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Arthur CHARPENTIER - Optimal reinsurance with ruin probability target
Optimal reinsurance with ruin probability target
Arthur Charpentier
7th International Workshop on Rare Event Simulation, Sept. 2008
http ://blogperso.univ-rennes1.fr/arthur.charpentier/
1

Ruin, solvency and reinsurance
“reinsurance plays an important role in reducing the risk in an insurance
portfolio.”
Goovaerts & Vyncke (2004). Reinsurance Forms in Encyclopedia of Actuarial
Science.
“reinsurance is able to oﬀer additional underwriting capacity for cedants, but also
to reduce the probability of a direct insurer’s ruin .”
Engelmann & Kipp (1995). Reinsurance. in Encyclopaedia of Financial
Engineering and Risk Management.
2

Proportional Reinsurance (Quota-Share)
• claim loss X : αX paid by the cedant, (1 − α)X paid by the reinsurer,
• premium P : αP kept by the cedant, (1 − α)P transfered to the reinsurer,
Nonproportional Reinsurance (Excess-of-Loss)
• claim loss X : min{X, u} paid by the cedant, max{0, X − u} paid by the
reinsurer,
• premium P : Pu kept by the cedant, P − Pu transfered to the reinsurer,
3

Proportional versus nonproportional reinsurance
claim 1 claim 2 claim 3 claim 4 claim 5
reinsurer
cedent
02468101214
Proportional reinsurance (QS)
claim 1 claim 2 claim 3 claim 4 claim 5
reinsurer
cedent
02468101214
Nonproportional reinsurance (XL)
Fig. 1 – Reinsurance mechanism for claims indemnity, proportional versus non-
proportional treaties.
4

Mathematical framework
Classical Cram´er-Lundberg framework :
• claims arrival is driven by an homogeneous Poisson process, Nt ∼ P(λt),
• durations between consecutive arrivals Ti+1 − Ti are independent E(λ),
• claims size X1, · · · , Xn, · · · are i.i.d. non-negative random variables,
independent of claims arrival.
Let Yt =
Nt
i=1
Xi denote the aggregate amount of claims during period [0, t].
5

Premium
The pure premium required over period [0, t] is
πt = E(Yt) = E(Nt)E(X) = λE(X)
π
t.
Note that more general premiums can be considered, e.g.
• safety loading proportional to the pure premium, πt = [1 + λ] · E(Yt),
• safety loading proportional to the variance, πt = E(Yt) + λ · V ar(Yt),
•
safety loading proportional to the standard deviation, πt = E(Yt) + λ · V ar(Yt),
• entropic premium (exponential expected utility) πt =
1
α
log E(eαYt
) ,
• Esscher premium πt =
E(X · eαYt
)
E(eαYt )
,
• Wang distorted premium πt =
∞
0
Φ Φ−1
(P(Yt > x)) + λ dx,
6

A classical solvency problem
Given a ruin probability target, e.g. 0.1%, on a give, time horizon T, ﬁnd capital
u such that,
ψ(T, u) = 1 − P(u + πt ≥ Yt, ∀t ∈ [0, T])
= 1 − P(St ≥ 0∀t ∈ [0, T])
= P(inf{St} < 0) = 0.1%,
where St = u + πt − Yt denotes the insurance company surplus.
7

A classical solvency problem
After reinsurance, the net surplus is then
S
(θ)
t = u + π(θ)
t −
Nt
i=1
X
(θ)
i ,
where π(θ)
= E
N1
i=1
X
(θ)
i and



X
(θ)
i = θXi, θ ∈ [0, 1], for quota share treaties,
X
(θ)
i = min{θ, Xi}, θ > 0, for excess-of-loss treaties.
8

Classical answers : using upper bounds
Instead of targeting a ruin probability level, Centeno (1986) and Chapter 9 in
Dickson (2005) target an upper bound of the ruin probability.
In the case of light tailed claims, let γ denote the “adjustment coefficient”,
defined as the unique positive root of
λ + πγ = λMX(γ), where MX(t) = E(exp(tX)).
The Lundberg inequality states that
0 ≤ ψ(T, u) ≤ ψ(∞, u) ≤ exp[−γu] = ψCL(u).
Gerber (1976) proposed an improvement in the case of finite horizon (T < ∞).
9

Classical answers : using approximations u → ∞
de Vylder (1996) proposed the following approximation, assuming that
E(|X|3
) < ∞,
ψdV (u) ∼
1
1 + γ
exp −
β γ µ
1 + γ
quand u → ∞
where
γ =
2µm3
3m2
2
γ et β =
3m2
m3
.
Beekman (1969) considered
ψB (u)
1
1 + γ
[1 − Γ (u)] quand u → ∞
where Γ is the c.d.f. of the Γ(α, β) distribution
α =
1
1 + γ
1 +
4µm3
3m2
2
− 1 γ et β = 2µγ m2 +
4µm3
3m2
2
− m2 γ
−1
10

R´enyi - see Grandell (2000) - proposed an exponential approximation of the
convoluted distribution function
ψR (u) ∼
1
1 + γ
exp −
2µγu
m2 (1 + γ)
quand u → ∞
In the case of subexponential claims
ψSE (u) ∼
1
γµ
µ −
u
0
F (x) dx
11

CL dV B R SE
Exponential yes yes yes yes no
Gamma yes yes yes yes no
Weibull no yes yes yes β ∈]0, 1[
Lognormal no yes yes yes yes
Pareto no α > 3 α > 3 α > 2 yes
Burr no αγ > 3 αγ > 3 αγ > 2 yes
12

