Conditional Tail Expectation Research Articles

Value-at-Risk, Tail Value-at-Risk and upper tail transform of the sum of two counter-monotonic random variables

ABSTRACT The Value-at-Risk (VaR) of comonotonic sums can be decomposed into marginal VaRs at the same level. This additivity property allows to derive useful decompositions for other risk measures. In particular, the Tail Value-at-Risk (TVaR) and the upper tail transform of comonotonic sums can be written as the sum of their corresponding marginal risk measures. The other extreme dependence situation, involving the sum of two arbitrary counter-monotonic random variables, presents a certain number of challenges. One of them is that it is not straightforward to express the VaR of a counter-monotonic sum in terms of the VaRs of the marginal components of the sum. This paper generalizes the results derived in [Chaoubi, I., Cossette, H., Gadoury, S.-P. & Marceau, E. (2020). On sums of two counter-monotonic risks. Insurance: Mathematics and Economics 92, 47–60.] by providing decomposition formulas for the VaR, TVaR and the stop-loss transform of the sum of two arbitrary counter-monotonic random variables.

Forecasting tail risk for Bitcoin: A dynamic peak over threshold approach

This paper employs a dynamic peak over threshold (PoT) model to measure and forecast both the lower and upper tail Value at Risks (VaRs) of Bitcoin returns, which offers a new perspective to investigate the tail risk dynamics for Bitcoin. We evaluate the VaR forecasting accuracy of this model compared with that of the GARCH-EVT models based on Student-t, skewed Student-t and Generalized error distribution. The empirical results illustrate that the dynamic PoT model exhibits superior out-of-sample VaR predictive ability, specifically for the lower tail VaR. Thus, this model can be a useful and reliable alternative for forecasting tail risk.

Smoothed Quantiles for Measuring Discrete Risks

Many risk measures can be defined through the quantile function of the underlying loss variable (e.g., a class of distortion risk measures). When the loss variable is discrete or mixed, however, the definition of risk measures has to be broadened, which makes statistical inference trickier. To facilitate a straightforward transition from the risk measurement literature of continuous loss variables to that of discrete, in this article we study smoothing of quantiles for discrete variables. Smoothed quantiles are defined using the theory of fractional or imaginary order statistics, which was originated by Stigler (1977). To prove consistency and asymptotic normality of sample estimators of smoothed quantiles, we utilize the results of Wang and Hutson (2011) and generalize them to vectors of smoothed quantiles. Further, we thoroughly investigate extensions of this methodology to discrete populations with infinite support (e.g., Poisson and zero-inflated Poisson distributions). Furthermore, large- and small-sample properties of the newly designed estimators are investigated theoretically and through Monte Carlo simulations. Finally, applications of smoothed quantiles to risk measurement (e.g., estimation of distortion risk measures such as Value at Risk, conditional tail expectation, and proportional hazards transform) are discussed and illustrated using automobile accident data. Comparisons between the classical (and linearly interpolated) quantiles and smoothed quantiles are performed as well.

Can Terahertz Provide High-Rate Reliable Low-Latency Communications for Wireless VR?

Wireless virtual reality (VR), a key 3GPP use case of emerging cellular systems, imposes new visual and haptic requirements directly linked to the Quality of Experience (QoE) of VR users. These QoE requirements can only be met by wireless connectivity that offers high-rate and high-reliability low-latency communications (HR2LLC), unlike the low rates commonly associated with ultrareliable low-latency communication. The high rates for VR over short distances can only be supported by an enormous bandwidth, available in the terahertz (THz)-frequency bands. To explore the potential of THz for meeting HR2LLC requirements, a quantification of the risk for an unreliable VR performance is conducted through a novel and rigorous characterization of the tail of the end-to-end (E2E) delay. Then, a thorough analysis of the Tail-Value-at-Risk (TVaR) is performed to concretely characterize the behavior of extreme wireless events crucial to the real-time VR experience. In particular, the probability distribution function of the THz transmission delay is derived and then used to infer the system reliability scenarios with guaranteed Line of Sight (LoS) as a function of THz network parameters. Numerical results show that abundant bandwidth and low molecular absorption are necessary to improve the reliability. However, their effect remains secondary compared to the availability of LoS, which significantly affects the THz HR2LLC performance. In particular, for scenarios with guaranteed LoS, a reliability of 99.999% (with an E2E delay threshold of 20 ms) for a bandwidth of 15 GHz along with data rates of 18.3 Gbps can be achieved by the THz network, compared to a reliability of 96% for twice the bandwidth, when blockages are considered.

