Reinsurance and Solvency
n a
Claramunt, M.M., Casta˜er, A. and M´rmol, M.
a o
Dept. Matem`tica Econ`mica, Financera i Actuarial, University of Barcelona,
Av. Diagonal, 696, 08034, Barcelona, Spain,
(e-mail: mmclaramunt@ub.edu)
(e-mail: acastaner@ub.edu)
(e-mail: mmarmol@ub.edu)
Abstract. This work is structured in two parts. In the first part we summarize the main results obtained
on the effect of reinsurance strategies on the solvency measures of the insurer. In the second part we
present a threshold proportional reinsurance strategy and we analyze the effect on ruin probability,
time of ruin and deficit at ruin. This dynamic reinsurance strategy assumes a retention level that is not
constant and depends on the level of the surplus. In a model with inter-occurrence times generalized
Erlang(n)-distributed we obtain the integro-differential equation for the Gerber-Shiu function. Then,
we present the solution for inter-occurrence times exponentially distributed and claim amount phase-
type(N ). Some examples for exponential and phase-type(2) claim amount are presented.
Keywords. Gerber-Shiu function, Generalized Erlang(n), Reinsurance strategy, Solvency measures.
1 Introduction to reinsurance
In a reinsurance contract, the insurer transfers a part of the risks to the reinsurer, and obviously,
also, a part of the premiums received from policyholders. Insurance companies can opt for
reinsurance contracts to assume greater risks and protect themselves from ruin.
In the reinsurance contract a retention function, h(X), is defined. This function determines
the amount of risk retained by the insurer. Then X − h(X) is the part of the risk that will be
paid by the reinsurer. The function h(X) satisfies the following properties (Melnikov (2003),
Kaas et al. (2001)):
1. h(X) and X − h(X) are no decreasing functions.
2. 0 ≤ h(X) ≤ X, h(0) = 0.
Basically, one can distinguish two groups of reinsurance: proportional and non-proportional
reinsurance. Proportional reinsurance includes quota-share and surplus. The former transfers
all the risks in the same proportion, whereas in the latter the ratio can vary.
From now on we will focus on quota-share reinsurance, which we call generically proportional
reinsurance. Therefore, it is considered that the retention function is
h(X) = kX,
being k ∈ [0, 1], the retention level of the insurer.
Centeno (1986, 2002) and Dickson and Waters (1996) studied the influence of proportional
reinsurance in ruin probability through the effect on the adjustment coefficient.
In this work, the Sparre Andersen model of risk theory is modified to incorporate a propor-
tional reinsurance contract.
In the Sparre Andersen model, the surplus process, R(t), at a given time t ∈ [0, ∞) is defined
as R (t) = u+ct−S (t), with u = R (0) ≥ 0 being the insurer’s initial surplus, S (t) the aggregate
claims and c the rate at which the premiums are received. S (t) is modeled as a compound process
where N (t), the number of claims occurring until time t, is an ordinary renewal process, and
2 Claramunt et al.
∞
the inter-occurrence times between claims, {Ti }i=1 , are modeled as a sequence of i.i.d. random
variables, where T1 denotes the time until the first claim and Ti , for i > 1, denotes the time
between the (i − 1)th and ith claim. The claims {Xi , i ≥ 1} are i.i.d. random variables with
density function f (x), and common expectation E [X] 0, is the penalty function, so that
φ(u) is the expected discounted penalty payable at ruin.
Depending on w(x, j), we can obtain different interpretations for the Gerber-Shiu function.
In this work we will consider the following possibilities,
φ(u) = E e−δT I (T 0 ⇒ φ(u) = E e−δT j m I (T 0, the ordinary moments of the deficit at ruin if ruin occurs, and
second, for w(x, j) = I (j ≤ y), the distribution function of the deficit at ruin if ruin occurs. For
w(x, j) = 1, we obtain the defective Laplace transform of the time of ruin being δ the parameter,
and then the moments of this important random variable can also be obtained.
2 Effect of the threshold proportional reinsurance strategy on
solvency measures
In this work we introduce a dynamic reinsurance strategy. It is assumed that the insurer agrees to
a contract for proportional reinsurance, where the retention level is not constant but it depends
on the level of the surplus. Thus, we define a threshold proportional reinsurance strategy: a
retention level k1 is applied if the reserves are below a certain threshold b, and a retention level
k2 otherwise. Threshold proportional reinsurance includes standard proportional reinsurance
(k1 = k2 = k) as a particular case, and also the option of not reinsuring (k1 = k2 = k = 1).
Graphically,
Reinsurance and Solvency 3
R(t)
k2 X
c2
b
k2 X
k1 X
u c1
k1 X
0 T1 T2 T4
T3
k
t
k1
k2
0
T1 T3 T4
T2
t
Fig. 1. Threshold Proportional Reinsurance
With this proportional reinsurance strategy, the Gerber-Shiu function behaves differently,
depending on whether its initial surplus u is below or above the level b,
φ1 (u) 0 ≤ u 1, the expressions are more complicated but numerical results
can always be obtained.
References
n a
Casta˜ er, A., Claramunt, M.M. and M´rmol, M. (2010): Deficit at ruin with threshold proportional rein-
surance. Insurance Markets and Companies: Analyses and Actuarial Computations, (To appear).
Centeno, L. (1986): Measuring the effects of reinsurance by the adjustment coefficient. Insurance: Math-
ematics and Economics, 5, 169–182.
Centeno, L. (2002): Measuring the effects of reinsurance by the adjustment coefficient in the Sparre
Anderson model. Insurance: Mathematics and Economics, 30, 37–49.
Dickson, D.C.M. and Waters, H.R. (1996): Reinsurance and ruin. Insurance: Mathematics and Eco-
nomics, 19, 61–80.
Gerber, H.U. and Shiu, E. (1998): On the time value of ruin. North American Actuarial Journal, 2,
48–78.
Gerber, H.U. and Shiu, E. (2005): The time value of ruin in Sparre Andersen model. North American
Actuarial Journal, 9, 49–84.
Kass, R., Goovaerts, M., Dhaene, J. and Denuit, M. (2001): Modern actuarial risk theory. Kluwer
Academic Publishers, Dordrecht.
Melnikov, A. (2003): Risk analysis in finance and insurance. Monographs and Surveys in Pure and
Applied Math. Chapman and Hall, Edmonton, Canada.