an optimal combination of several reinsurance protections...
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Optimal Reinsurance Programs
An optimal Combination of several ReinsuranceProtections on an heterogeneous
Insurance Portfolio
Robert Verlaak & Jan Beirlant
Aon Re Belgium & K.U. Leuven
IME 2002
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Table of Contents
1 Introduction..................................................................................................................................... 31.1 Proportional reinsurance ............................................................................................................ 41.2 Non-proportional reinsurance ..................................................................................................... 51.3 Combination of reinsurance protections in practice ...................................................................... 5
2 The optimisation criterion and reinsurance loading structure ............................................................... 63 Excess of loss after a surplus ............................................................................................................. 9
3.1 Special case A=0 ..................................................................................................................... 103.2 Special case X(k)=k Z.............................................................................................................. 113.3 Special case A=0 and X(k)=k Z................................................................................................. 113.4 QS as a special case of a Surplus............................................................................................... 123.5 Optimal solution: graphical presentation.................................................................................... 12
4 Excess of loss and quota share ........................................................................................................ 144.1 Excess of loss after quota share (QS-XL) .................................................................................. 154.2 Quota share after Excess of loss (XL-QS) ................................................................................. 164.3 Stop loss as a special case of Excess of loss............................................................................... 17
4.3.1 Stop loss after quota share (QS-SL)....................................................................................... 174.3.2 Quota share after stop loss (SL-QS)....................................................................................... 18
5 Surplus and quota share.................................................................................................................. 185.1 Surplus after quota share (QS-SPL) .......................................................................................... 195.2 Quota share after Surplus (SPL-QS) ......................................................................................... 20
6 Conclusions and general remarks .................................................................................................... 217 Annex ........................................................................................................................................... 22
7.1 Definitions, notations and well known results ............................................................................ 227.2 Some basic derivations ............................................................................................................. 24
7.2.1 Surplus & excess of loss ....................................................................................................... 247.2.1.1 General case.................................................................................................................. 247.2.1.2 Special case: ZkkX ⋅=)( ........................................................................................... 277.2.1.3 Optimal solution ............................................................................................................ 28
7.2.2 Surplus & individual loss...................................................................................................... 297.2.2.1 Special case: ZkkX ⋅=)( ........................................................................................... 30
7.2.3 Stop loss and excess of loss .................................................................................................. 308 References..................................................................................................................................... 31
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Abstract
In most practical cases the reinsurance protection of an insurance portfolio is not limited to one reinsurancetype such as quota share, surplus, excess of loss or stop loss, but is organised through a combination ofseveral methods of protection, a so-called reinsurance program.In this paper we will analyze optimal reinsurance programs for a given portfolio based on the “mean –variance” optimisation criterion.Special attention is given to a description of a "surplus reinsurance" in combination with an "excess of lossper risk protection" for an heterogeneous insurance portfolio.We derive the equations one needs to solve for finding the optimal solution for the following combinations:excess of loss after surplus, excess of loss after quota share, stop loss after quota share, quota share after stoploss, quota share after excess of loss, quota share before surplus and quota share after surplus.It turns out that the application order of the reinsurance protections has its importance.
Keywords
Proportional & non-proportional reinsurance; optimal reinsurance, mean variance criterion.
1 Introduction
In the actuarial literature most papers concerning optimal reinsurance are limited to one reinsuranceprotection given an optimisation criterion. Exceptions are Centeno (1985, 2002), Kaluszka (2001) andSchmitter (2001) who consider optimal solutions for a quota share - excess of loss combination.
In practice reinsurance however is much more complicated. Even for a relatively simple property portfolioone will construct reinsurance based on a combination of different types of protections. If we look in detail toa more or less traditional reinsurance construction one will detect that, among others,• A combination of proportional and non-proportional reinsurance protections is frequently used.• Many times the proportional reinsurance will include (or only exist of) a surplus protection. However the
optimal proportional reinsurance literature is essentially limited to quota share protections (a link tosurplus reinsurance exists via a variable quota share, but this is definitely not the same).
• There exists a combination of protections such as a "per risk" and "per event" protection (for example:conflagration, storm, earthquake, …) or a personal accident program. If both types of protections areindependent of each other (in the sense of the protection per risk will not influence the protection perevent), then one can describe the reinsurance by independent reinsurance protections (one for the per riskand one for the per event protection). Otherwise one needs a much more complicated model to describethe risk and the interactive reinsurance protections.
• Non-proportional protections as well as surplus reinsurance are mostly subdivided in several layers. Eachlayer will have its own conditions. These conditions will result to a sum of successive layers which is notequivalent to one layer (for example: the first layer will include an additional franchise (= AnnualAggregate Deductible = AAD); the different layers will include different co-reinsurance clauses; eachlayer will include a different number of reinstatements; each layer will include other types ofreinstatement clauses (pro rata claim; pro rata time; pro rata claim & time); each layer will include adifferent no-claim bonus; …).
• In non-proportional reinsurance one will cede many times the risk up to a certain level (= upper limit).The risk exceeding this upper limit goes back to the retention of the cedent. The same applies to the riskafter exhaustion of the limited number of reinstatements per layer.Note that this overflow risk will depend on the first retained risk, which makes the analyses of theretention much more complicated.However, if one treats the optimal reinsurance question, one will (nearly) never take this overflow risk
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into account. Nevertheless it is clear that for some extreme risks (for example: wind storm, earthquake,…) this overflow risk can strongly influence the analysis of the retained risk.
• Most portfolios will include a lot of heterogeneity. This heterogeneity will imply an additional deviationrisk one likes to cede to the reinsurance market.Many times this heterogeneity is an important reason for combining several reinsurance protections. Aclassical example is the heterogeneity induced by the difference in sum insured per policy, but manyother differences between policies will induce differences in exposure (for example: difference incoverage (theft, third party protection, …), difference in type (simple risks - commercial risks - industrialrisks - ...)).
In this paper we will not tackle all these problems but we will concentrate on 3 items:• Describing the combination of basic reinsurance protections. Basic stands here for: no
subdivision in layers, no subdivision of the principal protection in subcontracts at differentconditions, no overflow risk due to reinsurance limitations, ….
• Analysing and characterising the influence of the heterogeneity induced by the difference in riskexposure per policy (for example sum insured).
• Describing, discussing and applying of a practical optimisation criterion for reinsurance.
We will not deal explicitly with the typical long-tail risk items, but limit ourselves essentially to the so-calledshort-tail business (for example property, personal accident, …). However, if one ignores the time betweenthe loss occurrence (or the claims made date) and the loss payments, one can drop that limitation.
Classical reinsurance can be classified in 2 types: the proportional and the non-proportional (for a detaileddescription see for instance Carter (1995); Geratewohl (1980, 1992) and Kiln (1982)). To introduce somenotations and to summarise the main influence of these 2 methods, we consider a portfolio of n risks. Eachrisk i is characterised by an entity or risk iR , risk exposure iK (for example the sum insured value iC or
some Maximum Possible Loss value, in general all terms of the original policy), premium iP , individual
claims jiS , and claims per year & per entity ∑=
=in
jjii SS
1, for i = 1, .. , n and j = 1, .., in .
1.1 Proportional reinsurance
All the characteristics of each risk (= risk exposure, premium, claim) are proportionally shared betweencedent and reinsurer (for example: claim potential, risk of wrong tariff or conditions, timing risk, ). If weconsider a proportional cession per risk equal to ia−1 [ ]1,0∈ and consequentially a retention per risk equal
to ia , we obtain:
Retention Proportional cessionPremium
∑=
⋅n
iii Pa
1∑
=
⋅−n
iii Pa
1
)1(
Exposure∑
=
⋅n
iii Ka
1∑
=
⋅−n
iii Ka
1
)1(
Claim∑ ∑
= =
⋅
n
i
n
jjii
i
Sa1 1
, ∑=
⋅=n
iii Sa
1
∑ ∑= =
⋅−
n
i
n
jjii
i
Sa1 1
,)1( ∑=
⋅−=n
iii Sa
1
)1(
In practice one will use simple rules to define these cession proportions. Basically one uses two rules, if onemakes abstraction from the less essential differences, which will lead to a "surplus" and a "quota share" typeof reinsurance.
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For a surplus , one uses the non-negative real valued exposure function 0)(: ≥=→ iKiKiK todistribute proportionally the exposure to the surplus reinsurers in such a way that
<−
≥=
−=− +
i
i
i
i
i
ii Kline
K
lineKKline
K
lineKa if
)( if0
)(1 , with the so-called “line” the amount of retained
exposure by the cedent.
For a quota share the cedent cedes a fixed proportion for all risks, this means that aa i = for all i.
1.2 Non-proportional reinsurance
With this reinsurance method one tries to control the consequences of the portfolio by a direct attack on theclaim amounts. The reinsurer will take over the claim amounts that (in some sense) are too high. In practicethis means that the reinsurer takes for its account the claim amount that exceeds a franchise or priority.
Retention Non-proportional cessionFor each claim S Minimum of (S, priority) (S – priority)+
In practice one has different ways to define the "level" of a claim amount:• Per risk: +−= )( , prioritySS ji which corresponds to the so-called excess of loss per risk reinsurance.
• Per accident / event: +−= )( prioritySS g with ∑∑∈∈
==gji
jig SS),(
, if the claim j of the risk i is the
consequence of the event/accident g, which is the so-called excess of loss per event/accident reinsurance.
• Per year: += =
−= ∑∑ prioritySS
n
i
n
jji
i
)(1 1
, which defines the so-called stop loss reinsurance.
For a non-proportional cover the lot between cedent and reinsurer is (strongly) separated: good or bad resultsfor one party do not imply necessarily good or bad results for the other party implying that both results areonly correlated on a limited basis.In non-proportional covers the intervention of the reinsurer is often limited to a special part of the risk (e.g.limited to well defined events such as catastrophe risks like windstorm, earthquake, flood, …, limited coversuch as reinstatements, upper limit, …, timing of recovery such as after exhaustion of the priority, …,exclusion and so on), which will more strongly separate the results between cedent and reinsurer.
1.3 Combination of reinsurance protections in practice
To combine non-proportional reinsurance in a logical way, one needs to respect following order: first anexcess of loss per risk, secondly an excess of loss per event or accident and finally a stop loss (if one goes theother way around, one can not assign the recuperation of the n-th protection to the n-1-th without anysupplemental allocation rule).
A quota share can be ceded in any order, because the fixed proportion is always applicable to the portfolio asa whole and / or after any reinsurance protection.
