Tail risk in emerging markets of Southeastern Europe

This article examines the left-tail behaviour of returns on stocks in Southeastern Europe (SEE). We apply conditional extreme value theory (EVT) approach on daily returns of six stock market indices from SEE between 2004 and 2013. Predictive performance of value-at-risk (VaR) and expected shortfall (ES) based on EVT is compared against several alternatives, such as historical simulation and analytical approach based on GARCH with a single conditional distribution. Model backtesting with daily returns shows that EVT-based models provide more reliable VaR and ES forecasts than the alternative models in all six markets. Unlike the alternatives, the EVT-based models cannot be rejected as VaR confidence level is increased. This emphasizes the importance of extreme events in SEE markets and indicates that the ability of a model to capture volatility clustering accurately is not sufficient for a correct assessment of risk in these markets.     

A semiparametric approach to value-at-risk,expected shortfall and optimum asset allocation in stock–bond portfolios

This paper investigates stock–bond portfolios' tail risks such as value-at-risk (VaR) and expected shortfall (ES), and the way in which these measures have been affected by the global financial crisis. The semiparametric t-copulas adequately model stock–bond returns joint distributions of G7 countries and Australia. Empirical results show that the (negative) weak stock–bond returns dependence has increased significantly for seven countries after the crisis, except for Italy. However, both VaR and ES have increased for all eight countries. Before the crisis, the minimum portfolio VaR and ES were achieved at an interior solution only for the US, the UK, Australia, Canada and Italy. After the crisis, the corner solution was found for all eight countries. Evidence of “flight to quality” and “safety first” investor behaviour was strong, after the global financial crisis. The semiparametric t-copula adequately forecasts the outer-sample VaR. These findings have implications for global financial regulators and the Basel Committee, whose central focus is currently on increasing the capital requirements as a consequence of the recent global financial crisis.     

Pricing and hedging options with GARCH-stable proxy volatilities

This article considers modelling nonnormality in return with stable Paretian (SP) innovations in generalized autoregressive conditional heteroskedasticity (GARCH), exponential generalized autoregressive conditional heteroskedasticity (EGARCH) and Glosten-Jagannathan-Runkle generalized autoregressive conditional heteroskedasticity (GJR-GARCH) volatility dynamics. The forecasted volatilities from these dynamics have been used as a proxy to the volatility parameter of the Black–Scholes (BS) model. The performance of these proxy-BS models has been compared with the performance of the BS model of constant volatility. Using a cross section of S&P500 options data, we find that EGARCH volatility forecast with SP innovations is an excellent proxy to BS constant volatility in terms of pricing. We find improved performance of hedging for an illustrative option portfolio. We also find better performance of spectral risk measure (SRM) than value-at-risk (VaR) and expected shortfall (ES) in estimating option portfolio risk in case of the proxy-BS models under SP innovations.

Abbreviation: generalized autoregressive conditional heteroskedasticity (GARCH), exponential generalized autoregressive conditional heteroskedasticity (EGARCH) and Glosten-Jagannathan-Runkle generalized autoregressive conditional heteroskedasticity (GJR-GARCH)     

Incentive contracts in delegated portfolio management under VaR constraint

This paper studies the incentive effect of linear performance-adjusted contracts in delegated portfolio management under a value-at-risk (VaR) constraint. It is shown that a linear performance-based contract can provide incentives for the portfolio manager to work at acquiring private information under a VaR risk constraint. The expected utility and optimal effort of a risk-averse manager are increasing functions of the return sharing ratio in the contract. However, a risk constraint causes the portfolio manager to reduce effort in gathering private information, suggesting that the VaR constraint increases the moral hazard between the investor and the manager.     

Score-driven Markov-switching EGARCH models: an application to systematic risk analysis

We introduce new Markov-switching (MS) dynamic conditional score (DCS) exponential generalized autoregressive conditional heteroscedasticity (EGARCH) models, to be used by practitioners for forecasting value-at-risk (VaR) and expected shortfall (ES) in systematic risk analysis. We use daily log-return data from the Standard & Poor’s 500 (S&P 500) index for the period 1950–2016. The analysis of the S&P 500 is useful, for example, for investors of (i) well-diversified US equity portfolios; (ii) S&P 500 futures and options traded at Chicago Mercantile Exchange Globex; (iii) exchange traded funds (ETFs) related to the S&P 500. The new MS DCS-EGARCH models are alternatives to of the recent MS Beta-t-EGARCH model that uses the symmetric Student’s t distribution for the error term. For the new models, we use more flexible asymmetric probability distributions for the error term: Skew-Gen-t (skewed generalized t), EGB2 (exponential generalized beta of the second kind) and NIG (normal-inverse Gaussian) distributions. For all MS DCS-EGARCH models, we identify high- and low-volatility periods for the S&P 500. We find that the statistical performance of the new MS DCS-EGARCH models is superior to that of the MS Beta-t-EGARCH model. As a practical application, we perform systematic risk analysis by forecasting VaR and ES.

