Extreme Value Theory

Extreme Value Theory (EVT) is a statistical theory that evaluates extreme events like tail events or values lying beyond the range of traditional observations. Hence, it serves the purpose of providing a modeling framework and forecasting rare occasions occurring having low probabilities with significant effects.

Financial crises or stock market crashes are assessed, and extreme risks are managed using EVT. Its other uses lie in estimating the chances of extreme events, evaluating extreme quantiles, and creating risk metrics like Expected Shortfall (ES) & Value at Risk (VaR). Moreover, financial institutions utilize it in portfolio optimization, setting appropriate capital reserves, risk management and insurance payout.

• Extreme Value Theory (EVT) is a statistical concept that centers on examining severe occurrences, such as events in the tails of a distribution or values that fall outside the typical range of observations.
• Its main objective is to offer a structured approach for modeling and predicting infrequent events that have a low likelihood of happening but can have substantial consequences.
• It uses two methods to simulate and examine extreme occurrences: Block Maxima, which assesses the highest limit in non-overlapping data blocks, and Peaks Over Threshold, which focuses on events exceeding a high limit.
• It is a statistical theory focused on analyzing extreme events, while Value At Risk (VaR) is a risk measure that estimates the maximum potential loss.

Extreme Value Theory is defined as a statistical framework for measuring the likelihood of a highly improbable event related to a random variable. Therefore, EVT tries to comprehend extreme events' characteristics and statistical properties. However, Extreme value theory, as an introduction, originates from studying the likelihood of massive floods in the 20th century, which is used in engineering, finance, environmental science, and insurance.

Besides, this theory is based on three main results: the Fisher-Tippett-Gnedenko theorem, the generalized extreme value (GEV) distribution, and the generalized Pareto distribution (GPD).

Hence, the generalized extreme value (GEV) distribution assumption forms its basis. The theory comprises data fitting into the GEV distribution and determining parameters describing the extreme value distribution's shape, location, and scale. As a result, the estimation of probabilities and extreme quantiles gets done. Furthermore, EVT has specific pros, like risk measures, anomaly detection in streams with extreme value theory, extreme value theory Python, and robust inference within limited data.

It also has specific cons; its assumption based on asymptomatic independence may not always hold, it may be challenging to estimate extreme value parameters having incomplete data, and it becomes sensitive to choosing threshold values utilized in the evaluation.

Therefore, it has various financial implications, like setting appropriate capital reserves, risk management, and portfolio optimization. However, it also helps identify and manage extreme risks like insurance estimation, market movement, and financial crisis. Moreover, it has also helped in tail risk assessment, enhancing decision-making processes, and improving risk models used in the financial industry.

Hence, extreme evaluation theory has applications in finance, where it is used to estimate the value at risk (VaR) and expected shortfall (ES) of financial portfolios.

Multiple approaches are used by Extreme Value Theory (EVT) to simulate and examine extreme occurrences. The following are two typical methods employed in EVT:

• Block Maxima focuses on assessing the highest limit inside non-overlapping data blocks. Moreover, it supposes that the maximum value related to every block follows a GEV distribution. By doing so, this theory determines those parameters that tell about the extreme value distribution,, like shape, location, and scale. Hence, such an approach finds suitability when extreme events occur in a well-defined and infrequent manner.
• Peaks Over Threshold: Under this approach, EVT concentrates on those events that may exceed a particular high limit. Therefore, it presumes that generalized Pareto distribution (GPD) is followed for all the excesses beyond the threshold. EVT calculates various elements of the GPD, such as the threshold, size, and form parameters, by modeling the extras. This strategy is helpful whenever severe occurrences are less readily apparent or increasingly prevalent.

When modeling severe events, block maxima and peaks over the threshold have advantages and disadvantages. Block maxima offer robustness, whereas peaks over the threshold enable flexibility but may obliterate essential data. Researchers frequently mix various methods depending on the data and the study issue.

Let us use a couple of examples to understand the topic.

Example #1

Assuming, Jane is a risk manager at Black Rock, an investment firm. He is responsible for assessing the potential losses associated with a portfolio of stocks. He has a historical dataset of daily returns for the portfolio over the years.

To apply EVT to estimate the VaR of the portfolio, He follows these steps:

1. Data Preparation: Gather the daily return data for the portfolio, ensuring it covers a sufficiently long period and is reliable.
2. Select a Threshold: Determine an appropriate threshold that defines extreme returns. The choice of entry depends on the risk appetite and specific requirements of the investment firm. For this example, let's assume Jane chooses a threshold of -3% daily returns as the cutoff for extreme losses.
3. Peaks Over Threshold (POT) Approach: He identifies the exceedances below the chosen threshold (-3%) in the dataset. These are the extreme loss events that will be analyzed using EVT.
4. Calculate VAR: Use the estimated GPD parameters to evaluate the VaR at a desired confidence level. Hence, this VaR represents the maximum potential loss of the portfolio within a specified time horizon and confidence level.

Thus, by applying EVT to estimate the VaR of the portfolio, Jane gets an insight into the potential losses that could be incurred in extreme market conditions.

Example #2

Suppose, In the Enigma market, Leo Burnett offers applications for evaluating exceptional occurrences. Moreover, it may be used to model the behavior of unusual and spectacular market occurrences, including quantum market crashes or enormous value swings. Therefore, the company assists in evaluating the likelihood of exceptional market fluctuations and allows the creation of risk mitigation strategies, such as tail-risk insurance and strain testing of investment portfolios, by adjusting extraordinary value distributions to historic Enigma market data.

EVT and VaR have the potential to be used in conjunction with risk management; they are not complementary. VaR offers a precise measurement of prospective losses, whereas EVT aids with comprehension of the tail behavior of severe occurrences. However, let's use the table below to understand the differences between the two in detail:

This article has been a guide to what is Extreme Value Theory. Here, we explain its examples, comparison with value at risk, and approaches. You may also find some useful articles here -

• Modern Portfolio Theory
• Expected Utility Theory
• Arbitrage Pricing Theory