If one separates a specific part of the insured perils then it is possible to construct two (or more) parallelreinsurance protections: one for the separated insured dangers and another one reinsuring all the remainingdangers. In practice this is sometimes done for the natural catastrophic events, because they are almostexclusively protected by excess of loss per event covers.
A surplus protection should be the first in the series of the protection, if applicable, but as stated before aquota share can always precede.
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Note that a surplus protection after an excess of loss per risk will imply that the higher sum insured risks,which are systematically ceded to the surplus, have much more potential to recover from an excessintervention. So, one should recover the excess premium, but (much) more than on a pro rata basis from thesurplus reinsurers. In practice one will nearly always avoid such a difficulty.A surplus after an excess of loss per event is logically impossible, because the recovery from the excess isglobal and cannot be correctly distributed over the individual risks. However, as stated before, it is possibleto isolate some special events from the surplus cession (for example: windstorm and earthquake risks).
A typical reinsurance structure for property business is given in Table 1. One should note that a quota sharenormally once appears in the series of protection, a multiple occurrence is rare but not excluded.
2 The optimisation criterion and reinsurance loading structure
Several optimisation criteria exist for reinsurance in the literature. Best known is the "Mean - Variancecriterion”, which concerns the maximisation of the expected profit under the constraint of a fixed variance,or equivalent, the minimisation of the variance under the constraint of a fixed profit. Another popular methodis the “Probability of ruin criterion”, which concerns the maximisation of profit under the constraint of ruinprobability less then a fixed value, or equivalent, minimisation of the ruin probability under the constraint ofa minimal expected profit.Note that the finite or the infinite ruin probability can be chosen as a criterion but also the time model(discrete or continuous) will influence the solution. Sometimes the maximisation of the adjustmentcoefficient is used as the optimisation criterion due to the fact that this coefficient can be used to define anupper bound for the ruin probability with infinite time horizon.Furthermore also the dividend policy and the utility stabilisation criterion are described in the literature. Thefirst one is in some sense a generalisation of the ruin criterion and the second one of the mean varianceprinciple.For a detailed discussion of these optimisation criteria we refer to Bühlmann (1970), Daykin (1994),Goovaerts (1983, 1993, 1994), Kaas (1993), Klugman (1998), Straub (1988), Centeno (1985, 2002) andKaluszka (2001). In the context of optimal reinsurance with a combination of excess of loss and quota share,Kaluszka (2001) used the mean variance principle, while Centeno (2002) approach the problem via theadjustment coefficient.
In this paper we will use the mean-variance optimisation criterion. However we will introduce this criterionin a slightly different form, with the advantage of a more useful and more general interpretation.We will minimise the cost of the net retained risk (= retained risk after intervention of the whole reinsuranceprogram). The cost of retaining the risk will be defined as: the total of the reinsurance loading i.e.additional price to be paid above the expected risk ceded to the reinsurers plus a fixed multiple of theretained standard deviation.
We assume that the reinsurance loading is proportional with the expected risk ceded to the reinsurer,which corresponds to the expected value principle of (reinsurance) premium calculation. Thisproportion will vary per type of protection and can depend on the sequence order of the reinsuranceprotection.The multiple of the retained standard deviation is interpreted as the cost of capital allocation to carry the risk.We assume that the allocated capital is proportional to the standard deviation (for example a factor 4.5), butdue to compensation with other retained risks reduced with some factor (for example 0.80). This allocatedcapital should give a certain return (for example 25%). This implies an allocated capital cost that is alsoproportional to the retained standard deviation (for example 4.5 x 0.80 x 0.25 = 0.90).(Note that the assumption of an explicit link between standard deviation and allocated capital implies that werestrict ourselves to risks with bounded volatility).Important to note is that introducing this standard deviation constraint, implies that the cost of capitalallocation as a multiple of the standard deviation is not necessary to find an optimal reinsurance solution.This observation plus the fact that working with standard deviation or variation constraints is equivalent,implies that our "mean-variance criterion" drops down to the classical "mean-variance criterion".
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Formal description of the criterion and loading structure:• Premium income Cedent = PI (net of facultative reinsurance). Note that we don't suppose an explicit link
between S and PI, but in general PI > E(S), with E() the abbreviation for expected value.• Retained risk = )(Sℜ . This is the risk retained by the cedent after application of all the reinsurance
protections.• Allocated capital cost for the cedent = [ ])(SStd ℜ⋅α , with Std the abbreviation for standard deviation.
The parameter )0(≥α is supposed to be fixed.
• Reinsured or ceded risk = ∑=
n
ii ST
1
)( with iT one of the reinsurance protections (i.e. quota share, surplus,
stop loss or excess of loss).
• Reinsurance Loading cost = [ ]∑=
⋅n
iii STE
1
)(λ , with )0(≥iλ fixed. In general the loading factors iλ can
be subdivided in 2 parts. A first component really paid to the reinsurer (to finance his costs and grossmargin) and a second part as a (direct and indirect) cost to be borne by the cedent for managing andorganising the cessions of the reinsurance treaties.
• Note that following relations hold:
∑=
+ℜ=n
ii STSS
1
)()( , implying [ ] [ ]
+ℜ= ∑=
n
ii STESESE
1
)()( and
[ ] [ ]
+ℜ≤ ∑=
n
ii STStdSStdSStd
1
)()( .
We define the (gross) profit 1G of the cedent's (sub-) portfolio after reinsurance by "Premium Income minusthe retained risk minus the reinsurance cost minus the retained risk allocated capital cost" or
[ ] [ ])()()1()(1
1 SStdSTESPIGn
iii ℜ⋅−⋅+−ℜ−= ∑
=
αλ . Consequently the expected profit is equal to:
[ ] [ ] [ ] [ ]
[ ] [ ])()()(
)()()1()(
1
11
SStdSTESEPI
SStdSTESEPIGE
n
iii
n
iii
ℜ⋅−⋅−−=
ℜ⋅−⋅+−ℜ−=
∑
∑
=
=
αλ
αλ
We like to maximise this expected profit [ ]1GE while keeping the retained standard deviation [ ])(SStd ℜfixed. To optimise the profit problem we should use the Lagrange method of multipliers.With the formal notation jw for all the unspecified parameters which one needs for describing thereinsurance protections (for example: the quota share retention, surplus line, priority for excess of loss,
priority for stop loss), and *µ for the Lagrange multiplier, we need to solve the following equations for all j:
[ ] [ ] [ ] [ ][ ]
[ ]
=ℜ
ℜ∂∂−+
⋅+−ℜ−
∂∂=
∂∂= ∑
=
CSStd
SStdw
STESEPIww
GE
j
n
iii
jj
)(
)()()()1()(0 *
1
1 αµλ
Setting αµµ −= * , one sees immediately that this is the Lagrange formulation for the conditional
optimisation problem defined by: [ ]∑=
⋅−−=n
iii STESEPIG
1
)()( λ with [ ] 2)( CSVar =ℜ .
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Concluding, with [ ] [ ]GVarGE ⋅+=Ψ µ the Lagrange set of equations for the last formulation equals
[ ] [ ] 0=∂∂
⋅+∂∂
=∂
Ψ∂GVar
wGE
ww jjj
µ with [ ] 2)( CSVar =ℜ , or equivalently,
[ ] [ ] 0)()(1
=ℜ∂∂⋅+
∂∂⋅− ∑
=
SVarw
STEw j
n
ii
ji µλ with [ ] 2)( CSVar =ℜ
In the rest of the paper we will express the constraint on the variance by [ ] )()( SStdxSStd ⋅=ℜ with
[ ]1,0∈x (note that we assume that ∞<)(SStd ).This has the advantage that the possible conditional values are easier to control and the graphicalrepresentation of the optimal solutions becomes easier afterwards. The interpretation of the limited availableallocated capital is also easier to apply in practice. Even if one does not know the exact allocated capital it iseasier to discuss the consequences of the reinsurance program based on the retained volatility with theminimisation of the reinsurance cost, than focusing on profit margins (which are also not exactly known).
Remark on the reinsurance loading structure in practice:In the (limited) range of real existing protections one has the observation that the loadings (based on realisedtreaties after competitive negotiations) are not highly influenced by the protection parameters. So it seems tobe acceptable to work with constant loading percentages as a good first order approximation.At the other hand one observes that the loading percentages are more or less depending on: the protectedbranch, the covered peril (natural disaster reinsurance has other loading percentages than the basic fire risk),the country, the margins in the insurance tariff (especially for proportional reinsurance), …. This impliesthat, before analysing the optimisation question, one needs to know very well the reinsurance market inpractice. Also the reinsurance over different branches becomes more complex.
In the next table we give an overview of the notation for the loading factors we will use afterwards. Evenmore important is the application of the ordening of the reinsurance protections we assume.Further, we will postulate some ranges of values that are a first estimate of some market figures. Figures thatshould be understood as a compromise after negotiation in a competitive reinsurance market and which aremainly related to Belgian simple risk property business. As such they are just intended as examples ofpossible relevant values.
Order Reinsurance protection Notation Working values1 (*) Quota Share 1
qλ or 1qλ [ ] 0.125~15.0,10.0∈qλ2 Surplus
eλ [ ] 0.1625~20.0,125.0∈eλ3 (*) Quota Share 2
qλ or 2qλ See qλ4 Excess of loss per risk
xλ or xrλ [ ] 0.30~40.0,20.0∈xrλ5 (*) Quota Share 3
qλ or 3qλ See qλ6 Excess of loss per event
xλ or xeλ [ ] 0.40~50.0,30.0xeλ7 (*) Quota Share 4
qλ or 4qλ See qλ8 Stop loss
sλ [ ] 0.50~60.0,40.0∈sλ9 (*) Quota Share 5
qλ or 5qλ See qλ
Table 1: Loading factors for the reinsurance protections
In the last table we already used a natural ordening between these loading factors, which is given in nextequation.
054321 >>≥≥≥≥≥≥≥≥>>>>>>>>>>∞∞ qqqqqexrxes λλλλλλλλλ (Eq 1)
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Clearly non-proportional protections are, as a percentage of the expected ceded claim, (much) moreexpensive than proportional protections. In general one can state that the inequalities are strict betweendifferent protections, but among the different quota share loading factors 1qλ to 5qλ equalities are notexcluded.
Note however that in the case that no risk is retained by the cedent and the whole portfolio is ceded to onlyone reinsurance protection, all the loading factors should reduce to the same value (cfr line equal zero,priorities equal zero, retained proportion equal to zero).This observation implies some dependency of the loading factors from the parameters of the protection, butas stated before, we assume that the retained optimal solutions are close to the (undefined) set of real existingprotections.
