Expected Shortfall Risk Measures Research Articles

An evaluation of the adequacy of Lévy and extreme value tail risk estimates

This study investigates the simplicity and adequacy of tail-based risk measures—value-at-risk (VaR) and expected shortfall (ES)—when applied to tail targeting of the extreme value (EV) model. We implement Lévy–VaR and ES risk measures as full density-based alternatives to the generalized Pareto VaR and the generalized Pareto ES of the tail-targeting EV model. Using data on futures contracts of S&P500, FTSE100, DAX, Hang Seng, and Nikkei 225 during the Global Financial Crisis of 2007–2008, we find that the simplicity of tail-based risk management with a tail-targeting EV model is more attractive. However, the performance of EV risk estimates is not necessarily superior to that of full density-based relatively complex Lévy risk estimates, which may not always give us more robust VaR and ES results, making the model inadequate from a practical perspective. There is randomness in the estimation performances under both approaches for different data ranges and coverage levels. Such mixed results imply that banks, financial institutions, and policymakers should find a way to compromise or trade-off between “simplicity” and user-defined “adequacy”.

The Risk Measurement in the Skewed Student’s t Model

Throughout this paper, we discuss the problem of calculation of the value at risk and the expected shortfall risk measures in the skewed Student's t model. The investment portfolio which consists of assets with deterministic, inverse gamma and skewed Student's t returns is considered. Analytical results for the mentioned above monetary risk measures are derived basing on the formulas for the distribution functions of losses. The obtained formulas depend on the values of special mathematical functions including the generalized hypergeometric ones. Numerical examples of the computations are given.

Quantile-based individual risk measures for rockfall-prone areas

In mountain environments, precise rockfall risk evaluation is crucial to reduce death tolls and costs. However, to date, existing rockfall risk assessment procedures remain scarce, as they focus only on existing elements at risk and with the damage expectation as sole risk measure. Here, we propose an approach to evaluate the distribution of damages on an individual basis as a continuous function of space. Furthermore, rockfall risk is evaluated through (i) the damage expectation and based on (ii) the value-at-risk (VAR) and (iii) the expected shortfall (ES). VaR and ES risk measures allow better assessing the risk due to extreme events and accounting for various short-term/long-term constraints faced by stakeholders. This procedure is applied to Le Brocey slope (municipality of Crolles, French Alps), frequently affected by rockfall events, illustrating its potential for land-use planning. Notably, obtained individual risk values can be confronted with acceptability thresholds to perform legal zoning on sound basis. Also, they allow comparison of risk management strategies as function of different behaviors towards risk, budgetary constraints and/or temporal horizons. Hence, the approach provides valuable tools for future land-use planning and decision-making. It could easily be transferred to other hazards as a wider contribution to the determination of the best balance between safety and sustainability.

Static and dynamic risk capital allocations with the Euler rule

Risk capital allocations are of central importance in performance measurement. A popular solution concept in the academic literature is the Euler rule. This paper studies the volatility of the Euler rule for capital allocation in static and dynamic empirical applications with a simulated history. The Euler rule is not continuous with respect to small changes in the underlying risk capital allocation problem. We show that, when combined with value-at-risk, the Euler rule is very sensitive to empirical measurement error. The use of a known distribution with estimated parameters helps to reduce this error. The Euler rule with an expected shortfall risk measure is less volatile, but it is still more volatile than the proportional rule.

Bias-variance trade-off in portfolio optimization under expected shortfall with regularization

The optimization of a large random portfolio under the expected shortfall risk measure with an regularizer is carried out by analytical calculation for the case of uncorrelated Gaussian returns. The regularizer reins in the large sample fluctuations and the concomitant divergent estimation error, and eliminates the phase transition where this error would otherwise blow up. In the data-dominated region, where the number of different assets in the portfolio is much less than the length of the available time series, the regularizer plays a negligible role even if its strength is large, while in the opposite limit, where the size of samples is comparable to, or even smaller than the number of assets, the optimum is almost entirely determined by the regularizer. We construct the contour map of estimation error on the versus plane and find that for a given value of the estimation error the gain in due to the regularizer can reach a factor of about four for a sufficiently strong regularizer.

