IDEAS home Printed from https://ideas.repec.org/a/eee/insuma/v122y2025icp180-193.html
   My bibliography  Save this article

Pricing insurance contracts with an existing portfolio as background risk

Author

Listed:
  • De Vecchi, Corrado
  • Scherer, Matthias

Abstract

We develop and investigate a premium principle that explicitly takes into account the impact of a new risk on some insurer's existing portfolio. Specifically, we propose the notion of an indifference premium for a new risk conditioned on an existing portfolio acting as background risk. The resulting premium rule, which in our case depends on the joint distribution of the new risk and the existing portfolio, is analyzed in detail with respect to its mathematical properties. In order to underline the differences between our approach and the literature on law-invariant premium rules, special attention is given to the indifference premium behaviour with respect to some well-known dependence concepts. Axiomatic and continuity properties of the proposed indifference premium rule are also investigated. To demonstrate the practical relevance of our approach, we consider a portfolio of exchangeable risks and investigate the role of the portfolio's dimension on the price of a risk to be added. This illustrates the (limits of) diversification benefits under the flexible exchangeability assumption on the joint distribution of a sequence of risks.

Suggested Citation

  • De Vecchi, Corrado & Scherer, Matthias, 2025. "Pricing insurance contracts with an existing portfolio as background risk," Insurance: Mathematics and Economics, Elsevier, vol. 122(C), pages 180-193.
    • Handle: RePEc:eee:insuma:v:122:y:2025:i:c:p:180-193
    DOI: 10.1016/j.insmatheco.2025.03.001
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0167668725000381
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.insmatheco.2025.03.001?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to

    for a different version of it.