Abbreviation Single regime (SR); Markov-switching (MS); dynamic conditional score (DCS); exponential generalized autoregressive conditional heteroscedasticity (EGARCH); value-at-risk (VaR); expected shortfall (ES); Standard & Poor's 500 (S&P 500); exchange traded funds (ETFs); Skew-Gen-t (skewed generalized t); EGB2 (exponential generalized beta of the second kind); NIG (normal-inverse Gaussian); log-likelihood (LL); standard deviation (SD); partial autocorrelation function (PACF); likelihood-ratio (LR); ordinary least squares (OLS); heteroscedasticity and autocorrelation consistent (HAC); Akaike information criterion (AIC); Bayesian information criterion (BIC); Hannan-Quinn criterion (HQC).     

A novel nonlinear value-at-risk method for modeling risk of option portfolio with multivariate mixture of normal distributions

This paper proposes a novel nonlinear model for calculating Value-at-Risk (VaR) when the market risk factors of an option portfolio are heavy-tailed. A multivariate mixture of normal distributions is used to depict the heavy-tailed market risk factors and accordingly a closed form expression for the moment generating function that can reflect the change in option portfolio value can be derived. Moreover, in order to make use of the correlation between the characteristic function and the moment generating function, Fourier-Inversion method and adaptive Simpson rule with iterative algorithm of numerical integration into the nonlinear VaR model for option portfolio are applied for calculation of VaR values of option portfolio. VaR values of option portfolio obtained from different methods are compared. Numerical results of Fourier-Inversion method and Monte Carlo simulation method show that high accuracy VaR values can be obtained when risk factors have multivariate mixture of normal distributions than when they have normal distributions. Moreover, VaR values obtained by using the Fourier-Inversion method are not obviously different from VaR values obtained by using Monte Carlo simulation when market risk factors have normal distributions or multivariate mixture of normal distributions. However, the speed of computation is obviously faster when using Fourier-Inversion method, than when using Monte Carlo simulation method. Besides, Cornish Fisher method is faster and simpler than Monte Carlo simulation method or Fourier-Inversion method. However, this method does not offer high accuracy and cannot be used to calculate VaR values of option portfolio when market risk factors have heavy-tailed distributions.     

Expected Shortfall: A Natural Coherent Alternative to Value at Risk

We discuss the coherence properties of expected shortfall (ES) as a financial risk measure. This statistic arises in a natural way from the estimation of the 'average of the 100% worst losses' in a sample of returns to a portfolio. Here p is some fixed confidence level. We also compare several alternative representations of ES which turn out to be more appropriate for certain purposes
(J.E.L.: G20, C13, C14).     

A risk index to model uncertain portfolio investment with options

This paper models the portfolio investment performance with options by using a risk index, which is defined as the average loss below the risk-free interest rate. Using a risk-free interest rate as the uniform reference rate for all portfolios, the risk index offers an easier-to-compare loss value than the value-at-risk return, where portfolio specific references are used to calculate the average losses. Besides, uncertainty theory is used in the paper to derive the portfolio decision when stock prices are subject to experts' estimations. By analytical computation and empirical analysis, we find that portfolios considering options generate better return than the ones without options. The empirical analysis reveals that the options can effectively hedge the risk, and the call option with a higher exercise price offers higher return per unit of option premium. Furthermore, our proposed model produces higher expected return in most cases than the model where the risk is measured by the chance of the total return failing to reach the threshold level of return.     

Tail dependence and diversification benefits in emerging market stocks: an extreme value theory approach

This article examines tail dependence, the benefits of diversification and the relation between the two for emerging stock markets. We find most emerging equity markets are independent in limiting joint extremes. However, the dependence in finite levels of extremes is still much stronger than the dependence implied by multivariate normality. Therefore, simple correlation analysis can lead to gross underestimation of the chances of joint crashes in multiple markets. Assuming risk-averse investors guarding against extreme losses, diversification benefits are measured for each two-country optimal portfolio by the reduction in quantile risk measures such as value-at-risk and expected shortfall relative to an undiversified portfolio. It is shown that tail dependence measures developed from multivariate extreme value theory are negatively related to diversification benefits and more importantly can explain diversification benefits better than the correlation coefficient at the most extreme quantiles.     

