A note on weighted premium calculation principles

Insurance: Mathematics and Economics 51 (2012) 379–381

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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: Hw (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 ˜ z (X ) = E(Xz (X )), where z : [0, ∞) → premium of the form H (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 Eu(w − X + E(Xz (X ))), z ∈Z

∗

Corresponding author. E-mail address: [email protected] (M. Kaluszka).

0167-6687/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.insmatheco.2012.06.006

(1)

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) = eλx /EeλX , 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 nondecreasing (not necessarily concave or differentiable) function such that |Eu(W − X )| < ∞ with (possibly random) W being the insurer’s initial wealth. Then sup Eu(W − X + E(Xz (X ))) = Eu(W − X + sup X ),

(2)

z ∈Z

where sup X stands for the essential supremum of X with respect to the measure P and it is understood that u(∞) = limx→∞ u(x). Proof. Clearly, sup Eu(W − X + E(Xz (X ))) ≤ Eu(W − X + sup X ). z ∈Z

Fix c < sup X . Let zc be an arbitrary continuous and nondecreasing function such that zc (x) = 0 for x ≤ c and Ezc (X ) = 1, e.g., zc (x) = ac (x − c )+ for x ≤ c + c −1 and zc (x) = ac for x > c + c −1 , with ac being a norming constant. Define zc∗ (x) = (1 − ε)zc (x) + ε h(x),

380

M. Kaluszka et al. / Insurance: Mathematics and Economics 51 (2012) 379–381

where h is a fixed element from Z, ε = min(1/c , 1) if sup X = ∞, and ε = min(sup X − c , 1/2) if sup X < ∞. The function c → (1 −ε)c is increasing for sufficiently large c’s and (1 −ε)c → sup X as c → sup X . It is easy to verify that zc∗ ∈ Z for all c and Eu(W − X + E(Xzc∗ (X ))) ≥ Eu(W − X + (1 − ε)E(Xzc (X )))

Remark 3. Under the assumption that the premium cannot exceed a fixed value π ∈ [EX , sup X ), a solution to problem (1) is any (not unique) function z ∈ Z satisfying E(Xz (X )) = π . A similar conclusion can be drawn for the problem of determining z which maximizes a weighted sum of expected exponential utilities, i.e., max[α Eua (w − X + E(Xz (X ))) + Eub (w0 − E(Xz (X )))],

≥ Eu(W − X + (1 − ε)c Ezc (X )) = Eu(W − X + (1 − ε)c ).

z ∈Z

Set Yc = u(W − X + (1 − ε)c ) − u(W − X ) and observe that 0 ≤ Yc ≤ Yd for c < d. Since |Eu(W − X )| < ∞, by Lebesgue’s monotone convergence theorem (c → sup X ), sup Eu(W − X + E(Xzc (X ))) ≥ Eu(W − X + sup X ),

where uc (x) = (1 − e−cx )/c and w0 denotes the insured’s wealth. Indeed, z is not uniquely specified because the above problem is equivalent to the following:

(Ae−ah + ebh ),

min

EX ≤h≤sup X

∗

c
<sup>in which A is a constant and h = E(Xz (X )).

which completes the proof of (2). The supremum on the left-hand side of (2) is not attained. To induce attainability, one can extend Z to the set of all nondecreasing functions z such that Ez (X ) = 1. ˜ z (X ) = sup X and it is obtained for Then the optimal premium is H z (x) = 1(x = b)/P(X = b) when X takes values in the interval [0, b] and P(X = b) > 0. Hereafter, 1(a) = 1 if a is true and 0 otherwise. 

Observe that analysis similar to that in the proof of Proposition 1 shows that the maximal loss premium maximizes the insurer’s survival probability on condition that W > 0. Moreover, Proposition 1 remains true if we replace the expected utility Eu(Y ) by an arbitrary measure G(Y ) satisfying the following two conditions: C1. The measure G is monotone, i.e., G(X ) ≤ G(Y ), if X (ω) ≤ Y (ω) for all ω; C2. If (Yn (ω))∞ n=1 is a nondecreasing sequence for every ω , then limn→∞ G(Yn ) = G(limn→∞ Yn ). Many measures fulfill conditions C1–C2. By Lebesgue’s monotone convergence theory, they hold for spectral risk measures of the form

Remark 1. We recall that besides Bühlmann (1980) other important connections exist between the exponential premium (exponential utility), the Esscher premium (Esscher transform) and the relative entropy (Kullback–Leibler divergence). The equivalent utility premium for an insurer with exponential utility function takes the form λ1 log E exp(λX ). As is well known (see e.g. Goovaerts et al., 1984, 2004; Goovaerts and Laeven, 2008), the exponential(λ) premium can be viewed as a uniform mixture of Esscher premia and is bounded from above by the Esscher(λ) premium: 1

λ

log E exp(λX ) =

1

λ

λ

 0

E(X exp(bX )) E exp(bX )

db ≤

E(X exp(λX )) E exp(λX )

.

