Innovation methods, algorithms, and software for analysis of reinsurance contracts
A Monte Carlo based approach to evaluate and/or compare the riskiness of reinsurance treaties for both the ceding and the reinsurance companies is introduced. An experimental program system Reinsurance Analyser based on the indicated approach is presented. The program allows analyzing applications...
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irk-123456789-44672009-11-12T12:00:28Z Innovation methods, algorithms, and software for analysis of reinsurance contracts Silvestrov, D. Teugels, J. Masol, V. Malyarenko, A. A Monte Carlo based approach to evaluate and/or compare the riskiness of reinsurance treaties for both the ceding and the reinsurance companies is introduced. An experimental program system Reinsurance Analyser based on the indicated approach is presented. The program allows analyzing applications of a large set of reinsurance contracts under a variety of claim flow models. The effect of applications is compared by risk measures, provided that the parameters of the contracts are balanced by an average reinsurer’s load quantity. 2006 Article Innovation methods, algorithms, and software for analysis of reinsurance contracts / D. Silvestrov, J. Teugels, V. Masol, A. Malyarenko // Theory of Stochastic Processes. — 2006. — Т. 12 (28), № 3-4. — С. 203–238. — Бібліогр.: 23 назв.— англ. 0321-3900 http://dspace.nbuv.gov.ua/handle/123456789/4467 en Інститут математики НАН України |
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A Monte Carlo based approach to evaluate and/or compare the riskiness of reinsurance treaties for both the ceding and the reinsurance companies is introduced. An experimental program system Reinsurance Analyser based on the indicated approach is presented. The program allows analyzing
applications of a large set of reinsurance contracts under a variety of claim flow models. The effect of applications is compared by risk measures, provided that the parameters of the contracts are balanced by an average reinsurer’s load quantity. |
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Silvestrov, D. Teugels, J. Masol, V. Malyarenko, A. |
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Silvestrov, D. Teugels, J. Masol, V. Malyarenko, A. Innovation methods, algorithms, and software for analysis of reinsurance contracts |
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Silvestrov, D. Teugels, J. Masol, V. Malyarenko, A. |
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Innovation methods, algorithms, and software for analysis of reinsurance contracts |
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Innovation methods, algorithms, and software for analysis of reinsurance contracts |
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Innovation methods, algorithms, and software for analysis of reinsurance contracts |
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Innovation methods, algorithms, and software for analysis of reinsurance contracts |
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Innovation methods, algorithms, and software for analysis of reinsurance contracts |
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innovation methods, algorithms, and software for analysis of reinsurance contracts |
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Innovation methods, algorithms, and software for analysis of reinsurance contracts / D. Silvestrov, J. Teugels, V. Masol, A. Malyarenko // Theory of Stochastic Processes. — 2006. — Т. 12 (28), № 3-4. — С. 203–238. — Бібліогр.: 23 назв.— англ. |
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Theory of Stochastic Processes
Vol. 12 (28), no. 3–4, 2006, pp. 203–238
DMITRII SILVESTROV, JOZEF TEUGELS, VIKTORIYA MASOL,
AND ANATOLIY MALYARENKO
INNOVATION METHODS, ALGORITHMS, AND
SOFTWARE FOR ANALYSIS OF REINSURANCE
CONTRACTS1
A Monte Carlo based approach to evaluate and/or compare the riskiness
of reinsurance treaties for both the ceding and the reinsurance companies
is introduced. An experimental program system Reinsurance Analyser
based on the indicated approach is presented. The program allows ana-
lyzing applications of a large set of reinsurance contracts under a variety
of claim flow models. The effect of applications is compared by risk mea-
sures, provided that the parameters of the contracts are balanced by an
average reinsurer’s load quantity.
1. Introduction
The aim of reinsurance is to protect an insurance company, hereafter
called the ceding company or the first line insurer, against large losses caused
either by excessively large claims or by a large number of claims. A reinsur-
ance treaty determines the rules according to which claims are split between
the ceding and the reinsurance companies. One of the questions arising in
connection with reinsurance is which reinsurance treaty, applied to a given
claim flow, provides a less risky position to the reinsurer. It is therefore of
interest to develop methods that compare the riskiness of different treaties
and that select the optimal treaty. By way of procedure we take the point
of view of the reinsurer.
The most common measure for judging the riskiness of a reinsurance
treaty is the variance of the deductible or reinsured claim amount. In this
case, the optimality criterion consists in minimizing the variance while the
mean value of the claim amount under consideration is fixed. Minimization
of the variance of the deductible with respect to the classical reinsurance
treaties such as quota-share, excess-of-loss, stop-loss, surplus, or their com-
binations has been investigated by many authors (see, e.g., Pesonen (1984),
Daykin, Pentikäinen, and Pesonen (1993), Denuit and Vermandele (1998),
1Invited lecture.
2000 Mathematics Subject Classification. Primary 62P05; Secondary 62G32, 91B70.
Key words and phrases. Comparison of reinsurance treaties, large claims, heavy-tailed
distributions, stochastic modelling, program system.
This work is partly supported by the Perturbed Risk Processes, Extremal Processes
and Reinsurance Contracts Project financed by the Royal Swedish Academy of Sciences.
203
204 D. SILVESTROV, J. TEUGELS, V. MASOL, AND A. MALYARENKO
Gajek and Zagrodny (2000), Kaluszka (2001), Verlaak and Beirlant (2003),
Kaluszka (2004), and others). The ceding insurer’s and reinsurer’s optimal
choice of classical reinsurance was derived in Hesselager (1990). However,
this author used the minimization of the probability of ultimate ruin as
optimality criterion.
Not much is known on the optimal choice of a treaty when the set of
feasible treaties contains large claims reinsurance (ECOMOR, largest claims
reinsurance, etc.). In this case, the application of the “mean-variance”
criterion is complicated by the fact that no handy and/or general analytical
expressions exist for the expectation and the variance of the (re-)insured
amount. In 1972, Berliner showed a high degree of similarity between excess-
of-loss and largest claims reinsurance covers in terms of the correlation
coefficient of the reinsured claim amount for the Pareto-Poisson model. By
the latter we understand that the claim epochs follow a Poisson stream
while the claim sizes follow a Pareto distribution. In Kremer (1982), the
asymptotic equivalence of the net risk premium of the largest claims cover
and the net risk premium of the excess-of-loss treaty plus additive term
is established. This result was subsequently generalized in Kremer (1984)
to the case of a generalization of the largest claims cover. Further results
on the optimal choice of a treaty when large claims reinsurance is involved
can be found in Kremer (1990) and Kremer (1991). In the papers referred
to above, the largest claims cover is compared to the excess-of-loss treaty
with the help of either the asymptotic efficiency or the limiting efficiency.
It is shown that, from the insurer’s viewpoint, the excess-of-loss treaty is
better than the largest claims cover with respect to an asymptotic efficiency
measure; however the treaties are equivalent with regard to the limiting
efficiency.
The riskiness of a combination of quota-share and a large claim rein-
surance contract was studied in Ladoucette and Teugels (2006a). For a
survey of the analytical results concerning large claim reinsurance we refer
to Teugels (2003) and Ladoucette and Teugels (2006b).
In this paper we introduce a Monte Carlo based approach to evaluate
and/or compare the riskiness of reinsurance treaties for both the ceding
and the reinsurance companies. However, when making a decision on the
optimality and the expediency of a treaty, we restrict attention to the rein-
surer’s interest. Not only is this the main novelty in our approach, but
often first line insurance can be considered as a form of reinsurance where
the client accepts a certain franchise or deductible. Due to the Monte Carlo
technique the implementation of the indicated approach extends to a large
set of reinsurance treaties. In particular, simulation methods allow to cope
with the mathematical complexity of large claims reinsurance.
