Euro Stoxx 50 index

Extreme returns and tail modelling of the EURO STOXX 50 index for the European equity market

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Extreme returns and tail modelling of the EURO STOXX 50 index for the European equity market

Shengyu ZHENG

In this article, Shengyu ZHENG (ESSEC Business School, Grande Ecole Program – Master in Management, 2020-2024) describes the statistical behavior of extreme returns of the EURO STOXX 50 index for the European equity market and explains how extreme value theory can be used to model the tails of its distribution.

The EURO STOXX 50 index for the European equity market

The EURO STOXX 50 index stands as a benchmark of the European equity market, comprising 50 blue-chip stocks that collectively reflect the performance and market capitalization of leading companies across the Eurozone. Methodically constructed to represent diverse sectors, the index encapsulates the economic dynamics of 11 Eurozone nations. Established by STOXX Ltd., a subsidiary of Deutsche Börse Group, the selection of constituent stocks is governed by stringent criteria, including liquidity, free-float market capitalization, and sector representation. The objective is to provide investors with a comprehensive and representative gauge of the Eurozone’s equity markets.

The construction of the EURO STOXX 50 is rooted in a transparent and rules-based methodology. Component weights are determined by free-float market capitalization, a methodology designed to consider only the tradable shares of each company. This ensures that the index accurately reflects the economic significance of each constituent while preventing undue influence from large, non-tradable share blocks. Furthermore, the index is regularly reviewed and adjusted to accommodate changes in the market landscape, such as corporate actions, ensuring its relevance and accuracy in reflecting the dynamics of the Eurozone equities.

From an application standpoint, the EURO STOXX 50 serves as a valuable tool for market participants seeking exposure to the broader European equity markets. Investors and fund managers often utilize the index as a benchmark against which to measure the performance of their portfolios, assess market trends, and make informed investment decisions. Its widespread use as an underlying asset for financial products, such as exchange-traded funds (ETFs) and derivatives, underscores its significance as a reliable barometer of the Eurozone’s economic health and a foundational element in the global financial landscape.

In this article, we focus on the EURO STOXX 50 index of the timeframe from April 1st, 2015, to April 1st, 2023. Here we have a line chart depicting the evolution of the index level of this period.

Figure 1 below gives the evolution of the EURO STOXX 50 index from April 1, 2015 to April 1, 2023 on a daily basis.

Figure 1. Evolution of the EURO STOXX 50 index.
Evolution of the S&P 500 index
Source: computation by the author (data: Yahoo! Finance website).

Figure 2 below gives the evolution of the daily logarithmic returns of EURO STOXX 50 index from April 1, 2015 to April 1, 2023 on a daily basis. We observe concentration of volatility reflecting large price fluctuations in both directions (up and down movements). This alternation of periods of low and high volatility is well modeled by ARCH models.

Figure 2. Evolution of the EURO STOXX 50 index logarithmic returns.
Evolution of the S&P 500 index return
Source: computation by the author (data: Yahoo! Finance website).

Summary statistics for the EURO STOXX 50 index

Table 1 below presents the summary statistics estimated for the EURO STOXX 50 index:

Table 1. Summary statistics for the EURO STOXX 50 index.
summary statistics of the EURO STOXX 50 index returns
Source: computation by the author (data: Yahoo! Finance website).

The mean, the standard deviation / variance, the skewness, and the kurtosis refer to the first, second, third and fourth moments of statistical distribution of returns respectively. We can conclude that during this timeframe, the EURO STOXX 50 index takes on a slight upward trend, with relatively important daily deviation, negative skewness and excess of kurtosis.

Tables 2 and 3 below present the top 10 negative daily returns and top 10 positive daily returns for the index over the period from April 1, 2015 to April 1, 2023.

Table 2. Top 10 negative daily returns for the EURO STOXX 50 index.
Top 10 negative returns of the S&P 500 index
Source: computation by the author (data: Yahoo! Finance website).

Table 3. Top 10 positive daily returns for the EURO STOXX 50 index.
Top 10 positive returns of the S&P 500 index
Source: computation by the author (data: Yahoo! Finance website).

Modelling of the tails

Here the tail modelling is conducted based on the Peak-over-Threshold (POT) approach which corresponds to a Generalized Pareto Distribution (GPD). Let’s recall the theoretical background of this approach.

