Stable solutions for optimal reinsurance problems involving risk measures.

Innovative Applications of O.R.

Stable solutions for optimal reinsurance problems involving risk measures

Alejandro Balbás⇑, Beatriz Balbás, Antonio Heras

University Carlos III of Madrid, CL. Madrid, 126, 28903 Getafe, Madrid, Spain

University of Castilla la Mancha, Avda. Real Fábrica de Seda, s/n, 45600 Talavera, Toledo, Spain University Complutense of Madrid, Somosaguas-Campus, 28223 Pozuelo de Alarcón Madrid, Spain

a r t i c l e i n f o

Article history:

Received 3 May 2010 Accepted 19 May 2011 Available online 27 May 2011

Keywords:

Optimal reinsurance Risk measure Sensitivity Stable optimal retention Stop-loss reinsurance

a b s t r a c t

The optimal reinsurance problem is a classic topic in actuarial mathematics. Recent approaches consider a coherent or expectation bounded risk measure and minimize the global risk of the ceding company under adequate constraints. However, there is no consensus about the risk measure that the insurer must use, since every risk measure presents advantages and shortcomings when compared with others.

This paper deals with a discrete probability space and analyzes the stability of the optimal reinsurance with respect to the risk measure that the insurer uses. We will demonstrate that there is a ‘‘stable optimal retention’’ that will show no sensitivity, insofar as it will solve the optimal reinsurance problem for many risk measures, thus providing a very robust reinsurance plan. This stable optimal retention is a stop loss contract, and it is easy to compute in practice. A fast linear time algorithm will be given and a numerical example presented.

1. Introduction

Since the paper byArtzner et al. (1999)introduced the coherent measures of risk many authors have further extended the discus sion, which shows the importance that this topic is achieving in finance and insurance. Among others,Goovaerts et al. (2004)intro duced the consistent risk measures, also studied inBurgert and Rüschendorf (2006), Frittelli and Scandolo (2005)analyzed risk measures for stochastic processes, andRockafellar et al. (2006)de fined the deviations and the expectation bounded risk measures. Classical actuarial and financial problems have been then revis ited using risk measures beyond the variance. Among others, Nakano (2004) and Balbás et al. (2010)drew on risk measures when pricing in incomplete markets,Mansini et al. (2007) and Schied (2007)dealt with portfolio choice problems, andAnnaert et al. (2009)checked the efficiency of the classical portfolio insurance problem if the risk level is given by the value at risk (VaR) or the conditional value at risk (CVaR).

The optimal reinsurance problem is a main issue in actuarial science. A common approach attempts to minimize some measure of the first insurer risk after reinsurance. Seminal papers byBorch (1960) and Arrow (1963)used the variance as a risk measure and proved that the stop loss reinsurance minimizes the retained risk if

premiums are calculated following the expected value premium principle.

The subsequent research followed similar ideas and tried to take into account more general risk measures and premium princi ples, which may give optimal contracts other than stop loss. Recently,Gajec and Zagrodny (2004)considered more general symmetric and even asymmetric risk functions such as the abso lute deviation and the truncated variance of the retained loss, un der the standard deviation premium principle.Kaluszka (2005) studied reinsurance contracts with many convex premium princi ples (exponential, semi deviation and semi variance, Dutch, dis tortion, etc.). Other well known financial risk measures such as theVaRor the tail value at risk (TVaR) are also being considered. For example,Kaluszka (2005)uses theTVaRas a premium principle andCai and Tan (2007)calculate the optimal retention for a stop loss reinsurance by considering theVaRand the conditional tail expectation risk measures (CTE), under the expected value pre mium principle.

The most recent papers have finally incorporated coherent and/ or expectation bounded risk measures in the objective function to be minimized by the ceding company. Along with the paper ofCai and Tan (2007)above, other interesting examples areCai et al. (2008), Balbás et al. (2009)orBernard and Tian (2009). The differ ences among their approaches are caused by the insurer behavior. Very complete information may be found in the survey ofCenteno and Simoes (2009).

Despite the interest of the problem, as far as we know there are no analyses focusing on the stability of the optimal reinsurance. This should be an important topic since the optimality of many 0377-2217/$ - see front matterÓ2011 Elsevier B.V. All rights reserved.

doi:10.1016/j.ejor.2011.05.035

⇑Corresponding author at: University Carlos III of Madrid, CL. Madrid, 126, 28903 Getafe, Madrid, Spain.

E-mail addresses:[email protected](A. Balbás),[email protected]

(B. Balbás),[email protected](A. Heras).

Contents lists available atScienceDirect

European Journal of Operational Research

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / e j o r

reinsurance plans will critically depend on the risk measure and the pricing principle. There is no consensus about the risk measure that the insurer must use, since every risk measure presents advantages and shortcomings when compared with others.

This paper considers that the reinsurer’s premium principle is given by a convex function and deals with the optimal reinsurance problem if risk is measured by coherent and expectation bounded risk measures.1_{The focus is on the stability in the large of the opti}

mal retention plan with respect to the chosen risk measure. ‘‘Stabil ity in the large’’ is used in the sense ofSamuelson (1947),i.e., we will analyze whether the optimal contract remains constant as the risk measure becomes more and more risk adverse (the risk measure increases).