Pricing formulae for derivatives in insurance using Malliavin calculus

In this paper, we provide a valuation formula for different classes of actuarial and financial contracts which depend on a general loss process by using Malliavin calculus. Similar to the celebrated Black–Scholes formula, we aim to express the expected cash flow in terms of a building block. The former is related to the loss process which is a cumulated sum indexed by a doubly stochastic Poisson process of claims allowed to be dependent on the intensity and the jump times of the counting process. For example, in the context of stop-loss contracts, the building block is given by the distribution function of the terminal cumulated loss taken at the Value at Risk when computing the expected shortfall risk measure.

Static and Dynamic Risk Capital Allocations With the Euler Rule

Risk capital allocations are of central importance in performance measurement. A popular solution concept in the academic literature is the Euler rule. This paper studies the volatility of the Euler rule for capital allocation in static and dynamic empirical applications with a simulated history. The Euler rule is not continuous with respect to small changes in the underlying risk capital allocation problem. We show that, when combined with value-at-risk, the Euler rule is very sensitive to empirical measurement error. The use of a known distribution with estimated parameters helps to reduce this error. The Euler rule with an expected shortfall risk measure is less volatile, but it is still more volatile than the proportional rule.

Convex risk measures based on generalized lower deviation and their applications

Considering the implementability and the properties that a reasonable and realistic risk measure should satisfy, we propose a new class of risk measures based on generalized lower deviation with respect to a chosen benchmark. Besides convexity and monotonicity, our new risk measure can reflect the investor's degree of risk aversion as well as the fat-tail phenomenon of the loss distribution with the help of different benchmarks and weighted functions. Based on the new risk measure, we establish a realistic portfolio selection model taking market frictions into account. To examine the influence of the benchmarks and weighted functions on the optimal portfolio and its performance, we carry out a series of empirical studies in Chinese stock markets. Our in-sample and out-of-sample results show that the new risk measure and the corresponding portfolio selection model can reflect the investor's risk averse attitude and the impact of different trading constraints. Most importantly, with the new risk measure we can obtain an optimal portfolio which is more robust and superior to the optimal portfolios obtained with the traditional expected shortfall risk measures.

Bias-Variance Trade-Off in Portfolio Optimization under Expected Shortfall with <sub>2</sub> Regularization

The optimization of a large random portfolio under the Expected Shortfall risk measure with an l2 regularizer is carried out by analytical calculation. The regularizer reins in the large sample fluctuations and the concomitant divergent estimation error, and eliminates the phase transition where this error would otherwise blow up. In the data-dominated region, where the number of different assets in the portfolio is much less than the length of the available time series, the regularizer plays a negligible role, while in the opposite limit (which occurs much more frequently in practice), where the size of samples is comparable to, or even smaller than the number of assets, the optimum is almost entirely determined by the regularizer. Our results show that the transition region between these two extremes is relatively narrow, and it is only here that one can meaningfully speak of a trade-off between fluctuations and bias. The cause of both the anomalously large fluctuations and the limited usefulness of regularization is the unboundedness of Expected Shortfall as a loss function, a property it shares with all the other coherent measures, but also with downside risk measures in general, including Value at Risk.

On the Coherence of VAR Risk Measures for Levy Stable Distributions

The Value-at-Risk (VaR) risk measure has been widely used in finance and insurance for capital and risk management. However, in recent years it has fallen somewhat out of favour due to a seminal paper by Artzner et al. (1999) who showed that VaR does not in general have all the four coherence properties which are desirable for a risk measure. In particular, the violation of the sub-additive property discourages diversification and is counter-intuitive to risk finance. In this paper, it is proved (Theorem 3.1) that VaR for independent Levy-stable random variates is a coherent risk measure being translational invariant, monotonic, positively homogeneous and sub-additive. That is, Levy-stable distributions are VaR coherent. As Levy-stable distributions are a rich class of probability distributions, the VaR risk measure may still have widespread applications. A brief comparative discussion is also given for L-stable variates for the expected shortfall risk measure.

