Value-at-risk versus expected shortfall: A practical perspective

Value-at-Risk (VaR) has become a standard risk measure for financial risk management. However, many authors claim that there are several conceptual problems with VaR. Among these problems, an important one is that VaR disregards any loss beyond the VaR level. We call this problem the “tail risk”. In this paper, we illustrate how the tail risk of VaR can cause serious problems in certain cases, cases in which expected shortfall can serve more aptly in its place. We discuss two cases: concentrated credit portfolio and foreign exchange rates under market stress. We show that expected shortfall requires a larger sample size than VaR to provide the same level of accuracy.     

Optimal capital growth with convex shortfall penalties

The optimal capital growth strategy or Kelly strategy has many desirable properties such as maximizing the asymptotic long-run growth of capital. However, it has considerable short-run risk since the utility is logarithmic, with essentially zero Arrow–Pratt risk aversion. It is common to control risk with a Value-at-Risk (VaR) constraint defined on the end of horizon wealth. A more effective approach is to impose a VaR constraint at each time on the wealth path. In this paper, we provide a method to obtain the maximum growth while staying above an ex-ante discrete time wealth path with high probability, where shortfalls below the path are penalized with a convex function of the shortfall. The effect of the path VaR condition and shortfall penalties is a lower growth rate than the Kelly strategy, but the downside risk is under control. The asset price dynamics are defined by a model with Markov transitions between several market regimes and geometric Brownian motion for prices within a regime. The stochastic investment model is reformulated as a deterministic programme which allows the calculation of the optimal constrained growth wagers at discrete points in time.     

On the coherence of expected shortfall

Expected shortfall (ES) in several variants has been proposed as remedy for the deficiencies of value-at-risk (VaR) which in general is not a coherent risk measure. In fact, most definitions of ES lead to the same results when applied to continuous loss distributions. Differences may appear when the underlying loss distributions have discontinuities. In this case even the coherence property of ES can get lost unless one took care of the details in its definition. We compare some of the definitions of ES, pointing out that there is one which is robust in the sense of yielding a coherent risk measure regardless of the underlying distributions. Moreover, this ES can be estimated effectively even in cases where the usual estimators for VaR fail.     

Nonparametric Estimation of Expected Shortfall

The expected shortfall is an increasingly popular risk measurein financial risk management and it possesses the desired sub-additivityproperty, which is lacking for the value at risk (VaR). We considertwo nonparametric expected shortfall estimators for dependentfinancial losses. One is a sample average of excessive losseslarger than a VaR. The other is a kernel smoothed version ofthe first estimator (Scaillet, 2004 Mathematical Finance), hopingthat more accurate estimation can be achieved by smoothing.Our analysis reveals that the extra kernel smoothing does notproduce more accurate estimation of the shortfall. This is differentfrom the estimation of the VaR where smoothing has been shownto produce reduction in both the variance and the mean squareerror of estimation. Therefore, the simpler ES estimator basedon the sample average of excessive losses is attractive forthe shortfall estimation.     

Risk quantification for commodity ETFs: Backtesting value-at-risk and expected shortfall

This paper calibrates risk assessment of alternative methods for modeling commodity ETFs. We implement recently proposed backtesting techniques for both value-at-risk (VaR) and expected shortfall (ES) under parametric and semi-nonparametric techniques. Our results indicate that skewed-t and Gram-Charlier distributional assumptions present the best relative performance for individual Commodity ETFs for those confidence levels recommended by Basel Accords. In view of these results, we recommend the application of leptokurtic distributions and semi-nonparametric techniques to mitigate regulation concerns about global financial stability of commodity business.     

Estimation of risk contributions with MCMC

Determining risk contributions of unit exposures to portfolio-wide economic capital is an important task in financial risk management. Computing risk contributions involves difficulties caused by rare-event simulations. In this study, we address the problem of estimating risk contributions when the total risk is measured by value-at-risk (VaR). Our proposed estimator of VaR contributions is based on the Metropolis-Hasting (MH) algorithm, which is one of the most prevalent Markov chain Monte Carlo (MCMC) methods. Unlike existing estimators, our MH-based estimator consists of samples from the conditional loss distribution given a rare event of interest. This feature enhances sample efficiency compared with the crude Monte Carlo method. Moreover, our method has consistency and asymptotic normality, and is widely applicable to various risk models having a joint loss density. Our numerical experiments based on simulation and real-world data demonstrate that in various risk models, even those having high-dimensional (≈500) inhomogeneous margins, our MH estimator has smaller bias and mean squared error when compared with existing estimators.     