The POT approach takes into account all data entries above a designated high threshold u. The threshold exceedances could be fitted into a generalized Pareto distribution:

Illustration of the POT approach

An important issue for the POT-GPD approach is the threshold selection. An optimal threshold level can be derived by calibrating the tradeoff between bias and inefficiency. There exist several approaches to address this problematic, including a Monte Carlo simulation method inspired by the work of Jansen and de Vries (1991). In this article, to fit the GPD, we use the 2.5% quantile for the modelling of the negative tail and the 97.5% quantile for that of the positive tail.

Based on the POT-GPD approach with a fixed threshold selection, we arrive at the following modelling results for the GPD for negative extreme returns (Table 4) and positive extreme returns (Table 5) for the EURO STOXX 50 index:

Table 4. Estimate of the parameters of the GPD for negative daily returns for the EURO STOXX 50 index.
Modelling of Top 10 negative returns of the SX5E index
Source: computation by the author (data: Yahoo! Finance website).

Table 5. Estimate of the parameters of the GPD for positive daily returns for the EURO STOXX 50 index.
Modelling of Top 10 positive returns of the EURO STOXX 50 index
Source: computation by the author (data: Yahoo! Finance website).

Figure 3. GPD for the left tail of the EURO STOXX 50 index returns.
GPD for the left tail of the SX5E index returns
Source: computation by the author (data: Yahoo! Finance website).

Figure 4. GPD for the right tail of the EURO STOXX 50 index returns.
GPD for the right tail of the SX5E 500 index returns
Source: computation by the author (data: Yahoo! Finance website).

Applications in risk management

EVT as a statistical approach is used to analyze the tails of a distribution, focusing on extreme events or rare occurrences. EVT can be applied to various risk management techniques, including Value at Risk (VaR), Expected Shortfall (ES), and stress testing, to provide a more comprehensive understanding of extreme risks in financial markets.

Why should I be interested in this post?

The Extreme Value Theory is a useful tool to model the tails of the evolution of a financial instrument. In the ever-evolving landscape of financial markets, being able to grasp the concept of EVT presents a unique edge to students who aspire to become an investment or risk manager. It not only provides a deeper insight into the dynamics of equity markets but also equips them with a practical skill set essential for risk analysis. By exploring how EVT refines risk measures like Value at Risk (VaR) and Expected Shortfall (ES) and its role in stress testing, students gain a valuable perspective on how financial institutions navigate during extreme events. In a world where financial crises and market volatility are recurrent, this post opens the door to a powerful analytical framework that contributes to informed decisions and financial stability.

Download R file to model extreme behavior of the index

You can find below an R file (file with txt format) to study extreme returns and model the distribution tails for the EURO STOXX 50 index.

Download R file to study extreme returns and model the distribution tails for the Euro Stoxx 50 index

Related posts on the SimTrade blog

About financial indexes

   ▶ Nithisha CHALLA Financial indexes

   ▶ Nithisha CHALLA Calculation of financial indexes

   ▶ Nithisha CHALLA The Euro Stoxx 50 index

About portfolio management

   ▶ Youssef LOURAOUI Portfolio

   ▶ Jayati WALIA Returns

About statistics

   ▶ Shengyu ZHENG Moments de la distribution

   ▶ Shengyu ZHENG Mesures de risques

   ▶ Shengyu ZHENG Extreme Value Theory: the Block-Maxima approach and the Peak-Over-Threshold approach

   ▶ Gabriel FILJA Application de la théorie des valeurs extrêmes en finance de marchés

Useful resources

Academic resources

Embrechts P., C. Klüppelberg and T Mikosch (1997) Modelling Extremal Events for Insurance and Finance.

Embrechts P., R. Frey, McNeil A. J. (2022) Quantitative Risk Management, Princeton University Press.

Gumbel, E. J. (1958) Statistics of extremes. New York: Columbia University Press.

Longin F. (2016) Extreme events in finance: a handbook of extreme value theory and its applications Wiley Editions.

Other resources

Extreme Events in Finance

Chan S. Statistical tools for extreme value analysis

Rieder H. E. (2014) Extreme Value Theory: A primer (slides).

About the author

The article was written in October 2023 by Shengyu ZHENG (ESSEC Business School, Grande Ecole Program – Master in Management, 2020-2024).