Furthermore, we present an experimental software Reinsurance Analyser
and show results of experimental studies made with the help of the pro-
gram. The Reinsurance Analyser becomes an especially handy tool when
ANALYSIS OF REINSURANCE CONTRACTS 205
one compares reinsurance contracts of high mathematical complexity, typ-
ical for large claims reinsurance. Moreover, the use of computer programs
allows us to analyze applications of a reinsurance contract under a variety
of claim flow models.
Throughout the paper we assume that claim flows can be described by the
classical Sparre Andersen risk model where inter-claim times are assumed to
form a renewal process. The compound Poisson model is a Sparre Andersen
model where the inter-claim intervals are exponentially-distributed. The
distribution of claim sizes and the distribution of the inter-claim intervals are
modelled as mixtures of distributions chosen from a broad class containing
both light- and heavy-tailed distributions. Each distribution playing a role
in the mixture has a certain probability but the sum of the probabilities in
anyone mixture is equal to 1. Hence, these probabilities may be considered
as weights of the distributions in the mixture. The mixture approach for
modelling claim flows allows us to take account of claims of different nature
like frequent small, medium claims and rare extreme large claims. More
concretely, light-tailed distributions describing small and medium claims
have higher probabilities in the mixture while heavy-tailed distributions for
the large claims have lower probabilities.
The riskiness of a reinsurance treaty is evaluated on a specified time in-
terval called the evaluation interval by a set of risk measures (e.g., variance,
dispersion, coefficient of variation, value at risk, etc.) The evaluation inter-
val can be determined using different principles. In the present paper we
consider two such types, i.e., claim-type, containing a pre-specified number
of claims, and time-type. The claim amount retained by the first line in-
surer (respectively, reinsurer) within an evaluation interval will be referred
to as the interval deductible (interval reinsured claim amount).
In order to make a fair comparison between two reinsurance treaties by
any feasible risk measure, we need to calibrate them. We therefore choose
the parameters of the treaties in such a way that the average reinsurer’s
quota loads (i.e., the percentage of the interval claim amount covered by the
reinsurer in average) on the same evaluation interval are equal. Moreover
these balanced treaties are to be applied to the same claim flows. Note that
if the first or the second moment of the claim size distribution is infinite,
the same might happen to the corresponding moment of the distribution
of the reinsured amount. This is the case, for example, for excess-of-loss,
largest claims reinsurance treaties. It is therefore sensible to consider ratios
of the corresponding sample estimates of risk measures as they can display
asymptotic stabilization even in the case of heavy tailed distributions.
The approach to evaluate and compare reinsurance treaties as presented
in this paper has been used as a background for the experimental soft-
ware Reinsurance Analyser (ReAn). The program ReAn is written in the
programming language Java with the use of SSJ and JFreeChart class li-
braries. Due to the facilities of Java the software is platform independent
206 D. SILVESTROV, J. TEUGELS, V. MASOL, AND A. MALYARENKO
and is well-equipped for distributed calculations through the Internet. The
description of the program and the results of experimental studies are given
in the paper as well.
The remaining of the paper is structured as follows. The claim flows
modelling and the types of reinsurance contracts are described in Section 2.
Special attention is paid to the extreme value contracts. Section 3 treats
the evaluation of reinsurance treaties. In Sections 4 and 5 we formulate the
problems which can be solved with the help of the Reinsurance Analyser.
The algorithms for solving these problems are also included. In particu-
lar, the approach to compare reinsurance treaties is discussed in Section 5.
The description of the program Reinsurance Analyser is given in Section 6.
Finally, results of experimental studies are presented and commented in
Section 7.
2. Claim Flows and Reinsurance Treaties
Let T1, T2, . . . , be the inter-claim intervals (T1 is the moment when the
first claim arrives). If X1, X2, . . . , is the sequence of corresponding claim
sizes and N(t) = max{n : Tn ≤ t}, t ≥ 0, is the number of claims arrived
to an insurance company up to time t, then the total or aggregate claim
amount over that time period is
X(t) :=
N(t)∑
n=1
Xn, t ≥ 0,
and X(t) = 0 if N(t) = 0.
The moment of the appearance of the n-th claim is given by
(1) Zn = T1 + T2 + . . . + Tn, n = 1, 2, . . . .
Any reinsurance treaty is a rule according to which each individual claim
Xn is split into a “lower” and an “upper” part
Xn = XI
n + XR
n , n = 1, 2, . . . ,
where XI
n is the deductible payed by the first line insurer, and XR
n is the
reinsured amount. Consequently, the aggregate claim X(t) is split as
X(t) = XI(t) + XR(t) =
N(t)∑
n=1
XI
n +
N(t)∑
n=1
XR
n , t ≥ 0.
Figure 1 illustrates a reinsurance arrangement on a claim-by-claim basis.
ANALYSIS OF REINSURANCE CONTRACTS 207
Figure 1. The claim flow split between the ceding insurer and
the reinsurer.
2.1 Sparre Andersen model. We assume that for n = 1, 2, . . . ,
A: (Xn, Tn) are independent identically distributed random vectors with
non-negative components;
B: Xn and Tn are independent random variables which have finite ex-
pectations (but not necessarily finite variances) and distribution
functions F and G respectively.
As mentioned before, N(t) is a renewal process and the claim flow model
is called the Sparre Andersen model (see, e.g., Rolski, Schmidli, Schmidt,
and Teugels (1999)).
The Sparre Andersen model generalizes the classical compound Poisson
model where N(t) is the Poisson process, i.e., G(t) = 1−e−λt, an exponential
distribution function.
2.2 Claim size and inter-claim interval distributions. The distribu-
tion functions F and G are modelled as mixtures of several distributions
F (x) = p1F1(x) + . . . + pmFm(x),
G(t) = q1G1(t) + . . . + qkGk(t),
where p1 + . . . + pm = 1, q1 + . . . + qk = 1; Fi(x), i = 1, 2, . . . , m, and Gj(t),
j = 1, 2, . . . , k, are distribution functions from the following classes:
• Light-tailed distributions (Uniform(a, b), Exponential(λ), Gam-
ma(α ≤ 1, β), Weibull(λ, τ ≥ 1), etc.);
208 D. SILVESTROV, J. TEUGELS, V. MASOL, AND A. MALYARENKO
• Heavy-tailed distributions (Strict Pareto(α), Alternative Pare-
to(α, β), Weibull(λ, τ ≤ 1), Gamma(α ≥ 1, β), Reciprocal Gam-
ma(α, β), etc.);
• Super-heavy tailed distributions (Logarithmic (α), etc.).
The quantities p1, p2, . . . , pm, and q1, q2, . . . , qk will be referred to as the
probabilities or weights assigned to the distributions in the claim size and
inter-claim interval distribution mixtures, respectively. Intuitively, with
probability pi a particular claim size Xn arrives from the claim flow with
claim size distribution Fi, i = 1, 2, . . . , m. Alternatively, the probability pi
may be thought of as the proportion of claim sizes having distribution Fi,
i = 1, 2, . . . , m. Analogous reasonings can be made on a particular inter-
claim interval Tn with probabilities qj and inter-claim distributions given
by Gj , j = 1, 2, . . . , k.
The exponential and the strict Pareto distributions are used as bench-
marks to split distributions into light-, heavy-, and super-heavy tailed:
• if
lim supx→∞
F̄ (x)
e−λx
< ∞ for some λ > 0,
where F̄ (x) = 1− F (x) denotes the tail of the distribution function
F , we call F light-tailed;
• if
lim supx→∞
F̄ (x)
e−λx
> 0 for all λ > 0,
we call F heavy-tailed, and
• if
lim supx→∞
F̄ (x)
x−λ
> 0 for all λ > 0,
we say that F has a super-heavy tail.
According to these definitions, the exponential distribution itself is light-
tailed, the strict Pareto distribution given by F̄ (x) ∼ x−α, α > 0, is heavy-
tailed, and the logarithmic distribution given by F̄ (x) ∼ (ln x)−α, α > 0, is
super-heavy tailed.
