, 17:139 | Cite as

How retention levels influence the variability of the total risk under reinsurance

Original Paper


Recently, Escudero and Ortega (Insur. Math. Econ. 43:255–262, 2008) have considered an extension of the largest claims reinsurance with arbitrary random retention levels. They have analyzed the effect of some dependencies on the Laplace transform of the retained total claim amount. In this note, we study how dependencies influence the variability of the retained and the reinsured total claim amount, under excess-loss and stop-loss reinsurance policies, with stochastic retention levels. Stochastic directional convexity properties, variability orderings, and bounds for the retained and the reinsured total risk are given. Some examples on the calculation of bounds for stop-loss premiums (i.e., the expected value of the reinsured total risk under this treaty) and for net premiums for the cedent company under excess-loss, and complementary results on convex comparisons of discounted values of benefits for the insurer from a portfolio with risks having random policy limits (deductibles) are derived.


Excess-loss reinsurance Stop-loss reinsurance Stochastic directional convexity Directionally convex functions Increasing convex order 

Mathematics Subject Classification (2000)

60E15 62P05 


  1. Ammeter H (1964) The rating of “largest claim” reinsurance covers. Q Lett Algemeine Reinsur Co, Jubil Number 2:79–109 Google Scholar
  2. Balakrishnan N, Cramer E, Kamps U (2001) Bounds for means and variances of progressive type II censored order statistics. Stat Probab Lett 54:301–315 CrossRefGoogle Scholar
  3. Beirlant J, Goegebeur Y, Segers J, Teugels JL (2004) Statistics of extremes. Wiley, New York CrossRefGoogle Scholar
  4. Bühlmann H, Gagliardi B, Gerber H, Straub E (1977) Some inequalities for stop-loss premiums. ASTIN Bull IX:75–83 Google Scholar
  5. Cai J, Tan KT (2007) Optimal retention for a stop-loss reinsurance under the VaR and CTE risk measures. ASTIN Bull 37:93–112 CrossRefGoogle Scholar
  6. Chang C-S, Shanthikumar JG, Yao DD (1994) Stochastic convexity and stochastic majorization. In: Yao DD (ed) Stochastic modeling and analysis of manufacturing systems. Series in operations research. Springer, Berlin Google Scholar
  7. Courtois C, Denuit M (2007) Moment bounds on discrete expected stop-loss transforms, with applications. Methodol Comput Appl Probab 2:231–254 Google Scholar
  8. Denuit M, Müller A (2002) Smooth generators of integral stochastic orders. Ann Appl Probab 12:1174–1184 CrossRefGoogle Scholar
  9. Denuit M, Scaillet O (2004) Nonparametric tests for positive quadrant dependence. J Financ Econom 2:422–450 Google Scholar
  10. Denuit M, Vermandele C (1998) Optimal reinsurance and stop-loss order. Insur Math Econ 22:229–233 CrossRefGoogle Scholar
  11. Denuit M, Genest C, Mesfioui M (2006) Calcul de bornes sur la prime en excédent de perte de fonctions de risques dépendants en présence d’information partielle sur leurs marges. Ann Sci Math Qué 30:63–78 Google Scholar
  12. De Vylder F, Goovaerts M (1982) Analytical best upper bounds on stop-loss premiums. Insur Math Econ 1:163–175 CrossRefGoogle Scholar
  13. De Vylder F, Goovaerts M (1983) Best bounds on the stop loss premium in case of known range, expectation, variance and mode of the risk. Insur Math Econ 2:241–249 CrossRefGoogle Scholar
  14. Dhaene J, Denuit M, Goovaerts M, Kaas R, Vyncke D (2002) The concept of comonotonicity in actuarial science and finance: applications. Insur Math Econ 31:133–161 CrossRefGoogle Scholar
  15. Escudero LF, Ortega EM (2008) Actuarial comparisons of aggregate claims with randomly right truncated claims. Insur Math Econ 43:255–262 CrossRefGoogle Scholar
  16. Escudero LF, Ortega EM, Alonso J (2009) Variability comparisons for some mixture models with stochastic environments in biosciences and engineering. Stoch Environ Res Risk Assess. doi:10.1007/s00477-009-0310-6 Google Scholar
  17. Fang KT, Kotz S, Ng KW (1990) Symmetric multivariate and related distributions. In: Monographs on statistics and applied probability, vol 36. Chapman and Hall, London Google Scholar
  18. Genest C, Marceau E, Mesfioui M (2002) Upper stop-loss bounds for sums of possibly dependent risks with given means and variances. Stat Probab Lett 57:33–41 CrossRefGoogle Scholar
  19. Hua L, Cheung KC (2008) Stochastic orders of scalar products with applications. Insur Math Econ 42:865–872 CrossRefGoogle Scholar
  20. Hürlimann W (1996) Improved analytical bounds for some risk quantities. ASTIN Bull 26:185–199 CrossRefGoogle Scholar
