Estimation Of Expected Shortfall Research Articles

Joint value-at-risk and expected shortfall regression for location-scale time series models

. This article studies the joint value-at-risk (VaR) and expected shortfall (ES) regression for a wide class of location-scale time series models including autoregressive and moving average models with generalized autoregressive conditional heteroscedasticity errors. In contrast to the quasi-maximum likelihood estimation, we estimate the model parameters with the aim of more accurate VaR and ES estimation. Then, we show consistency and asymptotic normality for parameter estimators under weak regularity conditions. Finally, a simulation study and a real data analysis are shown to illustrate our results.

Hybrid Fourier asymmetric-garch estimation of value at risk and expected shortfall: Empirical evidence from crude oil prices

We propose a hybrid Fourier approximation in an autoregressive fractional integrated moving average (ARFIMA) model, to account for periodic unoberved components in financial time series. We apply this hybrid model on parametric estimation of value at risk (VaR) and expected shortfall (ES). Using crude oil returns, we show that Fourier approximation inclusion significantly accounts for unobserved periodic components in a VaR estimation using exponential generalized autoregressive conditional heteroscedasticity (EGARCH) model under generalized error distribution (GED). Similarly, in VaR estimation, Fourier approximation inclusion significantly accounts for unobserved periodic components using asymmetric power generalized autoregressive conditional heteroscedasticity (APARCH) model under skewed normal distribution (SNORM). For ES estimation, Fourier approximation inclusion only significantly accounts for unobserved periodic components of APARCH under SNORM.

Forecasting VaR and ES by using deep quantile regression, GANs-based scenario generation, and heterogeneous market hypothesis

Value at risk (VaR) and expected shortfall (ES) have emerged as standard measures for detecting the market risk of financial assets and play essential roles in investment decisions, external regulations, and risk capital allocation. However, existing VaR estimation approaches fail to accurately reflect downside risks, and the ES estimation technique is quite limited owing to its challenging implementation. This causes financial institutions to overestimate or underestimate investment risk and finally leads to the inefficient allocation of financial resources. The main purpose of this study is to use machine learning to improve the accuracy of VaR estimation and provide an effective tool for ES estimation. Specifically, this study proposes a VaR estimator by combining quantile regression with “Mogrifier” recurrent neural networks to capture the “long memory” and “clustering” properties of financial assets; while for estimating ES, this study directly models the quantile of assets and employs generative adversarial networks to generate future tail risk scenarios. In addition to the typical properties of financial assets, the model design is also consistent with heterogeneous market theory. An empirical application to four major global stock indices shows that our model is superior to other existing models.

Efficiency gains in value-at-risk and expected shortfall estimation by using copulas and full maximum likelihood

We provide Monte Carlo evidence on the efficiency gains obtained in GARCH-based estimations of value-at-risk (VaR) and expected shortfall (ES) by incorporating dependence information through copulas and subsequently using full maximum likelihood (FML) estimates. First, we consider an individual returns series; in this case, the efficiency gain stems from exploiting the relationship with another returns series using a copula model. Second, we consider a portfolio returns series obtained as a linear combination of returns series related through a copula model; in this case, the efficiency gain stems from using FML estimates instead of two-stage ML estimates. We consider three copulas models in order to analyze the effect of the type and degree of tail dependence on the results. Our results show that, in these situations, using copula models and FML leads to a substantial reduction in the mean squared error of the VaR and ES estimates when there is a relatively high degree of dependence between returns (up to 70% in the presence of lower-tail dependence) and a notable improvement in the performance of backtesting procedures.

Sparse estimation within Pearson's system, with an application to financial market risk

AbstractPearson's system is a rich class of models that includes many classical univariate distributions. It comprises all continuous densities whose logarithmic derivative can be expressed as a ratio of quadratic polynomials governed by a vector of coefficients. The estimation of a Pearson density is challenging, as small variations in can induce wild changes in the shape of the corresponding density . The authors show how to estimate and effectively through a penalized likelihood procedure involving differential regularization. The approach combines a penalized regression method and a profiled estimation technique. Simulations and an illustration with S&P 500 data suggest that the proposed method can improve market risk assessment substantially through value‐at‐risk and expected shortfall estimates that outperform those currently used by financial institutions and regulators.

Estimating Value-at-Risk and Expected Shortfall: Do Polynomial Expansions Outperform Parametric Densities?

