Insurer’s optimal reinsurance strategies
Lesław Gajek, Dariusz Zagrodny
∗Institute of Mathematics, Technical University of Łód´z, AL. Politechniki 11, 90-924 Łód´z, Poland
Received May 1999; accepted December 1999
Abstract
The problem of finding an optimal insurer’s strategy of purchasing reinsurance is considered under the standard deviation calculation principles. It is assumed that the strategy must satisfy several kinds of constraints. The optimal reinsurance contract is found out. © 2000 Elsevier Science B.V. All rights reserved.
MSC: 90A46, 62P05; 62C25, 49J55
Keywords: Principles of premium calculation; Optimal reinsurance
1. Introduction
LetY denote the total claim in a given time period andP be an amount of money which the insurer is ready to spend on the reinsurance. We shall assume thatY is a nonnegative random variable defined on a given probability space(,S,Pr). A reinsurer offers to cover a random quantityRofY, which is a measurable function ofY for a premiumPr. So the total claimYwould split into a part paid by the reinsurer,R(Y ), and the one paid by the insurer,
sayR(Y )˜ , in such a way thatY = ˜R(Y )+R(Y ). The premium paid by the insurer for the reinsurance arrangement is calculated according to the standard deviation principle (with a safety loading parameterβ > 0) and we will consider the set of all plausible reinsurance arrangements whose reinsurance premium is less than or equal toP, i.e. all the arrangements such that
P ≥Pr =ER(Y )+βDR(Y ), (1)
where DR(Y )=√VarR(Y ). Throughout the paper we assume that EY2 <∞, otherwise the premium for rein-surance would be infinite except for the caseβ =0, corresponding to the pure risk premium, which is of least interest to us. As pointed out by many authors (see, e.g. Bühlmann, 1970; Daykin et al., 1993; Deprez and Gerber, 1985) the standard deviation premium calculation principle (1) takes into account the variability of the reinsurer’s share so it is less problematic in practice than, e.g. pure risk premium. On the other hand, the insurer may still ask a question as to how to choose the reinsurance ruleRin a convenient way. Assume that the insurer is interested in minimizing the variance of the retained risk, measured by the variance ofR˜, in his own share. Additionally, the insurer is assumed to admit only those reinsurance arrangements which satisfy the following condition:
0≤ ˜R(Y )≤Y, (2a)
∗Corresponding author.
or equivalently,
0≤R(Y )≤Y. (2b)
Condition (2a) means that the insurer’s share is never larger than the total claim amount ((2a) is assumed, e.g. when finding the optimal reinsurance arrangement under the pure risk premium; see Daykin et al., 1993, p. 191). So the problem of finding the optimal insurer’s strategy of reinsuring relies on minimizing Var(Y−R(Y ))≡VarR˜over the space of all measurable functionsR:R+→R, whereR+=[0,∞), subject to the constraints (1) and (2b).
In Section 2 we prove that this minimization problem has a solutionR1of the form
R1(y)=
0 ifY ≤M≡change loss point,
(1−r)(Y −M) else, (3)
wherer(0≤ r < 1)andMdepend onP,β and truncated moments ofY. Since the rule is somewhat similar to the stop loss reinsurance we shall call it change loss reinsurance. Our optimality proof is based on the Lagrange multipliers method and uses Gâteaux derivatives of the variance and constraint functionals. A numerical example is provided to illustrate the advantage of using change loss reinsurance over the quota share reinsurance rule.
Another interesting result which concerns optimal purchasing of reinsurance was proven by Deprez and Gerber (1985). They called a reinsurance arrangementR optimal if it maximized the expectation of a given (risk averse) utility function, without imposing any restrictions onR, however. Strictly speaking, Deprez and Gerber (1985) considered a nonconstrained optimization problem with a target function different than ours.
2. Optimality of the change loss reinsurance contract
LetFdenote the distribution function of a total claim. We may assume that ER2(Y ) <∞(otherwise the premium
P could not be finite because of (1)) and let us denote the set of all measurable functionsR :R+ →Rsatisfying
this condition byR. ClearlyRis a Hilbert space with the normkRk =pER2(Y ). Given a fixed premiumP >0
and the safety loading coefficientβ >0, let us define functionalsg:R→RandV :R→Rby
g(R)=ER(Y )+βD(R(Y ))−P , (4)
V (R)=Var(Y−R(Y )). (5)
Both functionalsgandV are convex (see Deprez and Gerber, 1985) and Gâteaux differentiable. We recall that the Gâteaux derivative of a mappingQ:R→R(hereRandRstand for the Hilbert space defined above and for the set of real numbers, respectively) atR ∈Ris any linear continuous functionalx∗from the dual ofRsuch that
lim
t→0t
−1(Q(R
+tH)−Q(R))= hx∗, Hi
for anyH ∈ R. In the sequel we write∇Q(R)(H )instead ofhx∗, Hi. It is not difficult to check that for every
R1, H ∈R, we have
∇V (R1)(H )= −2E(Y−R1(Y ))H (Y )+2E(Y−R1(Y ))EH (Y ), (6)
and by the chain rule
∇g(R1)(H )=EH (Y )+β
ER1(Y )H (Y )−ER1(Y )EH (Y )
DR1(Y ) , (7)
whenever DR1(Y ) >0 andV,gare as in (4) and (5), respectively. Let us define a subsetC⊆Rby C= {R∈R|0≤R(y)≤ya.e.[F] onR+}.
