Paper: arXiv 2404.18029

Authors: Bingzhen Geng, Yang Liu, Yimiao Zhao

Abstract

The systemic risk measure plays a crucial role in analyzing individual losses conditioned on extreme system-wide disasters. In this paper, we provide a unified asymptotic treatment for systemic risk measures. First, we classify them into two families of Value-at-Risk- (VaR-) and expectile-based systemic risk measures. While VaR has been extensively studied, in the latter family, we propose two new systemic risk measures named the Individual Conditional Expectile (ICE) and the Systemic Individual Conditional Expectile (SICE), as alternatives to Marginal Expected Shortfall (MES) and Systemic Expected Shortfall (SES). Second, to characterize general mutually dependent and heavy-tailed risks, we adopt a modeling framework where the system, represented by a vector of random loss variables, follows a multivariate Sarmanov distribution with a common marginal exhibiting second-order regular variation. Third, we provide second-order asymptotic results for both families of systemic risk measures. This analytical framework offers a more accurate estimate compared to traditional first-order asymptotics. Through numerical and analytical examples, we demonstrate the superiority of second-order asymptotics in accurately assessing systemic risk. Further, we conduct a comprehensive comparison between VaR-based and expectile-based systemic risk measures. Expectile-based measures output higher risk evaluation than VaR-based ones, emphasizing the former’s potential advantages in reporting extreme events and tail risk. As a financial application, we use the asymptotic treatment to discuss the diversification benefits associated with systemic risk measures. The expectile-based diversification benefits consistently deduce an underestimation and suggest a conservative approximation, while the VaR-based diversification benefits consistently deduce an overestimation and suggest behaving optimistically.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 3.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents rigorous mathematical formulations of second-order asymptotics and introduces novel expectile-based systemic risk measures (ICE and SICE). However, the focus remains primarily theoretical and analytical, with limited real-world empirical validation or code provided for backtesting.

Research Flowchart

  flowchart TD
    A["Research Goal: Unified Asymptotic Treatment<br>for Systemic Risk & Diversification"] --> B["Inputs: Heavy-Tailed Multi-Asset Loss Vector<br>& Multivariate Sarmanov Distribution"]
    
    B --> C["Taxonomy & Proposal of Risk Measures"]
    
    C --> C1["VaR-Based Measures<br>(CoVaR, MES, SES)"]
    C --> C2["New Expectile-Based Measures<br>(ICE, SICE)"]
    
    C1 --> D["Derive Second-Order Asymptotics<br>(vs. First-Order Approximations)"]
    C2 --> D
    
    D --> E["Findings & Applications"]
    
    E --> E1["Expectile-Based Measures:<br>Higher Risk Output & Conservative Diversification"]
    E --> E2["VaR-Based Measures:<br>Lower Risk Output & Optimistic Diversification"]

Cited References (232)