Estimating Copula-Based Extension of Tail Value-at-Risk and Its Application in Insurance Claim

Dependent Tail Value-at-Risk, abbreviated as DTVaR, is a copula-based extension of Tail Value-at-Risk (TVaR). This risk measure is an expectation of a target loss once the loss and its associated loss are above their respective quantiles but bounded above by their respective larger quantiles. In this paper, we propose nonparametric estimators for DTVaR and establish their property of consistency. Moreover, we also propose the variability measure around this expected value truncated by the quantiles, called the Dependent Conditional Tail Variance (DCTV). We use this measure for constructing confidence intervals of the DTVaR. Both parametric and nonparametric approaches for DTVaR estimations are explored. Furthermore, we assess the performance of DTVaR estimations using a proposed backtest based on the DCTV. As for the numerical study, we take an application in the insurance claim amount.

Asymptotic risk decomposition for regularly varying distributions with tail dependence

In this paper we investigate the limiting behaviour of Conditional Tail Expectation (CTE) and its decomposition for a sum of real-valued tail-dependent random variables with regularly varying distributions. Asymptotic proportions to the corresponding Value at Risk (VaR) measures are obtained for a flexible dependence structure. For a certain practical case considering an investment portfolio exact formulas are derived and sensitivity to model parameters is analysed. We also carry out a simulation study verifying our results and revealing the speed of convergence for different values of parameters.

On Moments of Folded and Doubly Truncated Multivariate Extended Skew-Normal Distributions

ABSTRACT This article develops recurrence relations for integrals that relate the density of multivariate extended skew-normal (ESN) distribution, including the well-known skew-normal (SN) distribution introduced by Azzalini and Dalla-Valle and the popular multivariate normal distribution. These recursions offer a fast computation of arbitrary order product moments of the multivariate truncated extended skew-normal and multivariate folded extended skew-normal distributions with the product moments as a byproduct. In addition to the recurrence approach, we realized that any arbitrary moment of the truncated multivariate extended skew-normal distribution can be computed using a corresponding moment of a truncated multivariate normal distribution, pointing the way to a faster algorithm since a less number of integrals is required for its computation which results much simpler to evaluate. Since there are several methods available to calculate the first two moments of a multivariate truncated normal distribution, we propose an optimized method that offers a better performance in terms of time and accuracy, in addition to consider extreme cases in which other methods fail. Finally, we present an application in finance where multivariate tail conditional expectation (MTCE) for SN distributed data is calculated using analytical expressions involving normal left-truncated moments. The R MomTrunc package provides these new efficient methods for practitioners. Supplementary files for this article are available online.

Tail risk measures with application for mixtures of elliptical distributions

<abstract><p>In this paper we derive explicit formulas of tail conditional expectation ($ \text{TCE} $) and tail variance ($ \text{TV} $) for the class of location-scale mixtures of elliptical distributions, which includes the generalized hyper-elliptical ($ \text{GHE} $) distribution. We also develop portfolio risk decomposition with $ \text{TCE} $ for multivariate location-scale mixtures of elliptical distributions. To illustrate our findings, we focus on the generalized hyperbolic ($ \text{GH} $) family which is a popular subclass of the $ \text{GHE} $ for stocks modelling.</p></abstract>

Family of extended mean mixtures of multivariate normal distributions: Properties, inference and applications

<abstract><p>A new class of skewed distributions, with a matrix skewness parameter, called extended mean mixtures of multivariate normal (EMMN) distributions, is constructed. The family of EMMN distributions includes the SN and MMN distributions as special cases. Some basic properties of this family, such as characteristic function, moment generating function, affine transformation and canonical forms of the distributions are derived. An EM-type algorithm is developed to carry out the maximum likelihood estimation of the parameters. Two special cases of this family are studied in detail. A simulation is carried out to examine the performance of the estimation method, and the flexibility is illustrated by fitting a special case of this family to a real data. Finally, the theoretical formula of the multivariate tail conditional expectation of the EMMN distribution is derived.</p></abstract>

Optimal Reciprocal Reinsurance under VaR or TVaR Constraint

Abstract Most existing researches on optimal reinsurance contract are based on an insurer’s viewpoint. However, the optimal reinsurance contract for an insurer is not necessarily to be optimal for a reinsurer. Hence, this study aims to develop the optimal reciprocal reinsurance which satisfies the benefits of both the insurer and reinsurer. Additionally, due to legislative restriction or risk management requirement, the wealth of insurer and reinsurer are frequently imposed upon a VaR (Value-at-Risk) or TVaR (Tail Value-at-Risk) constraint. Therefore, this study develops an optimal reciprocal reinsurance contract which maximizes the common benefits (evaluated by weighted addition of expected utilities) of the insurer and reinsurer subject to their VaR or TVaR constraints. Furthermore, for avoiding moral hazard problem, the developed contract is additionally restricted to a regular form or incentive compatibility (both indemnity schedule and retained loss schedule are continuously nondecreasing).