Energy portfolio risk management using time-varying extreme value copula methods

This work is concerned with the statistical modeling of the dependence structure between three energy commodity markets (WTI crude oil, natural gas and heating oil) using the concept of copulas and proposes a method for estimating the Value at risk (VaR) of energy portfolio based on the combination of time series models with models of the extreme value theory before fitting a copula. Each return series is modeled by AR-(FI) GARCH univariate model. Then, we fit the GPD distribution to the tails of the residuals to model marginal residuals distributions. The extreme value copula to the iid residuals is fitted and we simulate from it to construct N portfolios and estimate VaR. As a first step, the method is applied to a two-dimensional energy portfolio. In second step, we extend method in trivariate context to measure VaR of three-dimensional energy portfolio. Dependences between residuals are modeled using a trivariate nested Gumbel copulas. Methods proposed are compared with various univariate and multivariate conventional VaR methods. The reported results demonstrate that GARCH-t, conditional EVT and FIGARCH extreme value copula methods produce acceptable estimates of risk both for standard and more extreme VaR quantiles. Generally, copula methods are less accurate compared with their predictive performances in the case of portfolio composed of exchange market indices.     

Testing the Gaussian and Student's t copulas in a risk management framework

This paper introduces a semiparametric framework for selecting either a Gaussian or a Student's t copula in a d-dimensional setting. We compare the two models using four different approaches: (i) four goodness-of-fit graphical plots, (ii) a bootstrapped correlation matrix generated in each scenario with the empirical correlation matrix used as a benchmark, (iii) Value-at-Risk (VaR) and Expected Shortfall (ES) as risk measures, and (iv) co-Value-at-Risk (CoVaR) and Marginal Expected Shortfall (MES) as co-risk measures. We illustrate this four-step procedure using a portfolio of daily returns of six international stock indices. The VaR results confirm that the t-based copula model is an attractive alternative to the Gaussian. The ES analysis is less conclusive, and indicates that risk managers should jointly use the risk measure as well as the copula model. The results highlight the importance of promoting stress testing rather than ES in the risk management industry, particularly in the aftermath of a financial crisis.     

Long range dependence in an emerging stock market’s sectors: volatility modelling and VaR forecasting

This study evaluates the sector risk of the Qatar Stock Exchange (QSE), a recently upgraded emerging stock market, using value-at-risk models for the 7 January 2007–18 October 2015 period. After providing evidence for true long memory in volatility using the log-likelihood profile test of Qu and splitting the sample and dth differentiation tests of Shimotsu, we compare the FIGARCH, HYGARCH and FIAPARCH models under normal, Student-t and skewed-t innovation distributions based on in and out-of-sample VaR forecasts. The empirical results show that the skewed Student-t FIGARCH model generates the most accurate prediction of one-day-VaR forecasts. The policy implications for portfolio managers are also discussed.     

Did financial deregulation help consumers? Access to market-yield instruments

The maturity effect (ME) of futures prices postulated by Samuelson (1965) is re-examined using three nonparametric tests. The consistent entropy asymmetry test by Racine and Maasoumi (2007) indicates that variance is an appropriate risk or uncertainty measure for ME, and value-at-risk and expected shortfall are also adopted. The Kolmogorov–Smirnov dominance test and Wilcoxon rank sum and signed rank test are employed to rank the estimates of the three risk measures under a moving-window framework. The testing outcomes are contingent on futures type, testing method and risk measures. The testing outcomes show mild support for ME.     

Fuzzy possibilistic portfolio selection model with VaR constraint and risk-free investment

We propose a possibilistic portfolio model with VaR constraint and risk-free investment based on the possibilistic mean and variance, while assuming that the expected rate of returns is a fuzzy number. The model shows more clearly that, in the financial market affected by several non-probabilistic factors, risk-averse investors wish not only to reach the expected rate of returns in their actual investment, but also to assure that the maximum of their possible future risk is lower than an expected loss. Under the condition that the expected rate of returns is a normal distribution fuzzy variable, we proposed a theorem as the solution, and derive a crisp equivalent form of the possibilistic portfolio under constraints of VaR and risk-free investment. This model is an expansion of the fuzzy possibilistic mean–variance model by Zhang (2007). Finally, an empirical study is carried out using the data concerning some stocks of various industries listed at the Shanghai Stock Exchange. A conclusion is reached that the investors are able to choose a portfolio more suitable to them under the VaR constraint.     

Backtesting VaR in consideration of the higher moments of the distribution for minimum-variance hedging portfolios

The higher moments of a distribution often lead to estimated value-at-risk (VaR) biases. This study's objective is to examine the backtesting of VaR models that consider the higher moments of the distribution for minimum-variance hedging portfolios (MVHPs) of the stock indices and futures in the Greater China Region for both short and long hedgers. The results reveal that the best backtesting VaR for the MVHP considered both the higher moments of the MVHP distribution and the asymmetry in volatility, cross-market asymmetry in volatility, and level effects in the covariance matrix of assets in the MVHP. These empirical results provide references for investors in risk management.     

