OPTIMIZATION OF SOME RISK MEASURES IN STOP-LOSS REINSURANCE
WITH MULTIPLE RETENTION LEVELS
SILVIA DEDU
In this paper we propose a new stop-loss reinsurance model with multiple reten- tion levels. We determine the analytical form of the stop-loss reinsurance premium with multiple retention levels and derive the Value-at-Risk and Conditional Tail Expectation measures corresponding to the aggregate loss. We build optimiza- tion problems based on minimizing some risk measures in stop-loss reinsurance with multiple retention levels and study the existence of the optimal solution.
We provide necessary and sufficient conditions for the existence of the optimal retention.
AMS 2010 Subject Classification: 62G32, 65C50, 90B50.
Key words: optimization, risk measure, stop-loss reinsurance, retention, Value-at- Risk, Conditional Tail Expectation.
1. INTRODUCTION
Recently the study of various optimization problems with applications in insurance has received a special interest among researchers: Cai and Tan [1], Dedu and Ciumara [2], Krokhmal, Palmquist and Uryasev [5], Preda and Ciumara [8]. Reinsurance is a risk management technique used by an insurance company to protect itself against the risk of losses, transferring the risk to a second insurance carrier. The former part is called the cedent or the insurer and the latter is the reinsurer. For a fixed reinsurance premium, stop-loss is the optimal solution among a wide arrays of reinsurance, in the sense that it gives the smallest variance of the insurer’s retained risk. The stop-loss agreement states that reinsurer will pay the cedent’s losses to the extent that losses exceed a specified amount, called retention. The problem of finding the optimal retention plays an important role in reinsurance.
The paper is organized as follows. In Section 2 we introduce the stop- loss reinsurance model with multiple retention levels and derive the stop-loss reinsurance premium corresponding to this model. In Section 3 we provide the representation for the survival function of the loss random variable in
MATH. REPORTS14(64),2 (2012), 131–139
a stop-loss reinsurance with multiple retention levels. We derive the Value- at-Risk (VaR) measure corresponding to the total cost of the insurer in a stop-loss reinsurance with multiple retention levels. We define a VaR-based optimization problem for this model and we find necessary and sufficient con- ditions for the existence of the optimal solution. In Section 4 we derive the Conditional Tail Expectation (CTE) measure corresponding to the total cost of the insurer in a stop-loss reinsurance with multiple retention levels. We define a CTE-based optimization problem for this model and find necessary and sufficient conditions for the existence of the optimal solution. Section 5 summarizes the conclusions.
2. THE STOP-LOSS REINSURANCE MODEL WITH MULTIPLE RETENTION LEVELS
We will model the aggregate loss of an insurance portfolio using a nonne- gative random variable X, with cumulative distribution functionFX, survival functionSX and mean E(X).We assume that X has a one-to-one continuous distribution function on [0,∞).LetXI andXR be, respectively, the loss ran- dom variables corresponding to the insurer and to the reinsurer in the presence of a stop-loss reinsurance model. Under stop-loss agreement with multiple re- tention levels, the reinsurer pays the part of X that exceeds the retention limit, while the insurer is effectively protected from a potential large loss by limiting the liability to the retention levels. Let di >0,i= 1, n,n∈N,n≥2 be the retention levels, 0< di < di+1,i= 1, n−1. The random variables XI, XR and X can be expressed as
XI =
⎧⎨
⎩
X, X≤d1,
di, di ≤X < di+1, i= 1, n−1, dn, X > dn,
and
XR=
⎧⎨
⎩
0, X≤d1,
X−di, di ≤X < di+1, i= 1, n−1, X−dn, X > dn.
Theorem 2.1. The stop-loss reinsurance premium with multiple reten- tion levels has the representation
(2.1) π(d1, d2, . . . , dn) = ∞
d1
SX(x)dx−n−1
i=1
(di+1−di)SX(di+1).
Proof. Using the mean value principle, we have π(d1, d2, . . . , dn) =E(XR) =
=
n−1
i=1
di+1
di
(x−di) dFX(x) + ∞
dn
(x−dn) dFX(x) =
=−
n−1
i=1
di+1
di
(x−di) dSX(x)− ∞
dn
(x−dn) dSX(x) =
=−n−1
i=1
(x−di)SX(x)di+1
di −(x−dn)SX(x)di+1
di
+
n−1
i=1
di+1
di
SX(x)dx+
+ ∞
dn
SX(x)dx= ∞
d1
SX(x)dx−
n−1
i=1
(di+1−di)SX(di+1).