Risk management with expectiles

Expectiles (EVaR) are a one-parameter family of coherent risk measures that have been recently suggested as an alternative to quantiles (VaR) and to expected shortfall (ES). In this work we review their known properties, we discuss their financial meaning, we compare them with VaR and ES and we study their asymptotic behaviour, refining some of the results in Bellini et al. [(2014). “Generalized Quantiles as Risk Measures.” Insurance: Mathematics and Economics, 54:41–48]. Moreover, we present a real-data example for the computation of expectiles by means of simple Garch(1,1) models and we assess the accuracy of the forecasts by means of a consistent loss function as suggested by Gneiting [(2011). “Making and Evaluating Point Forecast.” Journal of the American Statistical Association, 106 (494): 746–762]. Theoretical and numerical results indicate that expectiles are perfectly reasonable alternatives to VaR and ES risk measures.

Empirical analysis of long memory, leverage, and distribution effects for stock market risk estimates

In this study, eight generalized autoregressive conditional heteroskedasticity (GARCH) types of variance specifications and two return distribution settings, the normal and skewed generalized Student's t (SGT) of Theodossiou (1998), totaling nine GARCH-based models, are utilized to forecast the volatility of six stock indices, and then both the out-of-sample-period value-at-risk (VaR) and the expected shortfall (ES) are estimated following the rolling window approach. Moreover, the in-sample VaR is estimated for both the global financial crisis (GFC) period and the non-GFC period. Subsequently, through several accuracy measures, nine models are evaluated in order to explore the influence of long memory, leverage, and distribution effects on the performance of VaR and ES forecasts. As shown by the empirical results of the nine models, the long memory, leverage, and distribution effects subsist in the stock markets. Moreover, regarding the out-of-sample VaR forecasts, long memory is the most important effect, followed by the leverage effect for the low level, whereas the distribution effect is crucial for the high level. As for the three VaR approaches, weighted historical simulation achieves the best VaR forecasting performance, followed by filtered historical simulation, whereas the parametric approach has the worst VaR forecasting performance for all the levels. Furthermore, VaR models underestimate the true risk, whereas ES models overestimate the true risk, indicating that the ES risk measure is more conservative than the VaR risk measure. Additionally, based on back-testing, the VaR provides a better risk forecast than the ES since the ES highly overestimates the true risk. Notably, long memory is important for the ES estimate, whereas both the long memory and the leverage effect are crucial for the VaR estimate. Finally, via in-sample VaR forecasts in regard to the low level, it is found that long memory is important for the non-GFC period, whereas the distribution effect is crucial for the GFC period. On the other hand, with regard to the high level, the distribution effect is crucial for both the non-GFC and the GFC period. These results seem to be consistent with those found in the out-of-sample VaR forecasts. In accordance with these results, several important policy implications are proposed in this study.

Efficient VaR and Expected Shortfall computations for nonlinear portfolios within the delta-gamma approach

We present four numerical methods to compute the Value-at-Risk and Expected Shortfall risk measure values of portfolios with financial options. The numerical methods are based on either wavelets or Fourier cosine approximations and belong to the class of Fourier inversion methods. We show that the risk measures can be efficiently calculated in terms of accuracy and CPU time. Besides, we provide a theoretical result about the shape of the resulting probability density. This a priori knowledge, allows us to enhance the efficiency and effectiveness of the proposed methods. Finally, we assess the accuracy of the approach in the presence of convexity or concavity properties of the financial portfolios.

Coherent risk measures by pricing functionals

The aim of this paper is to introduce a new class of Coherent Risk Measures in Banach lattices, whose dual representation is consistent to the pricing functionals of an incomplete market. As a sub-class, we introduce the corresponding Expected Shortfall Risk Measures.

Optimal risk transfers in insurance groups

Optimal risk transfers are derived within an insurance group consisting of two separate legal entities, operating under potentially different regulatory capital requirements and capital costs. Consistent with regulatory practice, capital requirements for each entity are computed by either a value-at-risk or an expected shortfall risk measure. The optimality criterion consists of minimising the risk-adjusted value of the total group liabilities, with valuation carried out using a cost-of-capital approach. The optimisation problems are analytically solved and it is seen that optimal risk transfers often involve the transfer of tail risk (unlimited reinsurance layers) to the more weakly regulated entity. We show that, in the absence of a capital requirement for the credit risk that specifically arises from the risk transfer, optimal risk transfers achieve capital efficiency at the cost of increasing policyholder deficit. However, when credit risk is properly reflected in the capital requirement, incentives for tail-risk transfers vanish and policyholder welfare is restored.

