Heterogeneity and the need for capital in the individual model. (English) Zbl 1142.91039

Denuit, Michel; Frostig, Esther

Scand. Actuar. J. 2006, No. 1, 42-66 (2006).

The authors consider the individual model of risk theory, where the claim amount for policy \(i, i=1,\dots,n,\) has the form \(X_{i}=J_{q_{i}}C_{i}\). The random variable \(J_{q_{i}}\) is Bernoulli distributed with the mean \(q_{i}\); it indicates whether at least one claim occurred for policy \(i\). The random variable \(C_{i}\) is then the total cost of all these claims. The impact of the heterogeneity is studied with the help of various stochastic orders. The concept of majorization arising as a measure of diversity of the components of a vector is described. The authors discuss various stochastic orders and related risk measures. The main results of this paper are devoted to the study of the impact of an increase in the heterogeneity of the portfolio either at the occurrence level \(q_{i}\) or at the cost level \(C_{i}\), or both.

Reviewer: Aleksandr D. Borisenko (Kyïv)

MSC:

91B30

Risk theory, insurance (MSC2010)

Keywords:

majorization; Schur-increasing functions; stochastic dominance; stop-loss order; convex order; Laplace transform order

References:

[1]	DOI: 10.1016/S0167-6687(97)00039-5 · Zbl 0986.62085
[2]	Marshall A. W., Inequalities: Theory of Majorization and its Applications (1979) · Zbl 0437.26007
[3]	DOI: 10.1016/0024-3795(94)90342-5 · Zbl 0796.60021
[4]	Spreeuw , J . (1998) . Majorization order applied to a system of mortality profit distribution . Paper presented at the Second International Congress on Insurance: Mathematics and Economics.
[5]	DOI: 10.1080/034612302320179872 · Zbl 1039.91037
[6]	DOI: 10.1016/j.insmatheco.2003.08.001 · Zbl 1103.91360
[7]	DOI: 10.1080/03461230410020383 · Zbl 1144.91024
[8]	DOI: 10.1016/j.insmatheco.2004.06.003 · Zbl 1075.62094
[9]	Arnold B. C., Majorization and the Lorenz Order:A Brief Introduction (1987)
[10]	DOI: 10.1002/0470016450
[11]	DOI: 10.1016/S0167-6687(01)00075-0 · Zbl 1074.62527
[12]	DOI: 10.2307/1911158 · Zbl 0616.90005
[13]	DOI: 10.2143/AST.26.1.563234
[14]	DOI: 10.2307/253675
[15]	DOI: 10.2143/AST.32.2.1027 · Zbl 1090.91555
[16]	Frostig E, Monotonicity results for portfolios with heterogeneous claims arrival processes (2005) · Zbl 1168.91412