  1. Risk management with expectiles (2017)
  2. Estimation of the H aezendonck- G oovaerts risk measure for extreme risks (2021)
  3. Properties of second-order regular variation and expansions for risk concentration (2012)
  4. Extreme Value Theory: An Introduction (2006)
  5. Bias-corrected peaks-over-threshold estimation of the CVaR (2021)
  6. Weighted premium calculation principles (2008)
  7. Decision theoretic foundations of credibility theory (1989)
  8. Risk capital allocation and cooperative pricing of insurance liabilities (2003)
and 224 more
  1. Agricultural insurance ratemaking: Development of a new premium principle (2019)
  2. Optimal reinsurance under the risk-adjusted value of an insurer’s liability and an economic reinsurance premium principle (2017)
  3. Extreme versions of Wang risk measures and their estimation for heavy-tailed distributions (2017)
  4. Reinsurance premium principles based on weighted loss functions (2019)
  5. Coherent measures of risk (1999)
  6. On dependence consistency of $\text CoVaR $ and some other systemic risk measures (2014)
  7. Systemic risk measurement: Multivariate GARCH estimation of $\text CoVaR $ (2013)
  8. Extremes for coherent risk measures (2016)
  9. Best attainable rates of convergence for estimators of the stable tail dependence function (1998)
  10. Statistics of extremes: theory and applications (2004)
  11. Estimation of the marginal expected shortfall: the mean when a related variable is extreme (2015)
  12. Estimation of tail risk based on extreme expectiles (2018)
  13. Asymptotics for risk capital allocations based on conditional tail expectation (2011)
  14. Systemic risk: an asymptotic evaluation (2018)
  15. Asymptotics for systemic risk with dependent heavy-tailed losses (2021)
  16. How useful is the Marginal Expected Shortfall for the measurement of systemic exposure? A practical assessment (2014)
  17. Empirical estimation of risk measures and related quantities (2003)
  18. Risk measures, distortion parameters, and their empirical estimation (2007)
  19. Asymptotic risk decomposition for regularly varying distributions with tail dependence (2022)
  20. Generalized quantiles as risk measures (2014)
  21. Risk concentration under second order regular variation (2020)
  22. Risk concentration based on E xpectiles for extreme risks under $\text FGM $ copula (2015)
  23. Measuring systemic risk (2017)
  24. Representation of associative functions (1965)
  25. Generalized regular variation of second order (1996)
  26. Hidden regular variation, second order regular variation and asymptotic independence (2002)
  27. Diversification of aggregate dependent risks (2004)
  28. Additivity properties for $\text Value-at-Risk $ under $\text Archimedean $ dependence and heavy-tailedness (2009)
  29. Asymptotic expansions of generalized quantiles and expectiles for extreme risks (2015)
  30. Asymmetric least squares estimation and testing (1987)
  31. Quantiles, expectiles and splines (2009)
  32. Assessing value at risk with $\text CARE $, the C onditional A utoregressive E xpectile M odels (2009)
  33. Making and evaluating point forecasts (2011)
  34. Coherence and elicitability (2016)
  35. On elicitable risk measures (2015)
  36. Copules archim 'e diennes et families de lois bidimensionnelles dont les marges sont donn 'e es (1986)
  37. Statistical inference procedures for bivariate Archimedean copulas (1993)
  38. Multivariate models and multivariate dependence concepts (1997)
  39. Archimedean copulae and positive dependence (2005)
  40. Convolution bounds on quantile aggregation (2025)
  41. Systemic Risk Asymptotics in a Renewal Model with Multiple Business Lines and Heterogeneous Claims — → 2410.00158
  42. A Framework for Measures of Risk under Uncertainty (2024)
  43. Ordering and Inequalities of Mixtures on Risk Aggregation (2022)
  44. DIVERSIFICATION IN CATASTROPHE INSURANCE MARKETS (2021)
  45. Diversification quotients based on $\text VaR $ and $\text ES $ (2023)
  46. Diversification quotients: Quantifying diversification via risk measures (2025)
  47. An unexpected stochastic dominance: Pareto distributions, dependence, and diversification (2025)
  48. Scenario-based risk evaluation (2021)
  49. Systemic risk: Conditional distortion risk measures (2022)
  50. An introduction to copulas (2007)
  51. Approximations of the tail probability of the product of dependent extremal random variables and applications (2013)
  52. Tail behavior of the product of two dependent random variables with applications to risk theory (2013)
  53. Second-order properties of tail probabilities of sums and randomly weighted sums (2015)
  54. Sarmanov family of bivariate distributions for multivariate loss reserving analysis (2016)
  55. Second-order properties of risk concentrations without the condition of asymptotic smoothness (2013)
  56. Asymptotic expansions of convolutions of regularly varying distributions (2005)
  57. Optimal capital allocation principles (2012)
  58. Properties and applications of the $\text Sarmanov $ family of bivariate distributions (1996)
  59. Continuous Multivariate Distributions, Volume 1: Models and Applications (2004)
  60. Generalized normal correlation and two-dimensional \text Fr 'e chet classes (1966)
