Finance experts and statisticians still have considerable difficulties to understand extremal movements in stock prices. Basically, there are two approaches to shed some light on this question: 1. Tail inference based on full parametric assumptions 2. "Letting the tails speak for themselves" This paper discusses both approaches, the stable Paretian distribution serving as a conceptual framework for the analysis.
Table of Contents
1. INTRODUCTION: THE EMPIRICAL DISTRIBUTION OF STOCK RETURNS
1.1 STOCK RETURNS, VOLATILITY, AND ASSET ALLOCATION
1.2 STYLISED FACTS OF STOCK RETURN DISTRIBUTIONS
2. MODELS FOR STOCK RETURN DISTRIBUTIONS
2.1 AN OVERVIEW OF FULL PARAMETRIC RETURN DISTRIBUTION MODELS
2.2 BASIC APPROACHES TO INFERENCE ABOUT EXTREMAL RETURNS
3. TECHNICAL BACKGROUND FOR TAIL INFERENCE
3.1 EXTREME VALUE THEORY (EVT)
3.2 THE STABLE PARETIAN MODEL
4. ESTIMATION OF THE STABLE PARETIAN INDEX α
4.1 DESIRABLE PROPERTIES OF AN ESTIMATOR AND EVALUATION CRITERIA
4.2 ESTIMATION METHODOLOGIES AND PREREQUISITES
4.3 TAIL ESTIMATORS
4.3.1 Intuition behind tail estimation
4.3.2 Where does the tail start?
4.3.3 Hill estimator (1975)
4.3.4 Modifications of the Hill estimator
4.3.5 Generalisations of the Hill estimator
4.3.6 Pickands estimator (1975)
4.3.7 Generalisations of the Pickands estimator
4.3.8 De Haan-Resnick estimator (1980)
4.3.9 De Haan-Pereira estimator (1999)
4.4 ESTIMATION VIA THE PEAKS OVER THRESHOLD (POT) METHOD
4.4.1 ML techniques
4.4.2 Method of probability-weighted moments (PWM)
4.4.3 Falk-Marohn estimator (1999)
4.4.4 Choice of the threshold level
4.5 CHARACTERISTIC FUNCTION TECHNIQUES
4.5.1 Method of Moments estimators (MME)
4.5.2 Minimum Distance estimators (MDE)
4.5.3 Regression-type estimators
4.6 MAXIMUM LIKELIHOOD ESTIMATORS (MLE)
4.6.1 Algorithms for computation of the stable Paretian PDF
4.6.2 The ML estimation procedure
4.7 QUANTILE-BASED ESTIMATORS
4.7.1 Quantile estimator by Fama and Roll (1971)
4.7.2 McCulloch estimator (1986)
4.8 OTHER APPROACHES
4.9 WHICH ESTIMATOR IS THE BEST ONE?
5. AN EMPIRICAL COMPARISON OF ESTIMATORS
5.1 SIMULATION STUDY OF TAIL ESTIMATORS
5.1.1 Performance with Pareto data and small sample properties
5.1.2 Performance with Fréchet data
5.1.3 Performance with stable data
5.2 CONSIDERATION OF MODIFIED TAIL ESTIMATORS
5.3 APPLICATION TO STOCK RETURN DATA
5.3.1 Daily stock returns
5.3.2 Weekly stock returns
6. CONCLUSION & SUMMARY
APPENDIX TO MAIN TEXT
Objectives & Core Topics
The primary objective of this thesis is to provide a rigorous analysis of the tail behavior of stock return distributions, specifically focusing on the estimation of the stable Paretian index alpha. The research investigates which estimators are best suited to assess the heaviness of tails, given the observed limitations of traditional normal distribution assumptions in financial modeling.
- Theoretical evaluation of stable Paretian distributions and Extreme Value Theory (EVT).
- Comprehensive analysis of various tail index estimation methodologies.
- Empirical simulation study to compare the performance of different estimators under known distributional assumptions.
- Application of these estimation techniques to real-world financial data, including major stock indices and blue-chip equities.
Excerpt from the Book
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).
Summary of Chapters
1. INTRODUCTION: THE EMPIRICAL DISTRIBUTION OF STOCK RETURNS: This chapter introduces the context of financial asset risk and the importance of modeling extremal events, establishing the necessity of looking beyond normal distributions.
2. MODELS FOR STOCK RETURN DISTRIBUTIONS: This section reviews parametric models used for stock returns, discusses the limitations of the normality assumption, and introduces the stable Paretian distribution as a flexible alternative.
3. TECHNICAL BACKGROUND FOR TAIL INFERENCE: This chapter establishes the theoretical framework for tail analysis, covering Extreme Value Theory (EVT) and the technical properties of the stable Paretian model.
4. ESTIMATION OF THE STABLE PARETIAN INDEX α: This core chapter provides a comprehensive overview and classification of various tail index estimation methodologies, from classic Hill estimators to PoT and ML techniques.
5. AN EMPIRICAL COMPARISON OF ESTIMATORS: This section details the simulation-based comparison of the introduced estimators and applies them to empirical daily and weekly financial market data.
6. CONCLUSION & SUMMARY: The final chapter summarizes the research findings, confirming the suitability of tail-focused approaches for financial data and highlighting the trade-offs between different estimation methods.
Keywords
Stock Returns, Volatility, Asset Allocation, Stable Paretian Distribution, Extreme Value Theory, Tail Index, Hill Estimator, Value at Risk, Leptokurtosis, Fat Tails, Estimator Performance, Bias-Variance Trade-off.
Frequently Asked Questions
What is the primary subject of this thesis?
The work primarily focuses on the statistical modeling of extremal stock returns, aiming to accurately estimate the tail index alpha in a stable Paretian environment.
What are the key thematic areas covered?
The central themes include Extreme Value Theory, various parametric and semi-parametric tail index estimators, and the application of these models to mitigate risk in financial portfolios.
What is the main research question?
The thesis asks which estimators perform best for the index alpha under stable Paretian assumptions, and how to reliably infer tail shape when this model does not hold exactly.
Which scientific methodology is utilized?
The methodology combines a literature review of existing estimators with extensive simulation studies and the empirical application of these estimators to historic log-return data.
What is addressed in the main part of the document?
The main body systematically classifies and evaluates tail index estimators, discusses the bias-variance trade-off in finite samples, and performs comparative simulations under Pareto, Fréchet, and stable data scenarios.
Which keywords best characterize this work?
Key terms include Stable Paretian, Tail Index, Extreme Value Theory, Hill Estimator, Fat Tails, and Value at Risk.
How does the author evaluate the "Hill Estimator"?
The Hill estimator is identified as the most common, but the author concludes it performs unsatisfactorily without modifications, especially when the true tail index is near 2.
Why are weekly returns compared to daily returns?
The author analyzes both frequencies to demonstrate the stylized fact that weekly returns tend to exhibit a shift towards normality, contrasting with the heavier tails found in daily data.
- Quote paper
- Hendrik Kohleick (Author), 2003, Modelling extremal stock returns in a stable Paretian environment, Munich, GRIN Verlag, https://www.grin.com/document/34926