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.     

Backtesting for risk-based regulatory capital

In this paper we present a framework for backtesting all currently popular risk measurement methods for quantifying market risk (including value-at-risk and expected shortfall) using the functional delta method. Estimation risk can be taken explicitly into account. Based on a simulation study we provide evidence that tests for expected shortfall with acceptable low levels have a better performance than tests for value-at-risk in realistic financial sample sizes. We propose a way to determine multiplication factors, and find that the resulting regulatory capital scheme using expected shortfall compares favorably to the current Basel Accord backtesting scheme.     

Extreme spectral risk measures: An application to futures clearinghouse margin requirements

This paper applies the extreme-value (EV) generalised pareto distribution to the extreme tails of the return distributions for the S&P500, FT100, DAX, Hang Seng, and Nikkei225 futures contracts. It then uses tail estimators from these contracts to estimate spectral risk measures, which are coherent risk measures that reflect a user’s risk-aversion function. It compares these to VaR and expected shortfall (ES) risk measures, and compares the precision of their estimators. It also discusses the usefulness of these risk measures in the context of clearinghouses setting initial margin requirements, and compares these to the SPAN measures typically used.     

On the significance of expected shortfall as a coherent risk measure

This article shows that any coherent risk measure is given by a convex combination of expected shortfalls, and an expected shortfall (ES) is optimal in the sense that it gives the minimum value among the class of plausible coherent risk measures. Hence, it is of great practical interest to estimate the ES with given confidence level from the market data in a stable fashion. In this article, we propose an extrapolation method to estimate the ES of interest. Some numerical results are given to show the efficiency of our method.     

Risk management in the energy markets and Value-at-Risk modelling: a hybrid approach

This paper proposes a set of Value-at-Risk (VaR) models appropriate to capture the dynamics of energy prices and subsequently quantify energy price risk by calculating VaR and expected shortfall measures. Amongst the competing VaR methodologies evaluated in this paper, besides the commonly used benchmark models, a Monte Carlo (MC) simulation approach and a hybrid MC with historical simulation approach, both assuming various processes for the underlying spot prices, are also being employed. All VaR models are empirically tested on eight spot energy commodities that trade futures contracts on the New York Mercantile Exchange (NYMEX) and the constructed Spot Energy Index. A two-stage evaluation and selection process is applied, combining statistical and economic measures, to choose amongst the competing VaR models. Finally, both long and short trading positions are considered as it is of utmost importance for energy traders and risk managers to be able to capture efficiently the characteristics of both tails of the distributions.     

An Analytical Evaluation Method of the Operational Risk Using Fast Wavelet Expansion Techniques

A financial institution that adopts an advanced measurement approach (AMA) as a method of computing operational risk capital has to measure 99.9 % value-at-risk (VaR) as the amount of an operational risk. The most popular method to satisfy the AMA standards requires the evaluation of aggregate (compound) loss distribution, which is called the loss distribution approach (LDA). The Monte Carlo (MC) method is a well known method for calculating VaR under the LDA. However, when using the MC method to calculate VaR, the statistical error of VaR for the fat-tailed distribution increases and the computation time increases in proportion to the expected value of frequency distribution. Since the MC method has these problems, this paper presents a new methodology to compute VaR under the LDA using fast wavelet expansion techniques. The key features of our algorithm are follows: (1) Scale transformation technique for loss distributions, (2) Double exponential transformation for oscillatory integrals, (3) Finite series expansion of the wavelet scaling coefficients, (4) Wynn’s epsilon algorithm to accelerate the convergence of those series, (5) Efficient cubic spline interpolation method to calculate the moment generating function. We illustrate the effectiveness of our algorithms through numerical examples.     

Policy uncertainty and the capital shortfall of global financial firms

This paper examines how economic policy uncertainty affects financial firms' capital shortages in the event of a new crisis. By employing a global economic policy uncertainty index, we show that an increase in policy uncertainty leads to future capital shortfall increases in the event of a severe market decline. This effect of policy uncertainty is of a similar magnitude for all firms across all regions and sectors, albeit with a few exceptions. As expected, well-capitalized financial firms are less affected. Our findings have important implications for policymakers and politicians, since if their response during a severe market decline is not timely and decisive, the delay will come at a cost. Further, in terms of the implications for firms' managers, we show that during periods of elevated policy uncertainty and a severe market downturn, firms will face additional, unexpected capital requirements.     