Furthermore, 1

λ



log E exp(λX ) = sup EQ X − Q ≪P

1

λ



E (Q |P) ,

Remark 2. We note that problem (1) is not equivalent to the problem max Eu(w − f (Y ) + E(f (Y )Y )) s.t. EY = 1,

(3)

f ∈F

in which F denotes the set of continuous and strictly increasing functions f : (0, ∞) → [0, ∞) (cf. Goovaerts et al., 1984, p. 84). Our proposed modification of problem (1) contained in Proposition 2 below is inspired by problem (3). We determine a solution to (3) under the assumptions that: (i) u is strictly increasing, differentiable and concave, (ii) for some c > 0, there exists a function h such that cY = u′ (w − h(Y ) + E(h(Y )Y ));

(4)

cf. (7) in the proof of Proposition 2 below. Since u is concave, for any f ∈ F , Eu(w − f (Y ) + E(f (Y )Y )) − Eu(w − h(Y ) + E(h(Y )Y ))

≤ E(u′ (w − h(Y ) + E(h(Y )Y )) × (E(f (Y )Y ) − E(h(Y )Y ) − (f (Y ) − h(Y )))) = 0, by (4) and EY = 1. This clearly means that h is a solution to (3). In particular, if w = 0 and u(x) = (1 − e−λx )/λ, h(Y ) = (ln Y − E(Y ln Y ))/λ when E(Y ln Y ) < ∞.

φ(p)FX−1 (p)dp, 0

with FX−1 (p) = inf{x ∈ R : P(X ≤ x) ≥ p} the lower p-quantile of the random variable X and φ a nondecreasing probability density function (see Acerbi, 2002; Föllmer and Schied, 2004). An example of a spectral risk measure is the Tail-Value-at 1 −1 1 F (t )dt. Tversky and Kahneman (1992) Risk: TVaRp (X ) = 1− p p X consider, for discrete random variables, the following measure (see also Quiggin, 1982; Yaari, 1987):



with E (·|·) the relative entropy; see Csiszár (1975). The supremum on the right-hand side is attained in the Esscher density, given by exp(λX )/E exp(λX ); see e.g., Laeven and Stadje (2011).

1



ρφ (X ) =

Egh Y =

∞

g (P(Y > t ))dt −

0

∞



h(P(−Y > t ))dt ,

(5)

0

with g , h arbitrary distortion functions. Specific choices of g (and h) lead to the Value-at-Risk, Gini’s, Denneberg’s and other premium principles (see e.g. Wang, 1996; Kaas et al., 2008). The functional Egh Y is monotone (see e.g. Kaluszka and Krzeszowiec, 2012, Lemma ∞ 1). Under the additional assumption that 0 h(P(−Y1 > t ))dt < ∞, Egh Yn → Egh Y for any nondecreasing sequence Yn ; this can be concluded by applying the Lebesgue monotone convergence theorem to the first integral in formula (5) and the Lebesgue dominated convergence theorem to the second integral. 2. A modification We propose a modification of problem (1), the solution to which is the Esscher premium. Proposition 2. Let Z = z (X ) be an arbitrary nonnegative random variable such that EZ = 1. Let R, 0 ≤ R ≤ X , be the insurer’s part of X , w be the insurer’s initial wealth and u(x) = (1 − e−λx )/λ, λ > 0, be the insurer’s utility function. If Z = eλr (X ) /Eeλr (X ) , then the contract R = r (X ) is a solution to the problem max Eu(w − R + E(RZ ));

(6)

0≤R≤X

˜ z (R) = E(ReλR )/EeλR . the premium for risk R is the Esscher premium H Moreover, if the contract R = r (X ) in which 0 < r (x) < x for all x > 0, is a solution to problem (6), then Z = eλr (X ) /Eeλr (X ) .

M. Kaluszka et al. / Insurance: Mathematics and Economics 51 (2012) 379–381

Proof. We prove that if

381

with the fact that

u (w + E(r (X )Z ) − r (X )) = Z Eu (w + E(r (X )Z ) − r (X )), ′

′

(7)

then R = r (X ) is a solution to problem (6) with an arbitrary increasing, concave and differentiable utility function u. By concavity of u and (7), for any reinsurance contract 0 ≤ R∗ ≤ X , we have

≤ E(u (w + E(RZ ) − R)(R − R + E(R Z ) − E(RZ ))) = 0. ∗

∗

Hence, the contract R is an optimal solution. (Note that (7) is satisfied under the stated assumptions.) Now we prove the second assertion. Let R = r (X ) and let Rt = (1 − t )R + tR∗ , where 0 ≤ t ≤ 1 and R∗ is an arbitrary part of X such that 0 ≤ R∗ ≤ X . Since R is a solution to problem (6), the function f (t ) = Eu(w − Rt + E(Rt Z )) attains its maximum at t = 0. As a consequence, the right derivative f+′ (0) ≤ 0 and for any R∗ , 0 ≤ R∗ ≤ X , E((u′ (w + E(RZ ) − R) − Z Eu′ (w + E(RZ ) − R))(R∗ − R))

≤ 0.