    References listed on IDEAS

    as
    1. Marco Frittelli & Giacomo Scandolo, 2006. "Risk Measures And Capital Requirements For Processes," Mathematical Finance, Wiley Blackwell, vol. 16(4), pages 589-612, October.
    2. Frittelli, Marco & Rosazza Gianin, Emanuela, 2002. "Putting order in risk measures," Journal of Banking & Finance, Elsevier, vol. 26(7), pages 1473-1486, July.
    3. Paul Embrechts, 2000. "Actuarial versus Financial Pricing of Insurance," Journal of Risk Finance, Emerald Group Publishing Limited, vol. 1(4), pages 17-26, March.
    4. Müller, Alfred & Scarsini, Marco, 2005. "Archimedean copulæ and positive dependence," Journal of Multivariate Analysis, Elsevier, vol. 93(2), pages 434-445, April.
    5. Eeckhoudt, Louis & Gollier, Christian & Schlesinger, Harris, 1996. "Changes in Background Risk and Risk-Taking Behavior," Econometrica, Econometric Society, vol. 64(3), pages 683-689, May.
    6. Deprez, Olivier & Gerber, Hans U., 1985. "On convex principles of premium calculation," Insurance: Mathematics and Economics, Elsevier, vol. 4(3), pages 179-189, July.
    7. Yaari, Menahem E, 1987. "The Dual Theory of Choice under Risk," Econometrica, Econometric Society, vol. 55(1), pages 95-115, January.
    8. Tsanakas, Andreas, 2008. "Risk measurement in the presence of background risk," Insurance: Mathematics and Economics, Elsevier, vol. 42(2), pages 520-528, April.
    9. Blier-Wong, Christopher & Cossette, Hélène & Marceau, Etienne, 2023. "Risk aggregation with FGM copulas," Insurance: Mathematics and Economics, Elsevier, vol. 111(C), pages 102-120.
    10. Carole Bernard & Ludger Rüschendorf & Steven Vanduffel, 2017. "Value-at-Risk Bounds With Variance Constraints," Journal of Risk & Insurance, The American Risk and Insurance Association, vol. 84(3), pages 923-959, September.
    11. Dhaene, Jan & Denuit, Michel, 1999. "The safest dependence structure among risks," Insurance: Mathematics and Economics, Elsevier, vol. 25(1), pages 11-21, September.
    12. M. Kaina & L. Rüschendorf, 2009. "On convex risk measures on L p -spaces," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 69(3), pages 475-495, July.
    13. Tversky, Amos & Kahneman, Daniel, 1992. "Advances in Prospect Theory: Cumulative Representation of Uncertainty," Journal of Risk and Uncertainty, Springer, vol. 5(4), pages 297-323, October.
    14. Goovaerts, M. J. & Dhaene, J., 1999. "Supermodular ordering and stochastic annuities," Insurance: Mathematics and Economics, Elsevier, vol. 24(3), pages 281-290, May.
    15. Landsman, Zinoviy & Sherris, Michael, 2001. "Risk measures and insurance premium principles," Insurance: Mathematics and Economics, Elsevier, vol. 29(1), pages 103-115, August.
    16. Alfred Müller & Marco Scarsini, 2001. "Stochastic Comparison of Random Vectors with a Common Copula," Mathematics of Operations Research, INFORMS, vol. 26(4), pages 723-740, November.
    17. John Quiggin, 2003. "Background risk in generalized expected utility theory," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 22(3), pages 607-611, October.
    18. Wang, Bin & Wang, Ruodu, 2011. "The complete mixability and convex minimization problems with monotone marginal densities," Journal of Multivariate Analysis, Elsevier, vol. 102(10), pages 1344-1360, November.
    19. Carole Bernard & Corrado De Vecchi & Steven Vanduffel, 2021. "When do two- or three-fund separation theorems hold?," Quantitative Finance, Taylor & Francis Journals, vol. 21(11), pages 1869-1883, November.
    20. Xiaosheng Mu & Luciano Pomatto & Philipp Strack & Omer Tamuz, 2024. "Background Risk and Small-Stakes Risk Aversion," American Economic Review: Insights, American Economic Association, vol. 6(2), pages 262-276, June.
    21. Denuit Michel & Dhaene Jan & Goovaerts Marc & Kaas Rob & Laeven Roger, 2006. "Risk measurement with equivalent utility principles," Statistics & Risk Modeling, De Gruyter, vol. 24(1), pages 1-25, July.
    22. Chudziak, J., 2020. "On positive homogeneity and comonotonic additivity of the principle of equivalent utility under Cumulative Prospect Theory," Insurance: Mathematics and Economics, Elsevier, vol. 94(C), pages 154-159.
    23. Jan Dhaene & Mark Goovaerts & Rob Kaas, 2003. "Economic Capital Allocation Derived from Risk Measures," North American Actuarial Journal, Taylor & Francis Journals, vol. 7(2), pages 44-56.
