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.     

Model risk and capital reserves

We propose a procedure to take model risk into account in the computation of capital reserves. This addresses the need to make the allocation of capital reserves to positions in given markets dependent on the extent to which reliable models are available. The proposed procedure can be used in combination with any of the standard risk measures, such as Value-at-Risk and expected shortfall.     

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.     

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.     

Robustness and sensitivity analysis of risk measurement procedures

Measuring the risk of a financial portfolio involves two steps: estimating the loss distribution of the portfolio from available observations and computing a ‘risk measure’ that summarizes the risk of the portfolio. We define the notion of ‘risk measurement procedure’, which includes both of these steps, and introduce a rigorous framework for studying the robustness of risk measurement procedures and their sensitivity to changes in the data set. Our results point to a conflict between the subadditivity and robustness of risk measurement procedures and show that the same risk measure may exhibit quite different sensitivities depending on the estimation procedure used. Our results illustrate, in particular, that using recently proposed risk measures such as CVaR/expected shortfall leads to a less robust risk measurement procedure than historical Value-at-Risk. We also propose alternative risk measurement procedures that possess the robustness property.     

Multivariate risks and depth-trimmed regions

We describe a general framework for measuring risks, where the risk measure takes values in an abstract cone. It is shown that this approach naturally includes the classical risk measures and set-valued risk measures and yields a natural definition of vector-valued risk measures. Several main constructions of risk measures are described in this axiomatic framework. It is shown that the concept of depth-trimmed (or central) regions from multivariate statistics is closely related to the definition of risk measures. In particular, the halfspace trimming corresponds to the Value-at-Risk, while the zonoid trimming yields the expected shortfall. In the abstract framework, it is shown how to establish a both-ways correspondence between risk measures and depth-trimmed regions. It is also demonstrated how the lattice structure of the space of risk values influences this relationship. I. Molchanov supported by Swiss National Science Foundation Grant 200020-109217.     

Nonlinearly weighted convex risk measure and its application

We propose a new class of risk measures which satisfy convexity and monotonicity, two well-accepted axioms a reasonable and realistic risk measure should satisfy. Through a nonlinear weight function, the new measure can flexibly reflect the investor’s degree of risk aversion, and can control the fat-tail phenomenon of the loss distribution. A realistic portfolio selection model with typical market frictions taken into account is established based on the new measure. Real data from the Chinese stock markets and American stock markets are used for empirical comparison of the new risk measure with the expected shortfall risk measure. The in-sample and out-of-sample empirical results show that the new risk measure and the corresponding portfolio selection model can not only reflect the investor’s risk-averse attitude and the impact of different trading constraints, but can find robust optimal portfolios, which are superior to the corresponding optimal portfolios obtained under the expected shortfall risk measure.     

Estimating Value at Risk and Expected Shortfall Using Expectiles

Expectile models are derived using asymmetric least squares.A simple formula has been presented that relates the expectileto the expectation of exceedances beyond the expectile. We usethis as the basis for estimating the expected shortfall. Ithas been proposed that the quantile be estimated by the expectilefor which the proportion of observations below the expectileis . In this way, an expectile can be used to estimate valueat risk. Using expectiles has the appeal of avoiding distributionalassumptions. For univariate modeling, we introduce conditionalautoregressive expectiles (CARE). Empirical results for thenew approach are competitive with established benchmarks methods.     

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.     

Semi-parametric Bayesian tail risk forecasting incorporating realized measures of volatility

Realized measures employing intra-day sources of data have proven effective for dynamic volatility and tail-risk estimation and forecasting. Expected shortfall (ES) is a tail risk measure, now recommended by the Basel Committee, involving a conditional expectation that can be semi-parametrically estimated via an asymmetric sum of squares function. The conditional autoregressive expectile class of model, used to implicitly model ES, has been extended to allow the intra-day range, not just the daily return, as an input. This model class is here further extended to incorporate information on realized measures of volatility, including realized variance and realized range (RR), as well as scaled and smoothed versions of these. An asymmetric Gaussian density error formulation allows a likelihood that leads to direct estimation and one-step-ahead forecasts of quantiles and expectiles, and subsequently of ES. A Bayesian adaptive Markov chain Monte Carlo method is developed and employed for estimation and forecasting. In an empirical study forecasting daily tail risk measures in six financial market return series, over a seven-year period, models employing the RR generate the most accurate tail risk forecasts, compared to models employing other realized measures as well as to a range of well-known competitors.     

Performance ratio-based coherent risk measure and its application

Utilizing a specific acceptance set, we propose in this paper a general method to construct coherent risk measures called the generalized shortfall risk measure. Besides some existing coherent risk measures, several new types of coherent risk measures can be generated. We investigate the generalized shortfall risk measure’s desirable properties such as consistency with second-order stochastic dominance. By combining the performance evaluation with the risk control, we study in particular the performance ratio-based coherent risk (PRCR) measures, which is a sub-class of generalized shortfall risk measures. The PRCR measures are tractable and have a suitable financial interpretation. Based on the PRCR measure, we establish a portfolio selection model with transaction costs. Empirical results show that the optimal portfolio obtained under the PRCR measure performs much better than the corresponding optimal portfolio obtained under the higher moment coherent risk measure.     