The results of simulations presented in Section 7, are obtained for a com-
pound Poisson model where the distribution F of the claim sizes is a mixture
of an exponential F1 and a reciprocal gamma F2 with probabilities p1 > p2.
Recall that a random variable X has a reciprocal (α, β)-gamma distribu-
tion if its density is given by
fX(x) =
1
Γ(α)
·
(
1
x
)α+1
βα exp {−β/x}, x ≥ 0, α > 0, β > 0,
where Γ(α) =
∫ ∞
0
xα−1 exp {−x}dx. Alternatively, Y = 1
X
has a classical
(α, β)-gamma distribution with density
fY (y) =
1
Γ(α)
· yα−1βα exp {−βy}, y ≥ 0.
ANALYSIS OF REINSURANCE CONTRACTS 209
The parameter α is known as the shape parameter while β is the scale
parameter.
The right tail of the reciprocal gamma distribution follows a power law
with exponent −(α + 1). Its expected value is
EX =
β
α − 1
, α > 1,
while the variance is
V arX =
β2
(α − 2)(α − 1)2
, α > 2.
Hence, for 1 < α < 2 the expected value is finite but the variance is not.
2.3 Classical reinsurance treaties. The classical reinsurance arrange-
ments such as quota-share, surplus, excess-of-loss, stop-loss, and their com-
binations are commonly used and well known. We will consider the following
two types in more detail.
(1) An excess-of-loss treaty is determined by a positive number M ≥
0 called the retention level (or the priority). With this type of treaty
the reinsurer pays that part of each claim that exceeds the retention
M ; as such
XR
n = {Xn − M}+, n = 1, 2, . . . ,
where {x}+ = max{x, 0}, x ∈ R. The ceding insurer covers claims
that do not exceed the retention M and pays the amount M when
Xn > M ; more specifically,
XI
n = min{Xn, M}, n = 1, 2, . . . .
In the following we will use the notation excess-of-loss [M] to refer
to this treaty.
(2) In a quota-share treaty, the reinsurer accepts a certain proportion
a of each individual claim, while the cedent retains the remaining
1 − a, 0 < a < 1; therefore
XR
n = a · Xn,
and
XI
n = (1 − a) · Xn.
The parameter a is called the proportionality factor.
210 D. SILVESTROV, J. TEUGELS, V. MASOL, AND A. MALYARENKO
2.4 Extreme value reinsurance. If one is interested in coverage against
excessive claims, then one should look for extreme value treaties based on
the largest claims in the portfolio. Such treaties are largest claims reinsur-
ance, generalized largest claims reinsurance, ECOMOR, drop-down excess-
of-loss, etc. The commonly used classical reinsurance treaties do not involve
largest claims. However, extreme value reinsurance is far less popular. One
of the main reasons is the technical complexity of the dependent order sta-
tistics
X∗
1 ≤ X∗
2 ≤ · · · ≤ X∗
N(t)
of the individual claims X1, X2, . . . , XN(t), that are used to define extreme
value reinsurance treaties.
For example, in a largest claims treaty the reinsurer covers the r,
r = 1, 2, . . . , N(t), entire largest claims from the sequence X1, X2, . . . , XN(t).
With the ECOMOR (excédent du coût moyen relatif) treaty intro-
duced in Thépaut (1950) the reinsurer pays only that part of the r largest
claims X∗
N(t)−r+1, . . . , X
∗
N(t), r = 1, 2, . . . , N(t)−1, that exceeds the (r +1)-
th largest claim X∗
N(t)−r; more specifically,
XR(t) =
r∑
i=1
{
X∗
N(t)−i+1 − X∗
N(t)−r
}
, r = 1, 2, . . . , N(t) − 1,
and XR(t) = 0 for r ≥ N(t). As such, the ECOMOR can be viewed as an
excess-of-loss with a random retention level determined by the (r + 1)-th
largest claim.
Having in mind to tackle the technical complexity of extreme value rein-
surance and simplify its application in practice, we introduce in this paper a
modified approach for determining extreme value treaties. In general, by an
extreme value reinsurance treaty we will understand a reinsurance arrange-
ment on a claim by claim basis according to which the reinsurer’s share XR
n
of an individual claim Xn is defined as a function of Xn and order statistics
of the l previous claims Xn−l, Xn−l+1, . . . , Xn−1, l = 1, 2, . . ., n = 1, 2, . . ..
The order statistics will be called the ordered past sample and denoted by
X∗
n−l ≤ X∗
n−l+1 ≤ · · · ≤ X∗
n−1.
The parameter l will be called the past-sample size. The notation EV[l] will
be used to refer to an extreme value reinsurance treaty satisfying the above
general definition.
The crucial point in the last definition is that the rule for splitting claims
between the cedent and the reinsurer is given for each individual claim and
involves the order statistics from a fixed past claim epoch. At the same
time, it does not change the essence of the extreme value treaties. For
example, the largest claims treaty can be presented in the form
(2) XR
n =
{
0, if Xn < X∗
n−r,
Xn, if Xn ≥ X∗
n−r, r = 1, 2, . . . , l,
ANALYSIS OF REINSURANCE CONTRACTS 211
Figure 2. LC[l, r, c] reinsurance contract: l = 3, r = 2, c = 0.75.
while the ECOMOR treaty introduced in Thépaut (1950) can be presented
as
(3) XR
n =
{
0, if Xn < X∗
n−r,
Xn − X∗
n−r, if Xn ≥ X∗
n−r,
and be still viewed as an excess-of-loss with a random retention, i.e.,
XR
n =
{
Xn − X∗
n−r
}
+
, r = 1, 2, . . . , l.
In this paper we consider a generalization of the contract (2) arranged in
the following way. The reinsurer pays a certain proportion c, 0 ≤ c ≤ 1, of
each individual claim Xn only if this claim overshoots the (r + 1)-th largest
claim among the l claims prior to Xn; more strictly
(4) XR
n =
{
0, if Xn < X∗
n−r,
c · Xn, if Xn ≥ X∗
n−r, r = 1, 2, . . . , l,
where c is called the proportionality factor. Hence, the first line insurer
covers
XI
n =
{
Xn, if Xn < X∗
n−r,
(1 − c) · Xn, if Xn ≥ X∗
n−r, r = 1, 2, . . . , l.
The role played by the continuous parameter c will be discussed in subsec-
tion 4.2.
Hereafter we will use the notation LC[l, r, c] or the prefix LC- to refer to
the largest claims reinsurance treaty defined by (4).
Figure 2 illustrates the application of the LC -contract where the past-
sample size l = 3, the number of the largest claim r = 2, and the propor-
tionality factor c = 0.75.
212 D. SILVESTROV, J. TEUGELS, V. MASOL, AND A. MALYARENKO
Remark 2.1. In analogy to the largest claim reinsurance, the ECOMOR
contract given by (3) can be generalized to an ECOMOR[l, r, c] where the
reinsured amount is equal to
XR
n =
{
0, if Xn < X∗
n−r,
c · (Xn − X∗
n−r
)
, if Xn ≥ X∗
n−r, r = 1, 2, . . . , l.
Remark 2.2. Note that a past sample can be determined not only by a
fixed number l of claims prior to the claim Xn, but also by a time interval
of a fixed length, say H > 0, counted back from the moment Zn, i.e., the
interval (Zn − H, Zn]. In this case one obtains an extreme value contract
with a random past-sample size.
3. Evaluation of Reinsurance Treaties
The effect of a reinsurance treaty is evaluated on certain intervals Im =
(t′m, t′′m], m = 1, 2, . . ., called the evaluation intervals. Evaluation intervals
may be determined in a variety of ways. Two of them, namely claim-
type and time-type, will be considered in detail while others will only be
mentioned as possible alternatives.
