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

Part of: Distribution theory - Probability Applications

Published online by Cambridge University Press:  31 July 2024

Jun Duan ,
Zinoviy Landsman Open the ORCID record for Zinoviy Landsman [Opens in a new window]  and
Jing Yao Open the ORCID record for Jing Yao [Opens in a new window]
Jun Duan*
Affiliation:
Chongqing Normal University.
Zinoviy Landsman*
Affiliation:
University of Haifa and Holon Institute of Technology
Jing Yao*
Affiliation:
Soochow University
*
*Postal address: School of Economics & Management, Chongqing Normal University, China. Email address: duanjun@cqnu.edu.cn
**Postal address: Actuarial Research Center, Department of Statistics, University of Haifa, Israel. Faculty of Sciences, Holon Institute of Technology, Israel. Email address: landsman@stat.haifa.ac.il
***Postal address: Center for financial engineering and Department of Mathematics, Soochow University, China. Email address: j.yao@suda.edu.cn
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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.

Keywords

Tail variancetail conditional expectationprobability inequalitiescapital adequacyasymptotics

MSC classification

Primary: 62P05: Applications to actuarial sciences and financial mathematics
Secondary: 60E15: Inequalities; stochastic orderings
Type
Original Article
Information
Advances in Applied Probability , First View , pp. 1 - 25
Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Applied Probability Trust

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