Tail Risk Measures Research Articles

Jump tail risk exposure and the cross-section of stock returns

We introduce a new jump tail risk measure retrieved from option prices. We examine the cross-sectional pricing of stocks according to their sensitivities to jump tail risk. We find a negative market price of jump tail risk. A high-low portfolio sorted by jump tail risk betas delivers a statistically and economically significant negative premium of -9.95% per year. Risk-adjusted returns are also negative and highly significant. We document that the negative jump tail risk premium is mainly driven by its downside jump tail risk component. On the contrary, the premium of the high-low portfolio sorted by upside jump tail risk betas is insignificant. The negative premium of downside jump tail risk is significant when controlling for various risk factor loadings and firm characteristics, and remains strong for large firms. Our results carry over to a predictive setting, in which we compare subsequent realized returns of the quintile portfolios sorted by downside jump tail risk betas estimated over the previous period.

Oil jump tail risk as a driver of inflation dynamics

In this paper, we look at the role of various oil jump tail risk measures as drivers of both U.S. headline and core inflation. Those measures are first computed from high-frequency oil future prices and are then introduced into standard regression models in order to (i) assess in-sample determinants of inflation, (ii) assess overtime the evolution of inflation drivers, (iii) estimate impulse response functions and (iv) forecast inflation out-of-sample for various horizons. Empirical results suggest that oil jump tail risk measures contain useful information to describe inflation dynamics, generally leading to upward inflationary pressures. Even after controlling from standard variables involved in a Phillips curve, goodness-of-fit measures show evidence of a gain, in particular for headline inflation. Overall, we observe that oil jump tail risk measures are contributing more to inflation dynamics since the Covid-19 crisis.

Tail variance and confidence of using tail conditional expectation: analytical representation, capital adequacy, and asymptotics

Abstract In this paper, we explore the applications of Tail Variance (TV) as a measure of tail riskiness and the confidence level of using Tail Conditional Expectation (TCE)-based risk capital. While TCE measures the expected loss of a risk that exceeds a certain threshold, TV measures the variability of risk along its tails. We first derive analytical formulas of TV and TCE for a large variety of probability distributions. These formulas are useful instruments for relevant research works on tail risk measures. We then propose a distribution-free approach utilizing TV to estimate the lower bounds of the confidence level of using TCE-based risk capital. In doing so, we introduce sharpened conditional probability inequalities, which halve the bounds of conventional Markov and Cantelli inequalities. Such an approach is easy to implement. We further investigate the characterization of tail risks by TV through an exploration of TV’s asymptotics. A distribution-free limit formula is derived for the asymptotics of TV. To further investigate the asymptotic properties, we consider two broad distribution families defined on tails, namely, the polynomial-tailed distributions and the exponential-tailed distributions. The two distribution families are found to exhibit an asymptotic equivalence between TV and the reciprocal square of the hazard rate. We also establish asymptotic relationships between TCE and VaR for the two families. Our asymptotic analysis contributes to the existing research by unifying the asymptotic expressions and the convergence rate of TV for Student-t distributions, exponential distributions, and normal distributions, which complements the discussion on the convergence rate of univariate cases in [28]. To show the usefulness of our results, we present two case studies based on real data from the industry. We first show how to use conditional inequalities to assess the confidence of using TCE-based risk capital for different types of insurance businesses. Then, for financial data, we provide alternative evidence for the relationship between the data frequency and the tail categorization by the asymptotics of TV.

Currency tail risk measurement and spillovers: An improved TENET approach

Based on an improved TENET approach, this paper analyses the tail risk of 32 major global currencies and measures the tail risk spillover among these currencies using daily data. We find that (i) The tail risk of USD, EUR, GBP and JPY is relatively high, while CNY shows low risk with a continuous upward trend; (ii) The total tail risk connectedness of currencies declines over time, but spikes during significant events in recent years; (iii) The tail risk spillover of currencies from developed economies is higher than developing economies. Moreover, European and North American currencies primarily exhibit tail risk spillover, while Asian and African (South African) currencies mainly absorb risk from other regions.

Tail risk intersection between tech-tokens and tech-stocks

This study investigates the interconnectedness of upside and downside tail risks between tech-industry tokens and equities, focusing on uncertainties and portfolio implications. We estimate tail-risk measures at various quantiles using the conditional autoregressive value-at-risk method and investigate selected assets' static and dynamic connectedness using a time-varying parameter-vector autoregression approach. The findings reveal a moderate level of time-varying tail-risk interconnectedness. Furthermore, we examine the role of global uncertainty factors in tail-risk contagion and identify the factors driving connectedness. Moreover, this study analyzes portfolio implications, offering insights into effective risk management strategies and the potential benefits of diversification. These findings have significant implications for investors, regulators, and policymakers in risk management in the technology industry.