The paper’s outline is as follows. Section2will present the basic conditions and properties of the risk measure

q

to be used. Section 3provides our general optimal reinsurance problem. We will pres ent the problem in a discrete probability space. Actually, this sim plifies the mathematical exposition, and every probability space admits a discrete approximation which achieves as much accuracy as needed. Many actuarial and financial analyses deal with discrete probability spaces (seeBenati, 2003; Konno et al., 2005; Mansini et al., 2007, orMiller and Ruszczynski, 2008, among many others), since this is not a restriction in practice. The proposed optimal reinsurance problem seems to be quite flexible and general, since it allows us to incorporate many particular situations such as bud get constraints, the maximization of the insurer expected wealth, etc. The most important results in Section3areTheorem 2 and Corollary 4, since they characterize the optimal retention by means of Karush Kuhn Tucker (KKT) like conditions and permit us to introduce the ‘‘stable optimal retention’’,2 _{which will solve the}

problem for all of the risk measures with a subgradient satisfying adequate properties. Therefore, the stable optimal retention may be understood as a robust optimal reinsurance plan.

Section4is devoted to computing in practice the stable optimal retention. Here we will assume that the reinsurer uses a linear value principle, containing the expected value premium principle as a particular case. Of course it is not necessary, since practical optimal ity conditions have been given in a much more general framework, but the specific solution of the optimization problem depends on the premium principle we take, and considering more than one would significantly enlarge the paper. As already indicated, previ ous literature measuring the insurer risk by a general risk measure is still limited, so it seems to be natural and of interest to analyze concrete problems by taking the most used premium principle.

The most important result of this section isTheorem 8, because it gives explicit expressions for the stable optimal retention and theKKTmultipliers of the problem. According toTheorem 8, the stable optimal retention is a stop loss reinsurance.

Theorem 8is used in Section5so as to introduce a fast algorithm that gives the stable optimal retention in numerical applications. The algorithm is not time consuming since there is a linear relation ship between its computational complexity and the complexity of the portfolio of insurance policies. An illustrative numerical exam ple is also provided, which clarifies how to use the algorithm in practice and shows the robustness of the given reinsurance, in the sense that most of the usual risk measures lead to this solution.

The last section of the paper summarizes the most important conclusions.

2. Preliminaries and notations

As usual, consider the probability spaceðX;F;PÞcomposed of the set of ‘‘states of the world’’X, the

r

algebraFand the proba bility measureP. As said above, we will be dealing with a discrete framework, soXwill be composed of a finite number of elements,

X f

x

1;

x

2;. . .;

x

ng: ð1Þ

We will consider the probability of every single event

pi Pð

x

iÞ>0; i= 1, 2,. . .,n.

Denote byEðyÞthe mathematical expectation of every random variabley, and denote byL2_{the Hilbert space ofRvalued random}

variablesyonXendowed with the norm

kyk₂ Eðjyj2Þ1=2

for everyy2L2_.3

Let [0,T] be a time interval. From an intuitive point of view, one can interpret that everyy2L2_{may represent the wealth atTof an}

arbitrary insurer. Let

q

:L2!R

be the general risk function that an insurer uses in order to control the risk level of his final wealth atT. Denote by

Dq nz2L2; EðyzÞ6

q

_ðy_Þ;8y2L2o_:

ð2Þ

We will assume thatDqis convex and compact, and

q

ðyÞ Max EðyzÞ:z2Dq; ð3Þ

holds for everyy2L2_{. Furthermore, we will also suppose that the}

constant random variablez= 1 is inDqand Dq z2L2;EðzÞ 1

n o

: ð4Þ

Summarizing, we have:

Assumption 1. The setDqgiven by(2)is convex and compact,(3)

holds for everyy2L2_{,z= 1is inD}

q, and(4)holds. h

The assumption above is closely related to the representation theorem of risk measures stated inRockafellar et al. (2006). Fol lowing their ideas, it is easy to prove that the fulfillment of Assumption 1holds if and only if

q

satisfies:

(a)

q

ðyþkÞ

q

ðyÞ k ð5Þ

for everyy2L2_andk2R.

(b)

q

ð

a

yÞ

aq

ðyÞ ð6Þ

for everyy2L2_and

a

_{> 0.}

(c)

q

ðy1þy2Þ6

q

ðy1Þ þ

q

ðy2Þ ð7Þ

for everyy1,y22L2.

(d)

q

ðyÞP EðyÞ ð8Þ

for everyy2L2_.4

1_{Insurance premiums are usually given by convex functions, see, for instance,}

Deprez and Gerber (1985).

2_{Actually, the givenKKT–likeconditions are not exactly the same as the standard}

KKTconditions of the problem, and that is the reason why we say ‘‘KKT–like’’. Nevertheless, they are necessary and sufficient optimality conditions, and are

generated by theKKTconditions of an equivalent optimization problem presented

inBalbás et al. (2009). Further details may be found in that paper.

3_{Actually,Xbeing discrete and containingn2}

Nelements the dimension ofL2_is finite and equalsn. Thus,L2_=Lp_{for everyp2[1,1] and the normkk}

2above is equivalent to the normkkp. Though we have chosenp= 2, everyp2[1,1] may play the same role.

4_{Actually, the properties above are almost similar to those used byRockafellar} et al. (2006)in order to introduce their expectation bounded risk measures.

It is easy to see that if

q

satisfies properties (a) (d) then it is also coherent in the sense ofArtzner et al. (1999)if and only if

DqL2þ z2L 2_;P

ðzP0Þ 1

n o

: ð9Þ

Particular interesting examples are the conditional value at risk (CVaR) ofRockafellar et al. (2006), the weighted conditional value at risk (WCVaR) ofCherny (2006), the dual power transform (DPT) ofWang (2000)and the Wang measure (Wang, 2000), among many others. Furthermore, following the original idea ofRockafellar et al. (2006)to identify their expectation bounded risk measures and their deviation measures, it is easy to see that

q

ðyÞ

r

ðyÞ EðyÞ; ð10Þ

satisfies (a) (d) if

r

:L2_!

