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Fat‐tailed distributions have been found in many financial and economic variables ranging from forecasting returns on financial assets to modeling recovery distributions in bankruptcies. They have also been found in numerous insurance applications such as catastrophic insurance claims and in value‐at‐risk measures employed by risk managers. Financial applications include:

Popular understandings of the financial crisis tend to focus on the rents extracted by elite personnel in the financial sector. Professional discussions, however, have addressed the faulty assumptions underlying theory and practice – in particular, the assumption that returns to financial assets follow the Gaussian distribution, in the face of much empirical evidence that these have power law distributions with far higher kurtosis. It turns out that the power law tails of returns to financial assets are also a feature of the distribution of company rates of profit, a discovery that stems from proposals to ‘dissolve’ the traditional transformation problem by abandoning the condition of a uniform rate of profit and instead considering its distribution.Marx himself was aware of the importance of considering the distributional properties of economic variables, based on his reading of Quetelet. In fact, heavy-tailed distributions characterise a wide range of variables in capitalist economies, the best-known probably being the Paretian tail component in distributions of income and wealth. Nor is this simply an empirical fact – such distributions emerge readily from a range of agent-based simulations.Capitalist economies are, in a particular technical sense, complex self-organising systems perpetually on the brink of crisis. This modern understanding is prefigured in Marx’s discussion of how the compulsive character of social relations emerges from the atomistic exercise of human free will in commercial society. The developing literature of probabilistic Marxism successfully applies these insights to the wider fields of econophysics and complexity, demonstrating the continuing relevance of Marx’s thought.

The last few years in the financial markets have shown great instability and high volatility. In order to capture the amount of risk a financial firm takes on in a single trading day, risk managers use a technology known as value‐at‐risk (VaR). There are many methodologies available to calculate VaR, and each has its limitations. Many past methods have included a normality assumption, which can often produce misleading figures as most financial returns are characterized by skewness (asymmetry) and leptokurtosis (fat‐tails). The purpose of this paper is to provide an overview of VaR and describe some of the most recent computational approaches.

This paper compares the Student‐t, autoregressive conditional heteroskedastic (ARCH) family of models, and extreme value theory (EVT) as a means of capturing the fat‐tailed nature of a returns distribution.

Recent research has utilized the third and fourth moments to estimate the shape index parameter of the tail. Other approaches, such as extreme value theory, focus on the extreme values to calculate the tail ends of a distribution. By highlighting benefits and limitations of the Student‐t, autoregressive conditional heteroskedastic (ARCH) family of models, and the extreme value theory, one can see that there is no one particular model that is best for computing VaR (although all of the models have proven to capture the fat‐tailed nature better than a normal distribution).

This paper details the basic advantages, disadvantages, and mathematics of current parametric methodologies used to assess value‐at‐risk (VaR), since accurate VaR measures reduce a firm's capital requirement and reassure creditors and investors of the firm's risk level.

Previous research suggests that monthly commodity futures returns are like equity returns and recommend long-only portfolio positions. A follow-up question is whether the distributions of daily returns on commodity futures are fat-tailed, just like equity returns. This question has important implication for commodity futures traders because futures trade positions are marked to the market daily. The Extreme Value Theory (EVT) is used to test whether the distributions of the commodity futures returns are fat-tailed with finite variance. The results suggest that not all commodity futures returns have a fat-tail distribution and the tails of the distributions of commodity futures returns generally are smaller than the tails of the distribution of equity returns.

In the real world, the variance of portfolio returns provides only a limited quantification of incurred risks, as the distributions of returns have “fat tails” and the dependence between assets are only imperfectly accounted for by the correlation matrix. Value‐at‐risk and other measures of risks have been developed to account for the larger moves allowed by non‐Gaussian distributions. In this article, the authors distinguish “small” risks from “large” risks, in order to suggest an alternative approach to portfolio optimization that simultaneously increases portfolio returns while minimizing the risk of low frequency, high severity events. This approach treats the variance or second‐order cumulant as a measure of “small” risks. In contrast, higher even‐order cumulants, starting with the fourth‐order cumulant, quantify the “large” risks. The authors employ these estimates of portfolio cumulants based on fat‐tailed distributions to rebalance portfolio exposures to mitigate large risks.

