 2.1 analyses how these empirical characteristics have been captured in different return distribution models, following a historical timeline. The family of stable laws is introduced, along with theoretical and empirical findings on the goodness-of-fit.

Section 2.2 then shifts over to extremal returns. A short introduction to different ways of making inference about extremal returns is given, focused on parametric models and models based on EVT (which are most relevant here).

Section 3 provides the technical framework for the estimation of the tail index. Whilst section 3.1 gives necessary basics of EVT, section 3.2 introduces relevant technical background on the stable Paretian model.

Sections 4 and 5 are the central parts of this paper. Section 4 aims to give a com-prehensive overview of estimators of the tail index, based on the definition of desirable properties (4.1) and estimation methodologies and prerequisites (4.2). Common estim-ators are described and evaluated; previous theoretical and practical evidence on the performance provided (4.3-4.8). To conclude and summarise the theoretical part, the question whether there is a ‘best’ estimator of the tail index is addressed in paragraph 4.9.

Section 5 provides results of empirical studies conducted. In the simulation part (5.1), the focus is on evaluation of estimator performance when sampling from a known distribution with given tail index, whilst paragraph 5.2 contains applications of estimators to empirical data. Here, the aim will be to give an indication of how fat-tailed empirical stock return distributions actually are – broken down into upper and lower tails as well as daily and weekly returns.

Section 6 concludes by summarising the main findings and providing a future outlook.

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).

[...]