3 Excess of loss after a surplus
As noted before an excess of loss will generally be applied after a surplus retention. The other way aroundexists but is essentially linked to facultative reinsurance, which implies an individual protection at individualconditions. So the retained risks after facultative reinsurance can, without any loss of generality, be seen as adirectly underwritten risk at retained conditions.
We assume a portfolio risk described by a random variable S defined by a compound process ∑=
=N
iiXS
0
with N the random variable of the number of claims and iX i allfor X~ , the random variable of individualclaims, which depends on an exposure value k (for example sum insured) characterised by a non-negative
random variable K with density function )(kc and a distribution function ∫=
=k
l
dllckC0
)()( . So we assume
the individual claim X is a mixture of possible claims X(k) with exposure k .
It is important to note that C(k) is the distribution function of the exposure values of the policies attacked bya claim and not the distribution function of exposure values of the insured policies. The first distributiondescribes the probabilities that a loss is associated with some exposure value; the second one is a staticdescription of the relative number of insured risks with a certain exposure value in the insurance portfolio.
We suppose a surplus reinsurance with a retained line L and an excess of loss priority R. We sometimesexpress the priority R as a proportion r of the line L: r = R/L.We denote the individual retained risk with exposure k after surplus line L equal to )(kX L , the mixture of
all the retained risks by LX and after excess of loss priority R by ))(( RkX L ∧ and )( RX L ∧ .
An overview of all used notations, definitions and basic derivations can be found in the annex.
The retained risk after surplus line L and excess of loss priority R is equal to [ ]∑=
∧=ℜN
iiL RXRL
0
)(),( ,
with [ ] 1:)()(),( DSXRXENERLE L =∧⋅=ℜ and
[ ] [ ]{ } 222 :)()()(),( DSXRXEARXENERLVar LL =∧⋅+∧⋅=ℜ with
[ ]),1(
)(
)()( ∞−∈−=NE
NENVarA .
The total reinsurance cost is equal to [ ] 3:)()()()1( DSXXEXENE Le =−⋅⋅+ λ for the surplus plus the part
for the excess of loss after surplus given by [ ] 4:)()()()1( DSXRXEXENE LLx =∧−⋅⋅+ λ .
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If we note [ ]4312),( DSXDSXDSXDSXRL −−−+⋅=Ψ µ , then the Lagrange equations become
[ ] RorLWwithDSXDSXDSXW
DSXW
RLW
==−−−∂∂+
∂∂=Ψ
∂∂
0),( 4312µ .
Solving these equations leads to the following relation between L and R (see paragraph 7.2.1.3 Optimalsolution):
[ ] [ ]
[ ] [ ]
[ ] [ ]{ }
[ ] [ ]{ }∫
∫
∫∫
∫∫
∞
=
∞
=
∞
=
∞
=
∞
=
∞
=
⋅⋅−∧
⋅∧⋅∧⋅+∧=
⋅−∧
∧⋅∧⋅+∧=∧⋅+
Lk
Lk
L
LkLk
Lk
L
LkL
dkkckYEQrkYE
dkkcrkYErYEArkYE
dkkckYEQdkrkYEkc
dkrkYEkcrYEAdkrkYEkc
rYEAr
)()()(
)()()())((
)()()()(
)()()())(()(
)(
2
2
With )1,0(∈−=x
exQλ
λλ, LRr = ,
kkX
kY)(
)( = for each k>0 and L
XY L
L = for each L>0.
(Eq 2)
Note that for ∞→r the left hand side of (Eq 2) becomes ∞ and the right-hand side bounded (if Q<1),which implies that (for x<1) an excess of loss protection will always be used (except for L=0, implying that xshould also be zero).
However until now we did not use the variance constraint [ ] [ ]1,0 ),(),( 2 ∈=ℜ xSVarxRLVar orequivalently:
[ ][ ] [ ])()()()( 22222 XEAXExRXEARXE LL ⋅+⋅=∧⋅+∧ with [ ]1,0∈x , or equivalent
[ ] [ ])()()()( 22222 XEAXExrLXEArLXE LL ⋅+⋅=⋅∧⋅+⋅∧(Eq 3)
For a given line L this formula has a unique solution for R, if in case ∞=R the left hand side of the formulais greater than the right-hand side. Analogously one can prove that under a weak condition the formula has aunique solution for L, given R, if for ∞=L the left-hand side is greater than the right hand side. Seeequations (Eq 30), (Eq 31) and (Eq 32).
To find a feasible solution for L and R we used an iterative process:• Step 0: Choose a starting value L (for example by solving (Eq 3) for L with LR = or ∞=R ).• Step 1: For the resulting line L solve (Eq 2) for r.• Step 2: For the resulting priority r solve (Eq 3) for L. If L becomes ∞ then solve (Eq 3) for R and
stop the iteration.• Step 3: Repeat until convergence in R and L.
It is not clear if these equations will generate an unique solution, or if local optima are possible. Probablyone needs to add some conditions on the behaviour of X(k) or Y(k) in relation to K.
3.1 Special case A=0
Note that the Poisson process belongs to this case for which (Eq 2) reduces to
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[ ]
[ ] [ ]{ }∫
∫∞
=
∞
=
⋅⋅−∧
⋅∧=
Lk
Lk
dkkckYEQrkYE
dkkcrkYE
r
)()()(
)())(( 2
(Eq 4)
Remark that this equation in r is independent on the claim number process, but stays depending of the line Land the exposure distribution K.
Note also that 0>r implies that [ ] [ ]∫∫∞
=
∞
=
>∧LkLk
dkkckYEQdkkcrkYE )()()()( should hold.
The variance constraint here reduces to [ ] )()( 222 XExrLXE L ⋅=⋅∧ with [ ]1,0∈x .
3.2 Special case X(k)=k Z
This implies that Zk
kXkY ~
)()( = for all k and (Eq 2) reduces to (see (Eq 33) & (Eq 34))
[[ ]] [[ ]][[ ]] [[ ]]ZEQrZE
rZErYEArZErYEAr L
L ⋅⋅−−∧∧∧∧⋅⋅∧∧⋅⋅++∧∧==∧∧⋅⋅++ )()(
)(2
(Eq 5)
or equivalently
[[ ]] [[ ]][[ ]] [[ ]]ZEQrZE
ZEQrYEArZEr L
⋅⋅−−∧∧⋅⋅⋅⋅∧∧⋅⋅++∧∧== )()( 2
(Eq 6)
Remark that this equation in r stays dependent on the claim number process through the factor A, but also ofthe line L and the exposure distribution K (both through the factor )( rYE L ∧ ).
Note that if 0)()( >⋅−∧ ZEQrZE then r>0. This condition is true if we assume that ( )∞∈ ,0r because
)( rYEAr L ∧∧⋅⋅++ is always positive (for 0≥A is this obvious, but is also true for [ )0,1−∈A because
rrZErYE L ≤≤∧∧≤≤∧∧ )()( (see (Eq 33)), which implies that the left hand side of (Eq 5) and also thenumerator of the right hand side of (Eq 5) will be positive. We can conclude that the denominator shouldalways be positive.
The variance constraint further reduced, but is not fundamentally changed to
[[ ]] [[ ]])()()()()()( 2222222 KEZEAKEZExRXEARXE LL ⋅⋅⋅⋅++⋅⋅⋅⋅==∧∧⋅⋅++∧∧
3.3 Special case A=0 and X(k)=k Z
[[ ]][[ ]] [[ ]]ZEQrZE
rZEr
⋅⋅−−∧∧∧∧==
2)((Eq 7)
Remark that this equation in r is independent on the line L, independent on the claim number process andalso independent on the exposure distribution K.
Once the priority r is known one can solve for L with the help of the variance constraint. This constraintreduces further a little, but the most important advantage is that both equations separately of each other can
be solved. The variance constraint becomes: [[ ]] )()()( 2222 KEZExRXE L ⋅⋅⋅⋅==∧∧ for [ ]1,0∈x .
RVE - Art01_TOT.doc - 27/06/2002 12/32
3.4 QS as a special case of a Surplus
If we define an exposure measure k equal to 1 for all the risks in the insured portfolio and the line L a valuebelonging to the interval [0,1], then the corresponding surplus reinsurance is a quota share with L theretained proportion for all risks.For this exposure measure we become that: Y(k)=X(k)~X for all k , XLX L ⋅= , XYL ~ , Rr = and
0)( =kc for all k except for 1=k 1)1( =c . This reduces (Eq 2) to
[ ])()(
)()()(
22
XEQRXE
RXEARXERXEAR
⋅−∧∧⋅+∧+∧⋅−= with 10 <
−=<
x
qxQλ
λλ(Eq 8)
The variance constraint [ ] [ ]1,0),(),( 2 ∈=ℜ xSVarxRLVar becomes:
[ ][ ] [ ])()()()( 222222 XEAXExRXEARXEL ⋅+⋅=∧⋅+∧⋅ with [ ]1,0∈x and ]1,[xL ∈ (Eq 9)
3.5 Optimal solution: graphical presentation
To illustrate the optimal solution for a surplus-excess of loss combination in practice, we will use an exampleof the special case X(k)=k Z with A different from 0.In Table 2 we give an overview of the most important kernel-figures, which will characterise our claimstatistic. These numbers and distribution fits are based on a practical case-study in fire insurance (handled byAon Re Belgium).We will use two lognormal distributions to describe the exposure value K (= sum insured values of theclaims) and the claim severity per unit of exposure value Z. Afterwards we will compare some of the resultswith a double Burr model to illustrate the impact of model choice.
N K Z X S
Mean 1,225.000 125.000 0.0180 2.250 2,756.250Std 52.500 200.000 0.0900 21.530 762.738
Cv 0.04286 1.60000 5.00000 9.56870 0.27673Skewness
BurrLognormal
7.27.2
8.896
280280140
Value A 1.25Value Q (0.30-0.1625)/0.30 = 11/24 = 0.45833
Table 2: Description of the claim statistic
In Figure 1, we bring together the most important results. As a function of the retained proportion of standarddeviation x, one has the optimal line (= L_i) and the optimal priority (= R_i). To compare the optimalpriority with an excess of loss protection without surplus reinsurance the corresponding priority R0 is alsoplotted. These values are plotted on a logarithmic scale on the left axes. The plot illustrates clearly that with alow-level of reinsurance both priorities are nearly the same, indicating that the surplus protection has nearlyno influence on the retention.On the right axes the ratio is (= M_i = r) between the priority R_i and the line L_i plotted and compared withthe constant solution corresponding with A equal to zero (=M0A). This ratio increases for decreasingproportion of retained risk, implying that the surplus reinsurance becomes more important for high-levelreinsurance.