3.1 Claim-type evaluation intervals. Claim-type evaluation intervals
are formed in such a way that each of them contains a certain number, say
k = 1, 2, . . ., of successive claims
Im =
(
Z(m−1)·k, Zm·k
]
, m = 1, 2, . . . ,
where Zn is the moment when the n-th claim occurs.
Note that the distribution of the length of a claim-type interval is de-
termined by the inter-claim interval distribution G, and does not therefore
depend on k. Its actual length Zm·k−Z(m−1)·k, m = 1, 2, . . ., can be different
as shown in Figure 3.
3.2 Time-type evaluation intervals. Time-type evaluation intervals are
determined by a certain time period, say h > 0,
Im = ((m − 1) · h, m · h] , m = 1, 2, . . . .
The number of claims in one time-type evaluation interval may be dif-
ferent from the number of claims in another one as seen in Figure 4. This
time it is the distribution of this number that depends on the inter-claim
interval distribution G.
3.3 Money-type evaluation intervals. Money-type evaluation intervals
are determined by the amount of money, say e > 0, that the ceding (or,
reinsurance) company should pay within each interval such that the last
claim in the interval is the one that first overshoots the amount e.
ANALYSIS OF REINSURANCE CONTRACTS 213
Figure 3. Claim-type evaluation intervals: k = 3.
Figure 4. Time-type evaluation intervals: h = 4 (time units).
3.4 The average reinsurer’s quota load and contract characteris-
tics. For any evaluation interval Im = (t′m, t′′m], m = 1, 2, . . ., denote by X̃m
the aggregate interval claim amount
X̃m =
∑
n:t′m<Zn≤t′′m
Xn, m = 1, 2, . . . .
Then the interval reinsured claim amount can be obtained as
X̃R
m =
∑
n:t′m<Zn≤t′′m
XR
n , m = 1, 2, . . . ,
214 D. SILVESTROV, J. TEUGELS, V. MASOL, AND A. MALYARENKO
while the interval deductible equals
X̃I
m = X̃m − X̃R
m, m = 1, 2, . . . .
Denote by M the set of contract characteristics such as
• expectation,
• median,
• 25% and 75% quantiles,
• value at risk,
• variance,
• dispersion,
• coefficient of variation,
• skewness,
• kurtosis, etc.
of the interval deductible and interval reinsured claim amount.
Having a sample of interval deductibles X̃I
1 , . . . , X̃I
N and a sample of inter-
val reinsured amounts X̃R
1 , . . . , X̃R
N (where N is the sample size), we apply
Monte Carlo methods to estimate the characteristics from the set M and the
average reinsurer’s quota load QR. The estimates are then used to evaluate
and analyse reinsurance treaties.
Below, we will consider the reinsurance treaties described in subsec-
tions 2.3 and 2.4, i.e., the excess-of-loss, the quota-share, and the largest
claims treaty defined by (4), when the underlying claim process is of Sparre
Andersen type satisfying conditions (A) and (B) of subsection 2.1.
The average reinsurer’s quota load QR is the percentage of the aggregate
interval claim amount that the reinsurer covers on average. It is calculated
as
QR = P-limN→∞
X̃R
1 + · · ·+ X̃R
N
X̃1 + · · ·+ X̃N
· 100% =
(5) =
P-limN→∞
XR
1 +···+XR
N
N
P-limN→∞X1+···+XN
N
· 100% =
E
∗X̃R
1
E∗X̃1
· 100%,
where E
∗X̃R
1 (E∗X̃1) is the expected value of the reinsured (aggregate) claim
amount on the evaluation interval I1 in the “stationary” regime. If the
evaluation intervals are of claim-type then
(6) E
∗X̃R
1 = k · E
∗XR
1 ;
and if evaluation intervals are of time-type then
(7) E
∗X̃R
1 = h · E
∗XR
1
ET1
.
Thus, in both cases
(8) QR =
E
∗XR
1
E∗X1
· 100%.
ANALYSIS OF REINSURANCE CONTRACTS 215
Note that for the reinsurance contracts with simple structure where the
reinsurer’s share of a claim is defined by the size of this claim only (e.g.,
excess-of-loss, quota-share), E
∗XR
1 = EXR
1 and E
∗X1 = EX1 in formulas
(5)–(8). In this case formulas (5)–(8) can be proved with the use of classical
results of renewal theory, given, for example, in Feller (1966) or in Sil-
vestrov (1980a). Indeed, under these treaties, the sequences (X1, X2, . . .)
and
(
XR
1 , XR
2 , . . .
)
are sequences of independent and identically distrib-
uted random variables. However, under an LC-contract, the random se-
quence
(
XR
1 , XR
2 , . . .
)
belongs to the class of so-called sequences of (l + 1)-
dependent random variables. The last model can also be considered as a
particular case of a Markov renewal process, since the random sequence
X̄ l
n = (Xn−l, Xn−l+1, . . . , Xn−1, Xn) is a homogeneous Markov chain. The
generalization of the law of large numbers for such models as given, for in-
stance in Shurenkov (1989) or in Silvestrov (1980b), can be applied to prove
the relations (5)–(8) in this case as well.
An additional theoretical problem arises when the claim size distribution
mixture contains heavy-tailed distributions with infinite variance. In this
case, one can observe a “masking” effect of quasi-stability of estimates
X̃R
1 + · · ·+ X̃R
N
X̃1 + · · ·+ X̃N
of the quantity QR for large values of N . It turns out that the estimates
of QR are not stable and may have sudden jumps due to the appearance of
outliers in the simulated samples. This effect can appear when the claim
size distribution is modelled by a mixture
F = p1F1 + p2F2,
where F1 is a light-tailed distribution, F2 is a heavy-tailed distribution,
p1 � p2, while
p1 · m1 ≈ p2 · m2,
where mi denotes the mean value of the distribution function Fi, i = 1, 2.
In view of the above discussion, the estimate N̂ of the sample size N that
is sufficient to provide a reliable estimate for the average reinsurer’s quota
load QR does require additional theoretical investigation, especially for the
EV[l]-type contracts and LC-contracts.
It is also of interest to estimate the distribution functions of the inter-
val deductible and the interval reinsured amount. One can plot histograms
and logarithmic histograms of the distributions in question with the help
of the Reinsurance Analyser. The (ordinary) histograms of the interval de-
ductible and the reinsured claim amount are based on samples X̃I
1 , . . . , X̃
I
N
and X̃R
1 , . . . , X̃R
N , respectively. The log-histograms are plotted on the ba-
sis of the samples obtained from the initial data by the transformation
X̃ ′ = ln
(
X̃ + 1
)
. A log-histogram turns out to be more illustrative in
216 D. SILVESTROV, J. TEUGELS, V. MASOL, AND A. MALYARENKO
cases where the underlying distributions have a large or even infinite vari-
ance or/and are very skewed.
4. Direct and Inverse Problems
As mentioned in the Introduction, the prime use of the Reinsurance
Analyser is to compare reinsurance treaties. In addition, the program can
be used as a tool for solving the following two problems.
4.1 Problem 1 (Direct: estimation of the contract characteristics).
Given:
(a) the claim size and inter-claim interval distribution functions F and
G respectively,
(b) the reinsurance treaty R and its parameters,
(c) the sample size N and the type of evaluation intervals Im, m =
1, 2, . . . , N ,
estimate the average reinsurer’s quota load QR and contract characteristics
from the set M.
The solution algorithm is based on Monte Carlo simulations and consists
of the following steps
(A) simulate samples X̃R
1 , . . . , X̃R
N , and X̃I
1 , . . . , X̃I
N of the chosen sample
size N , namely,
(A1) simulate T1, T2, . . ., and X1, X2, . . . from the given claim flow
distribution,
(A2) split claim flow into deductibles XI
1 , XI
2 , . . ., and reinsured amo-
unts XR
1 , XR
2 , . . . according to the rules of the reinsurance treaty
R,
(A3) form samples X̃R
1 , . . . , X̃R
N , and X̃I
1 , . . . , X̃
I
N according to the
type of the evaluation intervals,
(B) estimate the average reinsurer’s quota load QR,
(C) estimate contract characteristics from the set M,
(D) plot (logarithmic) histograms of the interval deductible and the rein-
sured claim amount.