Corollary 2.1.The total stop-loss reinsurance premium with multiple retention, levels can be expressed as
(2.2) δ(d1, d2, . . . , dn) = (1 +ρ) ∞
d1
SX(x)dx−
n−1
i=1
(di+1−di)SX(di+1) , where ρ >0 represents the safety loading coefficient.
3. OPTIMAL RETENTION USING VALUE-AT-RISK MEASURE
First, we recall the definiton of Value-at-Risk measure. LetX be a ran- dom variable with cumulative distribution function FX and survival function SX.Letα∈(0,1).
Definition 3.1. The α-Value-at-Risk (α-VaR) corresponding to the ran- dom variable X and the probability levelα is defined by
VaRX(α) = inf{x∈R|SX(x)≤α}.
Remark 3.1. If X has a one-to-one continuous distribution function on [0,∞),then VaRX(α) is the unique solution of the equation
VaRX(α) =SX−1(α).
We model the total cost of the insurer in a stop-loss reinsurance with multiple retention levels using a random variable T. The total costT has two components: the loss random variableXI corresponding to the insurer and the reinsurance premium δ(d1, d2, . . . , dn):
T =XI+δ(d1, d2, . . . , dn).
Let us denote by VaRT(d1, d2, . . . , dn, α) the Value-at-Risk of the to- tal cost of the insurer, corresponding to the retention levels (di)i=1,n and to the probability level α. Our goal is to determine the optimal retention levels d∗1, d∗2, . . . , d∗nthat minimize the measure VaRT(d1, d2, . . . , dn, α), which means solving the optimization problem
(3.1) VaRT(d∗1, d∗2, . . . , d∗n, α) = min
0<d1<d2<...<dn{VaRT(d1, d2, . . . , dn, α)}.
Next, we will derive the analytical form of the VaRT(d1, d2, . . . , dn, α) measure corresponding to the total cost of the insurer.
Proposition 3.1. The survival function corresponding to the loss ran- dom variable of the reinsurer in a stop-loss reinsurance with multiple retention levels can be expressed as
(3.2) SXI(x) =
SX(x) 0≤x < d1, 0 x≥d1.
Theorem 3.1.The Value-at-Risk measure corresponding to the total cost of the insurer in a stop-loss reinsurance with multiple retention levels has the representation
VaRT(d1, . . . , dn, α) = (3.3)
=d1+ (1 +ρ) ∞
d1
SX(x)dx−n−1
i=1
(di+1−di)SX(di+1) for 0< d1≤SX−1(α) and, respectively,
VaRT(d1, . . . , dn, α) = (3.4)
=SX−1(α) + (1 +ρ) ∞
d1
SX(x)dx−n−1
i=1
(di+1−di)SX(di+1) for d1 > SX−1(α).
Proof. Since the total costT of the insurer isT =XI+δ(d1, d2, . . . , dn), we have
(3.5) VaRT(d1, d2, . . . , dn, α) = VaRXI(d1, d2, . . . , dn, α) +δ(d1, d2, . . . , dn). For every x ≥ d1 we have SXI(x) = 0. Consequently, if SXI(d1) ≥ α > 0 or d1 ≤ SX−1(α), then for every x < d1 we have SXI(x) > SXI(d1) ≥ α and it follows VaRXI(d1, d2, . . . , dn, α) =d1.
IfSX(d1) < αor d1 > SX−1(α), then for every x < d1 we haveSXI(x) = SX(x) and, consequently, VaRXI(d1, d2, . . . , dn, α) = SX−1(α). We have ob- tained
(3.6) VaRXI(d1, d2, . . . , dn, α) =
d1 0< d1 ≤SX−1(α), SX−1(α) d1 > SX−1(α). Using (3.5), (3.6) and (2.2) the conclusion follows.
Theorem 3.2. The solution of the optimization problem (4.1) exists if and only if the system
(3.7)
⎧⎪
⎪⎪
⎪⎨
⎪⎪
⎪⎪
⎩
1−(1 +ρ)SX(d1)−fX(d1) = 0;
−SX(di) +fX(di) +SX(di+1) = 0, i= 2, n−1;
−SX(dn) +fX(dn) = 0;
0< d1 < SX−1(α);
di < di+1, i= 1, n−1 has a solution d= d1,d2, . . . ,dn
and dsatisfies the inequalities (1 +ρ)fX(d1)−fX (d1)>0;
(3.8)
fX(di)−fX (di)>0, ∀i= 2, n.
