Optimal reinsurance under mean-variance premium principles
Marek Kaluszka
Institute of Mathematics, Technical University of Lodz, ul. Zwirki 36, 90-924 Lodz, Poland
Received 1 June 2000; received in revised form 1 September 2000; accepted 4 October 2000
Abstract
We derive optimal reinsurance under premium principles based on the mean and variance of the reinsurer’s share of the total claim amount. Both global reinsurance and local reinsurance are studied. Examples considered include standard deviation principle and variance principle. © 2001 Elsevier Science B.V. All rights reserved.
MSC: Primary 90A46; Secondary 62P05
Keywords: Optimal reinsurance; Global reinsurance; Local reinsurance
1. Introduction
Reinsurance is the transfer of risk from a direct insurer, the cedent, to a second insurance carrier, the reinsurer. The best known examples of reinsurance are quota share and stop loss. In quota share reinsurance, the reinsurer’s share of the total claim amountXis given byRq(X)=aX, wherea∈(0,1)is a parameter. An easy computation shows that the minimum ofD2(X−R(X))subject to the constraintP =D2R(X)is attained atRq, whereP is the premium of the reinsurer andD2Umeans the variance of random variableU. We assume throughout this paper that P is a fixed positive number.
In stop loss reinsurance,Rs(X)=(X−b)+,bbeing a parameter. Here and subsequently,a+means max{a,0}. It is well known that the retention functionRsis a solution of the following minimization problem:
D2(X−R(X))=min
R , under the conditionP =(1+β)ER(X), whereβ >0 is a safety loading coefficient (cf. Daykin et al., 1994).
The aim of the paper is to derive optimal reinsurance under other premium principles based on the mean and vari-ance of the reinsurer’s share of the total claim amount. Two premium calculation principles are often used in practice:
P =ER(X)+βDR(X) and P =ER(X)+βD2R(X),
called the standard deviation and the variance principle, respectively. Other rules are:
1. the modified variance principleP =ER(X)+βD2R(X)/ER(X), 2. the mixed principleP =ER(X)+αDR(X)+βD2R(X)
E-mail address: [email protected] (M. Kaluszka).
(cf. Daykin et al., 1994; Rolski et al., 1998; Straub, 1988). Observe that the above-mentioned principles are of the form
ER(X)=f (P ,DR(X)),
wheref :(0,∞)×[0, t (P ))→(−∞,∞)withDX < t (P )≤ +∞, and the following conditions hold: (F1)f (P ,0)=P,
(F2)f (P , t )is nonincreasing, concave, and differentiable int.
Condition (F1) is related to the property called no unjustified safety loading. In fact, ifΠ (X)denotes a solution of the equationEX=f (t,DX)fort >0, then (F1) implies thatΠ (c)=cfor allc >0.
A retention functionRis said to be optimal if it is a solution of the following problem:
minimize D(X−R(X)) subject to ER(X)=f (P ,DR(X)), 0≤R(x)≤xfor allx≥0. In Section 2 we prove that optimal retention functions are of the following form:
R∗(x)=a(x−b)+ (1)
with 0 < a ≤1,b ≥0. The rule defined by (1) is called change loss reinsurance. In Section 3 optimal rules for local reinsurance are provided.
The results to be presented generalize work published in Gajek and Zagrodny (2000). Using convex programming methods, they demonstrated that the change loss reinsurance is optimal under the standard deviation principle. All proofs are straightforward, in contrast to those of Gajek and Zagrodny (2000).
2. Global reinsurance
LetXbe a nonnegative random variable on a probability space(Ω, S,P), called the risk. Suppose 0<D2X <∞. Define supX=sup{b:P(X > b) >0}. The reinsurer premium risk is given byR=R(X), whereR: [0,+∞)→ [0,+∞)is a retention function. We seek for a rule which minimizes the cedent’s risk, measured by the variance of X−R, subject to the constraintR∈R(f ), where
R(f )= {R|ER=f (P ,DR), 0≤R(x)≤x for everyx ≥0}. (2) Thus the following minimization problem arises:
min R∈R(f )D
2(X
−R). (3)
Theorem 1.
1. Suppose there exist realsa, bsuch that the following conditions hold:
(i) 0< a≤1, 0< b <supX,
(ii) −E(b−X)+f2′(P , aD(X−b)+)=(1−a)D(X−b)+, (iii) aE(X−b)+=f (P , aD(X−b)+),
wheref2′(P , t )=∂f (P , t )/∂t. ThenR∗(x)=a(x−b)+is a solution of the problem (3).