  61. Risk-sensitive reinforcement learning based on convex scoring functions — → 2505.04553
  62. Extreme conditional expectile estimation in heavy-tailed heteroscedastic regression models (2021)
  63. Second-order regular variation, convolution and the central limit theorem (1997)
  64. Convolutions of heavy-tailed random variables and applications to portfolio diversification and MA (1) time series (2000)
  65. An adaptive optimal estimate of the tail index for MA (l) time series (2000)
  66. Risk concentration and diversification: $\text Second $-order properties (2010)
  67. $\text Second-order expansions of the risk concentration based on CTE $ (2012)
  68. Expectile-based capital allocation (2023)
  69. What is the best risk measure in practice? $\text A $ comparison of standard measures (2015)
  70. Capital allocation for $\text Sarmanov $’s class of distributions (2017)
  71. Estimating value at risk and expected shortfall using expectiles (2008)
  72. On mixed E rlang reinsurance risk: A ggregation, capital allocation and default risk (2016)
  73. $\text TVaR $-based capital allocation for multivariate compound distributions with positive continuous claim amounts (2012)
  74. $\text On Sarmanov mixed Erlang risks in insurance applications $ (2015)
  75. Risk aggregation, dependence structure and diversification benefit (2008)
  76. Dual Representation of Expectile based Expected Shortfall and Its Properties (2019) — link
  77. Second order regular variation and conditional tail expectation of multiple risks (2011)
  78. Validation of aggregated risks models (2018)
  79. Explicit diversification beneift for dependent risks (2016)
  80. $\text CoVaR $ (2016)
  81. Extreme value analysis for financial risk management (2021)
  82. An asymptotic study of systemic expected shortfall and marginal expected shortfall (2022)
  83. Regular Variation (1989)
  84. Tail risk of multivariate regular variation (2011)
  85. Asymptotics of the risk concentration based on the tail distortion risk measure (2013)
  86. On the tail behavior of sums of dependent risks (2006)
  87. Extreme value behavior of aggregate dependent risks (2012)
  88. Analysis of the expected shortfall of aggregate dependent risks (2005)
  89. Extremes on the discounted aggregate claims in a time dependent risk model (2010)
  90. Relative bound and asymptotic comparison of expectile with respect to expected shortfall (2020)
  91. Dependence structure of risk factors and diversification effects (2010)
  92. Modelling extremal dependence for operational risk by a bipartite graph (2020)
  93. $\text Extremes and products of multivariate AC-product risks $ (2013)
  94. Multivariate extremes and the aggregation of dependent risks: E xamples and counter-examples (2009)
  95. Ruin probabilities with insurance and financial risks having an $\text FGM $ dependence structure (2014)
  96. Asymptotic tail probabilities of sums of dependent subexponential random variables (2009)
  97. Tail probability of randomly weighted sums of dependent subexponential random variables with applications to risk theory (2019)
  98. Sums of dependent nonnegative random variables with subexponential tails (2008)
  99. Approximation of the tail probability of randomly weighted sums and applications (2009)
  100. Approximation of the tail probability of randomly weighted sums of dependent random variables with dominated variation (2011)
  101. Randomly weighted sums of subexponential random variables with application to capital allocation (2014)
  102. Uniform tail asymptotics for the stochastic present value of aggregate claims in the renewal risk model (2010)
  103. Asymptotics in a time-dependent renewal risk model with stochastic return (2012)
  104. Asymptotic ruin probabilities for a bivariate L 'e vy-driven risk model with heavy-tailed claims and risky investments (2012)
  105. The finite-time ruin probability of a risk model with stochastic return and Brownian perturbation (2018)
  106. Asymptotic results for ruin probability in a two-dimensional risk model with stochastic investment returns (2017)
  107. Asymptotics for the ruin probability of a time-dependent renewal risk model with geometric L 'e vy process investment returns and dominatedly-varying-tailed claims (2014)
  108. Randomly weighted sums of dependent random variables with dominated variation (2014)
  109. A Kesten-type bound for sums of randomly weighted subexponential random variables (2020)
  110. On the Kesten-type inequality for randomly weighted sums with applications to an operational risk model (2021)
  111. Randomly weighted sums of subexponential random variables with application to ruin theory (2003)
  112. Randomly weighted sums of dependent subexponential random variables with applications to risk theory (2018)
  113. The finite-time ruin probability of the compound Poisson model with constant interest force (2005)
  114. The subexponentiality of products revisited (2006)
  115. An Introduction to heavy-tailed and subexponential distributions (2013)
  116. Modelling extremal events: for insurance and finance (1997)
  117. Stochastic Processes in Queueing Theory (1976)
  118. Subexponential tails of discounted aggregate claims in a time-dependent renewal risk model (2010)
