Insurer’s optimal reinsurance strategies
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This work was supported by KBN grant no. 2 H02B 011 24. Corresponding author. E-mail address: gal@p.lodz.pl (L. Gajek).
∗ ଝ
(1)
0167-6687/$ – see front matter © 2004 Published by Elsevier B.V. doi:10.1016/j.insmatheco.2003.12.002
+ ϕ2 (t) = [max(0, t)]2 .
Having chosen ϕ, we define the risk measure ρ(R) as the expected harm, i.e. ρ(R) = Eϕ(Y − R(Y) − E(Y − R(Y))). (2) A risk measure ρ will be called convex if the corresponding harm function ϕ is convex. Let π(R) denote the price of the contract R. Having available an amount of money P the insurer is interested to purchase a contract R∗ such that π(R∗ ) ≤ P and ρ(R∗ ) ≤ ρ(R) for all R ∈ R which have the price π(R) ≤ P , where R denotes the set of all admissible contracts. The reinsurance arrangement R∗ having the above property will be called optimal within the set R of admissible contracts under the risk measure (2). A usual class of admissible contracts is the class R0 of all measurable R : [0, ∞) → [0, ∞) such that 0 ≤ R(y) ≤ y for all y ≥ 0. However other classes might be also of great interest—as pointed out by Pesonen (1984)—there is no good reason for such restrictions on R. In general, given two measurable boundary functions R1 , R2 : [0, ∞) → (−∞, ∞) such that R1 (y) ≤ R2 (y) for all y ≥ 0, we define the class R(R1 , R2 ) of all measurable functions R : [0, ∞) → (−∞, ∞) for which the inequalities R1 (y) ≤ R(y) ≤ R2 (y), hold for all y ≥ 0. Thus R0 is defined by the boundary functions R1 (y) ≡ 0 and R2 (y) ≡ y. In Section 2 we provide a general sufficient condition that a given contract is optimal within the class R(R1 , R2 ) under a risk measure (2) and when every contract R is priced according to the standard deviation rule, i.e. π(R) = ER + β D R, where D R denotes standard deviation of R and β ≥ 0 is a safety loading coefficient. It should be stressed up that the only restrictions we impose on ϕ is that it is a measurable function. We neither assume that ϕ is smooth, nor that it is convex. Moreover, the result is applicable to any boundary functions R1 and R2 such that R1 (y) ≤ R2 (y) for all y ≥ 0. In Section 3 we apply a general theory of Section 2 in order to derive an optimal reinsurance contract under the absolute deviation risk measure ρ1 (R) = E|Y − R(Y) − E(Y − R(Y))|.
1. Introduction ˜ covered by an insurer and the rest R Reinsurance contract relies on dividing the total claim Y into a portion R covered by a reinsurance company, i.e. ˜ + R. Y =R Throughout the paper we assume that Y is a nonnegative random variable defined on a given probability space (Ω, S, P) and R is a measurable function of Y . The contract is priced according to some rule π and its price will be denoted by π(R). The insurer is interested to purchase as much of risk protection as possible at a price not exceeding a given limit price P . So in order to find an optimal contract for the insurer one must first determine a risk measure as well as the pricing rule of the contracts. In this paper we consider a quite general measure of the insurer’s risk. As usual we assume that the insurer’s risk is caused by a fluctuation of ˜ = Y − R(Y), R
228
L. Gajek, D. Zagrodny / Insurance: Mathematics and Economics 34 (2004) 227–240
the share of the total claim which is covered by the insurer, relatively to its expectation E(Y − R(Y)). However, it is usually more dangerous if Y − R(Y) exceeds its expectation than if the opposite relationship occurs. Therefore we introduce a harm function ϕ : R → R+ which measures the insurer’s loss due to any fluctuation of (1). So far a standard choice of ϕ has been ϕ2 (t) ≡ t 2 (see e.g. Daykin et al., 1993; Gajek and Zagrodny, 2000; Kaluszka, 2001). In + this paper we examine several other choices of ϕ: in Section 3 the harm functions ϕ1 (t) ≡ |t | and ϕ1 (t) = max(0, t) are studied while Section 4 is devoted to another nonsymmetric harm function
Abstract The paper concerns the problem of purchasing the best risk protection from a reinsurance company. The question of choosing the risk measure is discussed and several choices of nonsymmetric risk measures are examined. Sufficient conditions for optimality of a reinsurance contract are given for arbitrary risk measure within any restricted class of admissible contracts. Explicit forms of optimal contracts are derived in the case of absolute deviation and truncated variance risk measures. © 2004 Published by Elsevier B.V.
JEL classification: C 61; D 81; G 22 Subj. Class.: IM 30; IM 51; IM 52; IE 30; IE 43; IB 90; IB 92 Keywords: Reinsurance; Stop loss; Change loss; Quota share; Convex risk measures; Utility
Insurance: Mathematics and Economics 34 (2004) 227–240
Optimal reinsurance under general k a,∗ , Dariusz Zagrodny b
a b
Institute of Mathematics, Technical University of Lód´ z, Al. Politechniki 11, 90-924 Lód´ z, Poland Faculty of Mathematics, Cardinal Stefan Wyszy´ nski University, Dewajtis 5, 01-815 Warsaw, Poland Received May 2002; received in revised form December 2003; accepted 9 December 2003
∗ ଝ
(1)
0167-6687/$ – see front matter © 2004 Published by Elsevier B.V. doi:10.1016/j.insmatheco.2003.12.002
+ ϕ2 (t) = [max(0, t)]2 .