Our goal is to minimizeV (·)over sets
M≤= {R∈R|g(R)≤0 andR∈C},
and
M== {R∈R|g(R)=0 andR∈C},
respectively. These are nonlinear constrained optimization problems (the former one is convex) which we shall solve using a Lagrangian function
Lλ(R)=V (R)+λg(R)+9C(R),
whereλis a Lagrange multiplier and9C:R→R∪ {+∞}is defined by
9C(R)=
0 ifR∈C,
+∞ otherwise.
Under suitable assumptions minimizingV over the setM≤is equivalent to a nonconstrained minimization ofLλ
with some parameterλ≥0. This follows from the Karush–Kuhn–Tucker theorem (see, e.g. Peressini et al., 1988 or Ioffe and Tikhomirov, 1974). Since we are interested only in a sufficient condition for constrained optimality, the following simple lemma will be useful.
Lemma 2.1. LetR1∈Candλ≥0 be such that (i)λg(R1)=0,
(ii)Lλ(R)≥Lλ(R1)for everyR∈R.
ThenR1minimizes V over the setM≤= {R∈R|g(R)≤0 andR ∈C}. It also minimizes V overM=whenever
g(R1)=0 (since it minimizes V overM≤). Proof. Let us notice that for everyR∈R
V (R1)=Lλ(R1)≤Lλ(R)
due to (i) and (ii). IfR∈Candg(R)≤0,Lλ(R)≤V (R)soV (R1)≤V (R).
Remark 2.2. It should be pointed out at this moment that the necessary condition for the existence of minimum over M=does not guarantee thatλis nonnegative (see, e.g. Ioffe and Tikhomirov, 1974). However if we can find a realλ
for which (ii) holds, then of courseR1becomes the minimizer overM=, sinceLλ(R)=V (R)onM=in this case.
When we are dealing with the convex problem (we want to minimize V overM≤), then the Karush–Kuhn–Tucker
theorem (the necessary part) guarantees the existence of a nonnegativeλsatisfying (i) and (ii) of Lemma 2.1.
Though Lemma 2.1 seems to be very simple, the main difficulty when applying it relies on findingλ ≥ 0 for which (ii) holds. In order to preserve the existence of such lambdas we assume that the following conditions are satisfied:
EY −M− Z
[M,∞)
(y−M)dF + r β
s Z
[M,∞)
(y−M)2dF − Z
[M,∞)
(y−M)dF 2
=0, (8)
(1−r)
Z
[M,∞)
(y−M)dF +β s
Z
[M,∞)
(y−M)2dF − Z
[M,∞)
(y−M)dF 2
=P . (9)
solvability of (8) and (9). What is more,P ≥EY +βDY means that the insurer is ready, and has enough money, to transfer the whole portfolio to the reinsurer, which is the least interesting situation from both theoretical and practical points of view.
Lemma 2.3. Assume thatP >0,βDY >0 and
EY +βDY > P . (10)
Then there exist real numbersM >0 andr∈[0,1)satisfying Eqs. (8) and (9).
Proof. Let us notice that the set
A≡
because otherwise DY would be 0, a contradiction. We derive from (8) that
r=β
by the Cauchy–Schwartz inequality, and for allM > M1there must hold equalities in (13) by the definition ofM1.
This implies however that eitherF has one atom not smaller thanM1, a contradiction, or Pr(Y > M1)=0. In the
latter case
and again the first factor in the right-hand side of (12) tends to−∞asM ր M1. Eventually, there must exist
M ∈(0, M1)such thath(M)=P becauseh(·)is continuous on [0, M1). For this veryMthe correspondingris
smaller than 1 by the definition ofh(M)and (11).
Theorem 2.4. Assume thatP >0 andβDY >0. LetM≥0 andr∈[0,1)be such numbers that
is the optimal reinsurance arrangement for the insurer under restrictions (1) and (2b) (i.e. it gives the minimum value of V over the setM≤). The function V attains also its minimum overM=atR1, sinceP =ER1+βDR1, by (15).