  119. Interplay of subexponential and dependent insurance and financial risks (2017)
  120. Ruin under interest force and subexponential claims: a simple treatment (2000)
  121. Ruin probabilities in the presence of heavy-tails and interest rates (1998)
  122. The ruin probability of the renewal model with constant interest force and negatively dependent heavy-tailed claims (2007)
  123. On Cram 'e r-like asymptotics for risk processes with stochastic return on investments (2002)
  124. Approximerad framst "a llning av sannolikhetsfunktionen. aterf "o rs "a kring av kollektivrisker. Akad (1903)
  125. On the mathematical theory of risk. Skandia Jubilee Volume,(1930) Stockholm. Reprinted in: Martin-L "o f, A.(Ed.) Cram 'e r, H.(1994) Collected Works
  126. Ruin probabilities (2010)
  127. On the collective theory of risk in case of contagion between claims (1957)
  128. Random difference equations with subexponential innovations (2016)
  129. Ruin probabilities allowing for delay in claims settlement (1985)
  130. Ruin probabilities for time-correlated claims in the compound binomial model (2001)
  131. On pairwise quasi-asymptotically independent random variables and their applications (2013)
  132. On asymptotic finite-time ruin probability of a renewal risk model with subexponential main claims and delayed claims (2019)
  133. The asymptotic estimate for the sum of two correlated classes of discounted aggregate claims with heavy tails (2011)
  134. Uniform asymptotics for the finite-time ruin probabilities of two kinds of nonstandard bidimensional risk models (2013)
  135. Asymptotic ruin probabilities in a generalized bidimensional risk model perturbed by diffusion with constant force of interest (2014)
  136. Asymptotic finite-time ruin probability for a bidimensional renewal risk model with constant interest force and dependent subexponential claims (2014)
  137. Finite-time and infinite-time ruin probabilities in a two-dimensional delayed renewal risk model with $\text Sarmanov $ dependent claims (2016)
  138. Uniform asymptotic estimate for finite-time ruin probabilities of a time-dependent bidimensional renewal model (2015)
  139. Uniform asymptotics for finite-time ruin probability of a bidimensional risk model (2019)
  140. Asymptotic ruin probabilities for a bidimensional renewal risk model with constant interest rate and dependent claims (2015)
  141. Asymptotic ruin probabilities for a bidimensional renewal risk model (2017)
  142. A revisit to asymptotic ruin probabilities for a bidimensional renewal risk model (2018)
  143. Ruin estimates under interest force (1995)
  144. On the maximum of randomly weighted sums with regularly varying tails (2006)
  145. Tail probabilities of randomly weighted sums of random variables with dominated variation (2006)
  146. Tail probability of randomly weighted sums of subexponential random variables under a dependence structure (2012)
  147. Randomly weighted sums of dependent subexponential random variables (2011)
  148. A Note on the Kesten-Type Inequality for Sums of Randomly Weighted Dependent Sub-exponential Random Variables (2020)
  149. On ultimate ruin in a delayed-claims risk model (2005)
  150. On a discrete time risk model with time-delayed claims and a constant dividend barrier (2012)
  151. Finite-time ruin probability of a perturbed risk model with dependent main and delayed claims (2021)
  152. Asymptotics for a delay-claim risk model with diffusion, dependence structures and constant force of interest (2019)
  153. Some concepts of dependence (1966)
  154. A note on cumulative sums (1946)
  155. A uniform asymptotic estimate for discounted aggregate claims with subexponential tails (2008)
  156. Subexponentiality of the product of independent random variables (1994)
  157. Ruin probabilities and penalty functions with stochastic rates of interest (2004)
  158. Ruin probabilities for a L 'e vy-driven generalised Ornstein–Uhlenbeck process (2020)
  159. Integrated insurance risk models with exponential L 'e vy investment (2008)
  160. Risk theory in a stochastic economic environment (1993)
  161. L 'e vy processes and infinitely divisible distributions (1999)
  162. Distributions for the risk process with a stochastic return on investments (2001)
  163. Ruin probabilities for a risk process with stochastic return on investments (2004)
  164. Uniformly asymptotic behavior of ruin probabilities in a time-dependent renewal risk model with stochastic return (2015)
  165. Tail asymptotic for discounted aggregate claims with one-sided linear dependence and general investment return (2019)
  166. Levy processes and stochastic calculus (2004)
  167. Value-at-Risk-and Expectile-based Systemic Risk Measures and Second-order Asymptotics: With Applications to Diversification — → 2404.18029
  168. Heavy tails of discounted aggregate claims in the continuous-time renewal model (2007)
  169. Exponential behavior in the presence of dependence in risk theory (2006)
  170. Tail probabilities for non-standard risk and queueing processes with subexponential jumps (1999)
  171. Dependent risk models with bivariate phase-type distributions (2009)
  172. Impact of correlation crises in risk theory: Asymptotics of finite-time ruin probabilities for heavy-tailed claim amounts when some independence and stationarity assumptions are relaxed (2008)