Ris a deviation, that is, if

r

satisfies (b), (c),

(e)

r

ðyþkÞ

r

ðyÞ for everyy2L2_andk2

R, and (f)

r

ðyÞP0

for everyy2L2_.

Among many others, a particular example is the classical p deviation for everyp2[1,1), given by

r

pðyÞ EjEðyÞ yjp

1=p

;

or the downsidepsemi deviation, given by

r

pðyÞ EjMaxfEðyÞ y;0gj p

1=p

:

The classical separation theorems allow us to prove that there is a one to one mapping

q

MDqbetween the risk measures satisfying

Assumption 1that are coherent and the set of convex and compact subsets ofL2_{such thatz= 1 is inD}

q, and(4) and (9)hold. Further

more(3)shows that this mapping is increasing,i.e.,

q1

(y)6

q2

(y) holds for everyy2L2_{if and only if}

Dq1Dq2 holds. Accordingly, the maximum coherent risk measure satisfyingAssumption 1is thatCassociated with the set

DC z2L2þ:EðzÞ 1

n o

: ð11Þ

It is easy to see that the risk measureCis

CðyÞ Min yf ð

x

iÞ:i 1;2;. . .;ng ð12Þ

for everyy2L2_{. Similarly,y!}

E(y) is the minimum risk measure satisfying the conditions above, sinceDE f1g. Thus

CðyÞP

q

ðyÞP EðyÞ; ð13Þ

holds for everyy2L2_{and every coherent}

q

_{satisfyingAssumption 1.}

Finally, once again the separation theorems allow us to prove that every convex combination

q

X

m

i 1

wi

q

i;

of risk measures satisfying(5) (9)also satisfies(5) (9), and

Dq

Xm i 1

wiDqi; ð14Þ

holds.

3. Optimal reinsurance: General problem and optimality conditions

Consider that the insurance company receives the fixed amount S0(premium) and will have to pay the random variable y02L

2 þ

within a given period [0,T] (claims). Without loss of generality we will assume thatPðy0>0Þ 1, since the absence of claims is

an unrealistic situation in practice.

Suppose that a reinsurance contract is signed in such a way that the company will only payy2L2_{, whereas the reinsurer will pay}

y0 y. If the reinsurer premium principle is given by the convex

and increasing function,

p

:L2_!R;

such that

p

(0) = 0, andS1> 0 is the largest amount that the insurer

would like to pay for the contract, then the insurance company will choosey(optimal retention) so as to solve the bi criteria optimiza tion problem Min

q

0ðS0 y

p

ðy0 yÞÞ; MaxEðS0 y

p

ðy0 yÞÞ;

p

ðy0 yÞ6S1; 06y6y 0; 8 > > > < > > > : ð15Þ

q0

being a coherent risk measure that satisfiesAssumption 1. Con ditions

p

(0) = 0 andS1> 0 imply thaty=y0satisfies the constraint,

so(15)is always feasible (Theorem 2below will show that it is also bounded and solvable). Notice that, if desired, constraint

p

(y0 y)6S1may be removed without modifying(15), since

p

is

increasing and therefore it is sufficient to chooseS1>

p

(y0).

First of all let as see that the multiobjective optimization prob lem(15)is convex.

Lemma 1. With the notations of (15) we have that the three functions L2 3y!

q

₀ðS0 y

p

ðy0 yÞÞ 2R; L2 3y! EðS0 y

p

ðy0 yÞÞ 2R ð16Þ and L23y!

p

ðy0 yÞ 2R are convex.

Proof. Let us prove that(16)is convex since the remaining cases are analogous. Thus, suppose thaty1,y22L2and 06k61.

q

0ðS0 ðky1þ ð1 kÞy2Þ

p

ðy0 ðky1þ ð1 kÞy2ÞÞÞ

q

₀ðkðS0 y1Þ þ ð1 kÞðS0 y2Þ

p

ðkðy0 y1Þ þ ð1 kÞðy0 y2ÞÞÞ:

Since

p

is convex we have that

p

ðkðy0 y1Þ þ ð1 kÞðy0 y2ÞÞP k

p

ðy0 y1Þ ð1 kÞ

p

ðy0 y2Þ:

q0

is decreasing because it is coherent.5_Hence

q

0ðS0 ðky1S0 y

p

ðy0 yÞy2Þ

p

ðy0 ðky1þ ð1 kÞy2ÞÞÞ

6

q

0ðkðS0 y1Þ þ ð1 kÞðS0 y2Þ k

p

ðy0 y1Þ ð1 kÞ

p

ðy0 y2ÞÞ:

Finally, since

q0

is convex,

q

0ðS0 ðky1þ ð1 kÞy2Þ

p

ðy0 ðky1þ ð1 kÞy2ÞÞÞ

6k

q

0ðS0 y1

p

ðy0 y1ÞÞ þ ð1 kÞ

q

0ðS0 y2

p

ðy0 y2ÞÞ:

Since the multiobjective optimization problem(15)is convex, it may be solved by scalarization methods,i.e., in order to obtain Par eto solutions one can minimize a convex combination of

q0

and E. Accordingly, take w0 and w1 non negative and such that

w0+w1= 1, let

q

w0q0 w1E, and solve

5_{SeeArtzner et al. (1999), or verify thatq}

0is decreasing from(3) and (9).

Min

q

ðS0 y

p

ðy0 yÞÞ;

p

ðy0 yÞ6S1; 06y6y₀: 8 > < > : ð17Þ

Bearing in mind the ideas of the previous section,

q

satisfies Assumption 1and is coherent, since it is a convex combination of

q0

and E.

It is worth remarking that the first (second) objective of(15) may be removed and the problem still fits in(17), because one can takew0= 0 andw1= 1 (w0= 1 andw1= 0).

