Insurance: Mathematics and Economics 51 (2012) 379–381 Contents lists available at SciVerse ScienceDirect Insurance: Mathematics and Economics journal homepage: www.elsevier.com/locate/ime A note on weighted premium calculation principles M. Kaluszka a,∗ , R.J.A. Laeven b , A. Okolewski a a Institute of Mathematics, Lodz University of Technology, ul. Wolczanska 215, 93-005 Lodz, Poland b Amsterdam School of Economics, University of Amsterdam, Valckenierstraat 65–67, 1018 XE Amsterdam, The Netherlands article info Article history: Received January 2012 Received in revised form June 2012 Accepted 12 June 2012 Keywords: Esscher’s premium Weighted premium principles Insurance risk abstract A prominent problem in actuarial science is to determine premium calculation principles that satisfy certain criteria. Goovaerts et al. [Goovaerts, M. J., De Vylder, F., Haezendonck, J., 1984. Insurance Premiums: Theory and Applications. North-Holland, Amsterdam, p. 84] establish an optimality-type characterization of the Esscher premium principle, but unfortunately their result is not true. In this note we propose a modified statement of this result. © 2012 Elsevier B.V. All rights reserved. 1. Introduction Let X ≥ 0 be a loss random variable defined on a probability space (Ω, A, P). The class of weighted premium calculation principles is defined in Furman and Zitikis (2008) as follows: H w (X ) = E(X w(X )) Ew(X ) , with w :[0, ∞) →[0, ∞) belonging to the set of functions such that 0 < Ew(X )< ∞ and E(X w(X )) < ∞. The class purports to provide a unifying approach to premium calculation; it contains many existing premium calculation principles as special cases. An important member of this class, corresponding to w(x) = exp(λx), λ> 0, is the Esscher premium, considered already by Bühlmann (1980) in the context of optimal risk exchanges à la Borch (1962) with exponential utilities. Other examples of weighted premium calculation principles include the net premium, the modified variance premium, Kamps’s premium and the conditional tail expectation. Properties of weighted functionals are studied by Furman and Zitikis (2008, 2009), Tsanakas (2008), and Choo and de Jong (2009); see also the early Goovaerts et al. (1984). Goovaerts et al. (1984) consider the problem of determining a premium of the form ˜ H z (X ) = E(Xz (X )), where z :[0, ∞) → (0, ∞) is strictly increasing, continuous and such that Ez (X ) = 1, maximizing the insurer’s expected utility. More specifically, denoting the set of all such z ’s by Z, the objective of the insurer can be written as max z∈Z Eu(w − X + E(Xz (X ))), (1) ∗ Corresponding author. E-mail address: kaluszka@p.lodz.pl (M. Kaluszka). in which w is the insurer’s initial wealth and u is the insurer’s utility function such that the expected utility is finite. Goovaerts et al. (1984, p. 84) claim that for an exponential utility u(x) = (1 − e −λx )/λ with λ> 0, the solution to (1) is z (x) = e λx /Ee λX , and consequently an optimal premium is the Esscher premium (see also Kaas et al. (2008), Theorem 5.4.3, and Denuit et al. (2005), p. 83). Unfortunately, their result is not true. In Proposition 1 below, we prove, for general u, the optimality of the maximal loss premium. Heuristically, since the claim size X in (1) does not depend on z and only the premium does, it is optimal to charge the maximum premium, that is, the maximal loss premium. Proposition 1. Assume u is an arbitrary continuous and nondecreas- ing (not necessarily concave or differentiable) function such that |Eu(W − X )| < ∞ with (possibly random) W being the insurer’s initial wealth. Then sup z∈Z Eu(W − X + E(Xz (X ))) = Eu(W − X + sup X ), (2) where sup X stands for the essential supremum of X with respect to the measure P and it is understood that u(∞) = lim x→∞ u(x). Proof. Clearly, sup z∈Z Eu(W − X + E(Xz (X ))) ≤ Eu(W − X + sup X ). Fix c < sup X . Let z c be an arbitrary continuous and nondecreasing function such that z c (x) = 0 for x ≤ c and Ez c (X ) = 1, e.g., z c (x) = a c (x − c ) + for x ≤ c + c −1 and z c (x) = a c for x > c + c −1 , with a c being a norming constant. Define z ∗ c (x) = (1 − ε)z c (x) + εh(x), 0167-6687/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.insmatheco.2012.06.006