Hostname: page-component-848d4c4894-wg55d Total loading time: 0 Render date: 2024-05-01T11:04:11.077Z Has data issue: false hasContentIssue false
  • English
  • Français

Tail variance allocation, Shapley value, and the majorization problem

Part of: Game theory Linear inference, regression

Published online by Cambridge University Press:  06 June 2023

Marcello Galeotti  and
Giovanni Rabitti
Marcello Galeotti*
Affiliation:
University of Florence
Giovanni Rabitti*
Affiliation:
Heriot-Watt University
*
*Postal address: Department of Statistics, Informatics and Applications, University of Florence, Via delle Pandette 9, 50127 Florence, Italy. Email: marcello.galeotti@unifi.it
**Postal address: Department of Actuarial Mathematics and Statistics, Heriot-Watt University, and Maxwell Institute for Mathematical Sciences, Edinburgh, U.K. Email: g.rabitti@hw.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

With a focus on the risk contribution in a portofolio of dependent risks, Colini-Baldeschi et al. (2018) introduced Shapley values for variance and standard deviation games. In this note we extend their results, introducing tail variance as well as tail standard deviation games. We derive closed-form expressions for the Shapley values for the tail variance game and we analyze the vector majorization problem for the two games. In particular, we construct two examples showing that the risk contribution rankings for the two games may be inverted depending on the conditioning threshold and the tail fatness. Motivated by these examples, we formulate a conjecture for general portfolios. Lastly, we discuss risk management implications, including the characterization of tail covariance premiums and reinsurance pricing for peer-to-peer insurance policies.

Keywords

Covariance matrixsum of random variablesvector majorizationcooperative gamestail risk sharing

MSC classification

Primary: 91A12: Cooperative games
Secondary: 62J10: Analysis of variance and covariance
Type
Original Article
Information
Journal of Applied Probability , Volume 61 , Issue 1 , March 2024 , pp. 153 - 171
Check for updates
Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Applied Probability Trust