2. If EX > f (P ,DX), then there exist realsa, bsuch that conditions (i)–(iii) are satisfied.
Proof. Recall that
Observe that
Cov(X, R)=Cov(X−b, R)≤Cov((X−b)+, R)+[E(X−b)+−E(X−b)]ER. SinceE(X−b)+−E(X−b)=E(b−X)+, we have
Cov(X, R)≤Cov((X−b)+, R)+E(b−X)+ER (5)
and the equality in (5) holds ifR(x)=0 for 0≤x≤b. From (4), (5) and the Cauchy–Schwarz inequality it follows that for everyR∈R(f )
D2(X−R)≥D2X−2D(X−b)+DR+D2R−2E(b−X)+f (P ,DR). Puttingt =DR/D(X−b)+, we get
D2(X−R)≥D2X+(t2−2t )D2(X−b)+−2E(b−X)+f (P , tD(X−b)+). (6) The equality in (6) holds ifR(x)=c(x−b)+with a realcsuch thatR∈ R(f ). By (F2) and (ii), the right-hand side of (6) is a convex function intand attains its minimum att =a. By (i) and (iii),R∗∈R(f ), and consequently for everyR ∈R(f )
D2(X−R)≥D2(X−R∗), (7)
completing the proof of the first part of the theorem.
We now show that there exist realsa, bsuch that conditions (i)–(iii) are satisfied. Define
φ (s, t )=tE(X−s)+−f (P , tD(X−s)+), 0≤s≤b1, 0≤t≤1,
whereb1is a real such thatφ (b1,1)=0. Sinceφ (0,1) >0, limb→∞φ (b,1)= −f (P ,0)= −P <0 andφ (s,1) is continuous,b1exists and 0< b1<supX. By (F2), for every 0≤s≤b1, there is only one point, sayt (s), such thatφ (s, t (s))=0. Sinceφis continuous the functiont (s)is continuous. Define
γ (s, t )=D2X+(t2−2t )D2(X−s)+−2E(s−X)+f (P , tD(X−s)+)
for 0 ≤s ≤ b1, 0≤ t ≤1. Since for every 0≤ s ≤ b1the functionγ (s, t )is strictly convex int, there is only one point, sayτ (s), such thatγ (s, τ (s)) =min0≤t≤1γ (s, t ). Obviously,τ (0)=1. Moreover, the functionτ (s), 0 ≤s ≤b1, is continuous sinceγ2′(s, t )is continuous. We have 0 < t (0) <1 =τ (0)andt (b1)=1. Therefore there area, bsuch that 0< b≤b1, 0< a≤1 anda=t (b)=τ (b), completing the proof.
Remark 1. Observe that if conditions (i)–(iii) hold, and ifP(X≤b) >0, then it must holdEX > f (P ,DX). In fact,
EX≥aEX > aE(X−b)+=f (P , aD(X−b)+)≥f (P , aDX)≥f (P ,DX), sinceD2(X−b)+≤D2XandE(X−b)+≤EXfor everyb >0.
Remark 2. It is clear that the assumptions of Theorem 1 can be weaken but we do not go into details since both (F1) and (F2) are fulfilled in the most common models (see Examples 1–4 of the paper).
Remark 3. Important contribution to solving the problem of optimal purchase of reinsurance was given by Deperez and Gerber (1985). They called a retention function optimal if it maximized the expectation of the cedent’s utility function over the set of all retention rules, i.e. the following problem was studied:
maximize Eu(R−H (R)−X) over allR=R(X),
principle which is not Gâteaux differentiable. The difference between our approach and that of Deperez and Gerber (1985) is: (a) the use of different target function and (b) minimization of target function under constraints.
Example 1 (Expected value principle). LetP = (1+β)ER, where β > 0 is a safety loading coefficient. Of course,f (P , t )=P /(1+β). From Theorem 1 it follows thatR∗(x)=(x−b)+is an optimal retention function if(1+β)E(X−b)+=P, which is a well-known fact (cf. Pesonen, 1984).
Example 2 (Standard deviation principle). LetP =ER+βDRwithβ >0. Thenf (P , t )=P−βtand Theorem 1 shows that ifEX+βDX > P, then an optimal retention function is given byR∗(x)=a(x−b)+, wherea, b are reals such that
−βE(b−X)++(1−a)D(X−b)+=0, aE(X−b)++aβD(X−b)+=P .
This result is due to Gajek and Zagrodny (2000). Explicit formulae can be found fora, bin some models of risk. For instance, supposeXis a Bernoulli risk, i.e.Xis a random variable that takes the valueM >0 with probability p∈(0,1)and zero with probabilityq,q =1−p. Then
a= P{pP−β[M(p−qβ 2)
−P]√pq}
p[P2+2β2MPq−β2M2q(p−qβ2)], b=
[M(p−β2q)−P]√pq+qβP (p−β2q)(√pq+qβ) .