Next we will give necessary and sufficient Karush Kuhn Tucker optimality conditions.

Theorem 2. Problem(17) is bounded and solvable. Moreover, the

existence ofð

s

_{;zÞ 2RL}2_{satisfying the following Karush Kuhn}

Tucker like conditions is necessary and sufficient to guarantee the optimality of y⁄_2L2_. Eðy_zÞ6_{EðyzÞ; 8z2}Dq

s

p

_ðy 0 yÞ S1 ð Þ 0;

p

ðy₀ y_{Þ S} 160; Eðy_{zÞ þ ð1þ}

s

_Þ

p

_ðy 0 yÞ6EðyzÞ þ ð1þ

s

Þ

p

ðy0 yÞ; 806y6y0;

s

_2R;

s

_P0;06y6y 0;z2Dq 8 > > > > > > < > > > > > > : ð18Þ (

s

⁄_{, z}⁄_{) will be called KKT multiplier of(17).}

Proof.The dimension ofL2_{is finite due to the assumptions (Xis}

discrete and finite, see(1)andFootnote 3). Thus, the finite dimen sion ofL2_{and the convexity of}

p

:L2

!Rguarantees the continuity of

p

(Luenberger, 1969). Similarly(6) and (7)show that

q

is convex and therefore continuous. Besides, the last constraint of(17)shows that the feasible set is bounded, and therefore compact. Hence, the Weierstrass theorem shows that (17) is bounded and solvable. Finally, we will not prove the Karush Kuhn Tucker like conditions because an analogous proof may be found in Balbás et al. (2009). h

A first important consequence is that one can give conditions ensuring that the solution of(17)remains the same if

q

is replaced by a lower one.6_{Hence we can give the first result guaranteeing the}

stability of the optimal insurance (retention) with respect to the risk measure.

Corollary 3.Suppose that y⁄

2L2_{solves(17)and (}

s

⁄_{, z}⁄_{) is a KKT}

multiplier. Take the coherent risk measure

q

satisfyingAssumption1

and such that

q

6

q

. If z_2D~

q and

q

replaces

q

then y⁄2L2still

solves(17)and (

s

⁄_{, z}⁄_{) is still a KKT multiplier.}

Proof.On the one hand,y⁄_{and (}

s

⁄_,z⁄_{) satisfy(18). On the other} hand, according to that properties given in the previous section, DqDqbecause

q

6

q

. Thus,z2Dqimplies that(18)still holds

ifDqreplacesDq. h

Corollary 4. Suppose that y C2L

2_{solves(17)and} ð

s

C;zCÞis a KKT

multiplier for the risk measureCof (12) . Then y

C still solves(17)

andð

s

C;zCÞis still a KKT multiplier for every

q

with zC2Dq.

Proof.It trivially follows from the previous corollary and(13). h

Remark 1.With the notations ofCorollary 4, ifz

CRDqone still can

look for a risk measure

q

P

q

quite similar to

q

and such that z

C2Dq, and therefore yC still solves (17)and

s

C;zC

is still a KKTmultiplier if one considers

q

. Indeed, it is sufficient to take the following convex and compact set,7

Dq~ CoDq[ fzCg

;

obviously associated with the risk measure ~

q

ðyÞ Max

q

ðyÞ; Eðyz CÞ

ð19Þ for everyy2L2_{. For this reason hereafter the solutiony}

C2L 2_of(17)

for the risk measure C of (12) will be called ‘‘stable optimal retention’’. h

Remark 2. If the ceding company is also interested in maximizing the expected wealth and deals with problem(15), thenCmay be replaced by w0C w1E (with wiP0, i= 0, 1, and w0+w1= 1).

Indeed, in such a case,(11) and (14)show that

Dw0Cw1E z2L 2

;EðzÞ 1 andzPw1

n o

: ð20Þ

Obviously,Corollary 3proves that ify

w0C2L 2 _{solves(15)and}

s

w0C;z w0C

is a KKTmultiplier for the risk measurew0C w1E

above, theny

w0Cstill solves(15)and

s

w0C;z w0C is still aKKTmul tiplier for every

q

such thatz

w0C2Dw0qw1E. Furthermore, a new

comment similar toRemark 1applies. h

4. Characterizing and computing the stable optimal retention

Let us give properties making it easier to verify the fulfillment of the inequalities of(18). To this purpose, and taking into account Corollary 4,Remark 2and the first condition in(18), let us give an instrumental lemma.

Lemma 5. Suppose that 06y⁄_6y

0 and z2Dw0Cw1E (see(20)).

Eðy_zÞ6_Eðy_{zÞholds for every z2Dw}

0Cw1Eif and only if y_ð

x

jÞ Max yf ð

x

iÞ:i 1;2;. . .;ng; holds for every j = 1, 2,. . ., n such that z⁄₍

x

j)>w1.

Proof. The inequality above holds if and only ifz⁄_{solves the linear} optimization problem MaxPn i 1 y_ð

x

iÞzð

x

iÞpi; Pn i 1 zð

x

iÞpi 1; w16zð

x

iÞ; i 1;2;. . .;n: 8 > > > > > < > > > > > :

According to the classical Karush Kuhn Tucker conditions, this is equivalent to the existence of

l0

;

l1

;. . .;

l

n2Rsuch that

y_ð

x

iÞpiþ

l

0pi

l

i 0; i 1;2;. . .;n; Pn i 1 z_ð

x

iÞpi 1; z_ð

x

iÞ w1 ð Þ

l

i 0; i 1;2;. . .;n;

l

iP0; i 1;2;. . .;n; z_ð

x

iÞPw1; i 1;2;. . .;n: 8 > > > > > > > > < > > > > > > > > :

Hence, the result trivially follows if one takes

l

0 Max yf ð

x

iÞ:i 1;2;. . .;ng

6 _{With the notations of (15), notice thatqdecreases if so doesq} 0,i.e.,

q

0P

q

~0)w0

q

0 w1EPw0

q

~0 w1E: 7_{As usual,Co(A) denotes the convex hull of every setAL}2_.