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1.E-01
1.E+00
1.E+01
1.E+02
1.E+03
1.E+04
1.E+05
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Retained proportion of Std x
Lin
e -
Pri
ori
ty (
log
sca
le)
0.0%
2.0%4.0%
6.0%
8.0%10.0%
12.0%
14.0%16.0%
18.0%
20.0%22.0%
24.0%
r =
Pri
ori
ty/L
ine
L_i
R_i
R0
M0A
M_i
Figure 1: Optimal line & priority (lognormal model)
To illustrate the relative importance of the two protections, the reinsurance loadings per protection ( eλ , xλ )
and the expected claim transfer (= [ ] [ ] [ ]( )LXEXENE −⋅ and [ ] [ ] [ ]( )RXEXENE LL ∧−⋅ ) are plotted asa function of the retained proportion of the standard deviation (see Figure 2). Here it becomes clear that for ahigh-level of reinsurance one will use less excess of loss protection in favour of the surplus protection.
0.0E+00
5.0E+02
1.0E+03
1.5E+03
2.0E+03
2.5E+03
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Retained proportion of Std x
Exp
ecte
d c
laim
rei
nsu
ran
ce
Surplus
Excess of loss
Total
0.000
100.000
200.000
300.000
400.000
500.000
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Retained proportion of Std x
Lo
ad
ing
re
ins
ura
nc
e
Total
Surplus
Excess of loss
Figure 2: Reinsurance loading and expected claim transfer (lognormal model)
If we compare the results with two different models (which has the same first 2 moments, but a differentskewness), then one becomes different optimal solutions.In this example one can remark that:
• The proportion between priority and line (= r = M_i) is not so different, about 5%.• The value of the line (= L_i) is relatively strong influenced, the priority less.• The reinsurance loading increases especially for low-level reinsurance protections.• Also the relative subdivision between the two protections has strongly changed.
Double Lognormal model Double Burr model Burr / Lognormal
x 0,1 0,3 0,5 0,7 0,1 0,3 0,5 0,7 0,1 0,3 0,5 0,7
MOA 0,18998 0,18998 0,18998 0,18998 0,24333 0,24333 0,24333 0,24333 1,281 1,281 1,281 1,281
M_i 0,22031 0,20244 0,19410 0,19106 0,27068 0,25721 0,24908 0,24508 1,229 1,271 1,283 1,283
L_i 65,643 381,506 1.450,970 5.897,050 60,775 296,046 941,784 3.384,424 0,926 0,776 0,649 0,574
R_i 14,462 77,231 281,638 1.126,705 16,451 76,145 234,584 829,467 1,137 0,986 0,833 0,736
RVE - Art01_TOT.doc - 27/06/2002 14/32
Double Lognormal model Double Burr model Burr / Lognormal
x 0,1 0,3 0,5 0,7 0,1 0,3 0,5 0,7 0,1 0,3 0,5 0,7
Reinsurance loading
Surplus 272,107 67,733 7,926 0,241 296,068 96,555 15,873 0,741 1,088 1,426 2,003 3,070
XL 51,838 77,793 43,541 12,013 41,337 73,194 50,962 16,323 0,797 0,941 1,170 1,359
Total 323,945 145,526 51,467 12,254 337,405 169,750 66,835 17,064 1,042 1,166 1,299 1,392
SPL% 84,0% 46,5% 15,4% 2,0% 87,7% 56,9% 23,7% 4,3%
XL% 16,0% 53,5% 84,6% 98,0% 12,3% 43,1% 76,3% 95,7%
Expected claim ceded to reinsurer
Surplus 1.674,505 416,816 48,775 1,485 1.821,956 594,185 97,681 4,558 1,088 1,426 2,003 3,070
XL 172,793 259,310 145,138 40,043 137,791 243,981 169,873 54,410 0,797 0,941 1,170 1,359
Total 1.847,298 676,127 193,913 41,528 1.959,746 838,167 267,554 58,968 1,061 1,240 1,380 1,420
SPL% 90,6% 61,6% 25,2% 3,6% 93,0% 70,9% 36,5% 7,7%
XL% 9,4% 38,4% 74,8% 96,4% 7,0% 29,1% 63,5% 92,3%
This example shows that, in the context of optimal reinsurance, the model choice is quite an important issue.Of course, also the estimates of the parameters (model and reinsurance loading structure) will influence theoptimised results.
4 Excess of loss and quota share
Here the retained risk equals to ∑∑==
∧=⋅∧⋅=ℜN
ii
N
ii RXaRaXaRa
00
)()(),( , if we express the excess of
loss priority R in terms of the 100% claim size.
The left-hand side retained risk formula expresses that we firstly apply the quota share and secondly theexcess of loss with priority Ra ⋅ . The right-hand side formula expresses that we firstly apply the excess ofloss priority R and secondly the quota share with retention a.
It is clear that both applications will give the same retained risk for the cedent, but nevertheless thedifference in application order is fundamentally different.The first application (corresponding to the left-hand side) says that the quota share reinsurers are notparticipating in the excess of loss protection. Not participating means: not participating in the advantages (=recuperating the excess of loss intervention) and not participating in the disadvantages (= participating in theexcess of loss premium).The second application (corresponding to right-hand side) says that the quota share reinsurers also participatein the excess of loss protection. This should imply that they participate as proportional reinsurer in the excessof loss premium (implied by the fortune’s sharing principle of the quota share reinsurance), which in practiceis nearly always the case.
A remark concerning the impact of the fortune’s sharing principle is necessary, because it generates a seriouscontradiction with one of our basis assumptions.If the quota share reinsurer pays his part in the common excess of loss protection, then this will increase itscost structure. To compensate this additional cost he needs to increase the quota share-loading factor. To besure that he will obtain the same result, this increase should (nominally) be equal to the excess of lossloading (in theory this loading could be smaller due to less management costs and lower volatility, but wewill neglect this potential marginal reduction). But both loadings are (in the framework of this paper) not onthe same basis, which implies that for different excess of loss priorities one becomes different loadingfactors. This observation means that the "constant mean value premium principle" hypothesis is not valid anymore.Nevertheless, this problem can be avoided in an elegant way by assuming that the excess of loss premium is
RVE - Art01_TOT.doc - 27/06/2002 15/32
paid directly by the cedent. This instead of passing the excess of loss premium through the quota sharecontract and recuperate it in an indirect way through the loading structure. This interpretation exists (veryrarely) in practice but the reverse is the standard. However the alternative interpretation gives (in theframework of this paper) the same (theoretical) result as the logical loading correction gives, nearly the sameresult as in practice and coincides perfectly with the "constant mean value premium principle".
4.1 Excess of loss after quota share (QS-XL)
As stated before the retained risk is equal to ∑∑==
∧=⋅∧⋅=ℜN
ii
N
ii RXaRaXaRa
00
)()(),( , with
[ ] 1:)()(),( DQXRXENEaRaE =∧⋅⋅=ℜ and with [ ]
),1()(
)()( ∞−∈−=NE
NENVarA .
[ ] [ ][ ] 2222 )()()()(),( DQXRXENEARXENEaRaVar ==∧⋅⋅+∧⋅⋅=ℜ
The total reinsurance cost is equal to 3:)()()1()1( DQSXENEa q =⋅⋅+⋅− λ for the quota share plus the
part for the excess of loss after quota share given by [ ] 4)()()()1( QDXRaXaEXaENEx =⋅∧⋅−⋅⋅⋅+ λ .
From the preceding section it follows as a special case from the surplus formulae (see (Eq 8)) and byinterpreting the numerator as the so-called normalised portfolio variance (see (Eq 25)) that
)()(
)()(
XEQRXE
RXVrnRXEAR
⋅−∧∧=∧⋅+ with 10 <
−=<
x
qxQλ
λλ(Eq 10)
This equation (Eq 10) is quite interesting to compare with (Eq 2), (Eq 5) and (Eq 7). The nominator of theright part of equation (Eq 2) could be understood as some generalised weighted variance formula. Equation(Eq 5) is very well comparable with (Eq 10), the only difference is generated by the factor )( rYE L ∧ andthe fact that the role of X and R is replaced by Z and r. Anyway, it is remarkable how close these two totaldifferent reinsurance programs come together (see also special case (Eq 7)).
We can conclude that the optimal excess of loss priority is independent on the quota share protection and infirst instance also independent on the variance constraint (if R implies that 1≤a , see further).
Due to the fact that 0)( ≥∧ RXVrn and ),1(Afor )( ∞−∈∧⋅≥ RXEAR one has that a feasiblesolution R should fulfil the inequality 0)()( ≥⋅−∧ XEQRXE implying that )(XEQR ⋅≥ .
Note that for ∞→R the left-hand side of the equation becomes ∞ and the right hand side limited (if Q<1),which implies that (for x<1) an excess of loss protection will always be used.
The variance constraint [ ] [ ]1,0),(),( 2 ∈=ℜ xSVarxRaVar becomes:
[ ][ ] [ ])()()()( 222222 XEAXExRXEARXEa ⋅+⋅=∧⋅+∧⋅ with [ ]1,0∈x ,
implying [ ]1,xa∈ .(Eq 11)
Note that for a=1 (= no quota share protection) only the next equation has to be solved for R:
[ ][ ] [ ])()(
)()()()(2
22222
XVrnxRXVrn
orXEAXExRXEARXE
⋅=∧
⋅+⋅=∧⋅+∧(Eq 12)
To find a feasible solution for a and R one can:
RVE - Art01_TOT.doc - 27/06/2002 16/32
• Firstly solve (Eq 10) for R.• Secondly check with (Eq 11) if the corresponding a value is less than or equal to 1. If the answer is
positive one will retain the R(Eq 10) and a((Eq 10) & (Eq 11)) as the optimal combined solution.• Thirdly, if the check is negative one needs to solve (Eq 12) for R and retain R(Eq 12) together with
a=1 as the optimal combined solution.
4.2 Quota share after Excess of loss (XL-QS)
We illustrate for this combination the "constant mean value principle problem" explicitly. So, if the quotashare reinsurer increases his loading with the corresponding proportion in the excess of loss loading oneobtains the following reinsurance cost:
• For the excess of loss protection [ ])()()()1( RXEXENEx ∧−⋅⋅+λ• For the quota share protection )()()1()1( RXENEa q ∧⋅⋅+⋅− λ• From the excess of loss loading, the quota share reinsurers will directly recuperate the corresponding
proportional part [ ])()()()1( RXEXENEa x ∧−⋅⋅⋅− λ , but we assume that they will recuperatethis additional cost indirectly through the loading structure.
