On risk management problems related to a coherence property

Value at Risk has lost the battle against Expected Shortfall on theoretical grounds, the latter satisfying all coherence properties while the former may, on carefully constructed cases, lack the sub-additivity property that is in a sense, the most important property a risk measure ought to satisfy. While the superiority of Expected Shortfall is evident as a theoretical tool, little has been researched on the properties of estimators proposed in the literature. Since those estimators are the real tools for calculating bank capital reserves in practice, the natural question that one may ask is whether a given estimator of Expected Shortfall also satisfies the coherence properties. In this paper, we show that it is possible to have estimators of Expected Shortfall that do not satisfy the sub-additivity condition. This finding should motivate risk managers and quantitative asset managers to investigate further the properties of the estimators of the risk measures they are currently utilizing.     

On the role of the estimation error in prediction of expected shortfall

In the estimation of risk measures such as Value at Risk and Expected shortfall relatively short estimation windows are typically used rendering the estimation error a possibly non-negligible component. In this paper we build upon previous results for the Value at Risk and discuss how the estimation error comes into play for the Expected Shortfall. We identify two important aspects where it may be of importance. On the one hand there is in the evaluation of predictors of the measure. On the other there is in the interpretation and communication of it. We illustrate magnitudes numerically and emphasize the practical importance of the latter aspect in an empirical application with stock market index data.     

Risk Measures for Hedge Funds: a Cross‐sectional Approach

This paper analyses the risk‐return trade‐off in the hedge fund industry. We compare semi‐deviation, value‐at‐risk (VaR), Expected Shortfall (ES) and Tail Risk (TR) with standard deviation at the individual fund level as well as the portfolio level. Using the Fama and French (1992) methodology and the combined live and defunct hedge fund data from TASS, we find that the left‐tail risk captured by Expected Shortfall (ES) and Tail Risk (TR) explains the cross‐sectional variation in hedge fund returns very well, while the other risk measures provide statistically insignificant or marginally significant results. During the period between January 1995 and December 2004, hedge funds with high ES outperform those with low ES by an annual return difference of 7%. We provide empirical evidence on the theoretical argument by Artzner et al. (1999) that ES is superior to VaR as a downside risk measure. We also find the Cornish‐Fisher (1937) expansion is superior to the nonparametric method in estimating ES and TR.     

Pricing and managing risks of European-style options in a Markovian regime-switching binomial model

We discuss the pricing and risk management problems of standard European-style options in a Markovian regime-switching binomial model. Due to the presence of an additional source of uncertainty described by a Markov chain, the market is incomplete, so the no-arbitrage condition is not sufficient to fix a unique pricing kernel, hence, a unique option price. Using the minimal entropy martingale measure, we determine a pricing kernel. We examine numerically the performance of a simple hedging strategy by investigating the terminal distribution of hedging errors and the associated risk measures such as Value at Risk and Expected Shortfall. The impacts of the frequency of re-balancing the hedging portfolio and the transition probabilities of the modulating Markov chain on the quality of hedging are also discussed.     

Approximation methods for multiple period Value at Risk and Expected Shortfall prediction

In this paper, we are interested in predicting multiple period Value at Risk and Expected Shortfall based on the so-called iterating approach. In general, the properties of the conditional distribution of multiple period returns do not follow easily from the one-period data generating process, rendering this a non-trivial task. We outline a framework that forms the basis for setting approximations and study four different approaches. Their performance is evaluated by means of extensive Monte Carlo simulations based on an asymmetric GARCH model, implying conditional skewness and excess kurtosis in the multiple period returns. This simulation-based approach was the best one, closely followed by that of assuming a skewed t-distribution for the multiple period returns. The approach based on a Gram–Charlier expansion was not able to cope with the implied non-normality, while the so-called Root-k approach performed poorly. In addition, we outline how the delta-method may be used to quantify the estimation error in the predictors and in the Monte Carlo study we found that it performed well. In an empirical illustration, we computed 10-day Value at Risk’s and Expected Shortfall for Brent Crude Oil, the EUR/USD exchange rate and the S&P 500 index. The Root-k approach clearly performed the worst and the other approaches performed quite similarly, with the simulation based approach and the one based on the skewed t-distribution somewhat better than the one based on the Gram–Charlier expansion.     

Sensitivity analysis of VaR and Expected Shortfall for portfolios under netting agreements

In this paper, we characterize explicitly the first derivative of the Value at Risk and the Expected Shortfall with respect to portfolio allocations when netting between positions exists. As a particular case, we examine a simple Gaussian example in order to illustrate the impact of netting agreements in credit risk management. Collateral issues are also dealt with. For practical purposes we further provide nonparametric estimators for sensitivities and derive their asymptotic distributions. An empirical application on a typical banking portfolio is finally provided.     