Proof. First let us observe that Pr(Y > M) >0 because otherwise (15) leads to a contradiction. Now let us define
λ= 2r
We shall check if (i) of Lemma 2.1 holds. By (15),
g(R1)=(1−r)
in an obvious way). By the definition ofR1,
R(y)−R1(y)≥0 for all 0≤y < M. (17)
SinceR1∈Candλ≥0,Lλis differentiable atR1in the direction of any pointR ∈Cwith the directional derivative
equal to [∇V (R1)+λ∇g(R1)](R−R1), where∇V (R1)and∇g(R1)denote the Gâteaux derivatives ofV andg,
respectively. Keep in mind here thatR [0,∞)R
2
1(y)dF > ( R
[0,∞)R1(y)dF )
2because otherwise Pr(Y
≥M)=1 and consequently Pr(Y =const.)=1, which contradicts the assumption. It follows from (6) and (7) that
Lλ(R) −Lλ(R1)≥ ∇V (R1)(R−R1)+λ∇g(R1)(R−R1)
where the last equality holds because of (14) and (16). Now by (17) and (18) we eventually get
Lλ(R)−Lλ(R1)≥0 for allR∈R,
so the assumption (ii) of Lemma 2.1 is satisfied and the proof is complete.
Remark 2.5. Let us notice that the solutions M andβof Eqs. (14) and (15) depend onβin such a way that M is
bounded whenβ →0. Indeed, under assumptions of Theorem 2.4 ifM→ ∞, the left-hand side of (15) would tend
to 0, a contradiction. But then it follows from (14) thatr→0 asβ→0, which implies thatR1, defined by (3), tends
to a stop loss reinsurance contract. It is known that stop loss is an optimal reinsurance arrangement under the pure risk premium calculation (see Daykin et al., 1993) so our solution coincides with this special case optimality result.
Remark 2.6. For a givenβ >0 the optimal strategy could be treated as a combination of the quota share and stop
M is reinsured proportionally (i.e. the reinsurer company reimburses a constant percentage of the claim above M, exactly(1−r)(Y −M)).
Remark 2.7. A careful verification of the proof of Theorem 2.4 shows that we can admitr =0. In this case we
haveλ = 0 (see (16)). However, ifλ = 0, then Lλ = V on C, thus the explicit form of (ii) in Lemma 2.1 is:
V (R)≥V (R1)for everyR∈M≤. Hence getting (ii) we prove directly thatR1is a minimizer.
LetPbe fixed. Below we compare the optimal strategies described in Theorem 2.4 with the quote share reinsurance arrangement. Let us define the reinsurance risk reduction function,φ (R), as follows:
φ (R)=DY −D(Y −R(Y )).
It is interesting to see how many timesφ (R1)is larger thanφ (R2), where R1 the optimal reinsurance contract described in (3), whileR2is the quota share reinsurance rule. Of course the larger theφ (R1)/φ (R2), the betterR1
is compared withR2. For the quote share reinsurance we have ˜
R2(Y )=αY, 0≤α≤1,
andR2(Y )=(1−α)Y. Thus DR˜2(Y )=αDY,which gives φ (R2)=DY−D(R˜2(Y ))=(1−α)DY.
Now using (1) let us calculateα(we assume here that the whole sum of money is spent on the reinsurance for this type of reinsurance strategy). We have
P =ER2(Y )+βDR2(Y )=(1−α)(EY +βDY ).
Hence
1−α= P
EY +βDY, (19)
which implies
φ (R2)=
PDY
EY +βDY = P β
DY
DY +EY /β. (20)
One may ask what is the acceptable value ofβfor both the insurer and the reinsurer. We do not pretend to answer this important question here but assume for a while that both accept the probability 0.95 that the amountR(Y )of the total claimY (which is transferred to the reinsurer) is not greater than the premium amount. Using the normal law approximation, we getβ =1.64.
Example 2.8. Assume thatY has a normal distribution with EY =109USD and DY =108USD. Takeβ =1.64 andP =2.91×108USD. ThenM =8.506×108USD andr =0.052 are the solutions to (8) and (9). Hence ER1˜ =8.556×108USD and DR1˜ =0.125×108USD and therefore
φ (R1)=0.875×108USD.
Paying the same premiumP =2.91×108USD the insurer can buy a quota share contractR2withα=0.75
(keep in mind (19)). Then by (20) the risk reduction function amounts to
φ (R2)=0.250×108USD.
References
Bühlmann, H., 1970. Mathematical Methods in Risk Theory. Springer, New York.
Daykin, C.D., Pentikäinen, T., Pesonen, M., 1993. Practical Risk Theory for Actuaries. Chapman & Hall, London.
Deprez, O., Gerber, H., 1985. On convex principles of premium calculation. Insurance: Mathematics and Economics 4, 179–189. Ioffe, A.D., Tikhomirov, V.M., 1974. Theory of Extremal Problems. Nauka, Moscow (in Russian).