We assess Value-at-Risk (VaR) and Expected Shortfall (ES) estimates assuming different models for the standardized returns: distributions based on polynomial expansions such as Cornish-Fisher and Gram-Charlier, and well-known parametric densities such as normal, skewed-t and Johnson. This paper aims to analyze whether models based on polynomial expansions outperform the parametric ones. We carry out the model performance comparison in two stages: first, with a backtesting analysis of VaR and ES; and second, using loss functions. Our backtesting results show that all distributions, except for normal ones, perform quite well in VaR and ES estimations. Regarding the loss function analysis, we conclude that polynomial expansions (specifically, the Cornish-Fisher one) usually outperform parametric densities in VaR estimation, but the latter (specifically, the Johnson density) slightly outperform the former in ES estimation; however, the gains of using one approach or the other are modest.

Comparing riskiness of exchange rate volatility using the Value at Risk and Expected Shortfall methods

This paper uses theValue at Risk (VaR) and the Expected Shortfall (ES) to compare the riskiness of the two currency exchange rate volatility, namely BitCoin against the US dollar (BTC/USD) and the South African Rand against the US dollar (ZAR/USD). The risks calculated are tail-related measures, so the Extreme Value Theory is used to capture extreme risk more accurately. The Generalized Pareto distribution (GPD) is assumed under Extreme Value Theory (EVT). The family of Generalized Autoregressive Conditional Heteroscedasticity (GARCH) models was used to model the volatility-clustering feature. The Maximum Likelihood Estimation (MLE) method was used in parameter estimation. Results obtained from the GPD are compared using two underlying distributions for the errors, namely: the Normal and the Student-t distributions. The findings show that the tail VaR on the BitCoin averaging 1.6 and 2.8 is riskier than on South Africa’s Rand that averages 1.5 and 2.3 at 95% and 99%, respectively. The same conclusion is made about tail ES, the BitCoin average of 2.3 and 3.6 is higher (riskier) than the South African Rand averages at 2.1 and 2.9 at 95% and 99%, respectively. The backtesting results confirm the model adequacy of the GARCH-GPD in the estimation of VaR and ES, since all p-values are above 0.05.

Estimating tail-risk using semiparametric conditional variance with an application to meme stocks

In this paper, we propose using Mishra, Su, Ullah’s (2010) semiparametric variance to estimate Value at Risk (VaR) and Expected Shortfall (ES). Although the variance estimate is established in the literature, it has not been applied to VaR and ES estimation. Here the returns’ variance is estimated using Mishra, Su, Ullah’s (2010) conditional semiparametric estimator, and the standardized residuals’ distribution is fitted nonparametrically. In order to reduce possible misspecification bias, the proposed estimators decouple the assumption of the returns’ distribution and the variance estimation. Empirical and simulated data are used to compare the performance of the new estimators against existing parametric and nonparametric VaR and ES models, both conditional and unconditional. The estimators are further applied to daily returns of certain meme stocks to estimate tail risk and test the performance of the estimators during periods of extreme volatility. Compared to the models studied, the proposed conditional, semiparametric VaR model produces fewer violations, and violations without a recognizable pattern - upholding the regulatory requirements. The expected shortfall estimated by the conditional, semiparametric model is also closest to the observed mean of the violations.

Evaluating risk in precious metal prices with generalised lambda, generalised pareto and generalised extreme value distributions

In this study we investigate the performance of the generalised lambda distribution (GLD), the generalised Pareto distribution (GPD) and the generalised extreme value distribution (GEVD) in modelling daily platinum, gold and silver price log-returns. Our primary goal is to compare GLD against GPD, and GEVD, in the estimation of Value-at-Risk (VaR) and expected shortfall (ES) as per the international Basel regulatory framework. Our analyses show that GPD and GLD generally outperform GEVD for VaR and ES estimation for negative precious metal returns. For gold, the GPD stands out as the most suitable model. For platinum, GPD and GLD are equally adequate, especially at the 1% VaR level. For silver, GLD is the most suitable at 1% VaR level, whereas GPD is the best model at 0.1%. This study has shown that GLD is a suitable model for extreme risk in precious metal prices and can be used for the estimation of VaR and ES values. Keywords: Expected shortfall, Generalised extreme value, Generalised lambda, Generalised Pareto, Precious metal returns, Value-at-Risk

Backtesting the Expected Shortfall

Backtesting of risk measure estimates is an integral part for an effective risk management. With the growing importance of the Expected Shortfall (ES) to potentially replace the Value at Risk (VaR) as a primary measure for market risk this also calls for suitable backtesting solutions. Although a variety of approaches has been proposed in the past, there is still an on-going discussion whether the ES can be properly backtested. The thesis adds to this discussion in the following way. Five of the most promising backtests for the ES are implemented, compared based on theoretical properties like empirical size and power and tested against ES estimation models which are fitted to historical returns of the SP backtesting; risk measures; statistical test.