A. Balbás et al. / European Journal of Operational Research 214 (2011) 796–804

and

l

i

l

0 yð

x

iÞ

pi; i= 1, 2,. . .,n. h

Despite the fact that previous analyses are quite general, the solutions of(18)will depend on the specific assumptions one im poses. Henceforth we will assume that the reinsurer uses a linear premium principle. Actually, as indicated in the introduction, pre vious literature considering a general risk measure is scant, so it seems to be natural and of interest to analyze concrete problems by taking the most used premium principle, which is the expected value premium principle,i.e., there existsk> 1 such that

p

ðyÞ kEðyÞ ð21Þ

for everyy2L2_{. We will impose something strictly weaker, such as}

the existence ofzp2L2such that

Pðzp>0Þ 1; ð22Þ

EðzpÞ>1 ð23Þ

and

p

ðyÞ EðyzpÞ ð24Þ

for everyy2L2_.8

Assumption 2. Henceforth we will assume the existence ofzp2L2

such that(22) (24)hold. h

Nevertheless, it is worth pointing out that the previous develop ments are more general, and therefore they also apply to alterna tive premium principles.

FromAssumption 2the necessary and sufficient optimality con ditions(18)become Eðy_zÞ6_{EðyzÞ; 8z2}Dq;

s

_Eðy 0 yÞzp ð Þ S1 ð Þ 0; Eððy0 yÞzpÞ S160; Ey_{z ð1þ}

s

_Þz p ð Þ ð Þ6_{Eðy zð ð1þ}

s

_{ÞzpÞÞ; 80}6_y6_y 0;

s

_2R;

s

_{P0; 0}6_y6_y 0; z2Dq: 8 > > > > > > < > > > > > > : ð25Þ Next let us present two simple lemmas. The first one simplifies the fourth condition of(25). Lemma 6. Let z⁄ 2L2_{, y}⁄ 2L2_{with 06y}⁄_6y 0, and

s

2R. Then, Ey_{z ð1þ}

s

_Þz p ð Þ ð Þ6_{Eðy zð ð1þ}

s

_ÞzpÞÞ;

holds for every y2L2_{with 06y6y}

0 if and only if there exists a

partition X X1[X2[X3; such that z_ð

x

_Þ>ð1þ

s

_Þz p; yð

x

Þ 0; if

x

2X1; z_ð

x

_{Þ ð1þ}

s

_Þz p; if

x

2X2; z_ð

x

_Þ<ð1þ

s

_Þz p; yð

x

Þ y0ð

x

Þ; if

x

2X3: 8 > < > : ð26Þ

Proof. It is obvious if we realize that the solution of MinEy z _ð1þ

s

_Þz p ð Þ ð Þ; 06_y6_y 0;

must be as large as possible (i.e., must equal y0) whenever

z⁄ _{(1 +}

s

⁄_)z

p< 0 and as small as possible (i.e., zero) if

z⁄ _{(1 +}

s

⁄_)z

p> 0, whereas its value is not relevant at all if

z⁄ _{(1 +}

s

⁄_)z

p= 0. h

Lemma 7. y⁄_{= 0 does not solve(17).}

Proof. Ify⁄_{= 0 solved(17)then(26)would lead toz}⁄

P(1 +

s

⁄_)z

p.

Bearing in mind(4) and (23), and taking expectations, one has the contradiction 1Pð1þ

s

_ÞEðz

pÞ>1. h

As already said the stop loss reinsurance is often obtained as the optimal retention (Balbás et al., 2009). Recall thaty2L2_{and ly}

ing between 0 andy0is said to be a stop loss reinsurance if there

exists

a

P0 such that

y y0; y06

a

;

a

; y0>

a

:

ð27Þ Hereafter the random variable of(27)will be denoted byya

0. Corollary 4andRemark 2show the importance of solving(17) when

q

w0C w1E, since the solution will generate a very stable

optimal reinsurance contract.

Theorem 8.Consider problem (17) with the risk measure w0C w1E. Suppose thatPðzp>w1Þ 1.9

(a) There exists

a

⁄_{> 0 such thaty}a

0 solves(17). (b) Suppose thatya 0 solves(17),Pðy0

a

Þ 0 and

s

w0C;z w0C is aKKTmultiplier of(17). Then z w0C w1; if y₀<

a

_; ð1þ

s

w0CÞzp; if y0>

a

_: ( ð28Þ (c) Suppose thatya

0 solves(17), there is a unique

x

i02Xwith y0ð

x

i0Þ

a

_and

s

w0C;z w0C is a KKTmultiplier of (17). Then z w0Cð

x

Þ w1; y0<

a

; 1 P_y 0ðxÞ>að1þsw₀CÞzpðxÞ w1Py₀ðxÞ<að1þsw₀CÞ pi₀ ;

x x

i0; ð1þ

s

w0CÞzpð

x

Þ; y0>

a

8 > > > < > > > : ð29Þ and 1 Py0ðxÞ>að1þ

s

w0CÞzpð

x

i0Þ w1 P y0ðxÞ<að1þ

s

w0CÞ pi0 6_ð1þ

s

w0CÞzpð

x

i0Þ; ð30Þ hold. (d) Suppose thatya 0 solves(17)and

s

w0C;z w0C is aKKTmulti plier of (17). Suppose that

q

is coherent and satisfies Assumption 1. If z w0C2Dw0q w1E then ya 0 solves (17) for w0q w1E. Proof

(a) Take the solutiony⁄_{of(17)whose existence is guaranteed by} Theorem 2, and define

a

_{Max y}

x

i

ð Þ:i 1;2;. . .;n

f g:

8_{Notice that(21)is a particular case of(24)that arises ifz}

premains constant and

equalsk.