Distortion risk measures for tail-risk multicriteria decision-making based on a probabilistic linguistic environment

Probabilistic linguistic term sets (PLTSs) provide a flexible tool to express linguistic preferences, and several multi-criteria decision models based on PLTSs have been recently developed. In this framework, distortion risk measures are extensively used in finance and insurance applications, but are rarely applied in fuzzy systems. In this paper, distortion risk measures are applied to fuzzy tail decisions. In particular, three tail risk measurement methods are put forward, referred to as probabilistic linguistic VaR (PLVaR), expected probability linguistic VaR (EPLVaR), and Wang tail risk measure and extensively study their properties. Our novel methods help to clarify the connections between distortion risk measure and fuzzy tail decision-making. In particular, the Wang tail risk measure is characterized by consistency and stability of decision results. The criteria and expert weights are unknown or only partially known during the decision making process, and the maximising PLTSs deviations are showed how to determine them. The theoretical results are showcased on an optimal stock fund selection problem, where the three tail risk measures are compared and analyzed.

Research on Tail Risk Hedging in the Digital Asset Market

In recent years, the digital asset market has experienced rapid growth, but at the same time, its implicit tail risks have gradually raised concerns, especially amidst the global economic downturn caused by the pandemic. This paper aims to study tail risks and propose hedging strategies. Based on the GARCH and VaR models, this paper uses weekly return data from July 2017 to December 2021 for Bitcoin digital cryptocurrency and traditional non-digital asset market indices such as stocks, bonds, gold, and commodities. Utilizing EViews and Excel, the study explores tail risk measurement and portfolio diversification effects in the digital asset market. The main findings are as follows: (1) During the sample period, cryptocurrency returns exhibited significant volatility, with weak correlations with returns from traditional non-digital assets. Further analysis revealed strong hedging effects of digital assets against the bond market but weaker effects against the gold and stock markets, as well as commodity markets; (2) Cryptocurrency in traditional investment portfolios can enhance the risk-return ratio by adding it to the asset mix.

Statistical Inference and Application of Asymmetrical Generalized Pareto Distribution Based on Peaks-Over-Threshold Model

Generalized Pareto distribution (GPD), an asymmetrical distribution, primarily models exceedances over a high threshold in many applications. Within the peaks-over-threshold (POT) framework, we consider a new GPD parameter estimation method to estimate a common tail risk measure, the value at risk (VaR). The proposed method is more suitable for the POT framework and makes full use of data information. Specifically, our estimation method builds upon the generalized probability weighted moments method and integrates it with the nonlinear weighted least squares method. We use exceedances for the GPD, minimizing the sum of squared differences between the sample and population moments of a function of GPD random variables. At the same time, the proposed estimator uses three iterations and assigns weight to further improving the estimated performance. Under Monte Carlo simulations and with a real heavy-tailed dataset, the simulation results show the advantage of the newly proposed estimator, particularly when VaRs are at high confidence levels. In addition, by simulating other heavy-tailed distributions, our method still exhibits good performance in estimating misjudgment distributions.

Research on Risk Contagion in ESG Industries: An Information Entropy-Based Network Approach.

Sustainable development is a practical path to optimize industrial structures and enhance investment efficiency. Investigating risk contagion within ESG industries is a crucial step towards reducing systemic risks and fostering the green evolution of the economy. This research constructs ESG industry indices, taking into account the possibility of extreme tail risks, and employs VaR and CoVaR as measures of tail risk. The TENET network approach is integrated to to capture the structural evolution and direction of information flow among ESG industries, employing information entropy to quantify the topological characteristics of the network model, exploring the risk transmission paths and evolution patterns of ESG industries in an extreme tail risk event. Finally, Mantel tests are conducted to examine the existence of significant risk spillover effects between ESG and traditional industries. The research finds strong correlations among ESG industry indices during stock market crash, Sino-US trade frictions, and the COVID-19 pandemic, with industries such as the COAL, CMP, COM, RT, and RE playing key roles in risk transmission within the network, transmitting risks to other industries. Affected by systemic risk, the information entropy of the TENET network significantly decreases, reducing market information uncertainty and leading market participants to adopt more uniform investment strategies, thus diminishing the diversity of market behaviors. ESG industries show resilience in the face of extreme risks, demonstrating a lack of significant risk contagion with traditional industries.