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.     

Monte Carlo Option Pricing for Tempered Stable (CGMY) Processes

Lévy processes are popular models for stock price behavior since they allow to take into account jump risk and reproduce the implied volatility smile. In this paper, we focus on the tempered stable (also known as CGMY) processes, which form a flexible 6-parameter family of Lévy processes with infinite jump intensity. It is shown that under an appropriate equivalent probability measure a tempered stable process becomes a stable process whose increments can be simulated exactly. This provides a fast Monte Carlo algorithm for computing the expectation of any functional of tempered stable process. We use our method to price European options and compare the results to a recent approximate simulation method for tempered stable process by Madan and Yor (CGMY and Meixner Subordinators are absolutely continuous with respect to one sided stable subordinators, 2005).     

Is Bitcoin a better portfolio diversifier than gold? A copula and sectoral analysis for China

This paper aims to compare Bitcoin with gold in the diversification of Chinese portfolios using daily data over the 2010–2020 period. We propose a new development of copula-based joint distribution function of returns to simulate the Value-at-Risk and expected shortfall of portfolios including Bitcoin (or gold) and those without it. The stochastic dominance method is also used to compare the return distributions of the three types of portfolios. Empirical results show that gold is a better portfolio diversifier than Bitcoin as it helps better reduce the risk of portfolios. On the other hand, Bitcoin better increases the return but also increases the risk. The stochastic dominance results further show that portfolios diversified by gold dominate those diversified by Bitcoin. Based on these findings, we conclude that in China, gold is a better portfolio diversifier than Bitcoin for risk-averse investors. However, for risk-seeking investors, Bitcoin can be a better choice. This result is found to be robust to the time, frequency and currency effects.     

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.     

Risk measures based on behavioural economics theory

Coherent risk measures (Artzner et al. in Math. Finance 9:203–228, 1999) and convex risk measures (Föllmer and Schied in Finance Stoch. 6:429–447, 2002) are characterized by desired axioms for risk measures. However, concrete or practical risk measures could be proposed from different perspectives. In this paper, we propose new risk measures based on behavioural economics theory. We use rank-dependent expected utility (RDEU) theory to formulate an objective function and propose the smallest solution that minimizes the objective function as a risk measure. We also employ cumulative prospect theory (CPT) to introduce a set of acceptable regulatory capitals and define the infimum of the set as a risk measure. We show that the classes of risk measures derived from RDEU theory and CPT are equivalent, and they are all monetary risk measures. We present the properties of the proposed risk measures and give sufficient and necessary conditions for them to be coherent and convex, respectively. The risk measures based on these behavioural economics theories not only cover important risk measures such as distortion risk measures, expectiles and shortfall risk measures, but also produce new interesting coherent risk measures and convex, but not coherent risk measures.     

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.     

Reinsurance premium principles based on weighted loss functions

ABSTRACT

In this paper, we propose new reinsurance premium principles that minimize the expected weighted loss functions and balance the trade-off between the reinsurer's shortfall risk and the insurer's risk exposure in a reinsurance contract. Random weighting factors are introduced in the weighted loss functions so that weighting factors are based on the underlying insurance risks. The resulting reinsurance premiums depend on both the loss covered by the reinsurer and the loss retained by the insurer. The proposed premiums provide new ways for pricing reinsurance contracts and controlling the risks of both the reinsurer and the insurer. As applications of the proposed principles, the modified expectile reinsurance principle and the modified quantile reinsurance principle are introduced and discussed in details. The properties of the new reinsurance premium principles are investigated. Finally, the comparisons between the new reinsurance premium principles and the classical expectile principle, the classical quantile principle, and the risk-adjusted principle are provided.     

Convex risk measures based on generalized lower deviation and their applications

Considering the implementability and the properties that a reasonable and realistic risk measure should satisfy, we propose a new class of risk measures based on generalized lower deviation with respect to a chosen benchmark. Besides convexity and monotonicity, our new risk measure can reflect the investor's degree of risk aversion as well as the fat-tail phenomenon of the loss distribution with the help of different benchmarks and weighted functions. Based on the new risk measure, we establish a realistic portfolio selection model taking market frictions into account. To examine the influence of the benchmarks and weighted functions on the optimal portfolio and its performance, we carry out a series of empirical studies in Chinese stock markets. Our in-sample and out-of-sample results show that the new risk measure and the corresponding portfolio selection model can reflect the investor's risk averse attitude and the impact of different trading constraints. Most importantly, with the new risk measure we can obtain an optimal portfolio which is more robust and superior to the optimal portfolios obtained with the traditional expected shortfall risk measures.     

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.