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Indifference pricing of reinsurance with reinstatements using coherent monetary criteria


We consider the problem of indifference pricing of reinsurance contracts that contain a reinstatement clause. We define the indifference price relative to both a monetary utility function and a risk measure, to take into account both the risk reduction and the relief of capital immobilization provided by reinsurance. We characterize the indifference price as the unique solution to a fixed point equation and we bound the price by two easily computable values, if one has access to losses simulations. We illustrate our results on a European catastrophe insurance portfolio, and we conduct a simulation study for comparison and reproducibility purposes, where we include the case of dependence between claim arrivals, using Hawkes processes.

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  1. 1.

    Albrecher H, Beirlant J, Teugels JL (2017) Reinsurance: actuarial and statistical aspects. John Wiley & Sons, Hoboken

    Book  Google Scholar 

  2. 2.

    Albrecher H, Haas S (2011) Ruin theory with excess of loss reinsurance and reinstatements. Appl Math Comput 217(20):8031–8043

    MathSciNet  MATH  Google Scholar 

  3. 3.

    Anderson RR, Dong W (1998) Pricing catastrophe reinsurance with reinstatement provisions using a catastrophe model. Casualty Actuarial Society Forum, pp 303–322

  4. 4.

    Artzner P, Delbaen F, Eber J-M, Heath D (1999) Coherent measures of risk. Math Finance 9(3):203–228

    MathSciNet  Article  Google Scholar 

  5. 5.

    Barrieu P, El Karoui N (2009) Pricing, hedging and optimally designing derivatives via minimization of risk measures. In: Carmona R (ed) Volume on indifference pricing. Princeton University Press

  6. 6.

    Biagini S, Frittelli M (2009) On the extension of the Namioka-Klee theorem and on the Fatou property for risk measures. Optimality and risk-modern trends in mathematical finance. Springer, Berlin, pp 1–28

    MATH  Google Scholar 

  7. 7.

    Damodaran A (2020) Cost of capital by sector. Accessed 5 June 2020

  8. 8.

    Deelstra G, Plantin G (2014) Risk theory and reinsurance. Springer, Berlin

    Book  Google Scholar 

  9. 9.

    Delbaen F (2012) Monetary utility functions. Osaka University Press, Osaka

    Google Scholar 

  10. 10.

    El Karoui N, Ravanelli C (2009) Cash subadditive risk measures and interest rate ambiguity. Math Finance 19(4):561–590

    MathSciNet  Article  Google Scholar 

  11. 11.

    Föllmer H, Schied A (2011) Stochastic finance: an introduction in discrete time. Walter de Gruyter, Berlin

    Book  Google Scholar 

  12. 12.

    Frittelli M, Gianin ER (2002) Putting order in risk measures. J Bank Finance 26(7):1473–1486

    Article  Google Scholar 

  13. 13.

    Goovaerts M, De Vylder E, Haezendonck J (1984) Insurance premiums: theory and applications. North-Holland, Amsterdam, p 406

  14. 14.

    Hawkes AG (1971) Spectra of some self-exciting and mutually exciting point processes. Biometrika 58(1):83–90

    MathSciNet  Article  Google Scholar 

  15. 15.

    Hess KT, Schmidt KD (2004) Optimal premium plans for reinsurance with reinstatements. ASTIN Bull J IAA 34(2):299–313

    MathSciNet  Article  Google Scholar 

  16. 16.

    Hürlimann W (1998) On stop-loss order and the distortion pricing principle. ASTIN Bull J IAA 28(1):119–134

    Article  Google Scholar 

  17. 17.

    Hürlimann W (2005) Excess of loss reinsurance with reinstatements revisited. ASTIN Bull J IAA 35(1):211–238

    MathSciNet  Article  Google Scholar 

  18. 18.

    Jouini E, Schachermayer W, Touzi N (2006) Law invariant risk measures have the Fatou property. Advances in mathematical economics. Springer, Berlin, pp 49–71

    Chapter  Google Scholar 

  19. 19.

    Jouini E, Schachermayer W, Touzi N (2008) Optimal risk sharing for law invariant monetary utility functions. Math Finance 18(2):269–292

    MathSciNet  Article  Google Scholar 

  20. 20.

    KPMG (2020) Cost of capital study 2019. Accessed 5 June 2020

  21. 21.

    Laeven R, Goovaerts M (2014) Premium calculation and insurance pricing. In: Balakrishnan N, Colton T, Everitt B, Piegorsch W, Ruggeri F, Teugels JL (eds) Wiley StatsRef: Statistics Reference Online.

  22. 22.

    Mata AJ (2000) Pricing excess of loss reinsurance with reinstatements. ASTIN Bull J IAA 30(2):349–368

    MathSciNet  Article  Google Scholar 

  23. 23.

    Rytgaard M (1996) On calculating the risk premium for an excess of loss cover with an annual aggregate deductible and a limited number of reinstatements. Astin Colloquium, Copenhagen, Denmark 1, vol 27, pp 82–94

  24. 24.

    Sundt B (1993) On excess of loss reinsurance with reinstatements. Insur Math Econ 12(1):73

    MathSciNet  Article  Google Scholar 

  25. 25.

    Walhin J-F, Paris J (2000) The effect of excess of loss reinsurance with reinstatements on the cedent’s portfolio. Blätter der Deutschen Gesellschaft für Versicherungsmathematik 24:616–627

    MATH  Google Scholar 

  26. 26.

    Walhin J-F, Paris J (2001) Excess of loss reinsurance with reinstatements: premium calculation and ruin probability of the cedent. Blätter der DGVFM 25(1):1–12

    Article  Google Scholar 

  27. 27.

    Wang S (1996) Premium calculation by transforming the layer premium density. ASTIN Bull J IAA 26(1):71–92

    Article  Google Scholar 

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Correspondence to Nabil Kazi-Tani.

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The author gratefully acknowledges the support of the insurer Axa who made its data available, and thanks Thierry Cohignac, Guillaume Gorge and Jean-Sébastien Lagacé for fruitful discussions.

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Kazi-Tani, N. Indifference pricing of reinsurance with reinstatements using coherent monetary criteria. Eur. Actuar. J. 11, 161–183 (2021).

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  • Insurance premium calculation
  • Convex risk measures
  • Monetary utility functions
  • Reinsurance layers
  • Reinstatements

Mathematics Subject Classification

  • 91G05
  • 91B05
  • 91B16
  • 62P05