This study examines problems that may occur when conventional Value‐at‐Risk (VaR) estimators are used to quantify market risks in an agricultural context. For example, standard VaR methods, such as the variance‐covariance method or historical simulation, can fail when the return distribution is fat tailed. This problem is aggravated when long‐term VaR forecasts are desired. Extreme Value Theory (EVT) is proposed to overcome these problems. The application of EVT is illustrated by an example from the German hog market. Multi‐period VaR forecasts derived by EVT are found to deviate considerably from standard forecasts. We conclude that EVT is a useful complement to traditional VaR methods.

The purpose of this paper is to find and describe some stylized facts for foreign exchange and stock market returns, which are explored using statistical methods.

Formal statistics for testing presence of autocorrelation, asymmetry, and other deviations from normality are applied. Dynamic correlations and different kernel estimations and approximations to the empirical distributions are also under scrutiny. Furthermore, dynamic analysis of mean, standard deviation, skewness and kurtosis are also performed to evaluate time‐varying properties in return distributions.

The findings include: different types of non‐normality in both markets, fat tails, excess furtosis, return clustering and unconditional time‐varying moments. Identifiable volatility cycles in both forex and stock markets are associated to common macro financial uncertainty events.

The paper is the first work of this type in Peru.

This paper aims to evaluate the accuracy of a quantile estimate. Especially when estimating high quantiles from a few data, the quantile estimator itself is a random number with its own distribution. This distribution is first determined and then it is shown how the accuracy of the quantile estimation can be assessed in practice.

The paper considers the situation that the parent distribution of the data is unknown, the tail is modeled with the generalized pareto distribution and the quantile is finally estimated using the fitted tail model. Based on well-known theoretical preliminary studies, the finite sample distribution of the quantile estimator is determined and the accuracy of the estimator is quantified.

In general, the algebraic representation of the finite sample distribution of the quantile estimator was found. With the distribution, all statistical quantities can be determined. In particular, the expected value, the variance and the bias of the quantile estimator are calculated to evaluate the accuracy of the estimation process. Scaling laws could be derived and it turns out that with a fat tail and few data, the bias and the variance increase massively.

Currently, the research is limited to the form of the tail, which is interesting for the financial sector. Future research might consider problems where the tail has a finite support or the tail is over-fat.

The ability to calculate error bands and the bias for the quantile estimator is equally important for financial institutions, as well as regulators and auditors.

Understanding the quantile estimator as a random variable and analyzing and evaluating it based on its distribution gives researchers, regulators, auditors and practitioners new opportunities to assess risk.

The purpose of this paper is to explore the extent of the so-called “small-sample problem” within quantitative exchange-rate risk management.

The authors take a closer look at the frequency distribution of nominal price changes in the European foreign exchange markets.

The analysis clearly illustrates the risk of seriously underestimating the probability and magnitude of tail events when frequency distributions are derived from fairly short data samples.

The authors suggest that financial institutions and regulators should have an eye for the long-term historical perspective when designing sensitivity tests or “worst case” scenarios in relation to risk assessments and stress tests.

The authors add to the literature by analysing the distribution of nominal exchange-rate fluctuations on the basis of a unique quarterly data set for ten European exchange-rate pairs covering a time span of 273 years constructed by the authors. To the best of the authors' knowledge this is the first study on nominal exchange-rate changes for a large number of exchange-rate pairs based on quarterly data spanning almost three centuries.

The investigation of the fractal nature of financial data has been growing in the literature. The purpose is to investigate the multifractal behavior of frontier markets using multifractal detrended fluctuation analysis (MFDFA).

This study used daily closing prices of nine frontier stock markets up to 31-Aug-2020. A preliminary analysis reveals that these markets exhibit fat tails and clustering patterns. For a more robust analysis, a combination of Seasonal and Trend Decomposition using Loess (STL) and MFDFA has been employed. The former method is used to decompose daily stock returns, where later detected the long rang dependence in the series.

The results confirm varying degree of multifractality in frontier stock markets, implying that they exhibit long-range dependence. Based on these multifractality levels, Serbian and Romanian stock markets are the ones exhibiting least long-range dependence, while Slovenian and Mauritius stock markets indicating highest dependence in their series. Furthermore, the markets of Kenya, Morocco, Romania and Serbia exhibit mean reversion (anti-persistent) behavior while the remaining frontier markets show persistent behaviors.

The information given by the detection of the fractal measure of data can support for investment and policymaking decisions.

Frontier markets are of great potential from the perspective of international diversification. However, most of the research focused on other emerging and developed markets, especially in the context of multifractal analysis. This study combines the STL method and a physics-based robust technique, MFDFA to detect the multifractal behavior of frontier stock markets.