Inter-industry network and credit risk

As previous literature has documented cross-industry returns and tail risk predictability, especially during a financial crisis, this research investigates the effects of industries' position within an economy, inter-industry connectedness, and industry returns on credit risk using a reduced-form approach. We employ an aggregate measure of tail risk emitted from an industry to capture its outgoing connectedness with other industries, and our empirical results show that the outgoing connectedness of some highly central industries positively impacts all sample firms’ default probabilities, controlling for a variety of firm-specific and macroeconomic variables that are well-known to be related to corporate defaults. In sum, our empirical results support that industry network risk helps explain corporate default and improves default prediction accuracy.

An axiomatic approach to default risk and model uncertainty in rating systems

In this paper, we deal with an axiomatic approach to default risk. We introduce the notion of a default risk measure, which generalizes the classical probability of default (PD), and allows to incorporate model risk in various forms. We discuss different properties and representations of default risk measures via monetary risk measures, families of related tail risk measures, and Choquet capacities. In a second step, we turn our focus on default risk measures, which are given as worst-case PDs and distorted PDs. The latter are frequently used in order to take into account model risk for the computation of capital requirements through risk-weighted assets (RWAs), as demanded by the Capital Requirement Regulation (CRR). In this context, we discuss the impact of different default risk measures and margins of conservatism on the amount of risk-weighted assets.

Is tail risk priced in the cross-section of international stock index returns?

This study examines the predictive power of tail risk measures in stock indices returns using a comprehensive dataset covering 50 countries from 1926 to 2021. Our findings reveal that tail risk measures exhibit predictive power when considered independently. However, their forecasting abilities disappear when other risk and return factors are incorporated. This suggests that tail risk measures do not contain incremental information about the cross-section of stock returns beyond the commonly used global factors. Our findings are robust across various considerations, holding for alternative tail risk measure types, estimation periods, and different control variables subsets.

Does sharia-compliant debt financing reduce stock price crash risk?

PurposeThis paper examines the impact of sharia-compliant debt financing on stock price crash risk. Unlike those previous studies that took Sukuk or sharia-compliant firms, this study tests the impact of the proportion reported sharia-compliant debt financing in the balance sheet on the risk of price crash of a firm.Design/methodology/approachUsing the data from 2,752 firm-year observations of 344 Malaysian non-financial listed companies from 2012 to 2019, this article used a robust panel data estimation technique for statistical inferences. This study also employs panel GMM and quantile least squares as the robustness check.FindingsThis study established a negative relationship between sharia-compliant debt financing and stock price crash risk. The robustness checks with different estimation techniques confirm the results. It implies that firms with a more significant proportion of Sharia-compliant financing tend to have lower future stock price crash risk.Practical implicationsConsistent with the Islamic finance literature, the present study contributes to the existing literature on Islamic capital markets from the perspective of stock price crash risk because it is vital for risk management and investment decision-making as a measure of tail risk for stocks. The findings of this research will assist investors in developing portfolio strategies that incorporate firms with higher levels of sharia-compliant debt financing in their balance sheets. Additionally, the results of this study suggest that policymakers and regulatory bodies should consider revising their monitoring approaches for publicly listed firms.Originality/valueThis study is interesting and unique, as it is a pioneer in testing the impact of sharia-compliant debt financing on reducing stock price crash risk.

The fatter the tail, the shorter the sail

AbstractGuided by the extreme value theory, this study empirically investigates the impact of tail risk measures on financial distress of publicly traded bank holding companies (BHCs) in the United States. Our results show that tail risk measures namely, value‐at‐risk and expected shortfall, are significantly and positively related to banks distress risk. Implying that BHCs with more frequent extreme negative daily equity returns induce higher tail risks, thereby increasing their likelihood of experiencing financial distress. Our results also show that tail risk measures enhance the explanatory power of traditional models explaining banks distress risk based on accounting information. These results indicate that market discipline is generally beneficial in managing and regulating banks, bolstering claims of the importance of macro‐prudential supervision of financial institutions.

Soft splicing model: bridging the gap between composite model and finite mixture model

Considerations of both the heavy-tail phenomenon and multi-modality of a claim severity distribution have been challenging in the actuarial literature and practices. In this article, we develop a novel class of soft splicing models that bridges the gap between pre-existing methods for handling the issues above. The proposed method is flexible enough to incorporate tail-heaviness and multi-modality with computational efficiency and nests finite mixture models and splicing models as its special and/or limiting cases. The soft splicing model is also more robust in extrapolating the tail-heaviness of distribution subject to model contamination. According to simulation studies and real insurance claim data analyses, it is shown that the proposed soft splicing model provides superior goodness-of-fit and more accurate estimates of tail risk measures than both finite mixture and composite models.

