Narrow Results

Value-at-risk (VaR) is a crucial subject that researchers and practitioners extensively use to measure and manage uncertainty in financial markets. Although VaR is a standard risk control instrument, there are criticisms about its performance. One of these cases, which has been studied in this research, is the VaR underestimation during times of crisis. In these periods, the non-Gaussian behavior of markets intensifies, and the estimated VaRs by typical models are lower than the real values. A potential approach that can be used to describe the non-Gaussian behavior of return series is the Tsallis entropy framework and nonextensive statistical methods. This paper has used the nonextensive models for analyzing financial markets’ behavior during crisis times. By applying the q-Gaussian probability density function for emerging and mature markets over 20 years, we can see a better VaR estimation than the regular models, especially during crisis times. We have shown that the q-Gaussian models composed of VaR and Expected Shortfall (ES) estimate risk better than the standard models. By comparing the ES, VaR, $q$-VaR and $q$-ES for emerging and mature markets, we see in confidence levels more than 0.98, the outputs of q models are more real, and the $q$-ES model has lower errors than the other ones. Also, it is evident that in the mature markets, the difference of VaR between normal condition and nonextensive approach increases more than one standard deviation during times of crisis. Still, in the emerging markets, we cannot see a specific pattern. The findings of this paper are useful for analyzing the risk of financial crises in different markets.

In this paper the theory of coherent imprecise previsions is applied to risk measurement. We introduce the notion of coherent risk measure defined on an arbitrary set of risks, showing that it can be considered a special case of coherent upper prevision. We also prove that our definition generalizes the notion of coherence for risk measures defined on a linear space of random numbers, given in literature. Consistency properties of Value-at-Risk (VaR), currently one of the most used risk measures, are investigated too, showing that it does not necessarily satisfy a weaker notion of consistency called 'avoiding sure loss'. We introduce sufficient conditions for VaR to avoid sure loss and to be coherent. Finally we discuss ways of modifying incoherent risk measures into coherent ones.

In this paper, we generalize the parametric Δ-VaR method from portfolios with normally distributed risk factors to portfolios with elliptically distributed ones. We treat both the expected shortfall and the Value-at-Risk of such portfolios. Special attention is given to the particular case of a multivariate t-distribution.

The broad spectrum and the increased complexity of financial products that compose modern portfolios have forced credit and financial institutions to focus on innovative and more effective ways of estimating market risks. These new approaches, very often, prove to be more conservative compared to traditional approaches in terms of market risk quantification. On the other hand, according to the Basel Committee evaluation framework, this conservatism is rewarded with lower multiplication factors when calculations of capital requirements take place. The present study elaborates on the comparison of several Value-at-Risk (VaR) methodologies based on the capital requirements they provide according to the Basel Committee regulatory framework.

Financial risk management typically deals with low-probability events in the tails of asset return distributions. To better capture the behavior of these tails, several studies have clearly highlighted that one should rely on a methodology that directly focuses on the tails of the distribution rather than getting the tails as an outcome of modelling the entire density function. Traditional Extreme Value Theory (EVT) distributions, however, provide a good fit for the bulk of the extreme data but usually underestimate a small amount of observations considered as "outliers".

Since the main objective of risk management analysis is to estimate the size and probability of very large price movements, these "outliers" are by definition the very events we need to investigate. In this paper we suggest the use of a Two-Component Extreme Value (TCEV) distribution where a 'basic distribution' generates ordinary extremes (more frequent and less severe in the mean) while an "outlying distribution" generates rarer but severe extremes.

Goodness-of-fit tests show the superiority of this distribution to capture the extremes of eleven MSCI Indices of the Pacific-Basin region relative to traditional EVT densities. Measures of accuracy and efficiency used to assess the performance of VaR forecasts also indicate that the additional flexibility brought by the TCEV model provides strong improvements for risk management.

We examine the relative risk performance of the Dow Jones Islamic Index (DJIS) and find that the index outperforms the Dow Jones (DJIM) WORLD Index in terms of risk. Using the most recent Value-at-Risk (VaR) methodologies (RiskMetrics, Student-t APARCH, and skewed Student-t APARCH) on the 1996–2005 period, and assuming one-day holding period for both indices with a moving window of 500 day data, we show that the value of VaR is greater for DJIM WORLD than for DJIS Islamic. We interpret the results mainly to the profit-and-loss sharing principle of Islamic finance where banks share the profits and bear losses (Mudarabah) or share both profits and losses (Musharaka) with the firm.

We compare in a backtesting study the performance of univariate models for Value-at-Risk (VaR) and expected shortfall based on stable laws and on extreme value theory (EVT). Analyzing these different approaches, we test whether the sum–stability assumption or the max–stability assumption, that respectively imply α–stable laws and Generalized Extreme Value (GEV) distributions, is more suitable for risk management based on VaR and expected shortfall. Our numerical results indicate that α–stable models tend to outperform pure EVT-based methods (especially those obtained by the so-called block maxima method) in the estimation of Value-at-Risk, while a peaks-over-threshold method turns out to be preferable for the estimation of expected shortfall. We also find empirical evidence that some simple semiparametric EVT-based methods perform well in the estimation of VaR.