4.2 Problem 2 (Inverse: estimation of the contract parameters
and contract characteristics). Given:
(a) the claim size and inter-claim interval distribution functions F and
G respectively,
(b) the type of reinsurance treaty R,
(c) the value Q for the average reinsurer’s quota load,
(d) the sample size N and the type of evaluation intervals Im, m =
1, 2, . . . , N ,
ANALYSIS OF REINSURANCE CONTRACTS 217
find the parameter(s) of the reinsurance treaty R such that the average
reinsurer’s quota load QR is equal to the given value Q, and then estimate
characteristics of the reinsurance treaty R composing the set M.
The algorithm for solving Problem 2 can be divided into two parts
I: find the parameter(s) of the chosen treaty R such that QR = Q,
II: estimate characteristics from the set M with already known contract
parameters.
Part II repeats the algorithm for solving Problem 1 described above. Part
I is non-trivial and requires closer consideration. The main idea is to use a
dichotomy procedure to estimate the parameter(s) of the given treaty R.
In general, if a treaty has two or more parameters it is supposed that
only one is free while the others are fixed.
Let p denote the parameter to be estimated.
(i) Suppose, first, that p takes on non-negative real values p ≥ 0 (for
example, the excess-of-loss treaty with retention level M ≥ 0). In this case
Part I of the algorithm consists of the following steps:
(I.1) set p equal to an initial value p0;
(I.2) simulate samples X̃R
1 , . . . , X̃R
N , and X̃I
1 , . . . , X̃
I
N with respect to the
treaty R[p0];
(I.3) estimate the average reinsurer’s quota load QR[p0];
(I.4) compare QR[p0] to Q;
if |QR[p0] − Q| < ε, where ε is the admissible error, then proceed
with the treaty evaluation (Part II); otherwise
(I.5) find the next approximation to p using the dichotomy procedure and
repeat steps (I.2)–(I.4).
Figure 5 gives a graphical illustration to the estimation procedure for the
retention level M of the excess-of-loss treaty.
(ii) Next, suppose that the parameter p is a non-negative integer. In this
case, the inverse problem to estimate p given the value Q does not always
have a solution. To circumvent this shortcoming, an additional continuous
parameter has to be introduced such that the essence of the treaty is still
kept.
For example, the largest claims cover defined by (2) has an integer pa-
rameter r taking values r = 1, 2, . . .. The larger the value of r, the larger
the reinsurer’s quota load QR. When solving the inverse problem, it may
happen that the equality QR = Q can never be reached: for some integer
value r of r the reinsurer’s quota load QR[r] is less than Q, QR[r] < Q, while
for the next integer value of r holds QR[r+1] > Q (see, also, Figure 6(a)).
However, the introduction of a new continuous parameter c and generaliza-
tion of the treaty as given by (4) make the inverse problem solvable for any
given value Q and keep the essence of the initial treaty as well.
On the other hand, the appearance of a new parameter implies that the
inverse problem may have more than one solution.
218 D. SILVESTROV, J. TEUGELS, V. MASOL, AND A. MALYARENKO
Figure 5. Estimation of the retention level M of the Excess-of-loss [M].
For the LC-treaty, we suggest two alternative solving procedures to cope
with the disadvantage caused by the non-uniqueness of the solution. Let
the past-sample size l be fixed, then either
1) given the value of r, estimate the proportionality factor c following
(I.1)–(I.5) for p = c in which case there exists either a unique or no
solution,
or
2) find the value r of r such that QLC[l,r−1,1] < Q ≤ QLC[l,r,1], and
estimate the corresponding value c(r) of c such that the equality
QLC[l,r,c(r)] = Q holds.
Both alternatives are illustrated in Figure 6(b).
5. Comparison of Reinsurance Treaties
The Reinsurance Analyser allows us to compare the riskiness of reinsur-
ance treaties by a set of risk measures, i.e., the value-at-risk and variance-
based contract characteristics from the set M. It is sensible to consider
the variance and functions of both the variance and the mean value. The
latter takes into account both characteristics while the former completely
overlooks the expectation.
The conditions of fair comparison of a pair of reinsurance contracts as-
sume
C1: that the contracts are balanced by the average reinsurer’s quota load,
i.e., their parameters are chosen or estimated such that the average
reinsurer’s quota loads of both contracts are equal;
C2: the contracts are applied to the same simulated sample;
C3: the same evaluation intervals are used.
ANALYSIS OF REINSURANCE CONTRACTS 219
(a) Proportionality factor c = 1.
(b) Proportionality factor c, 0 ≤ c ≤ 1.
Figure 6. LC[l, r, c] with the fixed past sample size l.
On the basis of the above, let us compare R1 and R2, two different rein-
surance treaties.
Given:
(a) the claim size and inter-claim interval distribution functions F and
G,
(b) the value Q for the average reinsurer’s quota load,
(c) the sample size N and the type of evaluation intervals Im, m =
1, 2, . . . , N ,
then the treaties R1 and R2 are compared in a solution algorithm that
consists of the following steps:
220 D. SILVESTROV, J. TEUGELS, V. MASOL, AND A. MALYARENKO
(A) estimate the parameters of treaties R1 and R2 such that QR1 =
QR2 = Q,
(B) simulate samples X̃Ri
1 , . . . , X̃Ri
N , and X̃Ii
1 , . . . , X̃Ii
N , i = 1, 2, of the
chosen sample size N ,
(C) estimate characteristics from the set M for both the R1 and R2
contracts,
(D) plot (logarithmic) histograms of the interval deductibles and the
reinsured claim amounts,
(E) compare the contracts R1 and R2 by (C) and (D).
Since we allow claim flows to contain excessively large claims, we model
claim sizes using heavy-tailed distributions with an infinite second or, even
an infinite first moment. This might in turn imply the infiniteness of the
corresponding moment of the distribution of the interval reinsured claim
amount. This is the case for reinsurance arrangements like excess-of-loss,
largest claims, etc. In order to compare a pair of treaties in this case, we
consider the ratios of the estimates of the corresponding risk measure. For
example, we might take the ratio of the sample variances
(9)
SR1
N
SR2
N
,
where SRi
N is the sample variance of the interval reinsured claim amount with
respect to the contract Ri, i = 1, 2, based on a sample size N . As pointed out
in subsection 3.4, the sequences
(
XR1
1 , XR1
2 , . . .
)
and
(
XR2
1 , XR2
2 , . . .
)
used
in the reinsurance contracts with a simple structure (such as excess-of-loss,
quota-share, etc.) are sequences of independent and identically distributed
random variables. If the underlying claim size distribution F has a finite
variance then the quantity (9) converges weakly to a constant that equals
to the ratio of real variances
(10) P-limN→∞
SR1
N
SR2
N
=
V ar∗X̃R1
V ar∗X̃R2
,
where V ar∗X̃Ri, i = 1, 2, is the (finite) variance of the interval reinsured
claim amount with respect to the contract Ri, i = 1, 2.
However, the problem becomes non-trivial when the underlying claim size
distribution F is heavy-tailed with infinite variance. As was shown in Al-
brecher and Teugels (2006), even in this case one can observe a stabilization
of the ratios of type (10) based on the same simulated sample. The fact
that the ratios of type (10) are stable can be used for the comparison of
reinsurance contracts even when the claim size distribution is heavy-tailed.
In the general case one requires a deeper theoretical approach, especially
for the case of the EV[l]-type reinsurance contracts.
ANALYSIS OF REINSURANCE CONTRACTS 221
6. Description of the Program Reinsurance Analyser
In this section we describe the experimental program system Reinsurance
Analyser (ReAn) and give some advice on how to work with it.