(3.9)
Proof. i) Let 0< d1< SX−1(α).From the conditions ∂VaRT(d∂d1,...,dn,α)
i = 0,
∀i= 1, n we obtain the system
⎧⎪
⎨
⎪⎩
1−(1 +ρ)SX(d1)−fX(d1) = 0;
−SX(di) +fX(di) +SX(di+1) = 0, i= 2, n−1;
−SX(dn) +fX(dn) = 0. Let we denote
HVaRT(d1, d2, . . . , dn, α) =
∂2VaRT(d1, . . . , dn, α)
∂di∂dj
i,j=1,n
= (hij)i,j=1,n. Then we have h11= (1 +ρ)fX(d1)−fX (d1);hii=fX(d2)−fX (d2),∀i= 2, n; hi,i+1 = −fX(di+1), ∀i = 2, n−1; hij = 0, ∀i > j, i = 2, n, j = 1, n−1;
h12 = 0; hij = 0, ∀i < j −1, i = 1, n−2, j = 3, n. In case the system (3.7) has a solutiond= d1,d2, . . . ,dn
and conditions (3.8) and (3.9) are ful- filled, d1,d2, . . . ,dn
is a minimum point of the function VaRT(d1, . . . , dn, α), consequently, the optimization problem (4.1) has a solution.
ii) Letd1 > SX−1(α).From the conditions ∂VaRT(d∂d1,...,dn,α)
i = 0, ∀i= 1, n, we get the system
(3.10)
⎧⎪
⎨
⎪⎩
−(1 +ρ)SX(d1)−fX(d1) = 0;
−SX(di) +fX(di) +SX(di+1) = 0, i= 2, n−1;
−SX(dn) +fX(dn) = 0.
Since the first equation in (3.10) has no solution, there is no solution for the optimization problem (4.1) in the case d1 > SX−1(α).From i) and ii) the conclusion follows.
4. OPTIMAL RETENTION USING
CONDITIONAL TAIL EXPECTATION MEASURE
We recall the definiton of Conditional Tail Expectation measure. LetX be a random variable with cumulative distribution function FX and survival function SX.Letα∈(0,1).
Definition 4.1. The α-Conditional Tail Expectation (α-CTE) measure corresponding to the random variable X and the probability level α is de- fined by
CTEα(X) = E [X|X≥VaRα(X)].
Let us denote by CTET(d1, d2, . . . , dn, α) the Conditional Tail Expectation of the total cost of the insurer, corresponding to the retention levels (di)i=1,n and to the probability level α. We want to determine the optimal retention levels d∗1, d∗2, . . . , d∗n that minimize CTET(d1, d2, . . . , dn, α) measure, which means solving the optimization problem
(4.1) CTET(d∗1, d∗2, . . . , d∗n, α) = min
0<d1<d2<...<dn{CTET(d1, d2, . . . , dn, α)}.
We will derive an analytical expression of CTET(d1, d2, . . . , dn, α) mea- sure.
Theorem 4.1. For every d >0 and 0< α < SX(0), we have
CTET(d1, . . . , dn, α) =d1+ (1 +ρ)
∞ d1
SX(x)dx−n−1
i=1
(di+1−di)SX(di+1)
for 0< d1≤SX−1(α) and
CTET(d1, . . . , dn, α) =SX−1(α) + (1 +ρ)
∞ d1
SX(x)dx−
−n−1
i=1
(di+1−di)SX(di+1)
+ 1 α
d1
SX−1(α)SX(x)dx for d1 > SX−1(α).
Theorem 4.2. The optimization problem (4.1)has solution if and only if one of the following conditions is fulfilled:
i)the system
(4.2)
⎧⎪
⎪⎪
⎪⎪
⎪⎨
⎪⎪
⎪⎪
⎪⎪
⎩
1−(1 +ρ)SX(d1)−fX(d1) = 0;
−SX(di) +fX(di) +SX(di+1) = 0, i= 2, n−1;
−SX(dn) +fX(dn) = 0;
0< d1 < SX−1(α);
di < di+1, i= 1, n−1 has a solution d= d1,d2, . . . ,dn
and dsatisfies the inequalities
(4.3) (1 +ρ)fX(d1)−fX (d1)>0 and
(4.4) fX(di)−fX (di)>0, ∀i= 2, n or
ii) the system
(4.5)
⎧⎪
⎪⎪
⎪⎪
⎪⎨
⎪⎪
⎪⎪
⎪⎪
⎩ 1
α −(1 +ρ)
SX(d1)−fX(d1) = 0;
−SX(di) +fX(di) +SX(di+1) = 0, i= 2, n−1;
−SX(dn) +fX(dn) = 0;
0< d1 < SX−1(α);
di < di+1, i= 1, n−1 has a solution d= d1,d2, . . . ,dn
and dsatisfies the inequalities
(4.6)
(1 +ρ)− 1α
fX(d1)−fX (d1)>0 and
(4.7) fX(di)−fX (di)>0, ∀i= 2, n.