The retained risk is equal to ∑=
∧=ℜN
ii RXaRa
0
)(),( , which implies again the same formulas
11 : DXQDQX = and 22 : DXQDQX = as stated in section 4.1.
The reinsurance cost is equal to [ ] 3:)()()()1( DXQRXEXENEx =∧−⋅⋅+ λ for the excess of loss (before
quota share) plus for the quota share 4)()()1()1( DXQRXENEa q =∧⋅⋅+⋅− λ .
Instead of solving the Lagrange equations, we will derive the solution of the problem (just to illustrate a fullyequivalent way of working) by setting the derivative of the expected profit with respect to a equal to zerowhile keeping the variance constraint under control through the parameter R.
Note also that the profit function G(a,R) is defined as: [ ]∑=
⋅+−ℜ−=n
iii STESPIRaG
1
)()1()(),( λ
and [ ] [ ]{ })()()()1()()(),( RXEXERXEaNESEPIRaGE xq ∧−⋅+∧⋅−⋅⋅−−= λλ
From the variance equation [ ][ ] [ ])()()()( 222222 XEAXExRXEARXEa ⋅+⋅=∧⋅+∧⋅ , we derive that
(with )()(' aRa
aR∂∂= ):
[ ] [ ] ))(())(()(
))((1
))(())(()(
)()(
3
2'
aRFaRXEAaR
aRXVrn
aaRFaRXEAaR
XVrn
a
xaR
⋅∧⋅+∧−=
⋅∧⋅+−= (Eq 13)
[ ] { }
[ ] .)(
)()1()()(
)()()()()()1()(),( ''
∧⋅+∧+−∧⋅⋅=
⋅⋅+∧⋅+⋅⋅−⋅−⋅=∂
∂
RXEARRXVrn
aQ
RXENE
aRRFRXEaRRFaNEa
RaGE
q
xqq
λ
λλλ
This equation is equal to zero if
[ ][ ]2
22
)()(
)()(
RXERXER
RXEARXEQa
∧−∧⋅∧⋅+∧= with ),0( ∞∈
−=
q
qxQλ
λλ(Eq 14)
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(From [ ]2)()( RXERXER ∧≥∧⋅ follows that 0≥a .)
Note that for ∞→R the right-hand side of the equation becomes 0 and the left-hand side stays limited to[ )∞∈ ,0a (if Q>0), which implies that (for x<1) an excess of loss protection will always be used (except for
a=0, implying that x should also be 0).
If we combine last equation with the variance constraint (Eq 11), one finds
[ ][ ][ ][ ] [ ])()(
)()(
)()( 222
22
3222 XEAXEx
RXERXER
RXEARXEQ ⋅+=∧−∧⋅
∧⋅+∧.
Due to [ ] 0)()( 2 ≥∧−∧⋅ RXERXER and also the other terms are positive (see variance formula for acompound process) we can conclude that:
[ ] [ ][ ][ ] ⇒
⋅+∧⋅+∧=∧−∧⋅)()(
)()()()( 22
3222
XEAXERXEARXE
xQ
RXERXER
[ ] [ ][ ][ ]
),0( with ,)(
)(1
)()(
)(
)()(
)()(
)()(
)(22
3222
∞∈−
=
∧+∧∧+∧⋅−=
⋅+∧⋅+∧
∧⋅+
∧∧=
q
qxQXVrn
RXVrnxQ
RXERXVrn
RXEA
XEAXE
RXEARXE
RXEx
Q
RXE
RXER
λλλ
(Eq 15)
Note that this equation depends only on the unknown R and is independent on the unknown a.
To find a feasible solution for a and R one can:• Firstly solve (Eq 15) for R.• Secondly check with (Eq 11) if the corresponding a value is less than or equal to 1. If the answer is
positive one will retain the R(Eq 15) and a((Eq 11) & (Eq 15)) as the optimal combined solution.• Thirdly, if the check is negative one needs to solve (Eq 12) for R and retain R(Eq 12) together with
a=1 as the optimal combined solution.
4.3 Stop loss as a special case of Excess of loss
If the number of claims is fixed (which corresponds with 1−=A and nN = ), then the excess of lossprotection is reduced to a fixed sum of stop loss protections. For n=1 the excess of loss reduces to a stop loss.
As stated before the retained risk is equal to ∑∑==
∧=⋅∧⋅=ℜn
ii
n
ii RXaRaXaRa
00
)()(),( , with
[ ] )(),( RXEnaRaE ∧⋅⋅=ℜ and [ ] [ ][ ])()(),( 222 RXEnRXEnaRaVar ∧⋅−∧⋅⋅=ℜ .It is clear that both equations are a n-multiple of a stop loss formulation.
4.3.1 Stop loss after quota share (QS-SL)
Equation (Eq 10) reduces now to
10)()(
)()( ≤
−=≤
⋅−∧∧+∧=
s
qsQwithSEQPSE
PSVarPSEP
λλλ
(Eq 16)
This implies that the optimal stop loss priority is independent on the quota share protection and in firstinstance also independent on the variance constraint (if P implies that 1≤a , see further).
Note that for ∞→P the left-hand side of the equation becomes ∞ and the right-hand side limited (if Q<1),
RVE - Art01_TOT.doc - 27/06/2002 18/32
which implies that (for x<1) a stop loss protection will always be used.
The variance constraint (Eq 11) becomes
)(
)(
PSStd
SStdxa
∧= with [ ]1,0∈x but also [ ]1,xa ∈ . (Eq 17)
4.3.2 Quota share after stop loss (SL-QS)
Equation (Eq 14) reduces to
[ ]2)()(
)(
PSEPSEP
PSVarQa
∧−∧⋅∧= , with
q
qsQλ
λλ −= ( ∞<≤ Q0 ). (Eq 18)
From [ ] )()( 2 PSEPPSE ∧⋅≤∧ it follows that 0≥a .
Note that for ∞→P the right-hand side of the equation becomes zero and the left-hand side stays limited to[ )∞∈ ,0a (if Q>0), which implies that (for x<1) a stop loss protection will always be used (except for a=0,
implying that x should also be zero).
The equation (Eq 15) becomes
[ ]
∞<−
=≤
∧+∧∧+∧=
∧∧
+∧∧=
q
qsQwithSVar
PSVar
x
Q
PSE
PSVarPSE
orSVar
PSVarxQ
PSEPSEPSE
P
λλλ
0,)(
)(1
)(
)()(
)()(
)(1
)()( 32
(Eq 19)
Note that this equation depends only on the unknown P and is independent on the unknown a.
5 Surplus and quota share
Combining quota share and surplus protections is a combination of two proportional reinsurance protections.This implies that the remarks concerning the “constant mean value premium principle” hypothesis are notrelevant for this combination.
For the description of this combination one can choose between the “individual model” and the “collectivemodel”. In the context of this paper, we will limit us to the “collective model”.
Due to
⋅>⋅⋅⋅⋅⋅
⋅≤⋅⋅=
>⋅
≤⋅=⋅
LakaifkXaka
LaLakaifkXa
LkifkXk
La
LkifkXakXa L ))((
))((
)(
)()( for each a, k and L,
we can conclude that ( ) ( ) )()( kXakXa LaL ⋅⋅=⋅ and consequently ( )( )LaL XaXa ⋅⋅=⋅ .
The last expression explains that the order of application for a surplus quota share combination has noinfluence on the retained risk.
We suppose a surplus reinsurance with a retained line L (expressed in terms of 100% claim size) and a quotashare with retention a.We denote the individual retained risk with exposure k after surplus line L equal to )(kX L and the mixture
RVE - Art01_TOT.doc - 27/06/2002 19/32
of all the retained risks by LX . The retained portfolio risk after surplus line L and quota share with retention
a is, in the collective model, equal to [ ]∑∑∑=
⋅==
⋅=⋅==ℜN
iiLa
N
iiL
N
iiL XaXaXaLa
0)(
00
)()()(),( .
5.1 Surplus after quota share (QS-SPL)
[ ] 1:)()(),( DQSXENEaLaE L =⋅⋅=ℜ and with [ ]
),1()(
)()( ∞−∈−=NE
NENVarA .
[ ] [ ]{ } 2222 :)()()(),( DQSXEAXENEaLaVar LL =⋅+⋅⋅=ℜ . The total reinsurance cost is equal to
3:)()()1()1( DQSXENEa q =⋅⋅+⋅− λ for the quota share, plus for the part for the surplus after quota
share [ ] 4:)()()()1( DQSXEXENEa Le =−⋅⋅⋅+ λ .
If we note [ ]4312),( DQSDQSDQSDQSLa −−−+⋅=Ψ µ , then the Lagrange equations become:
[ ] LoraWwithDQSDQSDQSW
DQSW
LaW
==−−−∂∂+
∂∂=Ψ
∂∂
0),( 4312µ . Which implies
[ ] [ ]
[ ] dkkckYE
dkkckYE
XEQXE
XEAXEXEAL
Lk
Lk
L
LLL
)()(
)()(
)()(
)()()(
2
22
⋅
⋅
⋅−⋅++⋅−=
∫
∫∞
=
∞
= ,
with for all k kkX
kY)(
)( = and )1,0(∈−
=e
qeQλ
λλ
(Eq 20)
This implies that the optimal surplus priority is independent on the quota share protection and in firstinstance also independent on the variance constraint (if L implies that 1≤a , see further).
Remark that last equation has a strong similarity with the excess of loss case (Eq 10 (the role of the priority Ris largely taken over by the line L) if we interpret the numerator as the so-called normalised portfoliovariance after surplus protection. The most important difference is induced by the ratio factor in the right-hand side.
The variance constraint [ ] [ ]1,0),(),( 2 ∈=ℜ xSVarxLaVar becomes:
[ ][ ] [ ])()()()( 222222 XEAXExXEAXEa LL ⋅+⋅=⋅+⋅ with [ ]1,0∈x and [ ]1,0∈a .
or )()( 22 XVrnxXVrna L ⋅=⋅(Eq 21)
Note that for a=1 (= no quota share protection) only the next equation has to be solved for L:
[ ][ ] [ ] )()()()()()( 22222 XVrnXVrnorXEAXExXEAXE LLL =⋅+⋅=⋅+ (Eq 22)
The variance of the retained portfolio after a surplus protection is in general not a non-decreasing functionfor the line L, implying that (Eq 22) not always possesses a unique solution. However we will furthersuppose that for [ )0,1−∈A the (weak) condition (Eq 43) of positive derivative is always fulfilled. Thiscondition implies that (Eq 21) and (Eq 22) has a unique solution for L and from (Eq 20) follows that the set
of feasible solutions for L fulfil the condition that )()( XEQXE L ⋅> implying that LYE
XEQ <⋅
)(
)( (see (Eq
26)).
