Conditional expectiles, time consistency and mixture convexity properties

Bellini, F.; Bignozzi, Valeria; Puccetti, Giovanni

doi:10.1016/j.insmatheco.2018.07.001

Cited by 18 publications

(18 citation statements)

References 23 publications

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“…The points  where CVaR is not smooth correspond to the discontinuity of VaR. (see for example [11,20] This result is intuitively obvious, because the expectile (optimal value * C in (1)) divides the whole interval of a random variable into two subintervals with different weights. The same division should produce the optimal value   in the formula (10) or the optimal t  in the formula (9).…”

Section: Definitions Of Expectilementioning

confidence: 99%

“…j Cu and constraints (19), (20) are added to the optimization problem. After solving the optimization problem value of expectile should be calculated since the optimal value * C may be greater than expectile at the optimal point.…”

Section: Expectile Linearizationmentioning

confidence: 99%

“…It can be helpful to use a linear maximization problem to calculate expectile. Such problem can be obtained by reducing the linear-fractional problem (9) to a linear problem or by transforming a problem dual to (18) - (20). We formulate a linear maximization problem as This result corresponds to the formulation of the optimization problem in (9) and   is the optimal confidence level in (8).…”

Section: Expectile Linearizationmentioning

confidence: 99%

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A New Family of Expectiles and its Properties

Kuzmenko

2020

Cybernetics and Computer Technologies

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Introduction. This paper considers a risk measure called expectile. Expectile is a characteristic of a random variable calculated using the asymmetric least square method. The level of asymmetry is defined by a parameter in the interval (0, 1). Expectile is used in financial applications, portfolio optimization problems, and other applications as well as Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR). But expectile has a set of advantageous properties. Expectile is both a coherent and elicitable risk measure that takes into account the whole distribution and assigns greater weight to the right tail. The purpose of the paper. As a rule, expectile is compared with quantile (VaR). Our goal is to compare expectile with CVaR by introducing the same parameter – confidence level. To do this we first give a new representation of expectile using the weighted sum of mean and CVaR. Then we consider a new family of expectiles defined by two parameters. Such expectiles are compared with quantile and CVaR for different continuous and finite discrete distributions. Our next goal is to build a regular risk quadrangle where expectile is a risk function. Results. We propose and substantiate two new expressions that define expectile. The first expression uses maximization by varying confidence level of CVaR and varying coefficient before CVaR. It is specified for continuous and finite discrete distributions. The second expression uses minimization of the new error function of the new expectile-based risk quadrangle. The use of two parameters in expectile definition changes the dependence of expectile on its confidence level and generates a new family of expectiles. Comparison of new expectiles with quantile and CVaR for a set of distributions shows that the proposed expectiles can be closer to the quantile than the standard expectile. We propose two variants for expectile linearization and show how to use them with a linear loss function. Keywords: Expectile, EVaR, Quantile, Conditional Value-at-Risk, CVaR, Kusuoka representation, Fundamental Risk Quadrangle, Portfolio Safeguard package.

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Section: Definitions Of Expectilementioning

confidence: 99%

Section: Expectile Linearizationmentioning

confidence: 99%

Section: Expectile Linearizationmentioning

confidence: 99%

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A New Family of Expectiles and its Properties

Kuzmenko

2020

Cybernetics and Computer Technologies

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“…However, the literature on the theoretical properties of this model is still limited. We cite, for instance, [6] who generalized the mean regression to the expectile regression by means of the minimisation of an asymmetric quadratic loss function and presented their main properties. The theoretical and numerical results of the comparison study for these risk measures are given in [5] and indicate that the expectiles are perfectly reasonable alternatives to the Value-at-Risk (VaR) and expected shortfall (ES) risk measures.…”

Section: Introductionmentioning

confidence: 99%

“…Its main features are the subadditivity and the sensitivity to the magnitude of the extreme losses. In particular it is well documented that the expectile model is the only elicitable coherent risk measure (see, [6]). As far as we know, the problem that we consider is open up to now, which motivated us to provide a further study.…”

Section: Introductionmentioning

confidence: 99%

On the nonparametric estimation of the functional expectile regression

Mohammedi¹,

Bouzebda

Laksaci³

2020

Comptes Rendus. Mathématique

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Real-time changepoint detection in a nonlinear expectile model

Ciuperca

Maciak

Pešta

2023

Metrika

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An online changepoint detection procedure based on conditional expectiles is introduced. The key contribution is threefold: nonlinearity of the underlying model improves the overall flexibility while a parametric form of the unknown regression function preserves a simple and straightforward interpretation; The conditional expectiles, well-known in econometrics for being the only coherent and elicitable risk measure, introduce additional robustness—especially with respect to asymmetric error distributions common in various types of data; The proposed statistical test is proved to be consistent and the distribution under the null hypothesis does not depend on the functional form of the underlying model nor the unknown parameters. Empirical properties of the proposed real-time changepoint detection test are investigated in a simulation study and a practical applicability is illustrated using the Covid-19 prevalence data from Prague.

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Conditional expectiles, time consistency and mixture convexity properties

Cited by 18 publications

References 23 publications

A New Family of Expectiles and its Properties

A New Family of Expectiles and its Properties

On the nonparametric estimation of the functional expectile regression

Real-time changepoint detection in a nonlinear expectile model

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