The prime use of the ReAn is to compare the riskiness of reinsurance
contracts. Also, it can be used as a tool to estimate
1) the average reinsurer’s quota load QR for the given treaty (Problem
1, Direct);
2) one or several parameters of the reinsurance treaties, given the av-
erage reinsurer’s quota load QR (Problem 2, Inverse).
The approach used in the program is based on global stochastic modelling
of various flows of claims described in Section 2 and involves the method
for evaluating reinsurance treaties presented in Section 3.
The input information required is
• sample size;
• claim size and inter-claim interval distributions;
• type of evaluation intervals;
• type(s) of reinsurance treaty(-ies);
• parameters of the treaty or the average reinsurer’s quota load;
• premium schemes (optional).
The output information is then
• an estimate of the average reinsurer’s quota load or estimates of the
parameters of the reinsurance treaties;
• estimates of characteristics from the set M;
• (logarithmic) histograms of the distributions of the interval deduc-
tible and the reinsured claim amount, etc.
6.1 Program characteristics and user’s interface. As mentioned in the
Introduction, the program is written in the programming language Java with
the use of SSJ and JFreeChart class libraries. Thanks to the flexibility of
Java the software is platform independent and well-equipped for distributed
calculations through the Internet.
The ReAn has a standard Swing graphical user’s interface (see, e.g., Fig-
ure 7). The main interactive element of the program is Contract window.
The Contract window consists of two panels, the upper tabbed panel and
the lower common panel. There are four tabs on the upper panel called,
subsequently, Claim Flow, Contract 1, Contract 2, and Functional
Characteristics. The lower panel contains information and facilities such
as money and time units, sample size, Start button and execution progress
bar, which are common to all four tabs in the Contract window. The panel
is called the Common panel and remains accessible independently of the
currently activated tab.
222 D. SILVESTROV, J. TEUGELS, V. MASOL, AND A. MALYARENKO
6.2 Claim flow tab. Claim size and inter-claim interval distributions are
described on the Claim flow tab. Recall that each of these two distributions
is modelled as a mixture of several distributions, where each entry in the
mixture has a certain probability. The mixtures are formed by the user in,
respectively, Claim size distribution and Inter-claim interval distribution
tables on the Claim flow tab.
Initially both tables are empty. To add a new distribution, the user in-
vokes Distribution club dialog by clicking Add... button and picks out a
distribution from the list, as shown in Figure 7. For convenience, compre-
hensive information about each distribution is available in the program’s
Help System. The distributions listed in the dialog are divided into three
groups according to their tails: light, heavy and super-heavy. Option Data-
base should be chosen if one has a database of real-life past claims and
would like to model a claim flow with the same distribution as the flow in
the database.
Remark 6.1. In the Distribution Club dialog, the items marked by black
colour in the list are already realised in the program while the items marked
by grey are reserved for future development.
Add... and Delete buttons, and Save/Change combo box serve for editing
already formed mixtures. Using them, one can add or delete distributions,
change their parameters or weights (provided that the sum of the weights
in one mixture remains equal 1). The load and total mean are automatically
calculated by the program, the former as the product of the mean value and
the corresponding assigned probability (weight), and the latter as the sum
of loads in one mixture (see Figure 7).
6.3 Contract 1 tab. The Contract 1 tab contains information about the
first reinsurance treaty, the type(s) of evaluation intervals, the average rein-
surer’s quota load and the cedent’s and reinsurer’s premium calculation
schemes. It further displays some results of simulations. The user decides
here which of the three problems formulated in Sections 4–5 he would like
to solve with the help of the program.
Contract 1 tab contains information that is classified as input or output
depending on the problem chosen by the user. In the subsequent subsec-
tions, we will consider the peculiarities of each of the three cases. However,
in any case the following data should be specified on the Contract 1 tab by
the user:
(A) type of reinsurance treaty;
(B) type of evaluation intervals;
(C) premium calculation principles with safety coefficients (optional).
A. The type of reinsurance treaty is picked out from the list of available
treaties in the Contract Club Dialog (see Figure 8). Treaties in the list are
grouped according to their structure, namely:
1) proportional reinsurance;
ANALYSIS OF REINSURANCE CONTRACTS 223
Figure 7. Distribution Club Dialog. Claim flow tab.
2) non-proportional reinsurance;
3) combination of treaties.
For convenience, the rule for calculating the reinsurer’s share is given as a
tool-tip near the name of the corresponding reinsurance contract.
Remark 6.2. As for the Distribution Club, the contracts marked by grey
colour in the Contract Club dialog are reserved for future development.
B. In order to specify the type of evaluation intervals, the user acti-
vates the proper check box and defines the corresponding parameter (for
example, the number of claims k for claim-type evaluation intervals) on
the Insurer’s/Reinsurer’s quota by panel. Note that the time parameter h,
defined in subsection 3.2, and the money parameter e, defined in subsec-
tion 3.3, are measured in time and money units, respectively, displayed on
the Common panel.
C. The user has the additional options to involve premiums in the analysis
and comparison of the contracts. In order to exclude them from the con-
sideration, the Ignore premiums check box should be activated. Otherwise,
one defines premium calculation principles along with the safety coefficients
for both ceding and reinsurance companies. The list of available premium
calculation principles contains
1) expected value principle
P = (1 + a)EX;
224 D. SILVESTROV, J. TEUGELS, V. MASOL, AND A. MALYARENKO
Figure 8. Contract Club Dialog. Contract 1 tab.
2) mean value principle
P = (EX2 + V arX)1/2;
3) standard deviation principle
P = EX + aDX;
4) variance principle
P = EX + aV arX;
5) modified variance principle
P = EX + a
V arX
EX
,
where P is the premium, a, 0 ≤ a ≤ 1, is the safety coefficient. Further, X
denotes either the total claim amount, in the case of the ceding company’s
premium, or the reinsured amount, in the case of the reinsurer’s premium.
Also, the user can extend the list of available contract characteristics by
four additional quantiles (see Figure 8).
6.4 Problem 1. Assume one would like to solve Problem 1 (Direct) for-
mulated in subsection 4.1 using the ReAn program. Then, in addition to
(A)–(C), one defines the parameter(s) of the chosen reinsurance treaty and
switches the indicator Single contract/ Compare contracts to the first posi-
tion Single contract.
ANALYSIS OF REINSURANCE CONTRACTS 225
Note that the ReAn program allows to solve Problem 1 for several types
of evaluation intervals at the same time. To realise this option, the user
just has to activate all the proper check boxes on the Insurer’s/Reinsurer’s
quota by panel and define the parameters of the chosen types of evaluation
intervals.
6.5 Problem 2. Suppose now that Problem 2 (Inverse) formulated in sub-
section 4.2 has to be solved. Then, in addition to (A)–(C), the user defines
the average reinsurer’s quota load QR in the text field under the check box
corresponding to the type of evaluation intervals specified earlier (e.g., in
the text field R1 if claim-type evaluation intervals are chosen), and activates
the check box next to the treaty’s parameter to be estimated, as shown in
Figure 9(a). The indicator Single contract/ Compare contracts should be
switched to position Single contract. In contrast to the direct problem,
only one type of evaluation intervals can be chosen when solving the inverse
problem.
In general, only one contract parameter can be estimated at a time. How-
ever, for the LC-treaty, the user can choose to estimate either one (propor-
tionality factor c) or two (the number of largest claims r and proportionality
factor c) parameters. Both alternatives were described in detail in subsec-
tion 4.2. Here we only recall that the program finds the smallest possible
value of r in the latter case. In order to realise the second alternative the
user activates two proper check boxes as shown in Figure 9(b).