Implementation of a Commitment Machine for an Adaptive and Robust Expected Shortfall Estimation.

This study proposes a metaheuristic for the selection of models among different Expected Shortfall (ES) estimation methods. The proposed approach, denominated “Commitment Machine” (CM), has a strong focus on assets cross-correlation and allows to measure adaptively the ES, dynamically evaluating which is the most performing method through the minimization of a loss function. The CM algorithm compares four different ES estimation techniques which all take into account the interaction effects among assets: a Bayesian Vector autoregressive model, Stochastic Differential Equation (SDE) numerical schemes with Exponential Weighted Moving Average (EWMA), a Generalized AutoRegressive Conditional Heteroskedasticity (GARCH) volatility model and a hybrid method that integrates Dynamic Recurrent Neural Networks together with a Monte Carlo approach. The integration of traditional Monte Carlo approaches with Machine Learning technologies and the heterogeneity of dynamically selected methodologies lead to an improved estimation of the ES. The study describes the techniques adopted by the CM and the logic behind model selection; moreover, it provides a market application case of the proposed metaheuristic, by simulating an equally weighted multi-asset portfolio.

Quasi Maximum Likelihood Estimation of Value at Risk and Expected Shortfall

Quasi maximum likelihood estimation of Value at Risk (VaR) and Expected Shortfall (ES) is discussed. The reference likelihood is that of a location-scale asymmetric Laplace distribution, related to a family of loss functions that lead to strictly consistent scoring functions for joint estimation of VaR and ES. The case of zero mean processes is considered, where quasi maximum likelihood estimators (QMLE) are consistent and asymptotically normal, as well as the case of non-zero mean processes, where quasi maximum likelihood estimators lead to inconsistent estimates due to lack of identification. In the latter situation, the asymptotic properties of two stage quasi maximum likelihood estimators (2SQMLE) are derived. QMLE and 2SQMLE are related with sample and M-estimators and compared in terms of asymptotic efficiency. A simulation study investigates the finite sample properties of QMLE, 2SQMLE, sample and M-estimators of expected shortfall.

Optimizing Expected Shortfall under an ℓ1 Constraint-An Analytic Approach.

Expected Shortfall (ES), the average loss above a high quantile, is the current financial regulatory market risk measure. Its estimation and optimization are highly unstable against sample fluctuations and become impossible above a critical ratio , where N is the number of different assets in the portfolio, and T is the length of the available time series. The critical ratio depends on the confidence level , which means we have a line of critical points on the plane. The large fluctuations in the estimation of ES can be attenuated by the application of regularizers. In this paper, we calculate ES analytically under an regularizer by the method of replicas borrowed from the statistical physics of random systems. The ban on short selling, i.e., a constraint rendering all the portfolio weights non-negative, is a special case of an asymmetric regularizer. Results are presented for the out-of-sample and the in-sample estimator of the regularized ES, the estimation error, the distribution of the optimal portfolio weights, and the density of the assets eliminated from the portfolio by the regularizer. It is shown that the no-short constraint acts as a high volatility cutoff, in the sense that it sets the weights of the high volatility elements to zero with higher probability than those of the low volatility items. This cutoff renormalizes the aspect ratio , thereby extending the range of the feasibility of optimization. We find that there is a nontrivial mapping between the regularized and unregularized problems, corresponding to a renormalization of the order parameters.

Quasi Maximum Likelihood Estimation of Value at Risk and Expected Shortfall

Quasi maximum likelihood estimation of Value at Risk (VaR) and Expected Shortfall (ES) is discussed. The reference likelihood is that of a location-scale asymmetric Laplace distribution, related to a family of loss functions that lead to strictly consistent scoring functions for joint estimation of VaR and ES. The case of zero mean processes is considered, where quasi maximum likelihood estimators (QMLE) are consistent and asymptotically normal, as well as the case of non-zero mean processes, where quasi maximum likelihood estimators lead to inconsistent estimates due to lack of identification. In the latter situation, the asymptotic properties of two stage QMLE (2SQMLE) are derived. QMLE and 2SQMLE are related with sample and M-estimators and compared in terms of asymptotic efficiency. A simulation study investigates the finite sample properties of QMLE, 2SQMLE, sample and M-estimators of expected shortfall.