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abbasi, B. and Hosseinifard, S. Z. (2013). Tail conditional expectation for multivariate distributions: A game theory approach. Statist. Prob. Lett. 10, 2228–2235.CrossRefGoogle Scholar
Albrecher, H., Beirlant, J. and Teugels, J. L. (2017). Reinsurance: Actuarial and Statistical Aspects. Wiley, Chichester.CrossRefGoogle Scholar
Algaba, E., Fragnelli, V. and Sanchez-Soriano, J. (2019). The Shapley value, a paradigm of fairness. In Handbook of the Shapley Value, eds E. Algaba, V. Fragnelli, and J. Sanchez-Soriano, CRC Press, Boca Raton, FL.CrossRefGoogle Scholar
Asmussen, S., Blanchet, J., Juneja, S. and Rojas-Nandayapa, L. (2011). Efficient simulation of tail probabilities of sums of correlated lognormals. Ann. Operat. Res. 189, 5–23.CrossRefGoogle Scholar
Chen, Z., Hu, Z. and Tang, Q. (2020). Allocation inequality in cost sharing problem. J. Math. Econom. 91, 111–120.CrossRefGoogle Scholar
Chen, Z. and Xie, W. (2021). Sharing the value-at-risk under distributional ambiguity. Math. Finance 31, 531–559.CrossRefGoogle Scholar
Clemente, G. P. and Marano, P. (2020). The broker model for peer-to-peer insurance: An analysis of its value. Geneva Papers Risk Insurance Issues Practice 45, 457–481.CrossRefGoogle Scholar
Colini-Baldeschi, R., Scarsini, M. and Vaccari, S. (2018). Variance allocation and Shapley value. Methodology Comput. Appl. Prob. 20, 919–933.CrossRefGoogle Scholar
Cox, L. A. (1985). A new measure of attributable risk for public health applications. Manag. Sci. 31, 800–813.CrossRefGoogle Scholar
Denault, M. (2001). Coherent allocation of risk capital. J. Risk 4, 1–34.CrossRefGoogle Scholar
Denuit, M. (2020). Investing in your own and peers’ risks: The simple analytics of P2P insurance. Europ. Actuarial J. 10, 335–359.CrossRefGoogle Scholar
Dhaene, J., Henrard, L., Landsman, Z., Vandendorpe, A. and Vanduffel, S. (2008). Some results on the CTE-based capital allocation rule. Insurance Math. Econom. 42, 855–863.CrossRefGoogle Scholar
Dhaene, J., Tsanakas, A., Valdez, E. A. and Vanduffel, S. (2012). Optimal capital allocation principles. J. Risk Insurance 79, 1–28.CrossRefGoogle Scholar
Embrechts, P., Liu, H. and Wang, R. (2018). Quantile-based risk sharing. Operat. Res. 66, 936–949.CrossRefGoogle Scholar
Fallahgoul, H. and Loeper, G. (2021). Modelling tail risk with tempered stable distributions: An overview. Ann. Operat. Res. 299, 1253–1280.CrossRefGoogle Scholar
Furman, E. and Landsman, Z. (2006). Tail variance premium with applications for elliptical portfolio of risks. ASTIN Bull. 36, 433–462.CrossRefGoogle Scholar
Galeotti, M. and Rabitti, G. (2021). On the comparison of Shapley values for variance and standard deviation games. J. Appl. Prob. 58, 609–620.CrossRefGoogle Scholar
Grömping, U. (2007). Estimators of relative importance in linear regression based on variance decomposition. Amer. Statistician 61, 139–147.CrossRefGoogle Scholar
Hart, S. and Mas-Colell, A. (1989). Potential, value, and consistency. Econometrica 57, 589–614.CrossRefGoogle Scholar
Il Idrissi, M., Chabridon, V. and Iooss, B. (2021). Developments and applications of Shapley effects to reliability-oriented sensitivity analysis with correlated inputs. Environm. Modelling Software 143, 105–115.Google Scholar
Iooss, B, and Prieur, C. (2019). Shapley effects for sensitivity analysis with dependent inputs: Comparisons with Sobol’ indices, numerical estimation and applications. Internat. J. Uncert. Quant. 9, 493–514.Google Scholar
Kim, J. H. T. and Kim, S.-Y. (2019). Tail risk measures and risk allocation for the class of multivariate normal mean-variance mixture distributions. Insurance Math. Econom. 86, 145–157.CrossRefGoogle Scholar
Landsman, Z. (2010). On the tail mean-variance optimal portfolio selection. Insurance Math. Econom. 46, 547–553.CrossRefGoogle Scholar
Lemaire, J. (1984). An application of game theory: Cost allocation. ASTIN Bull. 14, 61–81.CrossRefGoogle Scholar
Lipovetsky, S. and Conklin, M. (2001). Analysis of regression in game theory approach. Appl. Stoch. Models Business Industry 17, 319–330.CrossRefGoogle Scholar
Liu, F. and Wang, R. (2021). A theory for measures of tail risk. Math. Operat. Res. 46, 1109–1128.CrossRefGoogle Scholar
McNeil, A. J., Frey, R. and Embrechts, P. (2015). Quantitative Risk Management: Concepts, Techniques and Tools, 2nd edn. Princeton University Press.Google Scholar
Overbeck, L. (2000). Allocation of economic capital in loan portfolios. In Measuring Risk in Complex Systems, eds W. Haerdle and G. Stahl. Springer, New York, pp. 15–30.CrossRefGoogle Scholar
Owen, A. B. (2014). Sobol’ indices and Shapley value. SIAM/ASA J. Uncert. Quant. 2, 245–251.CrossRefGoogle Scholar
Owen, A. B. and Prieur, C. (2017). On Shapley value for measuring importance of dependent inputs. SIAM/ASA J. Uncert. Quant. 5, 986–1002.CrossRefGoogle Scholar
Plischke, E., Rabitti, G. and Borgonovo, E. (2021). Computing Shapley effects for sensitivity analysis. SIAM/ASA J. Uncert. Quant. 9, 1411–1437.CrossRefGoogle Scholar
Rabitti, G. and Borgonovo, E. (2020). Is mortality or interest rate the most important risk in annuity models? A comparison of sensitivity analysis methods. Insurance Math. Econom. 95, 48–58.CrossRefGoogle Scholar
Risk, J. and Ludkovski, M. (2018). Sequential design and spatial modeling for portfolio tail risk measurement. SIAM J. Financial Math. 9, 1137–1174.CrossRefGoogle Scholar
Shapley, L. S. (1953). A value for n-person games. In Contributions to the Theory of Games, eds H. W. Kuhn and A. W. Tucker. Princeton University Press, pp. 307–317.CrossRefGoogle Scholar
Tasche, D. (2004). Allocating portfolio economic capital to sub-portfolios. In Economic Capital: A Practitioner Guide, ed Dev, A.. Risk Books, London, pp. 275–302.Google Scholar
Tsanakas, A. and Barnett, C. (2003). Risk capital allocation and cooperative pricing of insurance liabilities. Insurance Math. Econom. 33, 239–254.CrossRefGoogle Scholar
Valdez, E. A. (2004). On tail conditional variance and tail covariances. In Proc. 8th Int. Congress Insurance Math. Econom. 2004.Google Scholar
Valdez, E. A. (2005). Tail conditional variance for elliptically contoured distributions. Belg. Actuarial Bull. 5, 26–36.Google Scholar
Wang, M. (2014). Capital allocation based on the Tail Covariance Premium Adjusted. Insurance Math. Econom. 57, 125–131.CrossRefGoogle Scholar