Risk aggregation and capital allocation using a new generalized Archimedean copula

In this paper, we address risk aggregation and capital allocation problems in the presence of dependence between risks. The dependence structure is defined by a mixed Bernstein copula which represents a generalization of the well-known Archimedean copulas. Using this new copula, the probability density function and the cumulative distribution function of the aggregate risk are obtained. Then, closed-form expressions for basic risk measures, such as tail value at risk (TVaR) and TVaR-based allocations, are derived.

CONFIDENCE INTERVALS OF THE ADJUSTED TAIL CONDITIONAL EXPECTATION RISK MEASURE FOR A STATIONARY SERIE

In this paper we present a semi-parametric estimator of the adjusted tail conditional expectation risk measure based on the theory of extreme values for a stationary serie. We prove its asymptotic normality and we construct the confidence intervals. The accuracy of these intervals is evaluated through a simulation study.

RISIKO INVESTASI SAHAM SECOND LINER DENGAN TAIL VALUE AT RISK

This pandemic which has been going on for almost a year, is very influential in all fields. Economic growth and investment in all countries have been declined dramatically. Indonesian Composite Stock Price Index (CSPI) as an indicator of stock performance in Indonesia is weakening. Big caps stocks, which are usually the main driver for domestic bourses, fell sharply uncontrollably. However, second liner stocks managed to hold down the CSPI throughout 2020. This condition made many investors switch to stocks in this group. Low prices and high profit potential are attractive considerations. But high price fluctuations make investing riskier. Tail Value at Risk (T-VaR) as a measure of risk in a bad situation is the right measure so that investors can estimate the worst possibility and a larger reserve fund. The purpose of this study is to measure the investment risk in second liner stocks based on T-VaR. The six stocks of Pefindo 25 Index indicate that at a 95% significance level, the investment risk in second liner stocks will be in the range 6,14% - 8,22% of the investment.

Conditional Tail Expectation Decomposition and Conditional Mean Risk Sharing for Dependent and Conditionally Independent Losses

Conditional Tail Expectation Decomposition and Conditional Mean Risk Sharing for Dependent and Conditionally Independent Losses

Impacts on catastrophe risk assessments from multi-segment and multi-fault ruptures in the UCERF3 model

The Uniform California Earthquake Rupture Forecast Version 3 (UCERF3) relaxes fault segmentation, allowing multi-segment and multi-fault ruptures through fault-to-fault “jumps,” with lengths up to ∼1200 km along the San Andreas Fault. Local faults are also highly interconnected, including ruptures on the order of hundreds of kilometers. These prescribed long ruptures did not exist in older models. Longer ruptures produce larger aggregate loss estimates for geographically dispersed assets (portfolios) due to the wider areas that are affected by strong ground shaking. In this study, we model probabilistic earthquake losses of a hypothetical state-wide building portfolio in California. We develop risk deaggregation methods to identify multi-segment and multi-fault ruptures that contribute significantly to high portfolio-wide risks. Three risk measures that are commonly used in risk management decisions are examined: Average Annual Loss (AAL), Return Period Loss (RPLα), and Tail Conditional Expectation (TCEα), for an annual exceedance probability “α,” or corresponding return period “1/α.” Our results show that while the super long ruptures (>500 km) contribute modestly (∼7%) to the portfolio AAL estimate, they contribute more significantly to portfolio catastrophe risk estimates. Specifically, at a 250 year return period, these long ruptures contribute about 26% and 32% to RPL250 and TCE250 estimates, respectively. At a 500-year return period, the corresponding contributions reach about 35% and 39%. Ruptures that connect complex fault systems are also found to be highly influential to estimated portfolio risks. At a 500-year return period, a mere six rupture groups contribute nearly 70% to the RPL500 estimate. Due to the importance of the UCERF3 model to many risk management and public policy decisions, a critical examination of the limit and uncertainty of fault connectivity and rupture lengths of future earthquakes, as well as their impacts on catastrophe risk assessments, is warranted in future model updates.

