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.     

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.     

The Risk Map: A new tool for validating risk models

This paper presents a new method to validate risk models: the Risk Map. This method jointly accounts for the number and the magnitude of extreme losses and graphically summarizes all information about the performance of a risk model. It relies on the concept of a super exception, which is defined as a situation in which the loss exceeds both the standard Value-at-Risk (VaR) and a VaR defined at an extremely low probability. We then formally test whether the sequences of exceptions and super exceptions are rejected by standard model validation tests. We show that the Risk Map can be used to validate market, credit, operational, or systemic risk estimates (VaR, stressed VaR, expected shortfall, and CoVaR) or to assess the performance of the margin system of a clearing house.     

Portfolio credit-risk optimization

This paper evaluates several alternative formulations for minimizing the credit risk of a portfolio of financial contracts with different counterparties. Credit risk optimization is challenging because the portfolio loss distribution is typically unavailable in closed form. This makes it difficult to accurately compute Value-at-Risk (VaR) and expected shortfall (ES) at the extreme quantiles that are of practical interest to financial institutions. Our formulations all exploit the conditional independence of counterparties under a structural credit risk model. We consider various approximations to the conditional portfolio loss distribution and formulate VaR and ES minimization problems for each case. We use two realistic credit portfolios to assess the in- and out-of-sample performance for the resulting VaR- and ES-optimized portfolios, as well as for those which we obtain by minimizing the variance or the second moment of the portfolio losses. We find that a Normal approximation to the conditional loss distribution performs best from a practical standpoint.     

Predicting tail-related risk measures: The consequences of using GARCH filters for non-GARCH data

We investigate the consequences for Value-at-Risk and expected shortfall purposes of using a GARCH filter on various mis-specified processes. In general, we find that the McNeil and Frey (McNeil, A.J. and R. Frey, 2000, Estimation of Tail-Related Risk Measures for Heteroscedastic Financial Time Series: An Extreme Value Approach, Journal of Empirical Finance 7, 271–300.) two step procedure has very good forecasting properties. Using an unconditional non-filtered tail estimate also appears to perform satisfactorily for expected shortfall measurements but less so for VaR computations. Methods assuming specific densities such as the Gaussian or Student-t may yield wrong predictions. Thus, the use of an adequacy test for filtered data to given densities appears relevant. The paper builds on recent techniques to obtain thresholds for extreme value computations. Statistical tests for the expected shortfall, based on the circular bootstrap, are developed.     

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.     

Estimation of tail-related risk measures for heteroscedastic financial time series: an extreme value approach

We propose a method for estimating Value at Risk (VaR) and related risk measures describing the tail of the conditional distribution of a heteroscedastic financial return series. Our approach combines pseudo-maximum-likelihood fitting of GARCH models to estimate the current volatility and extreme value theory (EVT) for estimating the tail of the innovation distribution of the GARCH model. We use our method to estimate conditional quantiles (VaR) and conditional expected shortfalls (the expected size of a return exceeding VaR), this being an alternative measure of tail risk with better theoretical properties than the quantile. Using backtesting of historical daily return series we show that our procedure gives better 1-day estimates than methods which ignore the heavy tails of the innovations or the stochastic nature of the volatility. With the help of our fitted models we adopt a Monte Carlo approach to estimating the conditional quantiles of returns over multiple-day horizons and find that this outperforms the simple square-root-of-time scaling method.     

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.     

Fast simulations in credit risk

We consider the problem of simulating tail loss probabilities and expected losses conditioned on exceeding a large threshold (expected shortfall) for credit portfolios. Our new idea, called the geometric shortcut, allows an efficient simulation for the case of independent obligors. It is even possible to show that, when the average default probability tends to zero, its asymptotic efficiency is higher than that of the naive algorithm. The geometric shortcut is also useful for models with dependent obligors and can be used for dependence structures modeled with arbitrary copulae. The paper contains the details for simulating the risk of the normal copula credit risk model by combining outer importance sampling with the geometric shortcut. Numerical results show that the new method is efficient in assessing tail loss probabilities and expected shortfall for credit risk portfolios. The new method outperforms all known methods, especially for credit portfolios consisting of weakly correlated obligors and for evaluating the tail loss probabilities at many thresholds in a single simulation run.     

