VaR and expected shortfall: a non-normal regime switching framework

We have developed a regime switching framework to compute the Value at Risk and Expected Shortfall measures. Although Value at Risk as a risk measure has been criticized by some researchers for lack of subadditivity, it is still a central tool in banking regulations and internal risk management in the finance industry. In contrast, Expected Shortfall is coherent and convex, so it is a better measure of risk than Value at Risk. Expected Shortfall is widely used in the insurance industry and has the potential to replace Value at Risk as a standard risk measure in the near future. We have proposed regime switching models to measure value at risk and expected shortfall for a single financial asset as well as financial portfolios. Our models capture the volatility clustering phenomenon and variance-independent variation in the higher moments by assuming the returns follow Student-t distributions.     

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.     

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.     

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.     

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.     

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.     

Short-horizon regulation for long-term investors

We study the effects of imposing repeated short-horizon regulatory constraints on long-term investors. We show that Value-at-Risk and Expected Shortfall constraints, when imposed dynamically, lead to similar optimal portfolios and wealth distributions. We also show that, in utility terms, the costs of imposing these constraints can be sizeable. For a 96% funded pension plan, both an annual Value-at-Risk constraint and an annual Expected Shortfall constraint can lead to an economic cost of about 2.5–3.8% of initial wealth over a 15-year horizon.     

中国上市银行系统性风险度量——基于MES方法的分析

运用边际预期损失(MES)方法,通过DCC-GARCH模型和非参数估计计算我国14家上市银行的边际预期损失,并结合资产规模和杠杆率等因素度量各上市银行的系统性风险。研究结果表明,虽然资产规模、杠杆率和边际期望损失都是决定系统性风险的重要因素,但我国上市银行的系统性风险总体表现为:规模越大的银行,系统性风险也越大,即大型商业银行的系统性风险最大,股份制商业银行次之,城市商业银行的系统性风险最小。此外,三类商业银行的系统性风险随时间呈不同的变化趋势。     

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.     

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.     

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.     

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.     

Identifying systemically important financial institutions in Turkey

This paper examines the systemic risk of financial firms in Turkey. Using Component Expected Shortfall, we provide estimates of systemic risk in Turkey using daily data from 2005 to 2018 and a comprehensive data set encompassing 54 financial firms. Empirical results show that the preponderance of systemic risk in the sample in Turkey is due to large commercial banks. Top ten systemically important financial institutions dominate systemic risk measures in Turkey and account for more than 90 % of total risk over the sample. Consequently, the risk in the Turkish financial system is concentrated in specific financial institutions and makes close monitoring of the top firms essential. Historical incidence of systemic risk in the sample shows elevated levels of systemic risk correspond to well-known external events. Finally, a bivariate VAR model shows that systemic risk is correlated with measures of global financial risks and has significant negative effects on the real economy particularly on industrial production. This is important from a financial stability point of view in that close monitoring of the systemic risk is important in maintaining a healthy financial system and a well- functioning market economy.     

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.     

Liquidity,implied volatility and tail risk: A comparison of liquidity measures

Liquidity is easily perceived but not easily measured in financial markets. Researchers and practitioners develop and test new measures of liquidity which may be good candidates for measuring this elusive concept. In this study, we present a comparison of variables within two empirical exercises using up to eight traditional liquidity proxies and two proposed proxies based on semi-deviations in a sample of NYSE-listed stocks. The first empirical exercise analyzes the relationship between liquidity and implied volatility, showing that increases in implied volatility impacts increases illiquidity. Using a decomposition of the squared VIX, we show that both conditional variance and variance premium components affects liquidity. Our second empirical exercise investigates the relationship between common factors in liquidity and tail risk. Common factors increase individual stocks' Value-at-Risk and Expected Shortfall, although the effect is not significant for market risk. In both applications, most of the studied proxies present results aligned with the body of literature.     

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.     

Expected shortfall assessment in commodity (L)ETF portfolios with semi-nonparametric specifications

ABSTRACT

This paper studies the risk assessment of semi-nonparametric (SNP) distributions for leveraged exchange trade funds, (L)ETFs. We applied the SNP model with dynamic conditional correlations (DCC) and EGARCH innovations, and implement recent techniques to backtest Expected Shortfall (ES) to portfolios formed by bivariate combinations of major (L)ETFs on metal (Gold and Silver) and energy (Oil and Gas) commodities. Results support that multivariate SNP-DCC model outperforms the Gaussian-DCC and provides accurate risk measures for commodity (L)ETFs.     

Portfolio optimization under Expected Shortfall: contour maps of estimation error

The contour maps of the error of historical and parametric estimates of the global minimum risk for large random portfolios optimized under the Expected Shortfall (ES) risk measure are constructed. Similar maps for the VaR of the ES-optimized portfolio are also presented, along with results for the distribution of portfolio weights over the random samples and for the out-of-sample and in-sample estimates for ES. The contour maps allow one to quantitatively determine the sample size (the length of the time series) required by the optimization for a given number of different assets in the portfolio, at a given confidence level and a given level of relative estimation error. The necessary sample sizes invariably turn out to be unrealistically large for any reasonable choice of the number of assets and the confidence level. These results are obtained via analytical calculations based on methods borrowed from the statistical physics of random systems, supported by numerical simulations.     

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.