Having chosen ϕ, we define the risk measure ρ(R) as the expected harm, i.e. ρ(R) = Eϕ(Y − R(Y) − E(Y − R(Y))). (2) A risk measure ρ will be called convex if the corresponding harm function ϕ is convex. Let π(R) denote the price of the contract R. Having available an amount of money P the insurer is interested to purchase a contract R∗ such that π(R∗ ) ≤ P and ρ(R∗ ) ≤ ρ(R) for all R ∈ R which have the price π(R) ≤ P , where R denotes the set of all admissible contracts. The reinsurance arrangement R∗ having the above property will be called optimal within the set R of admissible contracts under the risk measure (2). A usual class of admissible contracts is the class R0 of all measurable R : [0, ∞) → [0, ∞) such that 0 ≤ R(y) ≤ y for all y ≥ 0. However other classes might be also of great interest—as pointed out by Pesonen (1984)—there is no good reason for such restrictions on R. In general, given two measurable boundary functions R1 , R2 : [0, ∞) → (−∞, ∞) such that R1 (y) ≤ R2 (y) for all y ≥ 0, we define the class R(R1 , R2 ) of all measurable functions R : [0, ∞) → (−∞, ∞) for which the inequalities R1 (y) ≤ R(y) ≤ R2 (y), hold for all y ≥ 0. Thus R0 is defined by the boundary functions R1 (y) ≡ 0 and R2 (y) ≡ y. In Section 2 we provide a general sufficient condition that a given contract is optimal within the class R(R1 , R2 ) under a risk measure (2) and when every contract R is priced according to the standard deviation rule, i.e. π(R) = ER + β D R, where D R denotes standard deviation of R and β ≥ 0 is a safety loading coefficient. It should be stressed up that the only restrictions we impose on ϕ is that it is a measurable function. We neither assume that ϕ is smooth, nor that it is convex. Moreover, the result is applicable to any boundary functions R1 and R2 such that R1 (y) ≤ R2 (y) for all y ≥ 0. In Section 3 we apply a general theory of Section 2 in order to derive an optimal reinsurance contract under the absolute deviation risk measure ρ1 (R) = E|Y − R(Y) − E(Y − R(Y))|.
1. Introduction ˜ covered by an insurer and the rest R Reinsurance contract relies on dividing the total claim Y into a portion R covered by a reinsurance company, i.e. ˜ + R. Y =R Throughout the paper we assume that Y is a nonnegative random variable defined on a given probability space (Ω, S, P) and R is a measurable function of Y . The contract is priced according to some rule π and its price will be denoted by π(R). The insurer is interested to purchase as much of risk protection as possible at a price not exceeding a given limit price P . So in order to find an optimal contract for the insurer one must first determine a risk measure as well as the pricing rule of the contracts. In this paper we consider a quite general measure of the insurer’s risk. As usual we assume that the insurer’s risk is caused by a fluctuation of ˜ = Y − R(Y), R
228
L. Gajek, D. Zagrodny / Insurance: Mathematics and Economics 34 (2004) 227–240
the share of the total claim which is covered by the insurer, relatively to its expectation E(Y − R(Y)). However, it is usually more dangerous if Y − R(Y) exceeds its expectation than if the opposite relationship occurs. Therefore we introduce a harm function ϕ : R → R+ which measures the insurer’s loss due to any fluctuation of (1). So far a standard choice of ϕ has been ϕ2 (t) ≡ t 2 (see e.g. Daykin et al., 1993; Gajek and Zagrodny, 2000; Kaluszka, 2001). In + this paper we examine several other choices of ϕ: in Section 3 the harm functions ϕ1 (t) ≡ |t | and ϕ1 (t) = max(0, t) are studied while Section 4 is devoted to another nonsymmetric harm function
Abstract The paper concerns the problem of purchasing the best risk protection from a reinsurance company. The question of choosing the risk measure is discussed and several choices of nonsymmetric risk measures are examined. Sufficient conditions for optimality of a reinsurance contract are given for arbitrary risk measure within any restricted class of admissible contracts. Explicit forms of optimal contracts are derived in the case of absolute deviation and truncated variance risk measures. © 2004 Published by Elsevier B.V.
JEL classification: C 61; D 81; G 22 Subj. Class.: IM 30; IM 51; IM 52; IE 30; IE 43; IB 90; IB 92 Keywords: Reinsurance; Stop loss; Change loss; Quota share; Convex risk measures; Utility
Insurance: Mathematics and Economics 34 (2004) 227–240
Optimal reinsurance under general k a,∗ , Dariusz Zagrodny b
a b
Institute of Mathematics, Technical University of Lód´ z, Al. Politechniki 11, 90-924 Lód´ z, Poland Faculty of Mathematics, Cardinal Stefan Wyszy´ nski University, Dewajtis 5, 01-815 Warsaw, Poland Received May 2002; received in revised form December 2003; accepted 9 December 2003