The sum of two independent polynomially-modified hyperbolic secant random variables with application in computational finance

In this paper, by reshaping the hyperbolic secant distribution using Hermite polynomial, we devise a polynomially-modified hyperbolic secant distribution which is more flexible than secant distribution to capture the skewness, heavy-tailedness and kurtosis of data. As a portfolio possibly consists of multiple assets, the distribution of the sum of independent polynomially-modified hyperbolic secant random variables is derived. In exceptional cases, we evaluate risk measures such as value at risk and expected shortfall (ES) for the sum of two independent polynomially-modified hyperbolic secant random variables. Finally, using real datasets from four international computers stocks, such as Adobe Systems, Microsoft, Nvidia and Symantec Corporations, the effectiveness of the proposed model is shown by the goodness of Gram–Charlier-like expansion of hyperbolic secant law, for performance of value at risk and ES estimation, both in and out of the sample period.

A simple and robust approach for expected shortfall estimation

In risk management, estimating expected shortfall, though important and indispensable, is difficult when the sample size is small. This paper suggests a recipe for meeting such a challenge. A tail-based normal approximation with explicit formulas is derived by matching a specific quantile and the mean excess square of the sample observations. To enhance the estimation accuracy, we propose an adjusted tail-based normal approximation based on the sample's tail weight. The adjusted expected shortfall estimator is robust and efficient in the sense that it can be applied to various heavy-tailed distributions, such as Student t, lognormal, Gamma and Weibull, and the errors are all small. Moreover, compared with two common expected shortfall estimators -- the arithmetic average of excessive losses and extreme value theory estimator -- the proposed estimator achieves smaller mean squared errors for small samples, especially at high confidence levels. The properties of linear transformations on the expected shortfall estimator are also investigated to ensure its practicality.

TERES: Tail Event Risk Expectile Shortfall

We propose a generalized risk measure for expectile-based expected shortfall estimation. The generalization is designed with a mixture of Gaussian and Laplace densities. Our plug-in estimator is derived from an analytic relationship between expectiles and expected shortfall. We investigate the sensitivity and robustness of the expected shortfall to the underlying mixture parameter specification and the risk level. Empirical results from the US, German and UK stock markets and for selected NASDAQ blue chip companies indicate that expected shortfall can be successfully estimated using the proposed method on a monthly, weekly, daily and intra-day basis using a 1-year or 1-day time horizon across different risk levels.

Estimating expected shortfall using a quantile function model

AbstractDistribution of financial returns defined by the existing GARCH models usually focus on the overall features such as the location, scale, skewness and kurtosis of the distribution. When using such GARCH models for expected shortfall (ES) estimation, it is difficult to consider specific information about the tails (such as the shape of the tails of the distribution), resulting in possible bias in ES estimation. We propose a quantile function threshold GARCH model to overcome some of the limitations of existing models. The model allows us to use the information including the skewness and tail shape of the distribution and the structure changes in the volatility of financial returns to obtain ES estimates. Our results show that the proposed model outperforms the benchmark models, confirming that tail shape of the distribution also plays an important role in ES estimation.

Idiosyncrasies of Intraday Risk in Emerging and Developed Markets: Efficacy of the MCS-GARCH Model and Extreme Value Theory

This article aims to study the effectiveness of the multi-step multiplicative component generalized autoregressive conditional heteroscedasticity (MCS-GARCH) model in forecasting intraday value at risk (VaR) and expected shortfall (ES) for both emerging and developed markets. The intraday VaR- and ES-forecasting performance of the MCS-GARCH model is compared with that of three other models. Five-minute returns data of stock market indices from four different countries are used in this study: India, China, the USA and the UK. The VaR and ES quantiles are estimated with the help of conditional extreme value theory (EVT) framework as well as with simple normal distribution assumption. The backtesting of VaR and ES suggest that the conditional EVT models provide superior forecasts compared to their GARCH counterparts. However, the accuracy of these models differs between emerging and developed markets. The multi-step MCS-GARCH or its EVT counterpart fails to outperform other competing models consistently, especially in emerging markets. The inability of the MCS-GARCH model in forecasting intraday risk accurately suggests that the multiple steps in the estimation of this model may lead to the errors-in-variables problem. The results of this study are unique in their suggestion that multi-step models may not be a better option in the estimation of VaR and ES.

Jump Driven Risk Model Performance in Cryptocurrency Market

This paper aims at identifying a validated risk model for the cryptocurrency market. We propose a stochastic volatility model with co-jumps in return and volatility (SVCJ) to highlight the role of jumps in returns and volatility in affecting Value-at-Risk (VaR) and Expected Shortfall (ES) in cryptocurrency market. Validation results based on backtesting show that SVCJ model is superior in terms of statistical accuracy of VaR and ES estimates, compared to alternative models such as TGARCH (Threshold GARCH) volatility and RiskMetrics models. The results imply that for the cryptocurrency market, the best performing model is a stochastic process that accounts for both jumps in returns and volatility.