Efficient VAR and Expected Shortfall Computations for Non-Linear Portfolios within the Delta-Gamma Approach

We present four numerical methods to compute the Value-at-Risk and Expected Shortfall risk measure values of portfolios with financial options. The numerical methods are based on either wavelets or Fourier cosine approximations and belong to the class of Fourier inversion methods. We show that the risk measures can be efficiently calculated in terms of accuracy and CPU time. Besides, we provide a theoretical result about the shape of the resulting probability density. This a priori knowledge, allows us to enhance the efficiency and effectiveness of the proposed methods. Finally, we assess the accuracy of the approach in the presence of convexity or concavity properties of the financial portfolios.

Optimal Risk Transfers in Insurance Groups

Optimal risk transfers are derived within an insurance group consisting of two separate legal entities, operating under potentially different regulatory capital requirements and capital costs. Consistently with regulatory practice, capital requirements for each entity are computed by either a Value-at-Risk or an Expected Shortfall risk measure. The optimality criterion consists of minimizing the risk-adjusted value of the total group liabilities, with valuation carried out using a cost-of-capital approach. The optimization problems are analytically solved and it is seen that optimal risk transfers often involve the transfer of tail risk (unlimited reinsurance layers) to the more weakly regulated entity. We show that, in the absence of a capital requirement for the credit risk arising from the risk transfer, optimal risk transfers achieve capital efficiency at the cost of increasing policyholder deficit. However, when credit risk is properly reflected in the capital requirement, incentives for tail-risk transfers vanish and policyholder welfare is restored.

Tail nonlinearly transformed risk measure and its application

This paper discusses risk measurement and portfolio selection under the mean-risk framework. Through introducing a nonlinear convex transformation to large losses, we propose a new class of risk measures which are convex and monotone. We demonstrate the new risk measure's favorable financial and mathematical properties, and consider its estimation in practice and relevant consistent and asymptotic issues. A realistic portfolio selection model based on the new risk measure is then established with typical market frictions taken into account simultaneously. Based on trade data from Chinese stock markets and American stock markets over stable and volatile periods, respectively, both in-sample and out-of-sample empirical studies are carried out. Theoretical and empirical results show that the new risk measure and the corresponding portfolio selection model can not only reflect different investors' risk averse attitudes and control the fat-tail phenomenon of the return distribution, but also find a realistic optimal portfolio with superior performance and robustness than the corresponding optimal portfolio obtained under the expected shortfall risk measure.

Nonlinearly weighted convex risk measure and its application

We propose a new class of risk measures which satisfy convexity and monotonicity, two well-accepted axioms a reasonable and realistic risk measure should satisfy. Through a nonlinear weight function, the new measure can flexibly reflect the investor’s degree of risk aversion, and can control the fat-tail phenomenon of the loss distribution. A realistic portfolio selection model with typical market frictions taken into account is established based on the new measure. Real data from the Chinese stock markets and American stock markets are used for empirical comparison of the new risk measure with the expected shortfall risk measure. The in-sample and out-of-sample empirical results show that the new risk measure and the corresponding portfolio selection model can not only reflect the investor’s risk-averse attitude and the impact of different trading constraints, but can find robust optimal portfolios, which are superior to the corresponding optimal portfolios obtained under the expected shortfall risk measure.

Semiparametric Estimation of Dynamic Conditional Expected Shortfall Models

The paper proposes a simple estimator for a class of Conditional Expected Shortfall risk measures. The estimator is semiparametric, in the sense that it does not require a full specification of the conditional distribution of the data, and it is very simple to compute, being a least squares estimator with a closed-form expression. We establish its consistency and asymptotic normality under mild regularity conditions. A simulation study provides evidence of the excellent finite-sample properties of the estimator and an application to some exchange rates highlights the semiparametric aspect of the new estimator.