Can a regulatory risk measure induce profit-maximizing risk capital allocations? The case of conditional tail expectation

Risk capital allocations (RCAs) are an important tool in quantitative risk management, where they are utilized to, e.g., gauge the profitability of distinct business units, determine the price of a new product, and conduct the marginal economic capital analysis. Nevertheless, the notion of RCA has been living in the shadow of another, closely related notion, of risk measure (RM) in the sense that the latter notion often shapes the fashion in which the former notion is implemented. In fact, as the majority of the RCAs known nowadays are induced by RMs, the popularity of the two are apparently very much correlated. As a result, it is the RCA that is induced by the Conditional Tail Expectation (CTE) RM that has arguably prevailed in scholarly literature and applications. Admittedly, the CTE RM is a sound mathematical object and an important regulatory RM, but its appropriateness is controversial in, e.g., profitability analysis and pricing. In this paper, we address the question as to whether or not the RCA induced by the CTE RM may concur with alternatives that arise from the context of profit maximization. More specifically, we provide exhaustive description of all those probabilistic model settings, in which the mathematical and regulatory CTE RM may also reflect the risk perception of a profit-maximizing insurer.

A class of generalised hyper-elliptical distributions and their applications in computing conditional tail risk measures

This paper introduces a new family of Generalised Hyper-Elliptical (GHE) distributions providing further generalisation of the generalised hyperbolic (GH) family of distributions, considered in Ignatieva and Landsman (2019). The GHE family is constructed by mixing an elliptical distribution with a Generalised Inverse Gaussian (GIG) distribution. We present an innovative theoretical framework where a closed form expression for the tail conditional expectation (TCE) is derived for this new family of distributions. We demonstrate that the GHE family is especially suitable for heavy-tailed insurance losses data. Our theoretical TCE results are verified for two special cases, Laplace - GIG and Student-t - GIG mixtures. Both mixtures are shown to outperform the GH distribution, providing excellent fit to univariate and multivariate insurance losses data. The TCE risk measure computed for the GHE family of distributions provides a more conservative estimator of risk in the extreme tail, addressing the main challenge faced by financial companies on how to reliably quantify the risk arising from extreme losses. Our multivariate analysis allows to quantify correlated risks by means of the GHE family: the TCE of the portfolio is decomposed into individual components, representing individual risks in the aggregate loss.

Tail Moments of Compound Distributions

In this article, we study the moment transform of both univariate and multivariate compound sums. We first derive simple explicit formulas for the first and second moment transforms when the (loss) frequency distribution is in the so-called class. Then we show that the derived formulas can be used to efficiently compute risk measures such as the tail conditional expectation (TCE), the tail variance (TV), and higher tail moments. The results generalize those in Denuit (North American Actuarial Journal, 24 (4):512–32, 2020).

The Combined Stop-Loss and Quota-Share Reinsurance: Conditional Tail Expectation-Based Optimization from the Joint Perspective of Insurer and Reinsurer

In the presence of reinsurance, an insurer may effectively reduce its (aggregated) loss by partially ceding such a loss to a reinsurer. Stop-loss and quota-share reinsurance contracts are commonly agreed between these two parties. In this paper, we aim to explore a combination of these contracts. The survival functions of the ceded loss and the retained loss are firstly investigated. Optimizing such a reinsurance design is then carried out from the joint perspective of the insurer and the reinsurer. Specifically, we explicitly derive optimal retentions under a criterion of minimizing a convex combination of conditional tail expectations of the insurer’s total loss and the reinsurer’s total loss. In addition, an estimation procedure and more explanations on numerical examples are also presented to find their estimated values.

Tail conditional risk measures for location-scale mixture of elliptical distributions

We present general results on the univariate tail conditional expectation (TCE) and multivariate tail conditional expectation (MTCE) for location-scale mixture of elliptical distributions. Examples include the location-scale mixture of normal distributions, location-scale mixture of Student- distributions, location-scale mixture of logistic distributions and location-scale mixture of Laplace distributions. We also consider portfolio risk decomposition with TCE for location-scale mixture of elliptical distributions. More specifically, we give MTCEs of generalized hyperbolic and slash distributions, and discuss the difference of MTCEs for generalized hyperbolic and slash distributions. As an illustrative example, we discuss the MTCE of five stocks including Amazon, Goldman Sachs, IBM, Google and Apple.