  173. On a risk model with dependence between interclaim arrivals and claim sizes (2006)
  174. On the compound Poisson risk model with dependence based on a generalized Farlie–Gumbel–Morgenstern copula (2008)
  175. Some results on ruin probabilities in a two-dimensional risk model (2003)
  176. On the ruin probabilities of a bidimensional perturbed risk model (2007)
  177. Asymptotics for the ruin probabilities of a two-dimensional renewal risk model with heavy-tailed claims (2011)
  178. Ruin probabilities of a bidimensional risk model with investment (2012)
  179. The infinite-time ruin probability for a bidimensional renewal risk model with constant force of interest and dependent claims (2017)
  180. Copula Methods in Finance (2004)
  181. Some asymptotic results of the ruin probabilities in a two-dimensional renewal risk model with some strongly subexponential claims (2016)
  182. Asymptotics for a bidimensional risk model with two geometric L 'e vy price processes (2019)
  183. On the first time of ruin in the bivariate compound Poisson model (2006)
  184. Asymptotic tail behavior of a random sum with conditionally dependent subexponential summands (2017)
  185. Asymptotics for ruin probabilities in L 'e vy-driven risk models with heavy-tailed claims (2018)
  186. The finite-time ruin probability in two non-standard renewal risk models with constant interest rate and dependent subexponential claims (2012)
  187. On a correlated aggregate claims model with P oisson and E rlang risk processes (2002)
  188. Asymptotic sum-ruin probability for a bidimensional risk model with common shock dependence (2021)
  189. A theorem on sums of independent positive random variables and its applications to branching random processes (1964)
  190. Asymptotic behaviour of the finite-time ruin probability under subexponential claim sizes (2007)
  191. Asymptotic behaviour of the finite-time ruin probability in renewal risk models (2009)
  192. Functions of probability measures (1973)
  193. Asymptotic methods in queuing theory (1984)
  194. Applied probability and queues (2003)
  195. Global and local asymptotics for the busy period of an M/G/1 queue (2010)
  196. On distribution tail of the maximum of a random walk (1997)
  197. Large deviations for random walks under subexponentiality: the big-jump domain (2008)
  198. Asymptotics for the moments of the overshoot and undershoot of a random walk (2009)
  199. Stochastic Processes for Insurance and Finance (1998)
  200. The class of subexponential distributions (1975)
  201. Limit theorems for probabilities of large deviations (1969)
  202. Subexponential distributions and dominated-variation tails (1978)
  203. Uniform asymptotics for the finite-time ruin probability with upper tail asymptotically independent claims and constant force of interest (2013)
  204. Finite-time ruin probability with NQD dominated varying-tailed claims and NLOD inter-arrival times (2009)
  205. A note on a dependent risk model with constant interest rate (2012)
  206. Asymptotics for ruin probability of some negatively dependent risk models with a constant interest rate and dominatedly-varying-tailed claims (2010)
  207. Uniform asymptotics for the finite-time ruin probability of a dependent risk model with a constant interest rate (2013)
  208. Uniformly asymptotic behavior for the tail probability of discounted aggregate claims in the time-dependent risk model with upper tail asymptotically independent claims (2016)
  209. Finite-time ruin probability with an exponential L 'e vy process investment return and heavy-tailed claims (2009)
  210. Uniform asymptotics for the ruin probabilities in a bidimensional renewal risk model with strongly subexponential claims (2019)
  211. On asymptotic finite-time ruin probabilities of a new bidimensional risk model with constant interest force and dependent claims (2021)
  212. Asymptotic finite-time ruin probabilities for a bidimensional delay-claim risk model with subexponential claims (2022)
  213. Asymptotics for a time-dependent by-claim model with dependent subexponential claims (2023) — link
  214. Asymptotic finite-time ruin probability of a bidimensional renewal risk model with subexponential main claims and delayed claims. (2023)
  215. Max-Sum local equivalence of random variables with Farlie–Gumbel–Morgenstern joint distribution (2016)
  216. An Asymptotic Result on Catastrophe Insurance Losses (2023)
  217. Second-order conditions of regular variation and Drees-type inequalities (2013)
  218. Correlation structure in F arlie- G umbel- M orgenstern distributions (1978)
  219. Correlation structure in iterated F arlie- G umbel- M orgenstern distributions (1984)
  220. On some generalized F arlie- G umbel- M orgenstern distributions-\text II regression, correlation and further generalizations (1977)
  221. Tail expectile process and risk assessment (2020)
  222. Asymptotics of Systemic Risk in a Renewal Model with Multiple Business Lines and Heterogeneous Claims — → 2410.00158
  223. When Are Expectiles Percentiles? (1993)
  224. Quantitative Risk Management: Concepts, Techniques and Tools (2015)

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