Extreme returns and tail modelling of the S&P 500 index for the US equity market

October 20, 2023 by Shengyu ZHENG

Extreme returns and tail modelling of the S&P 500 index for the US equity market

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 S&P 500 index for the US equity market and explains how extreme value theory can be used to model the tails of its distribution.

The S&P 500 index for the US equity market

The S&P 500, or the Standard & Poor’s 500, is a renowned stock market index encompassing 500 of the largest publicly traded companies in the United States. These companies are selected based on factors like market capitalization and sector representation, making the index a diversified and reliable reflection of the U.S. stock market. It is a market capitalization-weighted index, where companies with larger market capitalization represent a greater influence on their performance. The S&P 500 is widely used as a benchmark to assess the health and trends of the U.S. economy and as a performance reference for individual stocks and investment products, including exchange-traded funds (ETF) and index funds. Its historical significance, economic indicator status, and global impact contribute to its status as a critical barometer of market conditions and overall economic health.

Characterized by its diversification and broad sector representation, the S&P 500 remains an essential tool for investors, policymakers, and economists to analyze market dynamics. This index’s performance, affected by economic data, geopolitical events, corporate earnings, and market sentiment, can provide valuable insights into the state of the U.S. stock market and the broader economy. Its rebalancing ensures that it remains current and representative of the ever-evolving landscape of American corporations. Overall, the S&P 500 plays a central role in shaping investment decisions and assessing the performance of the U.S. economy.

In this article, we focus on the S&P 500 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. We can observe the overall increase with remarkable drops during the covid crisis (2020) and the Russian invasion in Ukraine (2022).

Figure 1 below gives the evolution of the S&P 500 index from April 1, 2015 to April 1, 2023 on a daily basis.

Figure 1. 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 S&P 500 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 S&P 500 index logarithmic returns.

Source: computation by the author (data: Yahoo! Finance website).

Summary statistics for the S&P 500 index

Table 1 below presents the summary statistics estimated for the S&P 500 index:

Table 1. Summary statistics for the S&P 500 index.
summary statistics of the S&P 500 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 S&P 500 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 S&P 500 index over the period from April 1, 2015 to April 1, 2023.

Table 2. Top 10 negative daily returns for the S&P 500 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 S&P 500 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 S&P 500 index:

Table 4. Estimate of the parameters of the GPD for negative daily returns for the S&P 500 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 S&P 500 index.

Source: computation by the author (data: Yahoo! Finance website).

Figure 3. GPD for the left tail of the S&P 500 index returns.

Source: computation by the author (data: Yahoo! Finance website).

Figure 4. GPD for the right tail of the S&P 500 index returns.

Source: computation by the author (data: Yahoo! Finance website).

Applications in risk management

Extreme Value Theory (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?

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 S&P 500 index.

Useful resources

Academic resources

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

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 business of financial indexes

March 18, 2023 by Nithisha CHALLA

The business of financial indexes

In this article, Nithisha CHALLA (ESSEC Business School, Grande Ecole Program – Master in Management, 2021-2023) explains the business of financial indexes.

Introduction

Indexes are frequently used in the financial sector to measure the evolution of market prices for a set of financial assets over time. These sets of assets can be defined to represent an asset class, country or geographical zone, or sector of the economy, and provide a comprehensive and accurate overview of the market.

Financial indexes serve as a benchmark for assessing the performance of an investor’s asset portfolio and give investors a way to monitor the performance of a given set of assets. By using financial indexes, investors can gain knowledge of market trends and conditions and make informed investment decisions. Index providers are responsible for creating and maintaining financial indexes.

Financial indexes can be developed to track particular geographical areas or market segments and can be created for a variety of asset classes, including equities, bonds, commodities, and currencies. Financial indexes are primarily provided by specialized companies with experience in data compilation and index value calculation, such as S&P Dow Jones Indices, MSCI, and FTSE Russell. Overall, the business of financial indexes is a critical component of the financial industry, providing valuable data and insights to investors.