  21. Hürlimann W (1998) On best stop-loss bounds for bivariate sums by known marginal means, variances and correlation. Mitt Verein Schweiz Versicherungmath 111–134 Google Scholar
  22. Hürlimann W (2000) Generalized algebraic bounds on order statistics function, with application to reinsurance and catastrophe risk. ASTIN Colloquium International Actuarial Association, Brussels, Belgium 2000: Porto Cervo, Italy Google Scholar
  23. Hürlimann W (2002) Analytical bounds for two value-at-risk functionals. ASTIN Bull 32:235–265 CrossRefGoogle Scholar
  24. Hürlimann W (2005) Excess of loss reinsurance with reinstatements revisited. ASTIN Bull 35:211–238 CrossRefGoogle Scholar
  25. Jansen K, Haezendonck J, Goovaerts M (1986) Upper bounds on stop-loss premiums in case of known moments up to the fourth order. Insur Math Econ 5:315–334 CrossRefGoogle Scholar
  26. Joe H (1997) Multivariate models and dependence concepts. Chapman and Hall, London Google Scholar
  27. Kaas R, Goovaerts M (1986) Bounds on stop-loss premiums for compound distributions. ASTIN Bull 16:13–17 CrossRefGoogle Scholar
  28. Kaas R, Dhaene J, Goovaerts M (2000) Upper and lower bounds for sums of random variables. Insur Math Econ 27:151–168 CrossRefGoogle Scholar
  29. Kaas R, Goovaerts M, Dhaene J, Denuit M (2001) Modern actuarial risk theory. Kluwer Academic, Dordrecht Google Scholar
  30. Kaluszka M, Okolewski A, Szymanska K (2005) Sharp bounds for L-statistics from dependent samples of random length. J Stat Plan Inference 127:71–89 CrossRefGoogle Scholar
  31. Krishnaiah PR (1985) Multivariate Gamma distributions. In: Encyclopedia of statistical sciences, vol 6. Wiley, New York, pp 63–66 Google Scholar
  32. Lai CD, Xie M (2006) Stochastic ageing and dependence for reliability. Springer, Berlin Google Scholar
  33. Marinacci M, Montrucchio L (2005) Ultramodular functions. Math Oper Res 30:311–332 CrossRefGoogle Scholar
  34. Meester LE, Shanthikumar JG (1993) Regularity of stochastic processes. A theory based on directional convexity. Probab Eng Inf Sci 7:343–360 CrossRefGoogle Scholar
  35. Meester LE, Shanthikumar JG (1999) Stochastic convexity on general space. Math Oper Res 24:472–494 CrossRefGoogle Scholar
  36. Merton R (1973) Theory of rational option pricing. Bell J Econ Manag Sci 4:141–184 CrossRefGoogle Scholar
  37. Müller A, Scarsini M (2000) Some remarks on the supermodular order. J Multivar Anal 73:107–119 CrossRefGoogle Scholar
  38. Müller A, Scarsini M (2001) Stochastic comparison of random vectors with a common copula. Math Oper Res 26:723–740 CrossRefGoogle Scholar
  39. Müller A, Scarsini M (2005) Archimedean copulae and positive dependence. J Multivar Anal 93:434–445 CrossRefGoogle Scholar
  40. Müller A, Stoyan D (2002) Comparison methods for stochastic models and risks. Wiley, Chichester Google Scholar
  41. Nelsen RB (1999) An introduction to copulas. Springer, New York Google Scholar
  42. Ortega EM, Escudero LF (2009) On expected utility for financial insurance portfolios with stochastic dependencies. Eur J Oper Res. doi:10.1016/j.ejor.2008.11.042 Google Scholar
  43. Pellerey F (2006) Personal communication Google Scholar
  44. Rychlik T (1993) Bounds for expectations of L-estimates for dependent samples. Statistics 24:1–7 CrossRefGoogle Scholar
  45. Rolski T, Schmidli H, Schmidt V, Teugels J (1999) Stochastic processes for insurance and finance. Wiley, Chichester CrossRefGoogle Scholar
  46. Rüschendorf L (2004) Comparison of multivariate risks and positive dependence. Adv Appl Probab 41:391–406 Google Scholar
  47. Shaked M, Shanthikumar JG (1990) Parametric stochastic convexity and concavity of stochastic processes. Ann Inst Stat Math 42:509–531 CrossRefGoogle Scholar
  48. Shaked M, Shanthikumar JG (2007) Stochastic orders. Springer, New York CrossRefGoogle Scholar
  49. Teugels JL (1985) Selected topics in insurance mathematics. University of Leuven, Belgium Google Scholar
  50. Wei G, Hu T (2002) Supermodular dependence ordering on a class of multivariate copulas. Stat Probab Lett 57:375–385 CrossRefGoogle Scholar
  51. Yaari ME (1987) The dual theory of choice under risk. Econometrica 55:95–115 CrossRefGoogle Scholar
  52. Young V (2004) Premium principles. In: Enclyclopedia of actuarial science. Wiley, New York, pp 1–9 Google Scholar

Copyright information

© Sociedad de Estadística e Investigación Operativa 2009

Authors and Affiliations

  1. 1.Dpto. Estadística e Investigación OperativaUniversidad Rey Juan CarlosMóstolesSpain
  2. 2.Centro de Investigación OperativaUniversidad Miguel HernándezOrihuelaSpain

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