(8)

Since 0 < r (x) < x for all x > 0, the condition (7) holds, and so Z = eλr (X ) /Eeλr (X ) . Observe that if r (x) can take values 0 or x, then (7) is satisfied only for ω’s such that 0 < r (X (ω)) < X (ω). If r (X (ω)) = 0, then from (8) it follows that u′ (w + E(r (X )Z ) − r (X (ω)))

≤ Z (X (ω))Eu′ (w + E(r (X )Z ) − r (X )). If r (X (ω)) = X (ω), the inequality opposite (9) is satisfied.

(9)

Remark 4. In the behavioral theory of choice some nondifferentiable utility functions are analyzed; see e.g., Gillen and Markowitz (2010) and Tversky and Kahneman (1992). If u is an arbitrary continuous, nondecreasing and left- and right- differentiable function, then a sufficient condition for R = r (X ) to be a solution to problem (6) is that Z = Y /EY , where Y is a selection of the subdifferential of u, i.e., u′− (w + E(RZ ) − R) ≤ Y ≤ u′+ (w + E(RZ ) − R), in which u′− and u′+ , respectively, denote the left and the right derivative of u. Proposition 2 can also be generalized to measures other than Eu(Y ). Let X denote a set of nonnegative random variables defined on Ω , called the insurable risks. Let Π be a mapping from X into the real numbers. We assume that for every X ∈ X there exists a random variable, say Π ′ (X ), such that for every Y ∈ X, (10)

Any random variable ω → Π (X )(ω) will be called a derivative of Π at X . Any functional Π satisfying (10) will be called a G-concave functional. We say that a functional Π is G-convex if −Π is G-concave. Many measures are G-concave or G-convex. ′

Example 1. Let v be any differentiable convex function. If Π (X ) = Ev(X − EX ), then

Π ′ (X ) = v ′ (X − EX ) − Ev ′ (X − EX ). Indeed, combining the inequality Ev(Y − EY ) ≥ Ev(X − EX )

+ E(v ′ (X − EX )(Y − EY − (X − EX )))

gives Π (Y ) ≥ Π (X ) + E(Π ′ (X )(Y − X )), so Π is G-convex.

Example 3. Let u be increasing, concave and differentiable, W be any random variable and H be a G-convex functional. Then the functional Π (X ) = Eu(W − H (X ) + X ) is G-concave and

Π ′ (X ) = u′ (W − H (X ) + X ) − H ′ (X )Eu′ (W − H (X ) + X ).

(11)

Other examples can be found in Kaluszka (2004). Arguing as in the proof of Proposition 2 one can show that if

Π ′ (E(r (X )Z ) − r (X )) = Z EΠ ′ (E(r (X )Z ) − r (X )),

(12)

then R = r (X ) maximizes the functional Π (−R + E(RZ )) subject to the constraint 0 ≤ R ≤ X . Moreover, if the functions t → Π (X + tY ) are differentiable for any X , Y ∈ X, condition (12) is also necessary for optimality of the contract R = r (X ), provided 0 < r (x) < x for all x > 0. Acknowledgments



The economic interpretation of Proposition 2 is that, given a premium of the form indicated (E(RZ ) with EZ = 1 or rather E(r (X )z (X )) with Ez (X ) = 1), the expected exponential(λ) utility maximizing insurer decides to insure the partial risk r (X ) if and only if the premium is the Esscher(λ) premium.

Π (Y ) ≤ Π (X ) + E(Π ′ (X )(Y − X )).

= E((v ′ (X − EX ) − Ev ′ (X − EX ))(Y − EY − (X − EX ))) = E((v ′ (X − EX ) − Ev ′ (X − EX ))(Y − X ))

Example 2. Let Π (X ) = Eu(W − X ), where u is an increasing, concave and differentiable function, and W is an arbitrary random variable. The functional Π is G-concave and Π ′ (X ) = −u′ (W − X ).

Eu(w − R∗ + E(R∗ Z )) − Eu(w − R + E(RZ )) ′

E(v ′ (X − EX )(Y − EY − (X − EX )))

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