Explaining intraday crude oil returns with higher order risk-neutral moments

High frequency crude oil option data is used to extract the higher order risk-neutral moments from the crude oil market. These risk-neutral moments include the variance, third central moment and the recently developed tail risk variation measures. We find it is beneficial to disaggregate these risk-neutral moments into their semi-moments, and to work with their log differences instead of the level. The log differences of the second and third semi-moments, and to a lesser extent, the log differences of the tail risk measures, are found to explain returns in the crude oil and S&P 500 futures at high frequency. We also provide evidence that the efficient market hypothesis holds at high frequency in these markets.

The Impact of ESG Scores on Risk Market Performance

Over the last two decades, there has been an increased attention to and awareness of corporate environmental, social, and governance (ESG) responsibilities. The asset allocation process has changed accordingly to consider these ESG responsibilities, and it has largely been recognized that private and institutional investors are sensitive to ESG factors when deciding on firms in which to invest. In addition to ESG factors, other key stock-related factors to which investors generally pay attention are risk-adjusted indicators, such as the Sharpe ratio (SR) and the Sortino index (SI), as well as tail risk measures, such as the Value-at-Risk (VaR) and the Expected Shortfall (ES). Overall, the SR, SI, VaR, and ES can provide a guide for investors concerning the risk market performance of a stock under investigation. In this context, the research question that arises is the following: are firms’ performances sensitive to ESG rates? The present contribution aims to answer this question. In particular, the SR, SI, VaR, and ES measures of a set of listed firms are calculated and evaluated. Among these, there are firms with low ESG grades and some with high ESG grades according to two ESG rate providers. The list of stocks under consideration consists of the first 25 constituents (by weight) of the S&P500 index in the period from 2020 and 2022. The empirical findings indicate that risk market performance does not properly depend on high or low ESG rates.

Risk of Stochastic Systems for Temporal Logic Specifications

The wide availability of data coupled with the computational advances in artificial intelligence and machine learning promise to enable many future technologies such as autonomous driving. While there has been a variety of successful demonstrations of these technologies, critical system failures have repeatedly been reported. Even if rare, such system failures pose a serious barrier to adoption without a rigorous risk assessment. This article presents a framework for the systematic and rigorous risk verification of systems. We consider a wide range of system specifications formulated in signal temporal logic (STL) and model the system as a stochastic process, permitting discrete-time and continuous-time stochastic processes. We then define the STL robustness risk as the risk of lacking robustness against failure . This definition is motivated as system failures are often caused by missing robustness to modeling errors, system disturbances, and distribution shifts in the underlying data generating process. Within the definition, we permit general classes of risk measures and focus on tail risk measures such as the value-at-risk and the conditional value-at-risk. While the STL robustness risk is in general hard to compute, we propose the approximate STL robustness risk as a more tractable notion that upper bounds the STL robustness risk. We show how the approximate STL robustness risk can accurately be estimated from system trajectory data. For discrete-time stochastic processes, we show under which conditions the approximate STL robustness risk can even be computed exactly. We illustrate our verification algorithm in the autonomous driving simulator CARLA and show how a least risky controller can be selected among four neural network lane-keeping controllers for five meaningful system specifications.

Portfolio optimization using cellwise robust association measures and clustering methods with application to highly volatile markets

This paper introduces the minCluster portfolio, which is a portfolio optimization method combining the optimization of downside risk measures, hierarchical clustering and cellwise robustness. Using cellwise robust association measures, the minCluster portfolio is able to retrieve the underlying hierarchical structure in the data. Furthermore, it provides downside protection by using tail risk measures for portfolio optimization. We show through simulation studies and a real data example that the minCluster portfolio produces better out-of-sample results than mean-variances or other hierarchical clustering based approaches. Cellwise outlier robustness makes the minCluster method particularly suitable for stable optimization of portfolios in highly volatile markets, such as portfolios containing cryptocurrencies.

Natural gas and the utility sector nexus in the U.S.: Quantile connectedness and portfolio implications

Given that natural gas is a vital input for the U.S. utility sector, this study empirically investigates the return connectedness between the natural gas and utility stocks in the U.S. market. Using the quantile connectedness approach, we show that the nexus between natural gas and utility stocks is more pronounced at the tails compared to the central of the conditional distribution. The return connectedness indices are time-varying with a net receiver role of natural gas and driven by various macro-variables. Finally, our portfolio implication analyses with alternative tail risk measures suggest that it can be more beneficial for risk-adverse investors to allocate substantial weights into the electricity utility stocks in normal market conditions. However, during the COVID-19- induced recession, it is critical to shift more fund to the natural gas futures to reduce tail risks.