What is the best risk measure in practice? A comparison of standard measures

Expected Shortfall (ES) has been widely accepted as a risk measure that is conceptually superior to Value-at-Risk (VaR). At the same time, however, it has been criticised for issues relating to backtesting. In particular, ES has been found not to be elicitable which means that backtesting for ES is less straightforward than, e.g., backtesting for VaR. Expectiles have been suggested as potentially better alternatives to both ES and VaR. In this paper, we revisit commonly accepted desirable properties of risk measures like coherence, comonotonic additivity, robustness and elicitability. We check VaR, ES and Expectiles with regard to whether or not they enjoy these properties, with particular emphasis on Expectiles. We also consider their impact on capital allocation, an important issue in risk management. We find that, despite the caveats that apply to the estimation and backtesting of ES, it can be considered a good risk measure. As a consequence, there is no sufficient evidence to justify an all-inclusive replacement of ES by Expectiles in applications. For backtesting ES, we propose an empirical approach that consists in replacing ES by a set of four quantiles, which should allow to make use of backtesting methods for VaR. Keywords: Backtesting; capital allocation; coherence; diversification; elicitability; expected shortfall; expectile; forecasts; probability integral transform (PIT); risk measure; risk management; robustness; value-at-riskComment: 27 pages, 1 tabl

Fair Estimation of Capital Risk Allocation

In this paper we develop a novel methodology for estimation of risk capital allocation. The methodology is rooted in the theory of risk measures. We work within a general, but tractable class of law-invariant coherent risk measures, with a particular focus on expected shortfall. We introduce the concept of fair capital allocations and provide explicit formulae for fair capital allocations in case when the constituents of the risky portfolio are jointly normally distributed. The main focus of the paper is on the problem of approximating fair portfolio allocations in the case of not fully known law of the portfolio constituents. We define and study the concepts of fair allocation estimators and asymptotically fair allocation estimators. A substantial part of our study is devoted to the problem of estimating fair risk allocations for expected shortfall. We study this problem under normality as well as in a nonparametric setup. We derive several estimators, and prove their fairness and/or asymptotic fairness. Last, but not least, we propose two backtesting methodologies that are oriented at assessing the performance of the allocation estimation procedure. The paper closes with a substantial numerical study of the subject

The Sign RCA Models: Comparing Predictive Accuracy of VaR Measures

Evaluating Value at Risk (VaR) methods of predictive accuracy in an objective and effective framework is important for both efficient capital allocation and loss prediction. From this reasons, finding an adequate method of estimating and backtesting is crucial for both the regulators and the risk managers’. The Sign RCA models may be useful to obtain the accurate forecasts of VaR. In this research one briefly describes the Sign RCA models, the Value at Risk and backtesting. We compare the predictive accuracy of alternative VaR forecasts obtained from different models. Empirical example is mainly related to the PBG Capital Group shares on the Warsaw Stock Exchange.Family of Sign RCA Models, Value at Risk, backtesting, loss function.

Early Detection Techniques for Market Risk Failure

The implementation of appropriate statistical techniques for monitoring conditional VaR models, i.e, backtesting, reported by institutions is fundamental to determine their exposure to market risk. Backtesting techniques are important since the severity of the departures of the VaR model from market results determine the penalties imposed for inadequate VaR models. In this paper we make six contributions to backtesting techniques. In particular, we show that the Kupiec test can be viewed as a combination of CUSUM change point tests; we detail the lack of power of CUSUM methods in detecting violations of VaR as soon as these occur; we develop an alternative technique based on weighted U-statistic type processes that have power against wrong specifications of the risk measure and early detection; we show these new backtesting techniques are robust to the presence of estimation risk; we construct a new class of weight functions that can be used to weight our processes; and our methods are applicable both under conditional and unconditional VaR settings

Expected Shortfall is jointly elicitable with Value at Risk - Implications for backtesting

In this note, we comment on the relevance of elicitability for backtesting risk measure estimates. In particular, we propose the use of Diebold-Mariano tests, and show how they can be implemented for Expected Shortfall (ES), based on the recent result of Fissler and Ziegel (2015) that ES is jointly elicitable with Value at Risk

Portfolio optimization for heavy-tailed assets: Extreme Risk Index vs. Markowitz

Using daily returns of the S&P 500 stocks from 2001 to 2011, we perform a backtesting study of the portfolio optimization strategy based on the extreme risk index (ERI). This method uses multivariate extreme value theory to minimize the probability of large portfolio losses. With more than 400 stocks to choose from, our study seems to be the first application of extreme value techniques in portfolio management on a large scale. The primary aim of our investigation is the potential of ERI in practice. The performance of this strategy is benchmarked against the minimum variance portfolio and the equally weighted portfolio. These fundamental strategies are important benchmarks for large-scale applications. Our comparison includes annualized portfolio returns, maximal drawdowns, transaction costs, portfolio concentration, and asset diversity in the portfolio. In addition to that we study the impact of an alternative tail index estimator. Our results show that the ERI strategy significantly outperforms both the minimum-variance portfolio and the equally weighted portfolio on assets with heavy tails.Comment: Manuscript accepted in the Journal of Empirical Financ