9_{(22)implies the fulfillment of this property wheneverw}

0= 1. Sincew061, the property holds if the reinsurer draws on the expected value premium principle, since thenzp=k> 1.

Lemma 7implies that

a

⁄_{> 0. Let us see thaty y}a

0. Indeed,

y⁄ _{being (17)feasible we have that y}⁄_6y

0, so

a

⁄6y0(

x

)

whenevery⁄₍

x

_{) =}

a

⁄_{. Besides, if y}⁄₍

x

_{) <}

a

⁄_and

s

w0C;z

w0C

is a KKTmultiplier (its existence follows fromTheorem 2), then the first condition in (25) and Lemma 5 lead to z

w0Cð

x

Þ w1. Hence, the fourth condition in(25), Expression (26)andzp>w1lead toy⁄(

x

) =y0(

x

), and thereforey ya

0.

(b) Ifya

0ð

x

Þ<

a

(ory ⁄₍

x

) <

a

⁄_{) then the equalityz}

w0Cð

x

Þ w1 may be proved with the same arguments. Suppose that ya

0ð

x

Þ

a

_{. Then, consider the partition of Lemma 6and}

obviously

x

2X2or

x

2X3, sinceya

0ð

x

Þ–0. But

x

2X3

would implyy0(

x

) =

a

⁄, which cannot hold.

(c) As in the proof of (b),z

w0Cð

x

Þ w1 whenevery a

0ð

x

Þ<

a

.

Suppose that ya

0ð

x

Þ

a

. Then, consider the partition of

Lemma 6 and obviously

x

2X2 or

x

2X3, since

ya

0ð

x

Þ–0. But

x

2X3implies thaty0(

x

) =

a

⁄, and therefore

x

x

i0. Thus, taking into account(4), we have(29). Finally, (30)comes from(26), because Pya

0 0 0 implies that z w0C6 1þ

s

w0C zp.

(d) It trivially follows fromCorollary 4andRemark 2. h

Remark 3.According to the previous theorem the ‘‘stable optimal retention’’ ofRemark 1is a stop loss reinsuranceya

0.Theorem 8

also provides the multiplierz

w0C(see(28)or(29)), so the condition z

w0C2Dw0q w1Eis very easy to verify in practical examples. Actu

ally, we will see in the next section that the assumptions of state ments 8b and 8c are usually fulfilled in practice. h

Remark 4. Rockafellar et al. (2006)introduced the risk measure CVaRl0;

l

02 ð0;1Þbeing the level of confidence.CVaRl0is becom ing very important and popular among practitioners and research ers for its interesting properties. Indeed, it is coherent and expectation bounded (Rockafellar et al., 2006), and compatible with the second order stochastic dominance and the classical util ity functions (Ogryczak and Ruszczynski, 2002).10_{Rockafellar et al.}

(2006)proved that DCVaRl0 z2L 2_;06z6 1 1

l

0 ;EðzÞ 1 : ð31Þ

Considerw0= 1 (the expected wealth is not optimized by the ceding

company). Thus, if

q

CVaRl₀ in problem(17), thenya0will solve

the problem (i.e.,(31)will contain the random variablez

C) as long as

1 1

l

0

Pz

C; ð32Þ

which clearly holds for

l0

close enough to 100%. Analogously, if the insurance company deals with problem(15)and

q0

CVaRl₀, then

the solutionya

0 of(15)forw0C w1Ewill be still the solution for

thew0CVaRl0 w1Eas long as w0

1

l

0

þw1Pz

w0C;

which is also obvious forw0> 0 and

l

0large enough. An illustrative

numerical example will be given in Section5. h

5. Algorithm and numerical experiment

Next let us point out that the conditions ofTheorem 8usually hold in practice, and the stable optimal retention ya

0 and the

KKT multiplier

s

C;zC

may be easily calculated by drawing on an appropriate algorithm. The algorithm just tests the fulfillment of Theorem 8.

First of all we will introduce the algorithm and then we will present a numerical example. In order to simplify the exposition, in this section we will assume thatw1= 0 (the expected wealth

is not maximized, and only the risk level is minimized), though the extension forw1> 0 is straightforward.

Notice that, according toTheorem 8,ya

0 and

s

C;zC

will be known once we compute

a

⁄_and

s

C,i.e., we only have to estimate

two real numbers.

In order to introduce the algorithm we will assume that

X f

x

1;

x

2;. . .;

x

ng R; ð33Þ

0<

x

1<

x

2< <

x

n ð34Þ andy0is the identity map, so

Pðy0

x

iÞ pi; ð35Þ

i= 1, 2,. . .,n. Actually, this is a particular framework strictly more restricted than that inTheorem 8, but this is a standard simplifica tion in the literature about the optimal reinsurance problem. See, for instance,Gajec and Zagrodny, 2004; Kaluszka, 2005; Cai et al., 2008, and many others, where the authors do not deal with the ori ginal probability spaceðX;F;PÞ, but with its image_ðR;B;P_Þbyy

0,

composed of the real line R, the Borel

r

algebraBofR, and the probability measureP_{given by}

PðBÞ Pð

x

2X;y0ð

x

Þ 2BÞ

for every Borel subsetB2B. In such a particular casey0is replaced

by the identity map. Besides, in practical situations insurers usually deal withðR;B;P_{Þ and the identity map too, which means that}

they do not distinguish different events leading to the same cost of claims. Finally, though the new setting (33)is much more re stricted than the original one, the simplification does not modify the computation of the solution of(17). Indeed,Theorem 8guaran tees that we are looking for a stop loss reinsurance, and there obvi ously exists a one to one mapping between the stop loss contracts of the initial probability spaceðX;F;PÞand the stop loss contracts of its imageðR;B;P_Þ.