The fluctuation of shipping freight rates (freight rate risk) is an important source of market risk for all participants in the freight markets including hedge funds, commodity and energy producers. We measure the freight rate risk by the Value-at-Risk (VaR) approach. A range of parametric and non-parametric VaR methods is applied to various popular freight markets for dry and wet cargoes. Backtesting is conducted in two stages by means of statistical tests and a subjective loss function that uses the Expected Shortfall, respectively. We find that the simplest non-parametric methods should be used to measure freight rate risk. In addition, freight rate risk is greater in the wet cargoes markets. The margins in the growing freight derivatives markets should be set accordingly.

In this paper, we study the Kelly criterion in the continuous time framework building on the work of E.O. Thorp and others. The existence of an optimal strategy is proven in a general setting and the corresponding optimal wealth process is found. A simple formula is provided for calculating the optimal portfolio in terms of drift, short term risk-free rate and correlations for a set of generic multi-dimensional diffusion processes satisfying some simple conditions. Properties of the optimal investment strategy are studied. The paper ends with a short discussion of the implications of these ideas for financial markets.

It has been empirically verified that the strength of dependence in stock markets usually rises with volatility. In this paper we exploit this stylized fact combined with local maximum likelihood estimation of copula models to analyze the dynamic joint behavior of series of financial log returns. Explanatory variables based on the estimated GARCH volatilities are considered as potential regressors for explaining the dynamics in the copula parameters. The proposed model can assess and discriminate how much of the strength of dependence is due just to the time-varying volatility. The final local-parametric estimates may be used to compute risk measures, to simulate portfolio behavior, and so on. We illustrate our methods using two American indexes. Results indicate that volatility does affect the strength of dependence. The in-sample Value-at-Risk based on the dynamic model outperforms those based on the empirical estimates.

I show that the structure of the firm is not neutral with respect to regulatory capital budgeted under rules which are based on the Value-at-Risk. Indeed, when a holding company has the liberty to divide its risk into as many subsidiaries as needed, and when the subsidiaries are subject to capital requirements according to the Value-at-Risk budgeting rule, then there is an optimal way to divide risk which is such that the total amount of capital to be budgeted by the shareholder is zero. This result may lead to regulatory arbitrage by some firms.

Over the past two decades Value-at-Risk (VaR) became arguably the most popular measure of financial risk. Major banks calculate VaR on daily basis in order to determine the amount of capital a bank needs to offset the market risk. Banks use calculation methods of their choice, and many estimations are based on the assumption that portfolio rates of return have normal distribution. The important question is whether the chosen method of VaR calculation is accurate. As the light-tail property of the normal distribution can cause significant underestimation of VaR, the Basel Committee suggested to calculate the amount of capital needed by multiplying the bank's internal estimate of VaR by the factor 3. It's also common to use the so-called "square root of time" rule when evaluating VaR over a longer time horizon. This article aims to refine Stahl's argument behind the "factor 3" rule and say a word of caution concerning the "square root of time" rule.

For purposes of Value-at-Risk estimation, we consider several multivariate families of heavy-tailed distributions, which can be seen as multidimensional versions of Paretian stable and Student's t distributions allowing different marginals to have different indices of tail thickness. After a discussion of relevant estimation and simulation issues, we conduct a backtesting study on a set of portfolios containing derivative instruments, using historical US stock price data.

We study a dynamic mean-variance portfolio selection problem subject to possible limit of market risk. Three measures of market risk are considered: value-at-risk, expected shortfall, and median shortfall. They are all calculated in a dynamic consistent sense. After applying the technique of delta-normal approximation, we can explicitly solve for the optimal solution and calculate the economic loss brought by the risk budget constraint. With the analytical results obtained, influential factors of economic loss are then explored by which some guidelines on trading practice are proposed. The guidelines are independent of risk measures, and are valuable to both institutions and regulators, for they suggest that an institutional investor would spontaneously obey good investment discipline to avoid potential impact of risk constraint. This result meets the purpose of external regulation from the perspective of market discipline.

Numerical calculations of risk measures and risk contributions in credit risk models amount to the evaluation of various forms of quantiles, tail probabilities and tail expectations of the portfolio loss distribution. Though the moment generating function of the loss distribution in the ${\text{CreditRisk}}^{+}$ model is available in analytic closed form, efficient, accurate and reliable computation of risk measures (Value-at-Risk and Expected Shortfall) and risk contributions for the ${\text{CreditRisk}}^{+}$ model poses technical challenges. We propose various numerical algorithms for risk measures and risk contributions calculations of the enhanced ${\text{CreditRisk}}^{+}$ model under the common background vector framework using the Johnson curve fitting method, saddlepoint approximation method, importance sampling in Monte Carlo simulation and check function formulation. Our numerical studies on stylized credit portfolios and benchmark industrial credit portfolios reveal that the Johnson curve fitting approach works very well for credit portfolios with a large number of obligors, demonstrating high level of numerical reliability and computational efficiency. Once we implement the systematic procedure of finding the saddlepoint within an approximate domain, the saddlepoint approximation schemes provide efficient calculation and accurate numerical results. The importance sampling in Monte Carlo simulation methods are easy to implement, but they compete less favorably in accuracy and efficiency with other numerical algorithms. The less commonly used check function formulation is limited to risk measures calculations. It competes favorably in accuracy and reliability, but an extra optimization algorithm is required.