For concreteness, letM = $106,p =10−4,β = 2.3 andP = $104. Thena = 0.433987,b = $2455 and the reinsurer is obliged to pay $432 921 providedX=$106.
Example 3 (Variance principle). LetP =ER+βD2Rwithβ >0. SupposeEX+βD2X > P. Then an optimal retention function is given byR∗(x)=a(x−b)+, wherea, bsatisfy the following equations:
−2aβE(b−X)++1−a =0, aE(X−b)++a2βD2(X−b)+=P .
IfXis a Bernoulli risk, then
a= [p(4qβP +p)] 1/2
p(2qβM+1) , b=
p+2pqβM−[p(4qβP +p)]1/2 2qβ[p(4qβP +p)]1/2 .
LetM=$106,p=10−4,β=1/$103andP =$104. In this case,a=0.316,b=$1082 and the reinsurer covers $315 743 ifX=$106.
Example 4 (Modified variance principle). SupposeP =ER+βD2R/ER, withβ >0. Thenf (P , t )= (P + (P2−4βt2)1/2)/2 for 0≤t ≤P /(2β1/2). If 2β1/2DX < P <EX+βD2X/EX, thenR∗(x)=a(x−b)
+is an optimal retention function witha, bdefined by
−2aβE(b−X)++(1−a)(P2−4βa2D2(X−b)+)1/2=0, aE(X−b)++aβD 2(X
−b)+
E(X−b)+ =P . SupposeXis a Bernoulli risk. Then
a= 2qβP +u(p+qβ)P
(p+qβ)(2qβM+uP), b=M−
2qβM+uP
2qβ+u(p+qβ)
withu=(1−4pqβ/(p+qβ)2)1/2. For concreteness, letM=$106,p=10−4,β
3. Local reinsurance
LetX, X1, X2, . . . be a sequence of identically distributed random variables defined on a common probability space(Ω, S,P). LetNbe an integer-valued random variable with 0<DN <∞. Throughout this section we assume that the random variablesN, X, X1, X2, . . . are independent. We useX1, X2, . . . as the sequence of successive claims occurring in a time interval.Nmodels the number of claims over that period. Consider local reinsurance with a common retention functionR, i.e. each claim is divided between the cedent and the reinsurer as follows: for the ith claim of sizeXithe partR(Xi)is carried by the reinsurer. The cedent wants to have a reinsurance arrangement which minimizes the variance of his payoff subject to the following constraints:
0≤R(X)≤X and ER(X)=f (P ,DR(X)). The corresponding minimization problem is then given by
min
whereR(f )is defined by (2). The problem (8) can be treated in a similar way as that of (3), since
D2
withR =R(X)(see e.g. Rolski et al. (1998, Corollary 4.2.1)). Put
γ (s, t )=EN{D2X+(t2−2t )D2(X−s)+−2E(s−X)+f (P , tD(X−s)+)}
+{EX−f (P , tD(X−s)+)}2D2N (10) for 0≤t ≤1,s≥0, and defineγ2′(s, t )=∂γ (s, t )/∂t.
Theorem 2.
1. Suppose there exista, bsuch that
(i) 0< a≤1, 0< b <supX,
Proof. Taking into account (9) and using the same line of argument as in the proof of Theorem 1, we get
and, in consequence, the functionγ (b, t )is convex int ≥0. By (12) and (ii), for everyR∈R(f ), of the second part, we use the same arguments as in the proof of Theorem 1 but with one subtle difference. Now τ (0)may be not equal to 1 and we assume thatt (0)≤τ (0), i.e.
R∈RE(f ),
D2
" N
X
i=1
(Xi−R(Xi))
#
≥EN (DX−DR)2+(EX−f (P ,DR))2D2N
≥(1−a)2END2X+(EX−f (P , aDX))2D2N =D2
" N X
i=1
(Xi−R∗∗(Xi))
#
. (20)
SinceR∗∗∈RE(f ), the proof is completed.
Example 6. LetP =ER+βDR. Theorem 3 implies that the optimal rule is given by R∗∗(x)=a(x−EX−βDX)+P ,
whereaminimizes(1−t )2END2X+(EX−P +tβDX)2D2Novert ∈[(P −EX)/(βDX), P /(βDX)].
Remark 4. An interesting problem seems to be the analysis of optimal reinsurance under alternative premium calculation rules like the exponential principle, the Esscher principle or the risk-adjusted principle (see Embrechts, 1996; Embrechts et al., 1997; Gerber, 1980; Rolski et al., 1998; Young, 1999; Wang et al., 1997; Wang, 1998).
Acknowledgements
I would like to thank the anonymous referee for the helpful suggestions. I also thank Prof. L. Gajek and Prof. D. Zagrodny for arousing my interest in this field.
References
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