    24. Rothschild, Michael & Stiglitz, Joseph E., 1970. "Increasing risk: I. A definition," Journal of Economic Theory, Elsevier, vol. 2(3), pages 225-243, September.
    25. Heilpern, S., 2003. "A rank-dependent generalization of zero utility principle," Insurance: Mathematics and Economics, Elsevier, vol. 33(1), pages 67-73, August.
    26. Furman, Edward & Zitikis, Ricardas, 2008. "Weighted premium calculation principles," Insurance: Mathematics and Economics, Elsevier, vol. 42(1), pages 459-465, February.
    27. Marco Scarsini & Pietro Muliere, 1987. "Characterization of a Marshall-Olkin type class of distributions," Post-Print hal-00542248, HAL.
    28. Wang, Shaun, 1996. "Premium Calculation by Transforming the Layer Premium Density," ASTIN Bulletin, Cambridge University Press, vol. 26(1), pages 71-92, May.
    29. Embrechts, Paul & Puccetti, Giovanni & Rüschendorf, Ludger, 2013. "Model uncertainty and VaR aggregation," Journal of Banking & Finance, Elsevier, vol. 37(8), pages 2750-2764.
    30. Wang, Shaun S., 2003. "Equilibrium Pricing Transforms: New Results Using Buhlmann’s 1980 Economic Model," ASTIN Bulletin, Cambridge University Press, vol. 33(1), pages 57-73, May.
    31. Kihlstrom, Richard E & Romer, David & Williams, Steve, 1981. "Risk Aversion with Random Initial Wealth," Econometrica, Econometric Society, vol. 49(4), pages 911-920, June.
    32. Dhaene, Jan & Goovaerts, Marc J., 1996. "Dependency of Risks and Stop-Loss Order1," ASTIN Bulletin, Cambridge University Press, vol. 26(2), pages 201-212, November.
    33. Dirk Tasche, 2007. "Capital Allocation to Business Units and Sub-Portfolios: the Euler Principle," Papers 0708.2542, arXiv.org, revised Jun 2008.
    34. Denuit, M. & Genest, C. & Marceau, E., 1999. "Stochastic bounds on sums of dependent risks," Insurance: Mathematics and Economics, Elsevier, vol. 25(1), pages 85-104, September.
    35. Muller, Alfred, 1997. "Stop-loss order for portfolios of dependent risks," Insurance: Mathematics and Economics, Elsevier, vol. 21(3), pages 219-223, December.
    36. Mildenhall, Stephen J., 2022. "Similar risks have similar prices: A useful and exact quantification," Insurance: Mathematics and Economics, Elsevier, vol. 105(C), pages 203-210.
    37. Gollier, Christian & Pratt, John W, 1996. "Risk Vulnerability and the Tempering Effect of Background Risk," Econometrica, Econometric Society, vol. 64(5), pages 1109-1123, September.
    38. Kaas, Rob & Laeven, Roger J.A. & Nelsen, Roger B., 2009. "Worst VaR scenarios with given marginals and measures of association," Insurance: Mathematics and Economics, Elsevier, vol. 44(2), pages 146-158, April.
    39. Luciano Pomatto & Philipp Strack & Omer Tamuz, 2020. "Stochastic Dominance under Independent Noise," Journal of Political Economy, University of Chicago Press, vol. 128(5), pages 1877-1900.
    40. Sordo, Miguel A. & Castaño-Martínez, Antonia & Pigueiras, Gema, 2016. "A family of premium principles based on mixtures of TVaRs," Insurance: Mathematics and Economics, Elsevier, vol. 70(C), pages 397-405.
    41. Bernard, Carole & Chen, Jit Seng & Vanduffel, Steven, 2015. "Rationalizing investors’ choices," Journal of Mathematical Economics, Elsevier, vol. 59(C), pages 10-23.
    42. Debora Daniela Escobar & Georg Ch. Pflug, 2020. "The distortion principle for insurance pricing: properties, identification and robustness," Annals of Operations Research, Springer, vol. 292(2), pages 771-794, September.
    43. Wang, Shaun S. & Young, Virginia R. & Panjer, Harry H., 1997. "Axiomatic characterization of insurance prices," Insurance: Mathematics and Economics, Elsevier, vol. 21(2), pages 173-183, November.
    44. Edward Furman & Ričardas Zitikis, 2009. "Weighted Pricing Functionals With Applications to Insurance," North American Actuarial Journal, Taylor & Francis Journals, vol. 13(4), pages 483-496.
    45. Balter, Anne G. & Chau, Ki Wai & Schweizer, Nikolaus, 2024. "Comparative risk aversion vs. threshold choice in the Omega ratio," Omega, Elsevier, vol. 123(C).