6.6 Contract 2 tab. When two reinsurance treaties have to be compared
by means of the ReAn, one of them is described on the Contract 1 tab
and the other is described on the Contract 2 tab. The Contract 2 tab has
the same interface as the Contract 1 tab, and it contains the same kind
of information as the Contract 1 tab but for the other treaty. The only
difference between these two tabs is that the data common for both treaties
(e.g., the average reinsurer’s quota load) can be defined and edited only on
the Contract 1 tab.
6.8 Problem 3. Comparison of treaties. In order to compare the risk-
iness of two reinsurance treaties, the user defines their types on the Con-
tract 1 and Contract 2 tab respectively. Further, he specifies the average
reinsurer’s quota load and type of evaluation intervals, and switches the
indicator Single contract/ Compare contracts to position Compare contracts
as shown in Figure 10. Recall, that to provide a fair comparison, the type
of evaluation intervals and the average reinsurer’s quota load should be the
same for both treaties. Thus, this data, defined on the Contract 1 tab, is
transmitted automatically to the Contract 2 tab without a possibility to be
edited there (compare Figure 10 and Figure 11). As an alternative, one can
define parameters of the first treaty instead of the reinsurer’s quota load.
Then the latter will be estimated by the program.
226 D. SILVESTROV, J. TEUGELS, V. MASOL, AND A. MALYARENKO
(a) Estimation of the proportionality factor c, given the past-sample size
l = 100 and the number of largest claims r = 8.
(b) Estimation of the number of largest claims r and the proportionality factor
c, given the past-sample size l = 100.
Figure 9. Inverse problem. Estimation of the parameter(s) of
the LC[l, r, c], given the average reinsurer’s quota load QR = 45%.
6.8 Program output. In order to start calculations, the user clicks Start
button. The estimates of the contract(s) characteristics are displayed in the
Contract characteristics and Risk measures table(s). The (log)histograms
of the distribution of the interval deductible and reinsured claim amount
can be obtained by clicking on (LogHist...) Hist... buttons.
In case the comparison problem has been solved, the user can invoke Com-
pare Dialog by choosing command Compare in the Tools Menu as shown in
Figure 11. The Compare Dialog displays the estimates of the risk measures
for both contracts along with the ratios of the corresponding estimates.
6.9 Functional Characteristics tab (optional). The results of the func-
tioning of both ceding and reinsurance companies are represented on the
Functional Characteristics tab. As input data the tab contains the initial
capital and ruin level for both ceding and reinsurance company, and the
time horizon T . Also, the premium calculation principles for either of the
companies can be changed here, and it will in turn imply the change of the
corresponding premium scheme defined on the Contract 1 tab. The out-
put information displayed on the Functional Characteristics tab is the ruin
probability of ceding and reinsurance company before time horizon T . This
tab is reserved for future development.
ANALYSIS OF REINSURANCE CONTRACTS 227
Figure 10. Comparison of LC[l, r, c] and Excess-of-loss [M].
Reinsurer’s quota load QR = 35%. Contract 1: LC[100, 7, c], c to
be estimated.
7. Experimental Studies
In this section we give some illustrations of the estimation of the average
reinsurer’s quota load, parameters and characteristics of reinsurance con-
tracts. We also present the results of the experimental studies made with
the help of the ReAn program on the comparison of the riskiness of a pair of
reinsurance contracts. In particular, we compare the classical excess-of-loss
cover to the largest claims reinsurance arrangement defined in (4), both
contracts being applied to claim flows modelled by means of heavy-tailed
distributions.
In the simulation experiments presented below, the sample size is N =
105 when the inverse problem for the largest claims reinsurance contract
is not involved and N = 106 otherwise, a monetary unit equal EUR 1000,
and a time unit equal 1 day. Evaluation intervals are formed by k = 100
claims, Im = (Z100(m−1), Z100m], m = 1, 2, . . . , N ; neither the cedent’s nor
the reinsurer’s premiums are involved.
228 D. SILVESTROV, J. TEUGELS, V. MASOL, AND A. MALYARENKO
Figure 11. Comparison of LC[l, r, c] and Excess-of-loss [M].
Reinsurer’s quota load QR = 35%. Contract 2: Excess-of-loss [M],
M to be estimated.
Under such conditions, it takes on an ordinary computer (e.g., Intel Pen-
tium 4, CPU 2.80 GHz, RAM 512 MB) about 10–20 seconds to solve Prob-
lem 1 and 1–2 minutes to solve Problem 2 for any reinsurance contract.
Solution of the comparison problem takes from 1 to 4 minutes.
The claim flows are modelled as follows (see, also, Figure 7):
(a) Inter-claim intervals T1, T2, . . . are independent and exponentially
distributed with parameter λ1 = 30, i.e., the expected length of an
inter-claim interval equals ET1 = ET2 = . . . = 1
30
≈ 0.033 of a day.
(b) Claim sizes X1, X2, . . . are independent random variables, and their
distribution is given as a mixture of an exponential with parameter
λ2 = 0.05 and a reciprocal gamma with parameters α = 1.4 and
β = 32. The light-tailed exponential distribution used for modelling
small and middle claims has the higher weight 0.8, while the heavy-
tailed reciprocal gamma used for modelling excessive claims has the
lower weight 0.2. The values of the parameters were selected such
that both distributions give the same load.
(c) (Tn, Xn), n = 1, 2, . . ., are independent random vectors.
ANALYSIS OF REINSURANCE CONTRACTS 229
Figure 12. Estimation of QR for Excess-of-loss [40].
Remark 7.1. The inter-claim interval distribution is not involved in the
modelling of the claim flows if the evaluation intervals are of claim-type.
Remark 7.2. The claim size model used in the simulations reflects a real-
istic situation when the majority, 80%, of claims arriving to the insurance
company are often claims of small or middle size, while the remaining 20%
are large.
7.1 Estimation of the average reinsurer’s quota load. Figure 12
shows the results of the evaluation of the excess-of-loss contract with re-
tention level M = 40. In particular, the estimated value of the reinsurer’s
quota load is 38.2%. The logarithmic histogram of the interval reinsured
claim amount is shown in Figure 13.
In Figures 14 and 15, the reader can see the effect of applying the LC[100;
7; 1] under the same conditions as the excess-of-loss cover in the previous
example. In particular, the estimated value of the reinsurer’s quota load is
42.8%.
Note that with both excess-of-loss and LC-treaty many claims are entirely
retained by the ceding insurer (in the examples given here, it is about 13%
and 48% respectively). As a consequence, both histograms of the interval
reinsured claim amount have an outlier at zero. In Figure 15(b), it is shown
230 D. SILVESTROV, J. TEUGELS, V. MASOL, AND A. MALYARENKO
Figure 13. Log-histogram of the interval reinsured claim
amount under the Excess-of-loss [40].
Figure 14. Estimation of QR load for LC[100; 7; 1].
ANALYSIS OF REINSURANCE CONTRACTS 231
(a) Including reinsurer’s zero-shares. (b) Excluding reinsurer’s zero-shares.
Figure 15. Log-histograms of the interval reinsured claim
amount under the LC[100, 7, 1].
how the same histogram for the LC-treaty would look after the exclusion of
the reinsurer’s zero shares from the sample, used for plotting the histogram.
7.2 Estimation of the parameters of reinsurance treaties. In this
subsection we present results of the estimation of the retention level M of
an excess-of-loss treaty, and the proportionality factor c of an LC[l, r, c],
given the average reinsurer’s quota load QR = 45%.
As seen in Figure 16, the estimated retention level of the excess-of-loss
equals M = 31.3 which is less than the retention level M = 40 providing
QR = 38.2% in the example given in Figure 12.
Figure 17 shows that the average reinsurer’s quota load QR = 45% can
also be reached by applying an LC[100, 8, 0.986] where the number of largest
claims r = 8 has been defined by the user, and the proportionality factor
c = 0.986 has been estimated by the program.
Note that the average reinsurer’s quota load QR = 45% cannot be at-
tained by applying an LC[l, r, c] with c = 1 and an integer value of r. This
confirms the sensibility of having an additional continuous parameter for
the largest claims contract (see also discussion in subsection 4.2).
