Insurance: Mathematics and Economics 27 2000 105–112
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
Let Y denote the total claim in a given time period and P be an amount of money which the insurer is ready to spend on the reinsurance. We shall assume that Y is a nonnegative random variable defined on a given probability
space , S, Pr. A reinsurer offers to cover a random quantity R of Y , which is a measurable function of Y for a premium P
r
. So the total claim Y would split into a part paid by the reinsurer, RY , and the one paid by the insurer, say ˜
RY , in such a way that Y = ˜
RY + RY . The premium paid by the insurer for the reinsurance arrangement
is calculated according to the standard deviation principle with a safety loading parameter β 0 and we will consider the set of all plausible reinsurance arrangements whose reinsurance premium is less than or equal to P ,
i.e. all the arrangements such that
P ≥ P
r
= ERY + βDRY , 1
where DRY =
√ VarRY . Throughout the paper we assume that EY
2
∞, otherwise the premium for rein- surance would be infinite except for the case β
= 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 rule R in a convenient way. Assume that the insurer is interested in minimizing the variance of the retained risk, measured by the variance of ˜
R, in his own share. Additionally, the insurer is assumed to admit only those reinsurance arrangements which satisfy the following condition:
≤ ˜ RY
≤ Y, 2a
∗
Corresponding author. 0167-668700 – see front matter © 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 1 6 7 - 6 6 8 7 9 9 0 0 0 6 3 - 3
106 L. Gajek, D. Zagrodny Insurance: Mathematics and Economics 27 2000 105–112
or equivalently, ≤ RY ≤ 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 VarY
− RY ≡ Var ˜ R over
the space of all measurable functions R : R
+
→ R, where R
+
= [0, ∞, subject to the constraints 1 and 2b. In Section 2 we prove that this minimization problem has a solution R
1
of the form R
1
y =
if Y ≤ M ≡ change loss point,
1 − rY − M else,
3 where r0
≤ r 1 and M depend on P , β and truncated moments of Y . 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 arrangement R optimal if it maximized the expectation of a given risk averse
utility function, without imposing any restrictions on R, 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