Key Players

Index providers

An index provider is a specialized business that specializes in developing and computing market indices as well as licensing its intellectual property to be used as the foundation of passive products. The index providers are essential to the investment professionals in charge of looking after these assets because they provide reliable data distribution, sound index construction, and strict index maintenance. The primary activities of an index provider are product development, licensing, distribution, and related service and support.

Index Industry Association (IIA)

The production of indexes has become an industry! And every industry has a professional association. The index industry is no exception. The Index Industry Association (IIA) was founded in 2012. Some of the founding members are MSCI and S&P Dow Jones Indexes.

As stated on the IIA website, the association mandate is “to educate investors on the attributes and role of indexes within the investment process, to advocate for the interests of index users and providers worldwide, and to push for industry standards of best practice, independence and transparency”.

Business models

Index providers typically employ one of the following business models to make money from their indexes: licensing, creating index-linked products, getting charged for index inclusion, and selling data for index-related research and analysis.

Licensing

Index providers make money by licensing financial institutions like asset managers, banks, and insurance companies to use their indexes. These financial institutions pay a fee to the index provider for the right to use the indexes as a benchmark for their investment products, such as exchange-traded funds (ETFs) and index funds.

Creation of index-linked products

Index providers make money by developing their own index-linked products, such as index funds and ETFs. The investors that are invested in the product pay a management fee to the index provider.

Selling data

By selling the data that has been produced from the history, research, and analysis, the index providers make money.

Regulation of indexes

Index providers build and maintain indexes. In order to ensure that the index accurately reflects the performance of the market or sector it is meant to represent, they are in charge of defining the methodology used to construct the index, choosing the stocks or bonds included in the index, and performing routine index rebalancing.

Beyond the activity of index providers, financial authorities play a role to authorize indexes. The main objective of authorization is to safeguard investors who use the index as a benchmark for their investment decisions and to make sure that the index accurately reflects the performance of the market or sector it is meant to represent. In the United States, the US Securities and Exchange Commission (SEC) has the power to approve specific indexes that serve as the foundation for exchange-traded funds (ETFs) and other investment products. This is done to make sure that these products operate in the best interests of investors and are compliant with SEC regulations.

Why should I be interested in this post?

A wide range of professionals, including portfolio managers, investment advisors, and financial analysts, use financial indexes, which are a crucial part of the financial sector. Financial indexes change over time to take into account adjustments to the economy and market conditions.

You can stay on top of the curve and adjust to changes in the industry by staying informed of the most recent financial index developments. So, in my opinion, studying the business of financial indexes can give business students useful skills and knowledge that they can use in a variety of fields and jobs.

Useful resources

Index Industry Association (IIA)

S&P Global Who’s Behind the Index?

Committee for Economic Development of The Conference Board (CED) The Financial Index Industry

K&L Gates SEC solicits comments on whether index providers, model portfolio providers, and pricing services are investment advisers: seeking a problem for a “solution”

About the author

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

The S&P 500 index

January 17, 2025March 18, 2023 by Nithisha CHALLA

The S&P 500 index

In this article, Nithisha CHALLA (ESSEC Business School, Grande Ecole Program – Master in Management, 2021-2023) presents the S&P 500 index and details its characteristics.

The S&P 500 index

The performance of 500 major capital companies listed on the US stock exchange is summarized by a financial index called the S&P 500 index. The stocks of the S&P 500 index are traded on the New York Stock Exchange and NASDAQ, which are the two major stock exchanges in the United States of America. This index serves as a benchmark for the American stock market and investors use it to monitor the performance of the market. The selection of 500 stocks only is deemed enough to represent the stock market (in terms of market capitalization).

The S&P 500 index was first established by Standard & Poor’s, a provider of financial services, on March 4, 1957. In order to provide a comprehensive assessment of the U.S. stock market, the index consists of a range of large-capital businesses from various industries and sectors. The S&P 500 index is currently managed by the index provider S&P Dow Jones Indices (a division of S&P Global).