Proof. Let
HVaRT(d1, d2, . . . , dn, α) =
∂2VaRT(d1, . . . , dn, α)
∂di∂dj
i,j=1,n
= (hij)i,j=1,n. i) For 0< d1 < SX−1(α) we have
h11= (1 +ρ)fX(d1)−fX (d1);
hii=fX(d2)−fX (d2), ∀i= 2, n; hi,i+1 =−fX(di+1), ∀i= 2, n−1;
hij = 0, ∀i > j, i= 2, n, j= 1, n−1;
h12= 0;
hij = 0, ∀i < j−1, i= 1, n−2, j = 3, n.
If the system (4.2) has a solution d1,d2, . . . ,dn
and conditions (4.3) and (4.4) are fulfilled, d1,d2, . . . ,dn
is a minimum point for the function CTET (d1, . . . , dn, α),consequently, the optimization problem (4.1) has solution.
ii) Ford1 > SX−1(α) we have
h11= (1 +ρ)fX(d1)−fX (d1);
hii=fX(d2)−fX (d2), ∀i= 2, n; hi,i+1 =−fX(di+1), ∀i= 2, n−1;
hij = 0, ∀i > j, i= 2, n, j= 1, n−1;
h12= 0;
hij = 0, ∀i < j−1, i= 1, n−2, j = 3, n.
If the system (4.5) has a solution (d1,d2, . . . ,dn) and conditions (4.6) and (4.7) are fulfilled, (d1,d2, . . . ,dn) is a minimum point for the function CTET(d1, . . . , dn, α), consequently, the optimization problem (4.1) has solution.
5. CONCLUSIONS
In this paper we propose a new model, based on stop-loss reinsurance with multiple retention levels and we derive the stop-loss reinsurance premium corresponding to this model. We provide the analytical form of Value-at-Risk and Conditional Tail Expectation measures corresponding to the aggregated loss and build optimization problems based on these risk measures. We estab- lish necessary and sufficient conditions for the existence of the optimal solution and we find out these conditions depend on the distribution of the loss random variable, the probability level and the safety loading coefficient.
Acknowledgement.This research work was supported by the National University Research Council CNCSIS-UEFISCSU, Project number 844 PN II–IDEI, code ID 1778/2008.
REFERENCES
[1] J. Cai and K.S. Tan,Optimal retention for a Stop-Loss reinsurance under the VaR and CTE risk measures. Astin Bull.37(2007),1, 93–112.
[2] S. Dedu and R. Ciumara, Restricted optimal retention in Stop-Loss reinsurance. Proc.
Rom. Acad. Ser A Math. Phys. Tech. Sci. Ing. Sci.11(2010),3, 213–217.
[3] R. Kaas, M. Goovaerts, M. Denuit and J. Dhaene, Modern Actuarial Risk Theory.
Kluwer, Boston, 2001.
[4] S.A. Klugman, H.H. Panjer and G.E. Willmot. Loss Models: from Data to Decisions (2nd Edition). Wiley, New York, 2004.
[5] P. Krokhmal, J. Palmquist and S. Uryasev, Portfolio optimization with Conditional Value-at-Risk objective and constraints. J. Risk Uncertain4(2002),2, 11–27.
[6] N. Miller and A. Ruszczynski,Risk-adjusted probability measures in portfolio optimiza- tion with coherent measures of risk. European J. Oper. Res.191(2008),1, 193–206.
[7] G. Pflug, Some remarks on the Value-at-Risk and the Conditional Value-at-Risk. In:
S. Uryasev (Ed.), Probabilistic Constrained Optimization: Methodology and Applica- tions, Kluwer, 2000.
[8] V. Preda and R. Ciumara, On insurer portfolio optimization. An underwriting risk model. Romanian J. Econom. Forecasting9(2008),1, 102–112.
[9] V. Preda, S. Dedu and R. Ciumara. Restricted optimal retention in Stop-Loss reinsu- rance under VaR risk measure. Proc. 12th WSEAS Math. Methods, Comput. Tech- niques, Intelligent Systems Internat. Conference, 2010, pp. 143–146.
Received 5 July 2010 Academy of Economic Studies
Department of Mathematics 6 Piata Romana 010374 Bucharest, Romania