The Euro Stoxx 50 index

by

The Euro Stoxx 50 index

Nithisha CHALLA

In this article, Nithisha CHALLA (ESSEC Business School, Grande Ecole Program – Master in Management, 2021-2023) presents the Euro Stoxx 50 index and details its characteristics.

The Euro Stoxx 50 index

The performance of 50 large-capital companies with headquarters in Eurozone nations is reflected in the Euro Stoxx 50 stock market index. On February 26, 1998, Stoxx Ltd., a partnership between Deutsche Börse AG, Dow Jones & Company, and SIX Group AG, launched it. Companies from a wide range of industries, including the financial, consumer goods, healthcare, and industrial sectors are all included in the index.

Stocks for the Euro Stoxx 50 index are chosen based on market capitalization, liquidity, and sector representation, among other things. Every year in September, the index’s composition is reviewed, and adjustments are made as needed to reflect the state of the market and the performance of the companies.

The free-float market-capitalization-weighted index known as the Euro Stoxx 50. This means that rather than stock price, the index weights each company according to its market capitalization. The index is made available to the investors and traders worldwide and is disseminated in real-time by several financial news outlets.

The Euro Stoxx 50 index’s ticker symbol in the financial sector is “STOXX50E.”

Table 1 below gives the Top 10 stocks in the Euro Stoxx 50 index in terms of market capitalization as of January 31, 2023.

Table 1. Top 10 stocks in the Euro Stoxx 50 index.
Top 10 stocks in the Euro Stoxx 50 index
Source: computation by the author (data: Yahoo! Finance website).

Table 2 below gives the sector representation of the Euro Stoxx 50 index in terms of number of stocks and market capitalization as of January 31, 2023.

Table 2. Sector representation in the Euro Stoxx 50 index.
Sector representation in the Euro Stoxx 50 index
Source: computation by the author (data: Yahoo! Finance website).

Calculation of the Euro Stoxx 50 index value

The performance of 50 sizable, blue-chip companies from 12 Eurozone nations, including France, Germany, Italy, and Spain, is tracked by the free-floating market-capitalization-weighted Euro Stoxx 50 index. The index, that includes a wide range of industries including financial services, energy, healthcare, consumer goods, and information technology, is intended to represent the performance of the most liquid and actively traded companies in Eurozone.

A free-float market-capitalization-weighted methodology is utilized to calculate the Euro Stoxx 50 index, which means that each company’s weight in the index is determined by its market capitalization adjusted for the shares that are actually traded in the secondary market (float).

The formula to compute the Euro Stoxx 50 index is given by

Float Adjusted Market Capitalization Index value

where I is index value, k a given asset, K the number of assets in the index, Pk the market price of asset k, Nk the number of issued shares for asset k, Fk the float factor of asset k, and t the time of calculation of the index.

In a float-adjusted market-capitalization-weighted index, the weight of asset k is given by formula can be rewritten as

Float Adjusted Market Capitalization Weighted Index Weight

Use of the Euro Stoxx 50 index in asset management

One of the significant indices in Europe, the Euro Stoxx 50 is quite famous and changes in it can have a big impact on market trends and investor sentiment. Investors and traders worldwide have access to index’s real-time values that are published and distributed by a number of financial news sources. The Euro Stoxx 50 is a crucial resource for investors putting efforts to understand the economic and political climate of the Eurozone and gain access to the equity market there. The index can be used by the asset managers as a benchmark to compare the performance of their portfolio to the overall market and to spot potential risk or opportunity areas.

Benchmark for equity funds

Investors and fund managers frequently use the Euro Stoxx 50 to track the health of the Eurozone economy and assess investment opportunities in the region. It is recognized as the top benchmark for the performance of the Eurozone equity market. It consists of businesses from a range of industries, including consumer goods, technology, and finance. The index is used by asset managers to monitor and assess performance of their portfolios in relation to the overall market.

Financial products around the Euro Stoxx 50 index

There are several financial products tracking performance of the Euro Stoxx 50 index. These products allow investors to get exposure to the European stock market.