To find a feasible solution for a and L one can:
RVE - Art01_TOT.doc - 27/06/2002 20/32
• First solve (Eq 20) for L within the set of { }0)()( ofsolution ; ** =⋅−> XEQXELLL L .• Secondly check with (Eq 21) if the corresponding a value is less than or equal to 1. If the answer is
positive one will retain the L(Eq 20) and a((Eq 20) & (Eq 21)) as the optimal combined solution.• Third, if the check is negative one needs to solve (Eq 22) for L and retain "L(Eq 22)" together with
a=1 as the optimal combined solution.
5.2 Quota share after Surplus (SPL-QS)
The retained risk is equal to [ ]∑∑∑=
⋅==
⋅=⋅==ℜN
iiLa
N
iiL
N
iiL XaXaXaLa
0)(
00
)()()(),( , which implies the
same formulas 11 : DSQDQS = and 22 : DSQDQS = as stated in section 5.1.
The reinsurance cost is equal to [ ] 3)()()()1( DSQXEXENE Le =−⋅⋅+ λ for the surplus (before quota
share) plus for the quota share 4:)()()1()1( DSQXENEa Lq =⋅⋅+⋅− λ .
If we note [ ]4312),( DSQDSQDSQDSQLa −−−+⋅=Ψ µ , then the Lagrange equations become:
[ ] LoraWwithDSQDSQDSQW
DSQW
LaW
==−−−∂∂+
∂∂=Ψ
∂∂
0),( 4312µ . Which implies
with ),0( ∞∈−
=q
qeQλ
λλ
[ ][ ]
[ ]
)1(
2
2
22 )()(
)()(
)()(
)()(
−
∞
=
∞
=
−⋅
⋅⋅⋅+⋅=
∫
∫LL
Lk
LkLL XEXE
dkkckYE
dkkckYE
LXEAXEQa (Eq 23)
The quota share retention a can never be negative, which implies that the line L should satisfy that
[ ]
[ ]0)()(
)()(
)()(2
2
>−⋅
⋅
∫
∫∞
=
∞
=LL
Lk
Lk XEXE
dkkckYE
dkkckYE
L . This inequality is stronger than the (weak) condition (Eq 43)
used in preceding paragraph, because )()( 22LL XEXE ≥ . From now on we will assume that this
(stronger) condition is fulfilled.The variance constraint [ ] [ ]1,0),(),( 2 ∈=ℜ xSVarxLaVar is equivalent with equation (Eq 21)Combining equation (Eq 21) and (Eq 23) and using the Vrn notation one becomes that
[ ]
[ ] )(
)()()(
)()(
)()(3
2
2
2
2
2
XVrn
XVrn
x
QXEXE
dkkckYE
dkkckYE
L LLL
Lk
Lk ⋅=
−⋅
⋅
∫
∫∞
=
∞
=
The expression between the brackets at left-hand side is by assumption positive, which implies
RVE - Art01_TOT.doc - 27/06/2002 21/32
[ ]
[ ]),0(,
)(
)(1
)(
)()(
)()(
)()(2
∞∈−
=
⋅+⋅+⋅−=
⋅
⋅
∫
∫∞
=
∞
=
q
qeL
L
LL
Lk
Lk QXVrn
XVrn
x
Q
XE
XVrnXEA
dkkckYE
dkkckYE
Lλ
λλ(Eq 24)
This equation is fully comparable with the excess of loss case, see (Eq 15). The role of the priority R islargely taken over by the line L, if we interpret the nominator as the so-called normalised portfolio varianceafter surplus protection. The most important difference is induced by the ratio factor in the left-hand side.
To find a feasible solution for a and L one can:• First solve (Eq 24) for L.• Secondly check with (Eq 23) if the corresponding a value is less than or equal to 1. If the answer is
positive one will retain the L(Eq 24) and a ((Eq 23) & (Eq 24)) as the optimal combined solution.• Third, if the check is negative one needs to solve (Eq 22) for L and retain L(Eq 22) together with
a=1 as the optimal combined solution.
6 Conclusions and general remarks
• For all the reinsurance combinations described in this paper, the optimal solutions depend on the loading
factors through a factor Q , which is of the form of α
βα
β
βαβα λ
λλλ
λλλλ
−−= or ),(Q . This implies that
the optimal solutions depend only on the relative loading structure, because),(),( βαβα λλλλλλ QQ ==⋅⋅⋅⋅ for all 0>>λ .
This is an advantage in practice, because it is much easier to obtain an estimate of the relative loadingstructure then of the absolute structure.
• In this paper we limit us to combinations of two reinsurance protections. A longer chain of reinsuranceprotections should receive more attention in the future. Also the optimal combination of an excess of losswith a stop loss protection is still an open problem.
• We derived only the results for the protection of one portfolio. However, the generalisation to moreindependent portfolios is straightforward (see for instance Bühlmann (1970) and Schmitter (2001)).
• From this paper is it clear that the order of the reinsurance protection chain is important if one uses aconstant loading proportional with the expected risk ceded to the reinsurers.This order dependency makes the question of optimal reinsurance more complex.Also the heterogeneity of the portfolio has an important impact on the optimisation question. At least itwill make the description and the computation much more complex. At the other hand it gives theopportunity to analyse a surplus treaty more efficiently.
• If we look to the combinations containing a quota share, then for a low level of reinsurance protection noquota share is needed.However, for the surplus - excess of loss combination both protections exist always together, but theimpact of the surplus is negligible for low levels of reinsurance protection and even zero if the exposurevalues K are limited (which is always the case in practice).
RVE - Art01_TOT.doc - 27/06/2002 22/32
7 Annex
7.1 Definitions, notations and well known results
• A random variable X assumes certain real numerical values in accordance with well-defined probabilities.We speak of the law of probability for a given random variable. This law can be described by means of the(cumulative) distribution function +ℜ∈≤= xwithxXxFX )(Prob)( (in this paper we limit us to non-negative random variables).
• If one can write ∫=x
XX dyyfF0
)( , one calls Xf the density function of X.
• If the context is clear we will use )(xF and )(xf instead of )(xFX and )(xf X .
• We note )(1)( xFxF −= .
• For a random variable X we define 0),,min()( ≥∀=∧ RRXRX .
Thus 0,)( ≥∀
>≤
=∧ RRXifR
RXifXRX .
It is the retained risk after an excess of loss (or stop loss) protection with priority R.
Note that
<≥
=≤∧=∧ xRif
xRifxFxRXxF X
RX 1
)()(Prob)()( and
<≥
=∧ xRifRF
xRifxfxf
X
XRX )(
)()()(
• For a random variable )(kX that depends of an exposure value +ℜ∈k (for example the insured value),
we define the random variable
>
≤=
LkifkXk
LLkifkX
kX L )(
)()( .
It is the retained risk after a surplus protection with a retention line equal to L +ℜ∈ .
Note that
<
≥=≤=
kLifxL
kF
kLifxFxkXxF
kX
kX
LkX L )(
)())((Prob)(
)(
)(
)( and
<
≥=
kLifxL
kf
L
kkLifxf
xfkX
kX
kX L )(
)()(
)(
)(
)(
• Suppose a non-negative exposure value K (for example sum insured) characterised by a density function
)(kc and a distribution function ∫=
=k
l
dllckC0
)()( .
Define X as the mixture of X(k) over K, the marginal distribution function of X is equal to
∫∞
=
⋅=0
)( )()()(k
kXX dkxFkcxF with density function ∫∞
=
⋅=0
)( )()()(k
kXX dkxfkcxf . Analogous is
∫∫∫∞
==
∞
=
⋅+⋅=⋅=Lk
kX
L
k
kX
k
kXX dkxL
kFkcdkxFkcdkxFkcxF
LL)()()()()()()( )(
0
)(
0
)( and
∫∫∫∫∫∫∞∞
====
∞∞
==
⋅⋅⋅⋅++⋅⋅==⋅⋅==Lk
kX
L
k
kX
k
kXX dkxL
kf
L
kkcdkxfkcdkxfkcxf
LL)()()()()()()( )(
0
)(
0
)( .
RVE - Art01_TOT.doc - 27/06/2002 23/32
For each k we define kkX
kY)(
)( = and Y as the mixture of Y(k) over K. Implying that
)()( )()( ykFyF kXkY ⋅= and )()( )()( ykfkyf kXkY ⋅⋅= , which results in
∫∫∞
=
∞
=
⋅⋅=⋅=0
)(
0
)( )()()()()(k
kX
k
kYY dkykFkcdkyFkcyF and
∫∫∞
=
∞
=
⋅⋅⋅=⋅=0
)(
0
)( )()()()()(k
kX
k
kYY dkykfkkcdkyfkcyf
• If we combine an "excess of loss reinsurance with priority R" after a "surplus protection with line L"one becomes the retained risk ),min()( LL XRRX =∧The corresponding distribution and density function have following expressions:
⋅+⋅=
>≤
= ∫∫∞
==∧
1
)()()()(1
)()( )(
0
)()( Lk
kX
L
k
kXXRX
dkxLk
FkcdkxFkcRxif
RxifxFxF L
L and
⋅+⋅
=
>=<
= ∫
∫∫∞
=
∞
==
∧
0
)(
)()()()(
0
)(
)(
)(
)(
0
)(
)(
Rx
X
Lk
kX
L
k
kX
X
X
RX dxxf
dkxLk
fLk
kcdkxfkc
Rxif
RxifRF
Rxifxf
xfLL
L
L
• Note that if for all k ZkkX ⋅=)( or ZkY =)( previous formula reduces to
)()()( kx
FxF ZkX = and kk
xfxf ZkX
1)()()( ⋅= . Which implies for Rx < :
)()()()(
)()()()()()()()()(
0
00
)(
LCLx
Fdkkx
Fkc
dkkcLx
Fdkkx
Fkcdkkx
Lk
Fkcdkkx
FkcxF
Z
L
k
Z
Lk
Z
L
k
Z
Lk
Z
L
k
ZRX L
⋅+⋅=
+⋅=⋅+⋅=
∫
∫∫∫∫
=
∞
==
∞
==∧
and
)()(
)()(
)()(1
)()(
)()(
)()(
)(
0
00
)(
Lx
fLLC
dkkx
fkkc
dkkcL
xf
Ldk
k
xf
k
kcdk
L
k
k
x
L
kf
k
kcdk
k
xf
k
kcxf
Z
L
k
Z
Lk
Z
L
k
Z
Lk
Z
L
k
ZRX L
⋅+⋅=
⋅+⋅=⋅⋅+⋅=
∫
∫∫∫∫
=
∞
==
∞
==∧
Note that these formulas not explicitly depend on R, which implies that they also correspond for a "surplusreinsurance without excess of loss protection" (the case ∞=R ).