Estimating financial risk measures for options

This paper proposes a simulation-lattice procedure to estimate financial risk measures for option positions. The framework proposed can be applied to many different kinds of options, including exotic and vanilla options; it can take account of early exercise features; heavy tails in underlying processes; estimate different risk measures, including VaR, Expected Shortfall and Spectral Risk Measures; and in a limited way it can be generalized to accommodate multiple-factors. It avoids many of the limitations of existing approaches and, in particular, avoids the problems associated approaches based on delta–gamma and similar approximations. It also generates some interesting results about the risk measures of some illustrative options positions.     

Measuring Value‐at‐Risk in Project Finance Transactions

Despite the remarkable importance of project finance in international financial markets, no quantitative models to measure and quantify the risk associated with a deal for the project's lenders have been developed yet. The topic has recently become crucial, since the New Basle Capital Accord gives banks a choice of whether to adopt simpler (but possibly higher) standard capital requirements or to develop internal rating models for project finance transactions. The paper proposes how Monte Carlo simulations may be used to derive a Value‐at‐Risk estimate for project finance deals and discusses the critical issues that must be considered when developing such a model.     

A comprehensive approach to measuring the relation between systemic risk exposure and sovereign debt

Using an integrated model to control for simultaneity, as well as new risk measurement techniques such as Adapted Exposure CoVaR and Marginal Expected Shortfall (MES), we show that the aggregate systemic risk exposure of financial institutions is positively related to sovereign debt yields in European countries in an episodic manner, varying positively with the intensity of the financial crisis facing a particular nation. We find evidence of a simultaneous relation between systemic risk exposure and sovereign debt yields. This suggests that models of sovereign debt yields should also include the systemic risk of a country's financial system in order to avoid potentially important mis-specification errors. We find evidence that systemic risk of a country's financial institutions and the risk of sovereign governments are inter-related and shocks to these domestic linkages are stronger and longer lasting than international risk spillovers. Thus, the channel in which domestic sovereign debt yields can be affected by another nation's sovereign debt is mostly an indirect one in that shocks to a foreign country's government finances are transmitted to that country's financial system which, in turn, can spill over to the domestic financial system and, ultimately, have a destabilizing effect on the domestic sovereign debt market.     

Coherent risk measures under filtered historical simulation

Recent studies have strongly criticised conventional VaR models for not providing a coherent risk measure. Acerbi provides the intuition for an entire family of coherent measures of risk known as “spectral risk measures” [Spectral measures of risk: A coherent representation of subjective risk aversion. Journal of Banking and Finance 26 (7) (2002) 1505–1518]. In this study we illustrate how the Filtered Historical Simulation [Barone-Adesi, G., Bourgoin, F., Giannopoulos, K., 1998. Don’t look back. Risk 11, 100–104; Barone-Adesi, Giannopoulos, K., Vosper, L., 1999. VaR without correlations for non-linear portfolios. Journal of Futures Markets 19, 583–602], can provide an improved methodology for calculating the Expected Shortfall. Thereafter, we prove that these new risk measures are spectral and are coherent as well, following Acerbi. Furthermore, we provide the statistical error formula that allows to calculate the error for our model.     

Can a Coherent Risk Measure Be Too Subadditive?

We consider the problem of determining appropriate solvency capital requirements for an insurance company or a financial institution. We demonstrate that the subadditivity condition that is often imposed on solvency capital principles can lead to the undesirable situation where the shortfall risk increases by a merger. We propose to complement the subadditivity condition by a regulator's condition. We find that for an explicitly specified confidence level, the Value‐at‐Risk satisfies the regulator's condition and is the “most efficient” capital requirement in the sense that it minimizes some reasonable cost function. Within the class of concave distortion risk measures, of which the elements, in contrast to the Value‐at‐Risk, exhibit the subadditivity property, we find that, again for an explicitly specified confidence level, the Tail‐Value‐at‐Risk is the optimal capital requirement satisfying the regulator's condition.     

Analytical Value-at-Risk and Expected Shortfall under regime-switching

It is well known that the use of Gaussian models to assess financial risk leads to an underestimation of risk. The reason is because these models are unable to capture some important facts such as heavy tails and volatility clustering which indicate the presence of large fluctuations in returns. An alternative way is to use regime-switching models, the latter are able to capture the previous facts. Using regime-switching model, we propose an analytical approximation for multi-horizon conditional Value-at-Risk and a closed-form solution for conditional Expected Shortfall. By comparing the Value-at-Risks and Expected Shortfalls calculated analytically and using simulations, we find that the both approaches lead to almost the same result. Further, the analytical approach is less time and computer intensive compared to simulations, which are typically used in risk management.     