Backtesting expected shortfall: evidence from European securitized real estate

Events such as the European sovereign debt crisis, terrorism and Brexit cause more uncertainty and volatility in capital markets. This encourages us to use both conditional and unconditional forecasts (backtests) for expected shortfall (ES) in 8 indices of listed European real estate securities and Real estate investment trusts (REITs). Using the method proposed by Du and Escanciano, we find that ES is generally superior to Value-at-Risk in describing and capturing risk during extreme events such as the financial crisis. Our results are important to regulators, risk managers and investors.     

Can cryptocurrencies be a safe haven: a tail risk perspective analysis

Cryptocurrencies are one of the most promising financial innovations of the last decade. Different from major stock indices and the commodities of gold and crude oil, the cryptocurrencies exhibit some characteristics of immature market assets, such as auto-correlated and non-stationary return series, higher volatility, and higher tail risks measured by conditional Value at Risk (VaR) and conditional expected shortfall (ES). Using an extreme-value-theory-based method, we evaluate the extreme characteristics of seven representative cryptocurrencies during 08 August 2015–01 August 2017. We find that during the sub-period of 01 August 2016–01 August 2017, there are finite loss boundaries for most of the selected cryptocurrencies, which are similar to the commodities, and different from the stock indices. Meanwhile, we find that left tail correlations are much stronger than right tail correlations among the cryptocurrencies, and tail correlations increased after August 2016, suggesting high and growing systematic extreme risks. We also find that cryptocurrencies to be both left tail independent, and cross tail independent with four selected stock indices, which implies part of the safe-haven function of the cryptocurrencies, indicating their ability to be a great diversifier for the stock market as gold, but not enough to be a tail hedging tool like gold.     

Forecasting the volatility of S&P depositary receipts using GARCH-type models under intraday range-based and return-based proxy measures

We employ four various GARCH-type models, incorporating the skewed generalized t (SGT) errors into those returns innovations exhibiting fat-tails, leptokurtosis and skewness to forecast both volatility and value-at-risk (VaR) for Standard & Poor's Depositary Receipts (SPDRs) from 2002 to 2008. Empirical results indicate that the asymmetric EGARCH model is the most preferable according to purely statistical loss functions. However, the mean mixed error criterion suggests that the EGARCH model facilitates option buyers for improving their trading position performance, while option sellers tend to favor the IGARCH/EGARCH model at shorter/longer trading horizon. For VaR calculations, although these GARCH-type models are likely to over-predict SPDRs' volatility, they are, nevertheless, capable of providing adequate VaR forecasts. Thus, a GARCH genre of model with SGT errors remains a useful technique for measuring and managing potential losses on SPDRs under a turbulent market scenario.     

Measuring and Optimizing Portfolio Credit Risk: A Copula‐based Approach*

In this work, we present a methodology for measuring and optimizing the credit risk of a loan portfolio taking into account the non‐normality of the credit loss distribution. In particular, we aim at modelling accurately joint default events for credit assets. In order to achieve this goal, we build the loss distribution of the loan portfolio by Monte Carlo simulation. The times until default of each obligor in portfolio are simulated following a copula‐based approach. In particular, we study four different types of dependence structure for the credit assets in portfolio: the Gaussian copula, the Student's t‐copula, the grouped t‐copula and the Clayton n‐copula (or Cook–Johnson copula). Our aim is to assess the impact of each type of copula on the value of different portfolio risk measures, such as expected loss, maximum loss, credit value at risk and expected shortfall. In addition, we want to verify whether and how the optimal portfolio composition may change utilizing various types of copula for describing the default dependence structure. In order to optimize portfolio credit risk, we minimize the conditional value at risk, a risk measure both relevant and tractable, by solving a simple linear programming problem subject to the traditional constraints of balance, portfolio expected return and trading. The outcomes, in terms of optimal portfolio compositions, obtained assuming different default dependence structures are compared with each other. The solution of the risk minimization problem may suggest us how to restructure the inefficient loan portfolios in order to obtain their best risk/return profile. In the absence of a developed secondary market for loans, we may follow the investment strategies indicated by the solution vector by utilizing credit default swaps.     

股票资产组合VaR研究基于混合D藤Copula模型

本文在对上证市场五种股票资产组合的风险分析中以VaR作为风险度量指标,采用基于Pair Copula高维建模理论的混合D藤Copula模型,建立了反应多个资产组合相关结构的联合分布模型。该模型对传统D藤Copula建模方法作了进一步的改进,通过一定的选择标准,确定了D藤中每个Pair Copula函数的最优函数族,这样使得所建立的模型不仅考虑到了资产维数的影响,而且还能捕捉到组合内部因子间相关结构的差异性,从而改进后的模型能更好地描述资产组合的相关结构,并且能更精确地反映资产组合收益的实际分布。最后,以混合D藤Copula模型为基础,利用Monte Carlo方法计算了上证市场五种股票资产组合的VaR,并通过实证研究进一步证明了该模型的有效性。