For Rx ≥ are the formulas straightforward and are left to the reader.
RVE - Art01_TOT.doc - 27/06/2002 24/32
• A compound process is a random sum ∑=
=N
iiXS
0
with for all X~to1fromi iX∞ , iid, X0 = 0 and X
independent on the non-negative random number N. We will call S the aggregate loss and Xi the (individual)loss.For the first 2 moments of a compound process one has the following well-known formulas:
)()()( XENESE ⋅= and
[ ][ ] ),1(
)(
)()(0)()()(
)()()()()()()()()()(
22
222
∞−∈−=≥⋅+⋅=
⋅−+⋅=⋅+⋅=
NE
NENVarAwithXEAXENE
XENENVarXENEXENVarXVarNESVar
)()()( XVarNESVar ⋅≥ and )()()( 2 XENVarSVar ⋅≥
(Eq 25)
We will use in this paper the abbreviation )()()( 22 XEAXEXVrn ⋅+= , which can be understood as astandard deviation of the portfolio normalised by the expected number of claims. Note that
)()( XVarXVrn ≥ .
7.2 Some basic derivations
For the application of the mean variance principle one needs to calculate the (partial) derivatives of thevariance and expected value of retained risk with respect to priority (in case of excess of loss and stop lossreinsurance), retained line (in case of surplus protection) and retained share (in case of quota share).To avoid a multiple derivation and to obtain the advantage of a general overview of these results we bringthem together in one subsection.
We suppose that all the random variables will have very nice mathematical properties such that integrals canbe changed of order, derivatives will exist and so on. We don't want to minimise the importance of all thesenecessary conditions (and the research behind it), but like firstly to concentrate on the results one canbecome for the optimisation problem.
7.2.1 Surplus & excess of loss
7.2.1.1 General caseFor an excess of loss protection after a surplus reinsurance we have that:
∫ ∫∫
∫ ∫∫
∫∫
∞
=
∞
==
=
∞
==
∞
==
⋅⋅+⋅
+
⋅⋅+⋅⋅=
+⋅=∧
Rx Lk
kX
L
k
kX
R
x Lk
kX
L
k
kX
Rx
X
R
x
XL
dxdkxLk
fLk
kcdkxfkcR
dxdkxL
kf
L
kkcdkxfkcx
dxxfRdxxfxRXELL
)()()()(
)()()()(
)()()(
)(
0
)(
0
)(
0
)(
0
Changing the integrals gives:
RVE - Art01_TOT.doc - 27/06/2002 25/32
⋅⋅++⋅⋅⋅⋅
++
++⋅⋅==∧∧
∫∫∫∫∫∫
∫∫∫∫∫∫∞∞
====
∞∞
==
∞∞
======
Rx
kX
R
x
kX
Lk
Rx
kX
R
x
kX
L
k
L
dxxL
kf
L
kRdxx
L
kf
L
kxdkkc
dxxfRdxxfxdkkcRXE
)()()(
)()()()(
)(
0
)(
)(
0
)(
0
With L
xky
⋅= and LRr = , we conclude:
[[ ]]
[[ ]]
[[ ]] [[ ]]
[[ ]] [[ ]]rYELdkrkYEkcL
dkrkYEkcLdkrkYEk
Lkkc
dkrkYEkcLdkkL
rkYEkkc
dyyfL
kRdyyyf
k
LdkkcdkRkXEkcRXE
k
Lk
L
k
Lk
L
k
L
Rky
kX
L
Rk
y
kX
Lk
L
k
L
∧∧⋅⋅==∧∧⋅⋅⋅⋅==
∧∧⋅⋅++∧∧⋅⋅⋅⋅≤≤
∧∧⋅⋅++
∧∧⋅⋅⋅⋅==
++++∧∧==∧∧
∫∫
∫∫∫∫
∫∫∫∫
∫∫∫∫∫∫∫∫
∞∞
==
∞∞
====
∞∞
====
∞∞
====
∞∞
====
0
0
0
)(
0
)(
0
)()(
(*) )()()()(
)()()()(
)()()()()()(
(Eq 26)
(*) follows from: for all Lk ≤≤ holds [ ]rkYEk
L
k
LrkYE ∧⋅≤
∧ )()( because both functions have the
same value for 0==r but the derivative of the left-hand side function is never greater than the derivative of
the right-hand side function ( )()( )()( rFk
L
k
L
k
LrF kYkY ⋅≤⋅ because r
k
Lr ≥≥ ).
Completely analogous one becomes that:
[[ ]] (( ))[[ ]] (( ))[[ ]]∫∫∫∫∞∞
====
∧∧++∧∧==∧∧Lk
L
k
L dkrkYEkcLdkkRkYEkkcRXE 22
0
222 )()()()()( (Eq 27)
For the derivatives of these formulas we need a less common known calculus result:
∫∫==
⋅+=
∂∂ L
k
L
k
dkLhLhkgLhLgdkLhkgL 0
''
0
)())(,())(,())(,( with ),(),(' LkgL
Lkg∂∂= and
)()(' LhL
Lh∂∂=
RVE - Art01_TOT.doc - 27/06/2002 26/32
[[ ]]
[[ ]]
∫∫∞∞
==⋅⋅⋅⋅−−∫∫
∞∞
==
∧∧
==
∫∫∞∞
== ∂∂
∧∧∂∂
++
∧∧
−−∫∫∞∞
==
∧∧
++∧∧⋅⋅==
∧∧
∂∂∂∂
++
∧∧⋅⋅
∂∂∂∂
==∂∂
∧∧∂∂∫∫∫∫∞∞
====
Lkdk
L
R
L
RkkXFkc
Lkdk
kL
RkkXE
kc
Lkdk
LLR
kkXE
kkc
L
LLR
LLXELcL
Lkdk
kLR
kkXEkcRLXELc
dkk
L
RkkXE
kcLL
dkRkXEkcLL
RXE
Lk
L
k
L
)()()()(
)(
)()(
)()(
)()()()(
)(
)()()()(
0
Conclusion
[[ ]] )(0)()()()(
)()()(
)()(
)(
)(
LRrwithdkkcrFrdkkcrkYE
dkLR
kFkcLR
dkk
LR
kkXEkcRXE
L
Lk Lk
kY
Lk
kX
Lk
L
==≥≥⋅⋅−−⋅⋅∧∧==
⋅⋅−−
∧∧
==∧∧∂∂∂∂
∫∫ ∫∫
∫∫∫∫∞∞
==
∞∞
==
∞∞
==
∞∞
== (Eq 28)
Completely analogous one becomes that:
[[ ]]
[[ ]] 0)()()())((2
,)(0)()(2
)())(()(
2
)()(2
))((
)(2)(
)(22
0
)(2
)(2
222
2
)(
2
2
2
2
≥≥
⋅⋅−−⋅⋅∧∧==
==≥≥
⋅⋅==
−−
∧∧==
−−
∧∧
==∧∧∂∂∂∂
∫∫∫∫
∫∫ ∫∫
∫∫
∫∫∫∫
∞∞
==
∞∞
==
∞∞
== ==
∞∞
==
∞∞
==
∞∞
==
Lk
kY
Lk
Lk
r
x
kY
Lk
kX
Lk
kX
Lk
L
dkkcrFrdkkcrkYEL
orLRrwithdkdyyfykcL
dkLR
kFLR
kLR
kkXEkkc
L
dkL
RkFkc
L
Rdk
kL
RkkXE
kcLRXEL
(Eq 29)
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( ) [ ]
[ ]
[ ] [ ]
[ ] 0 if 0)()()()(2
)()()()())(()(2
)()(2
)()(
)()()(
)(
2
2
221
≥≥⋅⋅∧⋅+⋅⋅
−
⋅∧⋅∧⋅+⋅∧⋅⋅=
∂
∧∂⋅∧⋅+∂
∧∂⋅=
∂
∧∂⋅+∂
∧∂⋅=∂
∧∂
∫
∫∫
∑
∞
=
∞
=
∞
=
=
AdkkcrFrRXEALrNE
dkkcrkYERXEAdkkcrkYELNE
L
RXERXEA
L
RXENE
L
RXEA
L
RXENE
L
RXVar
Lk
kYL
Lk
L
Lk
LL
L
LL
N
iiL
(Eq 30)
For [ ))0,1−∈A is this derivative positive if (we denote A for the absolute value of A)
∫ ∫∫ ∫∞
= =
∞
= =
⋅⋅∧⋅≥
⋅
Lk
r
x
kYL
Lk
r
x
kY dkdyyfykcL
RXEAdkdyyfykc
0
)(
0
)(2 )()(
)()()( (Eq 31)
From the equations (Eq 47), (Eq 48) and (Eq 51) we also known that:
0)()( ≥=∧∂∂
RFRXER LXL , [ ] 0)(2)( 2 ≥⋅=∧
∂∂
RFRRXER LXL and
( )[ ] 0)()()(21 ≥∧⋅+⋅⋅=
∂
∧∂ ∑
= RXEARRFNER
RXVar
LX
N
iiL
L
(Eq 32)
7.2.1.2 Special case: ZkkX ⋅=)(
Note that for all k ZkkX ⋅=)( or ZkY =)( previous formulas reduce to ( LRr = )
[[ ]] [[ ]]
[[ ]] [[ ]] [[ ]]rZELLCrZELdkrZEkL
kck
LCrZELdkkRZEkckRXE
L
k
L
k
L
∧∧⋅⋅==⋅⋅∧∧⋅⋅++∧∧⋅⋅⋅⋅≤≤
⋅⋅∧∧⋅⋅++∧∧⋅⋅⋅⋅==∧∧
∫∫
∫∫
==
==
)()((*)
)()()(
0
0(Eq 33)
For (*) see (Eq 26)
[[ ]] (( ))[[ ]] (( ))[[ ]] )()()( 22
0
222 LCrZELdkkRZEkkcRXEL
k
L ⋅⋅∧∧⋅⋅++∧∧⋅⋅⋅⋅==∧∧ ∫∫==
(Eq 34)
[ ][ ] ∫=
⋅=⋅⋅−∧=∧∂∂ r
z
ZZL dzzfzLCLCrFrrZERXEL 0
)()()()()( (Eq 35)
[[ ]] (( ))[[ ]][[ ]] ∫∫==
⋅⋅⋅⋅⋅⋅==⋅⋅⋅⋅−−∧∧⋅⋅==∧∧∂∂∂∂ r
z
ZZL dzzfzLCLLCrFrrZELRXEL 0
2222 )()(2)()(2)( (Eq 36)
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)()()()()()(0
LCLR
FdkkR
FkcRFRXER Z
L
k
ZXL L⋅+⋅==∧
∂∂
∫=
and [ ] )(2)( 2 RFRRXER LXL ⋅=∧
∂∂
(( ))
⋅⋅⋅⋅∧∧⋅⋅++⋅⋅⋅⋅⋅⋅⋅⋅==
∂∂
∧∧∂∂
∫∫∫∫∑∑
====
==r
z
ZL
r
z
Z
N
iiL
dzzfzRXEAdzzfzLLCNEL
RXVar
00
21 )()()()()(2
(( ))[[ ]]))()()()()()(2
0
1 RXEARLCrFdkk
RFkcNE
R
RXVar
LZ
L
k
Z
N
iiL
∧∧⋅⋅++⋅⋅
⋅⋅++⋅⋅⋅⋅==
∂∂
∧∧∂∂
∫∫∑∑
==
==
7.2.1.3 Optimal solution
As mentioned before (see paragraph 3) the Lagrange equations become 0),( =Ψ∂∂
RLW
with W=L or R,
and results in:
[ ] { } ⇒=⋅++−⋅⋅+∧⋅+⋅=Ψ∂∂
0)()1()()(1
)()()(2),( RFRFNERFRXEARNERLR LLL XxXXL λ
µ
xL RXEAR λµ2
1)( −=∧⋅+ (Eq 37)
[ ]
[ ]⇒
∂∂
−−∂
∧∂−=
∂∧∂
∧⋅+∂
∧∂
⇒=
∂∧∂−
∂∂⋅+−
∂∂++
∂∧∂−⋅⋅
+
∂
∧∂∧⋅+
∂∧∂
⋅=Ψ∂∂
L
XE
L
RXE
L
RXERXEA
L
RXE
L
RXE
L
XE
L
XE
L
RXENE
L
RXERXEA
L
RXENERL
L
Lex
Lx
LL
L
LLx
Le
L
LL
L
)()(
)(1)()(2
)(
0)()(
)1()(
)1()(
)(1
)()(2
)()(),(
2
2
λλλµ
λλµ
Denoting )1,0(∈−=x
exQλ
λλ
[ ]
−
⋅
−=
⋅∧⋅+
⋅
∫∫ ∫
∫ ∫∫ ∫∞
=
∞
= =
∞
= =
∞
= =
dkkck
kXEQdkdxxfx
k
kc
dkdxxfxk
kcRXEAdkdxxfx
k
kcL
LkLk
rk
x
kXx
Lk
rk
x
kXL
Lk
rk
x
kX
)()(
)()(
2
1
)()(
)()()(
0
)(
0
)(
0
)(2
2
λµ
(Eq 38)
It becomes also clear that the random variable )(kX divided by its exposure value k plays an important role.