Hence, assume(33) (35)and define

a

Max

x

n: Obviously, yaMax

0 y0 is (17)feasible because S1> 0 and

p

(0) = 0.

Due to (22), the premium principle of (24) generates a strictly increasing function

p

.11_{Consequently,}

p

_ðy

0 ya0Þstrictly decreases

as

a

grows. Consider a first case (Case_1) such that

p

(y0)6S1, which

implies thatya

0is(17)feasible for every

a

P0 and therefore we will

consider12

a

Min 0:

If

p

(y0) >S1then the continuity of

a

!

p

ðy0 ya0Þimplies the exis

tence of a unique

a

Min2(0,

a

Max) such that

p

ðy0 y aMin

0 Þ S1:

Let us distinguish two situations.Case_2 arises if

a

MinRX, in which case we will chosei0as the smallest subscript such that

p

ðy0

x

i0Þ<S1:

Case_3 holds if

a

Min

x

i0 12Xfor somei0. Obviously, for the three casesya

0is(17)feasible if and only if

a

Min6

a

6

a

Max:

10_{Recall that the standard deviation is not compatible with the second order}

stochastic dominance if asymmetries are involved (Ogryczak and Ruszczynski, 1999), and the stop-loss reinsurance obviously generates asymmetric results.

11_i.e.,p(y

1) <p(y2) whenevery16y2andy1–y2. 12_{Actually, Constraintp(y}

0y)6S1is redundant in this case, and may be removed in(17).

Algorithm 1.Suppose that Case_1 holds.Lemma 6implies that yaMin₀ does not solve(17), so the stable optimal retentionya

0 satisfies

p

y₀ ya

0

<S1and the second condition in(18)leads to

s

C 0.

Hence, we only have to estimate

a

⁄_. Step1. Define

a

1

x

1 2;

a

2

x

1;

a

3

x

1þ

x

2 2 ;

a

4

x

2;. . .;

a

2n 1

x

n 1þ

x

n 2 ;

a

2n

x

n: Step2. Forj= 1 toncheck whetherya2j1

0 and z 2j 1 0; if

x

<

a

2j 1; zp; if

x

P

a

2j 1;

satisfy(25) and (26). If these conditions are satisfied for someya2j1 0

then we will have the stable optimal retention and theKKTmulti plier. The algorithm can stop since the stable optimal retention has been found.

Notice that two different values ofjcannot satisfy(25) and (26), since(22)implies thatEðz

2j 1Þstrictly decreases withjand there

fore(4)cannot hold two times. Furthermore, if these conditions hold for somejthen every

a

⁄

2 (

a2

j 2,

a2

j) will generate a stop loss stable optimal retentionya

0, since the sameKKTmultipliersz2j 1

and

s

2j 1 0 will still apply.

Step3. Suppose thatStep2 did not lead to the stable optimal retention. Forj= 1 toncheck whetherya2j

0 and z 2jð

x

Þ 0;

x

<

a

2j; 1 P x>a2jzpðxÞ pj ;

x

a

2j; zpð

x

Þ;

x

>

a

2j; 8 > > < > > : ð36Þ

satisfy(25) and (26). Every time these conditions are satisfied we will have a solution of(17)for

q

=C. Notice that(26)will imply 1 Px>a2jzpð

a

2jÞ

pj

6_zpð

a

_{2jÞ: ð37Þ}

Algorithm 2. Suppose thatCase_2 holds. Then proceed as inAlgo rithm 1with minor modifications inSteps1 3. Now, inStep1 we must define

a

2i0 1

a

Minþ

x

i0 2 ;

a

2i0

x

i0;

a

2i0þ1

x

i0þ

x

i0þ1 2 ;. . .;

a

2n 1

x

n 1þ

x

n 2 ;

a

2n

x

n:

Obviously,Steps2 and 3 will start withj=i0rather thanj= 1.

Step4. IfSteps2 and 3 did not lead to the stable optimal reten tion then we must addressStep4 so as to check the opti mality ofyaMin

0 . In this case

s

C>0 may hold and we are in

the conditions ofTheorem 8b. We must verify whether yaMin0 ;

s

Cand z C 0; if y0<

a

Min; 1þ

s

C zp; if y0>

a

Min; ( ð38Þ satisfy(25) and (26)for some

s

CP0. Actually the only condition

one must check is(4),i.e., 1þ

s

C

X

xi>aMin

zpð

x

iÞ 1;

so the optimality ofyaMin0 holds if and only if

s

C 1 P xi>aMinzpð

x

iÞ 1P0: ð39Þ

Thus,Step4 reduces to the verification of the inequality in(39). If this inequality holds then the equality in(39)provides us with

s

C (38) provides us with z

C, and y aMin

0 is the stable optimal

retention. h

Algorithm 3. Suppose thatCase_3 holds. Then proceed as inAlgo rithm 2with a minor modification inStep1. Now we must modify

a2

i0 1according to

a

2i0 1

x

i0 1þ

x

i0 2 ;

a

2i0

x

i0;

a

2i0þ1

x

i0þ

x

i0þ1 2 ;. . .;

a

2n 1

x

n 1þ

x

n 2 ;

a

2n

x

n:

Once again,Step2 andStep3 will start withj=i0.