    46. Furman, Edward & Zitikis, Ricardas, 2008. "Weighted risk capital allocations," Insurance: Mathematics and Economics, Elsevier, vol. 43(2), pages 263-269, October.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Lauzier, Jean-Gabriel & Lin, Liyuan & Wang, Ruodu, 2023. "Pairwise counter-monotonicity," Insurance: Mathematics and Economics, Elsevier, vol. 111(C), pages 279-287.
    2. Alexandru V. Asimit & Raluca Vernic & Riċardas Zitikis, 2013. "Evaluating Risk Measures and Capital Allocations Based on Multi-Losses Driven by a Heavy-Tailed Background Risk: The Multivariate Pareto-II Model," Risks, MDPI, vol. 1(1), pages 1-20, March.
    3. Furman, Edward & Zitikis, Ricardas, 2008. "Weighted risk capital allocations," Insurance: Mathematics and Economics, Elsevier, vol. 43(2), pages 263-269, October.
    4. Gilles Boevi Koumou & Georges Dionne, 2022. "Coherent Diversification Measures in Portfolio Theory: An Axiomatic Foundation," Risks, MDPI, vol. 10(11), pages 1-19, October.
    5. Denuit Michel & Dhaene Jan & Goovaerts Marc & Kaas Rob & Laeven Roger, 2006. "Risk measurement with equivalent utility principles," Statistics & Risk Modeling, De Gruyter, vol. 24(1), pages 1-25, July.
    6. Shuo Gong & Yijun Hu & Linxiao Wei, 2022. "On conditional distortion risk measures under uncertainty," Papers 2211.00520, arXiv.org, revised Apr 2025.
    7. Choo, Weihao & de Jong, Piet, 2015. "The tradeoff insurance premium as a two-sided generalisation of the distortion premium," Insurance: Mathematics and Economics, Elsevier, vol. 65(C), pages 238-246.
    8. Furman, Edward & Landsman, Zinoviy, 2010. "Multivariate Tweedie distributions and some related capital-at-risk analyses," Insurance: Mathematics and Economics, Elsevier, vol. 46(2), pages 351-361, April.
    9. Tsanakas, Andreas, 2009. "To split or not to split: Capital allocation with convex risk measures," Insurance: Mathematics and Economics, Elsevier, vol. 44(2), pages 268-277, April.
    10. Kaluszka, M. & Laeven, R.J.A. & Okolewski, A., 2012. "A note on weighted premium calculation principles," Insurance: Mathematics and Economics, Elsevier, vol. 51(2), pages 379-381.
    11. Lin, Feng & Peng, Liang & Xie, Jiehua & Yang, Jingping, 2018. "Stochastic distortion and its transformed copula," Insurance: Mathematics and Economics, Elsevier, vol. 79(C), pages 148-166.
    12. Furman, Edward & Kuznetsov, Alexey & Zitikis, Ričardas, 2018. "Weighted risk capital allocations in the presence of systematic risk," Insurance: Mathematics and Economics, Elsevier, vol. 79(C), pages 75-81.
    13. Furman, Edward & Wang, Ruodu & Zitikis, Ričardas, 2017. "Gini-type measures of risk and variability: Gini shortfall, capital allocations, and heavy-tailed risks," Journal of Banking & Finance, Elsevier, vol. 83(C), pages 70-84.
    14. Ruodu Wang & Ričardas Zitikis, 2021. "An Axiomatic Foundation for the Expected Shortfall," Management Science, INFORMS, vol. 67(3), pages 1413-1429, March.
    15. Belles-Sampera, Jaume & Merigó, José M. & Guillén, Montserrat & Santolino, Miguel, 2013. "The connection between distortion risk measures and ordered weighted averaging operators," Insurance: Mathematics and Economics, Elsevier, vol. 52(2), pages 411-420.
    16. Dhaene, Jan & Laeven, Roger J.A. & Zhang, Yiying, 2022. "Systemic risk: Conditional distortion risk measures," Insurance: Mathematics and Economics, Elsevier, vol. 102(C), pages 126-145.
    17. Frédéric Godin & Van Son Lai & Denis-Alexandre Trottier, 2019. "A general class of distortion operators for pricing contingent claims with applications to CAT bonds," Scandinavian Actuarial Journal, Taylor & Francis Journals, vol. 2019(7), pages 558-584, August.
    18. Yuyu Chen & Ruodu Wang, 2024. "Infinite-mean models in risk management: Discussions and recent advances," Papers 2408.08678, arXiv.org, revised Oct 2024.
    19. Tsanakas, Andreas, 2008. "Risk measurement in the presence of background risk," Insurance: Mathematics and Economics, Elsevier, vol. 42(2), pages 520-528, April.
    20. Mickael Beaud & Marc Willinger, 2015. "Are People Risk Vulnerable?," Management Science, INFORMS, vol. 61(3), pages 624-636, March.

    More about this item

    Keywords

    ;
    ;
    ;
    ;
    ;

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:insuma:v:122:y:2025:i:c:p:180-193. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/locate/inca/505554 .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.