With proportional reinsurance, if 1 − α is the ceding ratio,
S
(α)
t = u + απt −
Nt
i=1
αXi = (1 − α)u + αSt
Reinsurance can always decrease ruin probability.
Assuming that there was ruin (without reinsurance) before time T, if the insurance had
ceded a proportion 1 − α∗
of its business, where
α∗
=
u
u − inf{St, t ∈ [0, T]}
,
there would have been no ruin (at least on the period [0, T]).
α∗
=
u
u − min{St, t ∈ [0, T]}
1(min{St, t ∈ [0, T]} < 0) + 1(min{St, t ∈ [0, T]} ≥ 0),
then
ψ(T, u, α) = ψ(T, u) · P(α∗
≤ α).
13

q
q
0.0 0.2 0.4 0.6 0.8 1.0
−4−2024
Time (one year)
Impact of proportional reinsurance in case of ruin
Fig. 2 – Proportional reinsurance used to decrease ruin probability, the plain line is
the brut surplus, and the dotted line the cedant surplus with a reinsurance treaty.
14

In that case, the algorithm to plot the ruin probability as a function of the reinsurance
share is simply the following
RUIN <- 0; ALPHA <- NA
for(i in 1:Nb.Simul){
T <- rexp(N,lambda); T <- T[cumsum(T)<1]; n <- length(T)
X <- r.claims(n); S <- u+premium*cumsum(T)-cumsum(X)
if(min(S)<0) { RUIN <- RUIN +1
ALPHA <- c(ALPHA,u/(u-min(S))) }
}
rate <- seq(0,1,by=.01); proportion <- rep(NA,length(rate))
for(i in 1:length(rate)){
proportion[i]=sum(ALPHA<rate[i])/length(ALPHA)
}
plot(rate,proportion*RUIN/Nb.Simul)
15

0.0 0.2 0.4 0.6 0.8 1.0
0123456
Cedent's quota share
Ruinprobability(in%)
Pareto claims
Exponential claims
Fig. 3 – Ruin probability as a function of the cedant’s share.
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0.0 0.2 0.4 0.6 0.8 1.0
020406080100
rate
Ruinprobability(w.r.t.nonproportionalcase,in%)
1.05 (tail index of Pareto individual claims)
1.25
1.75
3
Fig. 4 – Ruin probability as a function of the cedant’s share.
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Nonproportional reinsurance (QS)
With nonproportional reinsurance, if d ≥ 0 is the priority of the reinsurance contract,
the surplus process for the company is
S
(d)
t = u + π(d)
t −
Nt
i=1
min{Xi, d} where π(d)
= E(S
(d)
1 ) = E(N1) · E(min{Xi, d}).
Here the problem is that it is possible to have a lot of small claims (smaller than d), and
to have ruin with the reinsurance cover (since p(d)
< p and min{Xi, d} = Xi for all i if
claims are no very large), while there was no ruin without the reinsurance cover (see
Figure 5).
18

q
q
0.0 0.2 0.4 0.6 0.8 1.0
−2−1012345
Time (one year)
Impact of nonproportional reinsurance in case of nonruin
Fig. 5 – Case where nonproportional reinsurance can cause ruin, the plain line is
the brut surplus, and the dotted line the cedant surplus with a reinsurance treaty.
19

Proportional reinsurance (QS), homogeneous Poisson
0 5 10 15 20
0510152025
Deductible of the reinsurance treaty
Fig. 6 – Monte Carlo computation of ruin probabilities, where n = 100, 000 tra-
jectories are generated for each deductible.
20

Proportional reinsurance (QS), nonhomogeneous Poisson
0 5 10 15 20
0510152025
+10%
−10%
21

0 5 10 15 20
0510152025
22

0 5 10 15 20
0510152025
+20%
−20%
23

Références
[1] Asmussen, S. (2000). Ruin Probability. World Scientific Publishing Company.
[2] Beekmann, J.A. (1969). A ruin function approximation. Transactions of the Society
of Actuaries,21, 41-48.
[3] Bühlmann, H. (1970). Mathematical Methods in Risk Theory. Springer-Verlag.
[4] Burnecki, K. Mista, P. & Weron, A. (2005). Ruin Probabilities in Finite and
Infinite Time. in Statistical Tools for Finance and Insurance, C´ızek,P., Härdle, W.
& Weron, R. Eds., 341-380. Springer Verlag.
[5] Centeno, L. (1986). Measuring the Effects of Reinsurance by the Adjustment
Coefficient. Insurance : Mathematics and Economics 5, 169-182.
[6] Dickson, D.C.M. & Waters, H.R. (1996). Reinsurance and ruin. Insurance :
Mathematics and Economics, 19, 1, 61-80.
[7] Dickson, D.C.M. (2005). Reinsurance risk and ruin. Cambridge University Press.
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[8] Engelmann, B. & Kipp, S. (1995). Reinsurance. in Peter Moles (ed.) :
Encyclopaedia of Financial Engineering and Risk Management, New York &
London : Routledge.
[9] Gerber, H.U. (1979). An Introduction to Mathematical Risk Theory. Huebner.
[10] Grandell, J. (1991). Aspects of Risk Theory. Springer Verlag.
[11] Goovaerts, M. & Vyncke, D. (2004). Reinsurance forms. in Encyclopedia of
Actuarial Science, Wiley, Vol. III , 1403-1404.
[12] Kravych, Y. (2001). On existence of insurer’s optimal excess of loss reinsurance
strategy. Proceedings of 32nd ASTIN Colloquium.
[13] de Longueville, P. (1995). Optimal reinsurance from the point of view of the excess
of loss reinsurer under the ﬁnite-time ruin criterion.
[14] de Vylder, F.E. (1996). Advanced Risk Theory. A Self-Contained Introduction.
Editions de l’Universit de Bruxelles and Swiss Association of Actuaries.
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