In spite of the fact that both the excess-of-loss and largest claims treaties
provide an equal average reinsurer’s quota load in the examples above, they
are still not comparable. As was pointed out in Section 5, both treaties
should be applied to exactly the same claim flow. The corresponding ex-
amples are given in the next subsection.
7.3 Comparison of the riskiness of reinsurance treaties. In Fig-
ures 18–21 the reader can see the results of the comparison of the LC-
contract and excess-of-loss. The estimates of the proportionality factor c
232 D. SILVESTROV, J. TEUGELS, V. MASOL, AND A. MALYARENKO
Figure 16. Estimation of the retention level M of the Excess-
of-loss [M], given the average reinsurer’s quota load QR = 45%.
and contract characteristics of the LC[l, r, c] are given in Figure 18, and the
estimates of the retention level and contract characteristics of the excess-
of-loss are shown in Figure 19.
The estimates of the risk measures of both treaties and the ratios of the
corresponding estimates are displayed in the Compare Dialog shown in Fig-
ure 20. One can observe the difference in 0.065 units between the estimates
of the expected value of the interval reinsured claim amount obtained under
the LC-contract and the excess-of-loss. The difference is caused by the fact
that the admissible precision of the estimate was chosen to be five hundredth
percent of the expectation of the total interval claim.
Logarithmic histograms of the interval reinsured claim amount under the
LC-contract and the excess-of-loss are given in Figures 21(a) and 21(b) re-
spectively. (Note that both figures display the histograms after the exclusion
of the outliers at zero.)
If we compare the histograms, we see that in spite of some common fea-
tures, they are essentially different in shape and structure. The similarities
are that each distribution is very skewed; the (highest) peak is situated be-
tween 60% and 65% quantile values while the expectation is lying on the
right-hand side from the (highest) peak between 75% and 85% quantiles.
ANALYSIS OF REINSURANCE CONTRACTS 233
Figure 17. Estimation of the proportionality factor c of the
LC[100, 8, c], given the average reinsurer’s quota load QR = 45%.
At the same time, the distribution obtained under the LC-contract is
much more asymmetric. In particular, the left slope of the histogram is ex-
tremely steep so that its left foot is situated far from zero (one can see a gap
between the ordinate axis and the histogram), while the histogram based
on the excess-of-loss is symmetric in the neighbourhood of its peak and
its left slope is smoothly approaching zero. Furthermore, in contrast with
the excess-of-loss, the distribution regarding the LC-contract is bimodal,
indicating the underlying claim size mixture, and the expectation is lying
between the peak values.
The experimental studies on the comparison of a pair of reinsurance con-
tracts by means of the ReAn program showed that, despite the infinite vari-
ance of the claim size distribution, the ratio of the estimate of the variance-
based risk measure of one contract to the estimate of the corresponding
variance-based risk measure of the other is stable.
This result suggests that the ratios mentioned above can still be used
for the comparison of the riskiness of reinsurance contracts. An example
of such a comparison made for the excess-of-loss and the largest claims
reinsurance contract defined by (4) was given above. As seen in Figure 20,
the ratios, denoted by R1/R2, of the estimates of all risk measures of the
234 D. SILVESTROV, J. TEUGELS, V. MASOL, AND A. MALYARENKO
Figure 18. Comparison of LC[l, r, c] and Excess-of-loss [M].
Reinsurer’s quota load QR = 35%. Contract 1: LC[100; 7; 0.805],
c = 0.805 is estimated.
reinsured amount are less than 1. Hence, from the reinsurer’s point of view,
the application of the LC-contract to the given claim flows is less risky than
the application of the excess-of-loss.
8. Conclusions and prospective developments
I. In the present paper we have proposed an approach to analyse and
compare the riskiness of different reinsurance treaties. The key idea is to
use Monte Carlo simulations to calibrate the treaties so that their reinsurer’s
quota loads are equal, and to estimate the ratios of the corresponding risk
measures of the treaties.
The approach based on Monte Carlo methods allows to cope with the
mathematical complexity of the largest claims, ECOMOR, and other ex-
treme value reinsurance contracts which makes the last more attractive and
accessible to the practitioners.
II. The experimental software Reinsurance Analyser based on the ap-
proach indicated above can fruitfully be used for experimental studies on
the application of deferent reinsurance contracts to a variety of claims flows.
As such, the program may be appreciated as a flexible and handy tool for
comparing the riskiness of reinsurance contracts in the interactive regime.
ANALYSIS OF REINSURANCE CONTRACTS 235
Figure 19. Comparison of LC[l, r, c] and Excess-of-loss [M].
Reinsurer’s quota load QR = 35%. Contract 2: Excess-of-
loss [48.9], M = 48.9 is estimated.
Figure 20. Comparison of LC[l, r, c] and Excess-of-loss [M].
The Compare Dialog.
It can be used by actuaries and experts in reinsurance business for analysis
of treaties of high complexity.
III. Experimental studies performed by means of ReAn showed that the
ratios of the estimates of the variance-based risk measures are stable even
in the case when the variance of the underlying distribution is infinite. It
allowed us to use the ratios in question for the comparison of the pair of
236 D. SILVESTROV, J. TEUGELS, V. MASOL, AND A. MALYARENKO
(a) LC-contract. (b) Excess-of-loss.
Figure 21. Comparison of LC[l, r, c] and Excess-of-loss [M].
Log-histograms.
reinsurance treaties. In particular, it was established that the LC-contract
may be less risky to the reinsurer than the excess-of-loss when the contracts
are applied to the claim process with infinite variance.
In conclusion, we would like to outline possible directions in which further
research may develop.
(i) The asymptotic stability of the estimates of QR-type characteristics
regarding the EV[l]-type reinsurance contracts needs to be inves-
tigated. The case when the underlying claim size and inter-claim
interval distributions are heavy-tailed is of special interest.
(ii) The stability of the ratios of the risk measure estimates, established
in the experimental studies, requires a deeper theoretical investi-
gation in case of heavy-tailed claim size distribution with infinite
second moment.
(iii) The influence of the parameters of heavy-tailed distributions in-
volved for modelling claim flows on the stability of the estimate of
the average reinsurer’s quota load QR needs further analysis. In par-
ticular, theoretical results on the simulation sample size N needed
to acquire a given relative precision for the estimate pointed out
above should be derived. A facility for regulating the lower bound
of the sample size (defined by the user in the ReAn program) will
subsequently be added.
(iv) More complicated claim flow models allowing the dependence be-
tween claim sizes and inter-claim intervals can be investigated. The
Reinsurance Analyser will be implemented with algorithms for mod-
elling such claim flows. Furthermore, an algorithm for modelling
ANALYSIS OF REINSURANCE CONTRACTS 237
claim flows by means of parametric re-sampling from real-life histor-
ical data needs to be elaborated and implemented into the program.
(v) Alternative approaches for balancing and comparing reinsurance
treaties can be studied.
(vi) The lists of reinsurance treaties needs to be extended and new risk
measures (e.g., inter-quartile range, reduction effect, etc.) have to be
introduced. Finally, the full-scale development of the ReAn program
including a well-developed Help system, is envisaged in the future.
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Department of Mathematics and Physics, Mälardalen University, Box
883, SE-721 23 Väster̊as, Sweden
E-mail address: dmitrii.silvestrov@mdh.se
Katholieke Universiteit Leuven, B-3001 Leuven (Heverlee), Belgium
E-mail address: jef.teugels@wis.kuleuven.be
Department of Mathematics and Physics, Mälardalen University, Box
883, SE-721 23 Väster̊as, Sweden
E-mail address: vicamasol@bigmir.net
Department of Mathematics and Physics, Mälardalen University, Box
883, SE-721 23 Väster̊as, Sweden
E-mail address: anatoliy.malyarenko@mdh.se
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