This study extends the method of Guermat and Harris (2002), the Power EWMA (exponentially weighted moving average) method in conjunction with historical simulation to estimating portfolio Value-at-Risk (VaR). Using historical daily return data of three hypothetical portfolios formed by international stock indices, we test the performance of this modified approach to see if it can improve the precise forecasting capability of historical simulation. We explicitly highlight the extended Power EWMA owns privileged flexibilities to capture time-varying tail-fatness and volatilities of financial returns, and therefore may promote the quality of extreme risk management. Our empirical results, derived from the Kupiec (1995) tests and failure ratios, show that our proposed method indeed offers substantial improvements on capturing dynamic returns distributions, and can significantly enhance the estimation accuracy of portfolio VaR.

Since Value-at-Risk (VaR) disregards tail losses beyond the VaR boundary, the expected shortfall (ES), which measures the average loss when a VaR is exceeded, and the tail-risk-of-VaR (TR), which sums the sizes of tail losses, are used to investigate risks at the tails of distributions for major stock markets. As VaR exceptions are rare, we employ the saddlepoint or small sample asymptotic technique to backtest ES and TR. Because the two risk measures are complementary to each other and hence provide more powerful backtests, we are able to show that (a) the correct specification of distribution tail, rather than heteroscedastic process, plays a key role to accurate risk forecasts; and (b) it is best to model the tails separately from the central part of distribution using the Generalized Pareto Distribution (GPD). To sum up, we provide empirical evidence that financial markets behave differently during crises, and extreme risks cannot be modeled effectively under normal market conditions or based on a short data history.

This paper proposes a method that uses volatility index of US and six other markets of Pacific Basin, namely Hong Kong, Australia, India, Japan, Korea, and China, to provide value-at-risk (VaR) and expected shortfall (ES) forecasts. Empirical constants that are used to multiply the levels of volatility indexes for estimating VaR and ES of various significance levels for 1–22 days ahead, one by one, for seven market indexes have been statistically determined using daily data spanning from 4.75 to 16 years. It is because it would be inappropriate to simply scale up the one-day volatility by multiplying the square root of time (or the number of days) ahead to determine the risk over a longer horizon of $i$ days. Results show that the shift to ES approach generally increases the regulatory capital requirements from 2.09% of India market to 8.56% of Korea market except for the China market where ES approach yields an unexpected decrease of 3.21% of capital requirement.

This study investigates the volatility of the Euro-to-US Dollar exchange rate, specifically focusing on identifying long-memory characteristics. Through the analysis of daily data spanning from January 1, 2018, to January 10, 2023, the study uncovers a robust long-memory feature. Supporting this exploration, the study endorses the use of sophisticated models such as Fractionally Integrated Generalized Autoregressive Conditionally Heteroskedastic (FIGARCH) and Hyperbolic Generalized Autoregressive Conditionally Heteroskedastic (HYGARCH), incorporating both student and skewed student innovation distributions. The results underscore the superior performance of FIGARCH and HYGARCH models, particularly when coupled with a skewed student distribution. This collaborative approach enhances the predictability of crucial financial metrics, including Value at Risk (VaR) and Expected Shortfall (ESF), for both long and short trading positions. Significantly, the FIGARCH model, when utilizing a skewed student distribution, demonstrates exceptional predictive power. This outcome challenges the efficient market hypothesis and suggests the potential for generating outstanding returns. In light of these findings, this research contributes valuable insights for comprehending and navigating the intricacies of the Euro-to-US Dollar exchange rate, providing a forward-looking perspective for financial practitioners and researchers alike.

This paper studies the model risk; the risk of selecting a model for estimating the Value-at-Risk (VaR). By considering four GARCH-type volatility processes exponentially weighted moving average (EWMA), generalized autoregressive conditional heteroskedasticity (GARCH), exponential GARCH (EGARCH), and fractionally integrated GARCH (FIGARCH), we evaluate the performance of the estimated VaRs using statistical tests including the Kupiec's likelihood ratio (LR) test, the Christoffersen's LR test, the CHI (Christoffersen, Hahn, and Inoue) specification test, and the CHI nonnested test. The empirical study based on Shanghai Stock Exchange A Share Index indicates that both EGARCH and FIGARCH models perform much better than the other two in VaR computation and that the two CHI tests are more suitable for analyzing model risk.