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It is well known that the probability distribution of stock returns is non-Gaussian. The tails of the distribution are too “fat,” meaning that extreme price movements, such as stock market crashes, occur more often than predicted given a Gaussian model. Numerous studies have attempted to characterize and explain the fat-tailed property of returns. This is because understanding the probability of extreme price movements is important for risk management and option pricing. In spite of this work, there is still no accepted theoretical explanation. In this chapter, we use a large collection of data from three different stock markets to show that slow fluctuations in the volatility (i.e., the size of return increments), coupled with a Gaussian random process, produce the non-Gaussian and stable shape of the return distribution. Furthermore, because the statistical features of volatility are similar across stocks, we show that their return distributions collapse onto one universal curve. Volatility fluctuations influence the pricing of derivative instruments, and we discuss the implications of our findings for the pricing of options.

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.

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 investigate the effect of non-normality in returns and market capitalization of stock portfolios and stock indices on value at risk and conditional VaR estimation. It is a well-documented fact that returns of stocks and stock indices are not normally distributed, as Indian financial markets are more prone to shocks caused by regulatory changes, exchange rate fluctuations, financial instability, political uncertainty and inadequate economic reforms. Further, the relationship of liquidity represented by volume traded of stocks and the market risk calculated by VaR of the firms is studied.

In this paper, VaR is estimated by fitting empirical distribution of returns, parametric method and by using GARCH(1,1) with Student’s t innovation method.

It is observed that both the stocks, stock indices and their residuals exhibit non-normality; therefore, conventional methods of VaR calculation are not accurate in real word situation. It is observed that parametric method of VaR calculation is underestimating VaR and CVaR but, VaR estimated by fitting empirical distribution of return and finding out 1-a percentile is giving better results as non-normality in returns is considered. The distributions fitted by the return series are following Logistic, Weibull and Laplace. It is also observed that VaR violations are increasing with decreasing market capitalization. Therefore, we can say that market capitalization also affects accurate VaR calculation. Further, the relationship of liquidity represented by volume traded of stocks and the market risk calculated by VaR of the firms is studied. It is observed that the decrease in liquidity increases the value at risk of the firms.

This methodology can further be extended to other assets’ VaR calculation like foreign exchange rates, commodities and bank loan portfolios, etc.

This finding can help risk managers and mutual fund managers (as they have portfolios of different assets size) in estimating VaR of portfolios with non-normal returns and different market capitalization with precision. VaR is used as tool in setting trading limits at trading desks. Therefore, if VaR is calculated which takes into account non-normality of underlying distribution of return then trading limits can be set with precision. Hence, both risk management and risk measurement through VaR can be enhanced if VaR is calculated with accuracy.

This paper is considering the joint issue of non-normality in returns and effect of market capitalization in VaR estimation.

The purpose of this paper is to provide an analysis of the dependence structure between returns from real estate investment trusts (REITS) and a stock market index. Further, the aim is to illustrate how copula approaches can be applied to model the complex dependence structure between the assets and for risk measurement of a portfolio containing investments in REIT and equity indices.

The usually suggested multivariate normal or variance‐ covariance approach is applied, as well as various copula models in order to investigate the dependence structure between returns of Australian REITS and the Australian stock market. Different models including the Gaussian, Student t, Clayton and Gumbel copula are estimated and goodness‐of‐fit tests are conducted. For the return series, both the Gaussian and a non‐parametric estimate of the distribution is applied. A risk analysis is provided based on Monte Carlo simulations for the different models. The value‐at‐risk measure is also applied for quantification of the risks for a portfolio combining investments in real estate and stock markets.

The findings suggest that the multivariate normal model is not appropriate to measure the complex dependence structure between the returns of the two asset classes. Instead, a model using non‐parametric estimates for the return series in combination with a Student t copula is clearly more suitable. It further illustrates that the usually applied variance‐covariance approach leads to a significant underestimation of the actual risk for a portfolio consisting of investments in REITS and equity indices. The nature of risk is better captured by the suggested copula models.

To the authors', knowledge, this is one of the first studies to apply and test different copula models in real estate markets. Results help international investors and portfolio managers to deepen their understanding of the dependence structure between returns from real estate and equity markets. Additionally, the results should be helpful for implementation of a more adequate risk management for portfolios containing investments in both REITS and equity indices.

Application of financial risk models in the emerging markets poses special challenges. A fundamental challenge is to accurately model the return distributions which are particularly fat tailed and skewed. Value-at-Risk (VaR) measures based on the Extreme Value Theory (EVT) have been suggested, but typically data histories are limited, making it hard to test and apply EVT. The chapter addresses issues in (i) modeling the VaR measure in the presence of structural breaks in an economy, (ii) the choice of stable innovation distribution with volatility clustering effects, (iii) modeling the tails of the empirical distribution, and (iv) fixing the cut-off point for isolating extreme observations. Pakistan offers an instructive case since its equity market exhibits high volatility and incidence of extreme returns. The recent Global Financial Crisis has been another source of extreme returns. The confluence of the two sources of volatility provides us with a rich data set to test the VaR/EVT model rigorously and examine practical challenges in its application in an emerging market.

We extend earlier work on the NoVaS transformation approach introduced by Politis (2003a, 2003b). The proposed approach is model-free and especially relevant when making forecasts in the context of model uncertainty and structural breaks. We introduce a new implied distribution in the context of NoVaS, a number of additional methods for implementing NoVaS, and we examine the relative forecasting performance of NoVaS for making volatility predictions using real and simulated time series. We pay particular attention to data-generating processes with varying coefficients and structural breaks. Our results clearly indicate that the NoVaS approach outperforms GARCH model forecasts in all cases we examined, except (as expected) when the data-generating process is itself a GARCH model.

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.

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.

The purpose of this paper is to give a review of the standard approaches to extreme value theory. Special focus on the tail of the distribution is underlined, in particular concerning the fat‐tails phenomenon typical of financial returns. The core of the work is then represented by a survey of models which try to overtake some problems in determining the right shaping of extreme financial returns distribution.

The paper attempts to give a broad view of the theory about the Tail of distribution of financial market returns, with a special focus on bond returns. The aim of the core work is to find and explore via data, the best solution in order to give a right estimate of the higher moments of the distribution and of the Tail index associated with particular tail shape.

The EVT approach to VaR has certain advantages over traditional parametric and non‐parametric approaches to VaR. Parametric approaches estimate VaR by fitting some distribution to a set of observed returns while non‐parametric estimate VaR by reading off the VaR from an appropriate histogram of returns. Results show how EVT allows to overtake the problems of underestimation of risk typical of standard VaR measures. In particular the paper compares with historical simulation. The difference is quite evident showing a consistent improvement of the risk measurement performance.

It is necessary to underline how the result in the paper relies on very specific assumptions and dataset feature. Back to drawbacks of EVT, it is very important then to remind how the dataset is usually and necessarily limited to sporadic extreme events. Moreover, there is no mathematical safety of claiming robust result in the absence of normality.