Estimating portfolio risk for tail risk protection strategies

We forecast portfolio risk for managing dynamic tail risk protection strategies, based on extreme value theory, expectile regression, copula‐GARCH and dynamic generalized autoregressive score models. Utilizing a loss function that overcomes the lack of elicitability for expected shortfall, we propose a novel expected shortfall (and value‐at‐risk) forecast combination approach, which dominates simple and sophisticated standalone models as well as a simple average combination approach in modeling the tail of the portfolio return distribution. While the associated dynamic risk targeting or portfolio insurance strategies provide effective downside protection, the latter strategies suffer less from inferior risk forecasts, given the defensive portfolio insurance mechanics.     

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.     

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.     

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.     

Exposure-Based Cash-Flow-at-Risk: An Alternative to VaR for Industrial Companies

Cash-Flow-at-Risk (CFaR) is the cash flow equivalent of Value-at-Risk (VaR), a measure widely used as the basis for risk management in financial institutions. Whereas VaR-based systems specify the maximum amount of total value a firm is expected to lose under most foreseeable conditions (for example, with a 99% confidence level), CFaR-based systems determine the maximum shortfall of cash the firm is willing to tolerate. CFaR is gaining in popularity among industrial companies for much the same reasons VaR has succeeded with financial firms: it sums up all the company's risk exposures in a single number that can be used to guide corporate risk management decisions.
The authors describe a six-step process for calculating a measure they call "exposure-based CFaR" and then demonstrate its application to Norsk Hydro, the Norwegian industrial conglomerate. Exposure-based CFaR involves the estimation of a set of exposure coefficients that provide information about how various macroeconomic and market variables are expected to affect the company's cash flow, while also accounting for interdependencies among such effects. The resulting model enables management to estimate the variability in corporate cash flow as a function of various risks, and to predict how a hedging contract or a change in financial structure will alter the company's risk profile.     

Ethical and unethical investments under extreme market conditions

This study investigates the time-varying volatility and risk measures of ethical and unethical investments. We apply the Bayesian Markov-switching generalized autoregressive conditional heteroscedasticity (MS-GARCH) approach to compute the value-at-risk (VaR) and expected shortfall (ES) of ethical and unethical indices returns, which allows for detecting the differences between ethical and unethical investments. The innovative finding of our study is that ethical investments are less affected during global financial crises compared with unethical and conventional investments. The policy implication of this study is that investors should consider ethical investments as a hedging asset for their portfolios during extreme market conditions.     

Securitization and systemic risk: An empirical investigation on Italian banks over the financial crisis

This research examines the effects of securitization on the bank's risk exposure both in terms of individual expected shortfall and marginal expected shortfall as a measure of systemic risk. The relationship between securitization activity and tail risks is especially relevant in light of the consequences for financial stability, both for the individual securitizing banks and for the market as a whole, as the financial crisis 2007–2008 reveals. By using a sample of Italian listed banks over the period 2000–2009, we find that securitizing banks have, on average, higher expected losses in case of extreme events. This adds new evidence on the main findings in the literature that focused on the evidence that risk transfer through securitization is relatively insignificant compared to the risk retained by the originating bank. We show that this risk retention is in terms of an increase of tail risk. We also find that securitization increases the probability of banks to become “systemically” riskier, but we find no difference when comparing the pre-crisis with the post-crisis period. This suggests that the systemic exposures of Italian banks are still as high as before the crisis with severe implications for financial stability.     

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.     

Accrual mispricing,value-at-risk,and expected stock returns

We investigate the extent to which a parsimonious measure of maximum likely loss that captures the tail risk of returns—known as value-at-risk (VaR)—explains the relationship between accruals and the cross-sectional dispersion of expected stock returns. We construct portfolios based on Sloan’s (Account Rev 71(3):289–315, 1996) total accruals (TA) measure and individual asset-level VaR, which reflects the dynamic behavior of the asset distribution. We document that VaR is in congruence with portfolio-level accruals and that there is a significant positive relationship between VaR and the cross-section of portfolio returns. Allowing a double-sort involving VaR and TA further suggests that the spread between low- and high-TA portfolios is significantly attenuated after controlling for VaR. We also conduct a firm-level cross-sectional regression analysis and demonstrate that the TA- and VaR-based characteristics—but not the factor-mimicking portfolios—are compensated with higher expected returns, and that VaR neither subsumes nor is subsumed by TA. Finally, our cross-sectional decomposition analysis suggests that the firm-level VaR captures at least 7% of the accrual premium even in the presence of size and book-to-market. These findings lend support for the mispricing explanation of the accrual anomaly.

     

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.