Step5. We still have to check the optimality ofyaMin

0 y

xi0 1 0 . This

retention level is optimal if and only if we can find

s

CP0 such thatyxi0 1 0 ;

s

Cand z Cð

x

Þ 0;

x

<

x

i0 1; 1 P_x>x i0 1 1þs C ð ÞzpðxÞ pi_{0 1} ;

x

x

i0 1; 1þ

s

C zpð

x

Þ;

x

>

x

i0 1; 8 > > > < > > > : ð40Þ

satisfy(25) and (26). The existence of

s

Cis easy to verify, because,

bearing in mind the findings of Sections3 and 4, one only needs to check the conditions

061 X x>xi_{0 1} 1þ

s

C zpð

x

Þ; ð41Þ 1 Px>xi 0 1 1þ

s

C zp

x

i0 1 pi0 1 6 _1þ

s

C zp

x

i0 1 ð42Þ and 1 X x>xi0 1 1þ

s

C zpð

x

Þ þ 1þ

s

C X x>xi0 1 zpð

x

ÞPð

x

Þ 1: ð43Þ Equality(43)yields

s

C, and then the inequalities(41) and (42)are

equivalent to

s

C6 1 P x>xi0 1zpð

x

Þ 1 ð44Þ and

s

CP 1 zp

x

i0 1 pi0 1þ1 1; ð45Þ

respectively. Thus, Step 4 reduces to the computation of

s

C by

means of(43)and then the verification of the inequalities

s

CP0 (44) and (45). If the three inequalities hold then(40)provides us with the multiplier andyaMin0 is the stable optimal retention. h Remark 5. Notice that the existence of solution of(17)and the findings of Sections 3 and 4show that at least one of the three algorithms must generate a stable optimal retention. h

Remark 6. As said above, notice that the algorithm just tests the fulfillment ofTheorem 8, and consequently it is not very time con suming. Actually, it is a linear time algorithm, in the sense that there is a linear relationship between its computational complexity and the number of realizations of the global costy0. h

Next let us present a simple numerical example. Our only objective is to illustrate how to use the algorithm in practical situations.

Example 1. Suppose thaty0can reach the values 100, 200, 300,

400 and 500 with a similar probability 0.2. Suppose that the reinsurer uses the expected value premium principle with a price 80% higher than the expected claims,i.e.,

p

ðyÞ 1:8EðyÞ:

Suppose finally that the ceding company does not impose any bud get constraint,i.e., we are inCase_1 above. With the notations of Algorithm 1, define

a

1 50;

a

2 100;

a

3 150;

a

4 200;. . .;

a

9 450;

a

10 500:

InStep2 we have to check the optimality of five stop loss con tracts. The first one isy50

0. Consider z 1 0; if

x

<50; 1:8; if

x

P50: Obviously,z

1remains constant and equals 1.8, so it is not in the set

DCof(11). Then,y500 is not a stable optimal retention. If one repeats

the analysis with the four remaining ‘‘candidates’’ then similar re sults apply, so Step2 does not generate any stable optimal retention.

InStep3 we have to check the optimality of the remaining five stop loss contracts. The first one isy100

0 , and(36)gives z 2 0; if

x

<100; 2:2; if

x

100; 1:8; if

x

>100; 8 > < > :

which do not belong toDC. Repeat the exercise with the remaining

values of

a

, and for

a

= 200 we get z 4 0; if

x

<200; 0:4; if

x

200; 1:8; if

x

>200; 8 > < > :

which implies thaty200

0 is not a stable optimal retention either.

Analogously, for

a

= 300 we get

z 6 0; if

x

<300; 1:4; if

x

300; 1:8; if

x

>300 8 > < > : andy300

0 is the ‘‘stable optimal retention’’ we are looking for. It is

easy to check thaty400

0 andy5000 are not stable optimal retentions.

In fact, fory400 0 one obtains z 8 0; if

x

<400; 3:2; if

x

400; 1:8; if

x

>400 8 > < > :

and this multiplier is not feasible because(37)does not hold. An analogous caveat arises fory500

0 .

Reinsurancey300

0 will be the optimal retention for many risk

measures. For instance, if one considers

q

CVaRl0, according to (32)y300

0 solves the problem if

1 1

l

0

P1:8;

which holds for

l

0P0.45 (or

l

0P45%), and, in particular, for the

usual values of this parameter in the industry, which are higher than 90%. Finally, it is worthwhile to point out that the role of the CVaRl₀may be also played by many other important risk measures

in actuarial sciences, such as,WCVaR,DPT, Wang, etc. h

6. Conclusions

The optimal reinsurance problem is a classic topic in actuarial theory. Since coherent and expectation bounded risk measures

are becoming very important in Finance and Insurance, recent ap proaches deal with them so as to address the optimal reinsurance problem. However, there is no consensus about the risk measure that one must use, since every risk measure presents advantages and shortcomings when compared with others.

This article analyzes the ‘‘stability in the large’’ of the optimal reinsurance with respect to the risk measure that the insurer uses. It has been pointed out that there is a ‘‘stable optimal retention’’ that will show no sensitivity, insofar as it will solve the optimal reinsurance problem for many risk measures, providing a very ro bust reinsurance plan. For the expected value premium principle this stable optimal retention is a stop loss contract, and it is easy to compute in practice. A fast linear time algorithm has been given and a numerical example presented. The approach is general en ough. Actually, if desired, the analysis permits us to incorporate both budget constraints and the simultaneous maximization of the ceding company expected wealth.

Acknowledgments

Research partially supported by ‘‘RD_Sistemas SA’’, ‘‘Comunidad

Autónoma de Madrid’’ (Spain), GrantS2009/ESP 1594, and ‘‘MEyC’’

(Spain), GrantECO2009 14,457 C04.

The authors sincerely thank the reviewers for their useful comments.

The usual caveat applies.

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