Who makes the shortlist of the index and how the field is narrowed down? The S&P Dow Jones Indices oversees the selection procedure for index inclusion. The public float, financial viability, market capitalization, and a diverse representation of the US stock market—including technology, healthcare, financials, consumer goods, etc.—are some of the key criteria used to define the composition of the index.

How is the S&P 500 index represented in trading platforms and financial websites? The ticker symbol used in the financial industry for the S&P 500 index is “SPX”.

Table 1 gives the Top 10 stocks in the S&P 500 index in terms of market capitalization as of January 31, 2023.

Table 1. Top 10 stocks in the S&P 500 index.

Source: computation by the author (data: YahooFinance! financial website).

Table 2 gives the sector representation of the S&P 500 index in terms of number of stocks and market capitalization as of January 31, 2023.

Table 2. Sector representation in the S&P 500 index.

Source: computation by the author (data: YahooFinance! financial website).

Calculation of the S&P 500 index value

The S&P 500 index is a value-weighted index (also called a market-capitalization- weighted index). This means the larger companies have a greater impact on the index than the smaller companies.

At the end of each trading day the value of the S&P 500 index is determined in real-time and can be used as a benchmark for the performance of the index’s constituent companies’ current market prices.

The formula to compute the S&P 500 index is given by

SP500 Index value

where I is the index value, k a given asset, K the number of assets in the index, P^k the market price of asset k, N^k the number of issued shares for asset k, and t the time of calculation of the index.

In a S&P 500 index, the weight of asset k is given by formula can be rewritten as

SP500 Index Weight

which clearly shows that the weight of each asset in the index is its market capitalization of the asset divided by the sum of the market capitalizations of all assets.

The divisor, whose calculation is based on the number of shares, is typically adjusted for events such as stock splits and dividends. The divisor is used to ensure that the value of the index remains consistent over time despite changes in the number of outstanding shares.

Note that there are two versions of the S&P 500 index: one which includes the performance of the company as well as the dividends the companies pay (so it is a dividend included index), and another one which only considers the performance of the company but does not consider the dividends.

Use of the S&P 500 index in asset management

Given that the index is used for performance measuring it is widely used for constructing and analyzing investment portfolios. This index’s primary use is to create investment strategies, mitigate risk, and assess portfolio performance. Investors and asset managers utilize this index as an useful index to measure the overall performance of the market.

Benchmark for equity funds

There are a number of indices used as a benchmark for equity funds but the S&P 500 index particularly focuses on the large capped companies in the US market. It is mainly differentiated by the asset class the index is focusing on and the investment strategies followed by the companies. For Example: DJIA uses price weighted stock strategy for the top 30 blue chip companies, whereas the NASDAQ Composite Index uses market capitalization-weighted index of more than 3,000 stocks in the NASDAQ Composite.

Financial products around the S&P 500 index

There are a number of financial products that either provide exposure to the index or use information from the index. Not just the index funds but there are numerous ETFs and specific sector related indices which provide exposure to the S&P 500 index. Other financial products would be mutual funds, futures and options etc.

Historical data for the S&P 500 index

How to get the data?

The S&P 500 index is the most common index used in finance, and historical data for the S&P 500 index can be easily downloaded from the internet.

For example, you can download historical data for the S&P 500 index from December 30, 1927 on Yahoo! Finance (the Yahoo! code for S&P 500 index is ^GSPC).

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 S&P 500 index.

Data file

The R program that you can download above allows you to download the data for the S&P 500 index from the Yahoo! Finance website. The database starts on December 30, 1927. It also computes the returns (logarithmic returns) from closing prices.

Table 3 below represents the top of the data file for the S&P 500 index downloaded from the Yahoo! Finance website with the R program.

Table 3. Top of the data file for the S&P 500 index.
Top of the file for the S&P 500 index data
Source: computation by the author (data: Yahoo! Finance website).

Summary statistics for the S&P 500 index

The R program that you can download above also allows you to compute summary statistics about the returns of the S&P 500 index.