  • ETFs are investment funds traded on stock exchanges that are designed to track the performance of an index. Several ETFs track the Euro Stoxx 50 index, such as the iShares EURO STOXX 50 UCITS and the Amundi ETF EURO STOXX 50 UCITS
  • Index funds based on the Euro Stoxx 50 index also allow investors to track performance of the index. Examples of index funds tracking Euro Stoxx 50 index include the DWS Invest Euro Stoxx 50 Fund and the BNP Paribas Easy Euro Stoxx 50 UCITS ETF.
  • Futures and options contracts based on Euro Stoxx 50 index provide investors with the ability to speculate on future performance of the index. For example, Eurex offers futures contracts based on the Euro Stoxx 50 index.
  • Certificates are investment products that allow investors to gain exposure to Euro Stoxx 50 index. Societe Generale offers range of certificates linked to the Euro Stoxx 50 index, such as the EURO STOXX 50 Tracker Certificate.

Investors and asset managers may use these financial products to gain exposure to the Euro Stoxx 50 index and manage their portfolios’ risk and return.

Historical data for the Euro Stoxx 50 index

How to get the data?

The Euro Stoxx 50 index is the most common index used in finance, and historical data for the Euro Stoxx 50 index can be easily downloaded from the internet.

For example, you can download data for the Euro Stoxx 50 index from January 3, 1984 on Yahoo! Finance (the Yahoo! code for Euro Stoxx 50 index is ^STOXX50E).

Yahoo! Finance
Source: Yahoo! Finance.

You can also download the same data from a Bloomberg terminal.

R program

The R program below written by Shengyu ZHENG allows you to download the data from Yahoo! Finance website and to compute summary statistics and risk measures about the Euro Stoxx 50 index.

Download R file

Data file

The R program that you can download above allows you to download the data for the Euro Stoxx 50 index from the Yahoo! Finance website. The database starts on January 3, 1984. It also computes the returns (logarithmic returns) from closing prices.

Table 3 below represents the top of the data file for the Euro Stoxx 50 index downloaded from the Yahoo! Finance website with the R program.

Table 3. Top of the data file for the Euro Stoxx 50 index.
Top of the file for the Euro Stoxx 50 index data
Source: computation by the author (data: Yahoo! Finance website).

Evolution of the Euro Stoxx 50 index

Figure 1 below gives the evolution of the Euro Stoxx 50 index from January 3, 1984 to December 30, 2022 on a daily basis.

Figure 1. Evolution of the Euro Stoxx 50 index.
Evolution of the Euro Stoxx 50 index
Source: computation by the author (data: Yahoo! Finance website).

Figure 2 below gives the evolution of the Euro Stoxx 50 index returns from January 3, 1984 to December 30, 2022 on a daily basis.

Figure 2. Evolution of the Euro Stoxx 50 index returns.
Evolution of the Euro Stoxx 50 index return
Source: computation by the author (data: Yahoo! Finance website).

Summary statistics for the Euro Stoxx 50 index

The R program that you can download above also allows you to compute summary statistics about the returns of the Euro Stoxx 50 index.

Table 4 below presents the following summary statistics estimated for the Euro Stoxx 50 index:

  • The mean
  • The standard deviation (the squared root of the variance)
  • The skewness
  • The kurtosis.

The mean, the standard deviation / variance, the skewness, and the kurtosis refer to the first, second, third and fourth moments of statistical distribution of returns respectively.

Table 4. Summary statistics for the Euro Stoxx 50 index.
Summary statistics for the Euro Stoxx 50 index
Source: computation by the author (data: Yahoo! Finance website).

Statistical distribution of the Euro Stoxx 50 index returns

Historical distribution

Figure 3 represents the historical distribution of the Euro Stoxx 50 index daily returns for the period from January 3, 1984 to December 30, 2022.

Figure 3. Historical distribution of the Euro Stoxx 50 index returns.
Historical distribution of the daily Euro Stoxx 50 index returns
Source: computation by the author (data: Yahoo! Finance website).

Gaussian distribution

The Gaussian distribution (also called the normal distribution) is a parametric distribution with two parameters: the mean and the standard deviation of returns. We estimated these two parameters over the period from January 3, 1984 to December 30, 2022. The mean of daily returns is equal to 0.02% and the standard deviation of daily returns is equal to 1.37% (or equivalently 3.94% for the annual mean and 28.02% for the annual standard deviation as shown in Table 3 above).