Bank capital shortfall in the euro area

We quantify the bank capital shortfall that results from a financial crisis by estimating a macro-finance dynamic stochastic general equilibrium model that captures the interactions between the financial and real sectors of the euro-area economy. The introduction of both deposit and shadow banks captures several characteristics of the banking system and reveals a financial amplification mechanism. By using a combination of a large positive risk shock and a large negative investment shock, we show that a crisis similar to that observed in 2008 would generate a bank capital shortfall between 2.2% and 3% of euro-area GDP, which corresponds to approximately 207–282 billion euros.     

From BASEL III to BASEL IV and beyond: Expected shortfall and expectile risk measures

The article contributes to the ongoing search for a market risk measure that is both coherent and elicitable. We compare two traditional measures, namely Value-at-Risk and the expected shortfall, with another relatively novel one established on the expectile probability term. Our research is based on five models: Black–Scholes, exponential tempered stable, Heston, Bates and another stochastic volatility model with a tempered stable jump correction. We apply the general Fourier inversion formula to derive closed form formulas for calculating not only the expectile based risk measure but also the Value-at-Risk and the expected shortfall. These models are calibrated by combining nonlinear programming with simulated annealing at a moving window. Additionally, we compare the generated values of the risk measures with the real ones. Last but not least, we modify the expectile based risk measure as well as the expected shortfall by introducing correction coefficients.     

Capital adequacy rules, catastrophic firm failure, and systemic risk

This paper studies capital adequacy rules based on Value-at-Risk (VaR), leverage ratios, and stress testing. VaR is the basis of Basel II, and all three approaches are proposed in Basel III. This paper makes three contributions to the literature. First, we prove that these three rules provide an incentive to increase the probability of catastrophic financial institution failure. Collectively, these rules provide an incentive to increase (not decrease) systemic risk. Second, we argue that an unintended consequence of the Basel II VaR capital adequacy rules was the 2007 credit crisis. Third, we argue that to reduce systemic risk, a new capital adequacy rule is needed. One that is based on a risk measure related to the conditional expected loss given insolvency.     

基于VaR模型的商业银行体系系统性风险研究

本文以中国2016年之前上市商业银行作为中国银行业的代表,测算银行业系统性 风险VaR。整体来讲,我国银行业系统性风险较低,但VaR在2015年较高。虽如此,我国银行业资本持有量能够抵御银行体系的系统性风险。在系统性风险VaR贡献度方面,本文实证分析表明,在样本期间内,浦发银行、中国银行、农业银行、交通银行贡献度较高。银行体系系统 性风险VaR受GDP增长率和沪深300指数收益率的显著影响。     

A Bayesian approach for more reliable tail risk forecasts

This paper demonstrates that existing quantile regression models used for jointly forecasting Value-at-Risk (VaR) and expected shortfall (ES) are sensitive to initial conditions. Given the importance of these measures in financial systems, this sensitivity is a critical issue. A new Bayesian quantile regression approach is proposed for estimating joint VaR and ES models. By treating the initial values as unknown parameters, sensitivity issues can be dealt with. Furthermore, new additive-type models are developed for the ES component that are more robust to initial conditions. A novel approach using the open-faced sandwich (OFS) method is proposed which improves uncertainty quantification in risk forecasts. Simulation and empirical results highlight the improvements in risk forecasts ensuing from the proposed methods.     

论银行市场风险的资本计提——兼评内部模型法的适用性

市场风险及其监管资本要求的计量历来为业界和监管当局所关注。近期,次贷危机爆发导致的市场动荡使得全球银行业和监管当局开始重新审视其市场风险管理和监管资本要求。文章结合国际银行业和监管机构计量市场风险及其监管资本要求的当前做法,针对我国银行业的实际情况,重点探索了内部模型法在我国银行业的适用性,尤其是从方法论、特殊风险计量、验证等角度探讨了内部模型法的主要工具——风险价值体系在我国银行业计量市场风险及其监管资本要求的适用性,并从方法论和应用层面提出了相应的政策建议。