Table 4 below presents the following summary statistics estimated for the S&P 500 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 S&P 500 index.

Source: computation by the author (data: Yahoo! Finance website).

Evolution of the S&P 500 index

Figure 1 below gives the evolution of the S&P 500 index from December 30, 1927 to December 30, 2022 on a daily basis.

Figure 1. Evolution of the S&P 500 index.

Source: computation by the author (data: Yahoo! Finance website).

Figure 2 below gives the evolution of the S&P 500 index returns from December 30, 1927 to December 30, 2022 on a daily basis.

Figure 2. Evolution of the S&P 500 index returns.

Source: computation by the author (data: Yahoo! Finance website).

Statistical distribution of the S&P 500 index returns

Historical distribution

Figure 3 represents the historical distribution of the S&P 500 index daily returns for the period from December 30, 1927 to December 30, 2022.

Figure 3. Historical distribution of the S&P 500 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 December 30, 1927 to December 30, 2022. The mean of daily returns is equal to 0.02% and the standard deviation of daily returns is equal to 1.20% (or equivalently 5.88% for the annual mean and 19.38% for the annual standard deviation as shown in Table 3 above).

Figure 4 below represents the Gaussian distribution of the S&P 500 index daily returns with parameters estimated over the period from December 30, 1927 to December 30, 2022.

Figure 4. Gaussian distribution of the S&P 500 index returns.
Gaussian distribution of the daily S&P 500 index returns
Source: computation by the author (data: Yahoo! Finance website).

Risk measures of the S&P 500 index returns

The R program that you can download above also allows you to compute risk measures about the returns of the S&P 500 index.

Table 5 below presents the following risk measures estimated for the S&P 500 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 S&P 500 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 S&P 500 index while the study of the right tail is relevant for an investor holding a short position in the S&P 500 index.

Why should I be interested in this post?

For a number of reasons, ESSEC students should learn about the S&P 500 index. The performance of 500 large-cap American companies is tracked by this stock market index, which is first and foremost well-known and respected. Gaining a deeper understanding of the US stock market and the businesses that fuel its expansion requires knowledge of the S&P 500 index. Management students can assess the performance of their own investments and those of their organization by comprehending the S&P 500 index and its components. Last but not least, a lot of businesses base their mutual funds and exchange-traded funds (ETFs) on the S&P 500 index.

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.

Data: Yahoo! Finance

Yahoo! Finance

Yahoo! Finance Historical data for the S&P 500 index

Data: Bloomberg

Bloomberg

Bloomberg Data for the S&P 500 index

About the author

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

Extreme returns and tail modelling of the S&P 500 index for the US equity market

The S&P 500 index for the US equity market

Summary statistics for the S&P 500 index

Modelling of the tails

Applications in risk management

Why should I be interested in this post?

Download R file to model extreme behavior of the index

Related posts on the SimTrade blog

About financial indexes

About portfolio management

About statistics

Useful resources

Academic resources

Other resources

About the author

The business of financial indexes

Introduction

Key Players

Index providers

Index Industry Association (IIA)

Business models

Licensing

Creation of index-linked products

Selling data

Regulation of indexes

Why should I be interested in this post?

Related posts on the SimTrade blog

About financial indexes

Examples of financial indexes

Useful resources

About the author

The S&P 500 index

The S&P 500 index

Calculation of the S&P 500 index value

Use of the S&P 500 index in asset management

Benchmark for equity funds

Financial products around the S&P 500 index

Historical data for the S&P 500 index

How to get the data?

R program

Data file

Summary statistics for the S&P 500 index

Evolution of the S&P 500 index

Statistical distribution of the S&P 500 index returns

Historical distribution

Gaussian distribution

Risk measures of the S&P 500 index returns

Why should I be interested in this post?

Related posts on the SimTrade blog

About financial indexes

About other US financial indexes

About portfolio management

About statistics

Useful resources

Academic research about risk

Data: Yahoo! Finance

Data: Bloomberg

About the author