Figure 4 below represents the Gaussian distribution of the Euro Stoxx 50 index daily returns with parameters estimated over the period from January 3, 1984 to December 30, 2022.

Figure 4. Gaussian distribution of the Euro Stoxx 50 index returns.
Gaussian distribution of the daily Euro Stoxx 50 index returns
Source: computation by the author (data: Yahoo! Finance website).

Risk measures of the Euro Stoxx 50 index returns

The R program that you can download above also allows you to compute risk measures about the returns of the Euro Stoxx 50 index.

Table 5 below presents the following risk measures estimated for the Euro Stoxx 50 index:

  • The long-term volatility (the unconditional standard deviation estimated over the entire period)
  • The short-term volatility (the standard deviation estimated over the last three months)
  • The Value at Risk (VaR) for the left tail (the 5% quantile of the historical distribution)
  • The Value at Risk (VaR) for the right tail (the 95% quantile of the historical distribution)
  • The Expected Shortfall (ES) for the left tail (the average loss over the 5% quantile of the historical distribution)
  • The Expected Shortfall (ES) for the right tail (the average loss over the 95% quantile of the historical distribution)
  • The Stress Value (SV) for the left tail (the 1% quantile of the tail distribution estimated with a Generalized Pareto distribution)
  • The Stress Value (SV) for the right tail (the 99% quantile of the tail distribution estimated with a Generalized Pareto distribution)

Table 5. Risk measures for the Euro Stoxx 50 index.
Risk measures for the Euro Stoxx 50 index
Source: computation by the author (data: Yahoo! Finance website).

The volatility is a global measure of risk as it considers all the returns. The Value at Risk (VaR), Expected Shortfall (ES) and Stress Value (SV) are local measures of risk as they focus on the tails of the distribution. The study of the left tail is relevant for an investor holding a long position in the Euro Stoxx 50 index while the study of the right tail is relevant for an investor holding a short position in the Euro Stoxx 50 index.

Why should I be interested in this post?

For a number of reasons, management students (as future managers and individual investors) should learn about the Euro Stoxx 50 index. It is made up of businesses from 11 different Eurozone nations that operate in a variety of industries, including banking, technology, and healthcare. The Euro Stoxx 50 index is a key benchmark for the European equity market, which is one of the world’s largest market. Understanding how the index is constructed, how it performs, and the companies that make up the index is important for anyone studying finance or business in Europe or interested in investing in European equities. Students interested in careers in investment banking, asset management, or global business may find this information useful.

Individual investors can assess the performance of their own investments in the European equity market with the Euro Stoxx 50 index. Last but not least, a lot of asset management firms base their mutual funds and exchange-traded funds (ETFs) on the Euro Stoxx 50 index which can considered as interesting assets to diversify a portfolio. Learning about these products and their portfolio and risk management applications can be valuable for management students.

Related posts on the SimTrade blog

About financial indexes

   ▶ Nithisha CHALLA Financial indexes

   ▶ Nithisha CHALLA Calculation of financial indexes

   ▶ Nithisha CHALLA The business of financial indexes

   ▶ Nithisha CHALLA Float

Other financial indexes

   ▶ Nithisha CHALLA The S&P 500 index

   ▶ Nithisha CHALLA The FTSE 100 index

   ▶ Nithisha CHALLA The DAX 30 index

   ▶ Nithisha CHALLA The CAC 40 index

   ▶ Nithisha CHALLA The IBEX 35 index

About portfolio management

   ▶ Youssef LOURAOUI Portfolio

   ▶ Jayati WALIA Returns

About statistics

   ▶ Shengyu ZHENG Moments de la distribution

   ▶ Shengyu ZHENG Mesures de risques

Useful resources

Academic research about risk

Longin F. (2000) From VaR to stress testing: the extreme value approach Journal of Banking and Finance, N°24, pp 1097-1130.

Longin F. (2016) Extreme events in finance: a handbook of extreme value theory and its applications Wiley Editions.

Business

Wikipedia History of Euro Stoxx 50

Capital What is the Euro Stoxx Index Definition?

Deutsche Börse Xetra EURO STOXX 50® Index derivatives

Data

Yahoo! Finance

Yahoo Finance Euro Stoxx 50 index

About the author

The article was written in April 2023 by Nithisha CHALLA (ESSEC Business School, Grande Ecole Program – Master in Management, 2021-2023).