Variance of the CTE Estimator

Abstract

The Conditional Tail Expectation (CTE), also called Expected Shortfall or Tail-VaR, is a robust, convenient, practical, and coherent measure for quantifying financial risk exposure. The CTE is quickly becoming the preferred measure for statutory balance sheet valuation whenever real-world stochastic methods are used to set liability provisions. We look at some statistical properties of the methods that are commonly used to estimate the CTE and develop a simple formula for the variance of the CTE estimator that is valid in the large sample limit. We also show that the formula works well for finite sample sizes. Formula results are compared with sample values from realworld Monte Carlo simulations for some common loss distributions, including equity-linked annuities with investment guarantees, whole life insurance and operational risks. We develop the CTE variance formula in the general case using a system of biased weights and explore importance sampling, a form of variance reduction, as a way to improve the quality of the estimators for a given sample size. The paper closes with a discussion of practical applications.     

VaR and ES for linear portfolios with mixture of generalized Laplace distributions risk factors

In this paper, we propose an explicit estimation of Value-at-Risk (VaR) and Expected Shortfall (ES) for linear portfolios when the risk factors change with a convex mixture of generalized Laplace distributions (M-GLD). We introduce the dynamics Delta-GLD-VaR, Delta-GLD-ES, Delta-MGLD-VaR and Delta-MGLD-ES, by using conditional correlation multivariate GARCH. The generalized Laplace distribution impose less restrictive assumptions during estimation that should improve the precision of the VaR and ES through the varying shape and fat tails of the risk factors in relation with the historical sample data. We also suggested some areas of application to measure price risk in agriculture, risk management and financial portfolio optimization.     

How to choose the return model for market risk? Getting towards a right magnitude of stressed VaR

Value at Risk (VaR) and stressed value at Risk (SVaR) or expected shortfall are important risk measures widely used in the financial services industry for risk management and market risk capital computation. Fundamental to any (S)VaR model is the choice of the return type model for each risk factor. Because the resulting SVaR numbers are highly sensitive to the chosen return type model it is important to make a prudent choice on the return type modelling. We propose to estimate the return type model from historic data without making an a priori model assumption on the return model. We explain the fundamentals of return type modelling and how it impacts the magnitude of SVaR. We further show how to obtain a global return type model from a set of similar return type models by using geometric calculus. Numerical simulations and illustrations are provided. In this paper, we consider interest rate data, but the proposed methodology is general and can be applied to any other asset class such as inflation, credit spread, equity or fx.     

Optimal Portfolio Selection Based on Expected Shortfall Under Generalized Hyperbolic Distribution

This paper discusses optimal portfolio selection problems under Expected Shortfall as the risk measure. We employ multivariate Generalized Hyperbolic distribution as the joint distribution for the risk factors of underlying portfolio assets, which include stocks, currencies and bonds. Working under this distribution, we find the optimal portfolio strategy.     

Intraday risk management in International stock markets: A conditional EVT approach

The study compares the predictive ability of various models in estimating intraday Value-at-Risk (VaR) and Expected Shortfall (ES) using high frequency share price index data from sixteen different countries across the world for a period of seven and half months from September 20, 2013 to May 07, 2014. The main emphasis of the study has been given to Extreme Value Theory (EVT) and to evaluate how well Conditional EVT model performs in modeling tails of distributions and in estimating and forecasting intraday VaR and ES measures. We have followed McNeil and Frey's (2000) two stage approach called Conditional EVT to estimate dynamic intraday VaR and ES. We have compared the accuracy of Conditional EVT approach to intraday VaR and ES estimation with other competing models. The best performing model is found to be the Conditional EVT in estimating both the quantiles for the entire sample. The study is useful for market participants (such as intraday traders and market makers) involved in frequent intraday trading in such equity markets.     

Estimation of multiple period expected shortfall and median shortfall for risk management

With the regulatory requirements for risk management, Value at Risk (VaR) has become an essential tool in determining capital reserves to protect the risk induced by adverse market movements. The fact that VaR is not coherent has motivated the industry to explore alternative risk measures such as expected shortfall. The first objective of this paper is to propose statistical methods for estimating multiple-period expected shortfall under GARCH models. In addition to the expected shortfall, we investigate a new tool called median shortfall to measure risk. The second objective of this paper is to develop backtesting methods for assessing the performance of expected shortfall and median shortfall estimators from statistical and financial perspectives. By applying our expected shortfall estimators and other existing approaches to seven international markets, we demonstrate the superiority of our methods with respect to statistical and practical evaluations. Our expected shortfall estimators likely provide an unbiased reference for setting the minimum capital required for safeguarding against expected loss.     

Alternativen zu Value at Risk

The industry standard for the measurement of the downside risk of financial positions, Value at Risk (VaR), has serious deficiencies. In the current article we systematically discuss properties of risk measures and alternatives to VaR. It is demonstrated that utility-based shortfall risk (UBSR) possesses many desirable properties.