Premiums and reserves, adjusted by distortions.

*(English)*Zbl 1398.91352Summary: The net premium principle is considered to be the most genuine and fair premium principle in actuarial applications. However, actuarial due diligence requires additional caution in pricing of insurance contracts to avoid, for example, at least bankruptcy of the insurer. This paper addresses the distorted premium principle from various angles. Distorted premiums are typically computed by underweighting or ignoring low, but overweighting high losses. Dual characterizations, which are elaborated in a first part of the paper, support this interpretation. The main contribution consists in an opposite point of view – an alternative characterization – which leaves the probability measure unchanged, but modifies (increases) the outcomes instead in a consistent way. It turns out that this new point of view is natural in actuarial practice, as it can be used for premium calculations, but equally well to determine the reserve process in subsequent years in a time consistent way.

##### MSC:

91B30 | Risk theory, insurance (MSC2010) |

60E15 | Inequalities; stochastic orderings |

62P05 | Applications of statistics to actuarial sciences and financial mathematics |

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\textit{A. Pichler}, Scand. Actuar. J. 2015, No. 4, 332--351 (2015; Zbl 1398.91352)

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##### References:

[1] | Acerbi, C. (2002). Spectral measures of risk: a coherent representation of subjective risk aversion. Journal of Banking & Finance 26, 1505-1518. |

[2] | Artzner, P., Delbaen, F., Eber, J.-M. & Heath, D. (1999). Coherent measures of risk. Mathematical Finance 9, 203-228. · Zbl 0980.91042 |

[3] | Artzner, P., Delbaen, F. & Heath, D. (November 1997). Thinking coherently. Risk 10, 68-71. |

[4] | Balbás, A., Balbás, B. & Heras, A. (2009). Optimal reinsurance with general risk measures. Insurance: Mathematics and Economics 44, 374-384. · Zbl 1162.91394 |

[5] | Chi, Y. & Tan, K. S. (2011). Optimal reinsurance under VaR and CVaR risk measures. ASTIN Bulletin41(2), 487-509. · Zbl 1239.91078 |

[6] | Dana, R.-A. (2005). A representation result for concave Schur concave functions. Mathematical Finance 15, 613-634. · Zbl 1142.28001 |

[7] | Denneberg, D. (1990). Distorted probabilities and insurance premiums. Methods of Operations Research 63, 21-42. |

[8] | Ferguson, T. S. (1967). Mathematical statistics. New York: A Decision Theoretic Approach. Academic Press. · Zbl 0153.47602 |

[9] | Föllmer, H. & Schied, A. (2004). Stochastic finance: an introduction in discrete time. de Gruyter Studies in Mathematics 27. de Gruyter. Berlin, Boston: De Gruyter. · Zbl 1126.91028 |

[10] | Furman, E. & Zitikis, R. (2008). Weighted premium calculation principles. Insurance: Mathematics and Economics 42, 459-465. · Zbl 1141.91509 |

[11] | Goovaerts, M., Linders, D., Weert, K. V. & Tank, F. (2012). On the interplay between distortion, mean value and the Haezendonck-Goovaerts risk measures. Insurance: Mathematics and Economics 51, 10-18. · Zbl 1284.91235 |

[12] | Heras, A., Balbás, B. & Vilar, J. L. (2012). Conditional tail expectation and premium calculation. ASTIN Bulletin 42, 325-342. · Zbl 1277.91085 |

[13] | Hoeffding, W. (1940). Maßstabinvariante korrelationstheorie. Schriften des Mathematischen Instituts der Universität Berlin5, 181-233. In German. |

[14] | Jouini, E., Schachermayer, W. & Touzi, N. (2006). Law invariant risk measures have the Fatou property. Advances in Mathematical Economics 9, 49-71. · Zbl 1198.46028 |

[15] | Ko, B., Russo, R. P. & Shyamalkumar, N. D. (2009). A note on the nonparametric estimation of the CTE. ASTIN Bulletin 39(2), 717-734. · Zbl 1178.62025 |

[16] | Kusuoka, S. (2001). On law invariant coherent risk measures. Advances in Mathematical Economics 3, 83-95. · Zbl 1010.60030 |

[17] | Müller, A. & Stoyan, D. (2002). Comparison methods for stochastic models and risks. Chichester: Wiley series in probability and statistics. Wiley. |

[18] | Pflug, G. Ch. (2000). Some remarks on the value-at-risk and the conditional value-at-risk. In S. Uryasev editor, Probabilistic Constrained Optimization: Methodology and Applications, pp. 272-281. Dordrecht: Kluwer Academic Publishers. · Zbl 0994.91031 |

[19] | Pflug, G. Ch (2006). On distortion functionals. Statistics and Risk Modeling (formerly: Statistics and Decisions) 24, 45-60. · Zbl 1186.91125 |

[20] | Pflug, G. Ch. & Römish, W. (2007). Modeling. River Edge, NJ: Measuring and Managing Risk. World Scientific. |

[21] | Pichler, A. (2013). Evaluations of risk measures for different probability measures. SIAM Journal on Optimization 23(1), 530-551. · Zbl 1277.90083 |

[22] | Pichler, A. (2013). The natural Banach space for version independent risk measures. Insurance: Mathematics and Economics 53, 405-415. · Zbl 1304.91129 |

[23] | Rockafellar, R. T. & Uryasev, S. (2000). Optimization of conditional value-at-risk. Journal of Risk 2, 21-41. |

[24] | Rockafellar, R. T. & Uryasev, S. (2002). Conditional value-at-risk for general loss distributions. Journal of Banking and Finance 26, 1443-1471. |

[25] | Shaked, M. & Shanthikumar, J. G. (2007). Stochastic order. Springer Series in Statistics: Springer. |

[26] | Shapiro, A. (2012). On Kusuoka representations of law invariant risk measures. November: Mathematics of Operations Research. |

[27] | Shapiro, A., Dentcheva, D. & Ruszczyński, A. (2009). Lectures on stochastic programming. MOS-SIAM Series on Optimization. · Zbl 1183.90005 |

[28] | Valdez, E. A. & Xiao, Y. (2011). On the distortion of a copula and its margins. Scandinavian Actuarial Journal 4, 292-317. · Zbl 1277.62140 |

[29] | van der Vaart, A. W. (1998). Asymptotic Statistics. Cambridge, UK: Cambridge University Press. · Zbl 0910.62001 |

[30] | Villani, C. (2003). Topics in optimal transportation, volume 58 of graduate studies in mathematics. Providence, RI: American Mathematical Society. |

[31] | Wang, S. & Dhaene, J. (1998). Comonotonicity, correlation order and premium principles. Insurance: Mathematics and Economics 22, 235-242. · Zbl 0909.62110 |

[32] | Wang, S. S. (2000). A class of distortion operatiors for financial and insurance risk. The Journal of Risk and Insurance 67(1), 15-36. |

[33] | Wang, S. S. & Young, V. R. (1998). Risk-adjusted credibility premiums using distorted probabilities. Scandinavian Actuarial Journal 2, 143-165. · Zbl 1043.91512 |

[34] | Wang, S. S., Young, V. R. & Panjer, H. H. (1997). Axiomatic characterization of insurance prices. Insurance: Mathematics and Economics 21, 173-183. · Zbl 0959.62099 |

[35] | Young, V. R. |

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