We define k
kXkY
)()( == for each k>0 and
LX
Y LL = for each L>0.
RVE - Art01_TOT.doc - 27/06/2002 29/32
Using the notations CB, CB1, CB2, CC, CC1 and CD defined as follows
[ ] 21)()())(()(
)()()()(
)(22
0
)(2
0
)(2
2
CBCBdkkcrFrdkrkYEkc
dkdxyfykcdkdxxfxk
kcCB
Lk
kY
Lk
Lk
r
x
kY
Lk
rk
x
kX
+=⋅−∧=
⋅=
⋅=
∫∫
∫ ∫∫ ∫∞
=
∞
=
∞
= =
∞
= =
[ ]r
CBCCdkkcrFrdkrkYEkcdkdxxfx
k
kcCC
Lk
kY
LkLk
rk
x
kX
21)()()()()(
)()(
0
)( +=⋅−∧=
⋅= ∫∫∫ ∫
∞
=
∞
=
∞
= =
[ ] [ ] dkkckYEdkkck
kXECD
LkLk
)()()()(
∫∫∞
=
∞
=
==
We can rewrite (Eq 37) like { }CDQCCCCRXEACBL xL ⋅−−=⋅∧⋅+⋅ λµ2
1)( and combining with
(Eq 38) gives:
[ ]{ } ⇒⋅−∧⋅+=⋅∧⋅+⋅ CDQCCRXEARCCRXEACBL LL )()(
[ ] [ ] [ ]r
CBRXEARCDQCCRXEAR
r
CBRXEACBLCCRXEACBL
LL
LL
2)(1)(
2)(21)(1
∧⋅++⋅−⋅∧⋅+
=∧⋅+⋅+⋅∧⋅+⋅
This reduces nicely to [ ] [ ] 1)(11)( CCRXEACBLCDQCCRXEAR LL ⋅∧⋅+⋅=⋅−⋅∧⋅+ orequivalently
[ ] [ ]
[ ] [ ]
[ ] [ ]{ }
[ ] [ ]{ }∫
∫
∫∫
∫∫
∞
=
∞
=
∞
=
∞
=
∞
=
∞
=
⋅⋅−∧
⋅∧⋅∧⋅+∧=
⋅−∧
∧⋅∧⋅+∧=∧⋅+
Lk
Lk
L
LkLk
Lk
L
LkL
dkkckYEQrkYE
dkkcrkYErYEArkYE
dkkckYEQdkrkYEkc
dkrkYEkcrYEAdkrkYEkc
rYEAr
)()()(
)()()())((
)()()()(
)()()())(()(
)(
2
2
(Eq 39)
7.2.2 Surplus & individual loss
Based on (Eq 26) and (Eq 27)with ∞=R we become that the first 2 moments for the individual loss after asurplus protection:
[ ] [ ] )()()()()()(0
YELdkkckYELdkkckYEkXELk
L
k
L ⋅≤⋅+⋅⋅== ∫∫∞
==
(Eq 40)
This implies for ∞=L that [ ] [ ] dkkckYEkdkkckXEXEkk
)()()()()(00
⋅⋅=⋅= ∫∫∞
=
∞
=
.
RVE - Art01_TOT.doc - 27/06/2002 30/32
[ ] [ ] )()()()()()( 2222
0
222 YELdkkckYELdkkckYEkXELk
L
kL
⋅≤⋅+⋅⋅= ∫∫∞
==
(Eq 41)
This implies for ∞=L that [ ] [ ] dkkckYEkdkkckXEXEkk
)()()()()(0
2
0
22 ⋅⋅=⋅= ∫∫∞
=
∞
=
.
Their derivatives follow from (Eq 28), (Eq 29), (Eq 30):
[ ] 0)()()(
≥⋅=∂
∂∫∞
=
dkkckYEL
XE
Lk
L and [ ] 0)()(2)( 2
2
≥⋅=∂
∂∫∞
=
dkkckYELL
XE
Lk
L(Eq 42)
( )[ ] [ ]{ } 0A if 0)()()()()(2 21 ≥≥
⋅⋅⋅+⋅⋅=
∂
∂
∫∑ ∞
=
= dkkckYEXEAkYELNEL
XVar
Lk
L
N
iiL
The general condition to have a positive derivative for [ )0,1−∈A is given by
[ ]
[ ] L
AXE
dkkckYE
dkkckYE
L
Lk
Lk ⋅≥⋅
⋅
∫
∫∞
=
∞
= )(
)()(
)()(2
(with A denoting the absolute value of A)(Eq 43)
7.2.2.1 Special case: ZkkX ⋅=)(
Note that for all k ZkkX ⋅=)( or ZkY =)( previous formulas reduce to
)()()( LKEZEXE L ∧⋅= and [ ]222 )()()( LKEZEXEL
∧⋅= (Eq 44)
[ ] )()()()()( 2222 LKEZELKEZEXVar L ∧⋅−∧⋅= (Eq 45)
0)()()(
≥⋅=∂
∂LCZE
L
XE L and 0)()(2)( 2
2
≥⋅⋅=∂
∂LCZEL
L
XE L (Eq 46)
( )[ ] [ )
[ ][ ] 0)()()(2)()()()(2
)()()()()(2
,1,)()()()()(2
22
22
221
≥⋅⋅⋅⋅=⋅−⋅⋅⋅⋅≥
∧⋅−⋅⋅⋅⋅≥
∞−∈∧⋅⋅+⋅⋅⋅⋅=∂
∂ ∑=
ZVarLNELCLZEZELNELC
LKEZEZELNELC
ALKEZEAZELNELCL
XVarN
iiL
7.2.3 Stop loss and excess of loss
For an excess of loss or stop loss we have that:
RRFRxxdFxdFRxxdFRXERR
R
≤⋅+=+=∧ ∫∫ ∫∞
00
)()()()()( and )()( RFRRXE ⋅≥∧
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0)()()()()( ≥=+⋅−⋅=
∂∧∂
RFRFRfRRfRR
RXE(Eq 47)
2
0
22
0
222 )()()()()( RRFRxdFxxdFRxdFxRXERR
R
≤⋅+=+=∧ ∫∫ ∫∞
02)(2)(2)()()( 22
2
≥≥⋅=⋅+⋅−⋅=∂
∧∂RRFRRFRRfRRfR
RRXE (Eq 48)
[ ] 0)()(2)()()( 22
≥⋅∧−=∂
∧∂−∂
∧∂=∂
∧∂RFRXER
RRXE
RRXE
RRXVar (Eq 49)
[ ] [ ][ ] 22
22
)()()()()(
)()()()()(
RRFRRXERFRRXERFRR
RXERXERRXERXERXVar
≤⋅≤∧⋅⋅=∧⋅⋅−≤
∧⋅∧−≤∧−∧=∧(Eq 50)
( ) [ ]
[ ] 0)()()(2
)()()(
221
≥∧⋅+⋅=
∂
∧∂⋅+∂
∧∂⋅=∂
∧∂ ∑
=
RXEARRFNE
R
RXEA
R
RXENE
R
RXVarN
ii
(Eq 51)
which implies that the variance of a compound excess of loss contract is (strictly) increasing because1−≥A .
If the distribution LX a mixture is of )(kX L , with ( ]∞∈ ,0LX , over the exposure distribution K, we referto the corresponding equations in section 7.2.1.
8 References
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General Reinsurance Company Limited), Brentford, Middlesex, 1995.Centeno, M. L. Measuring the effects of reinsurance by the adjustment coefficient in the
Sparre Anderson Model, Insurance: Mathematics and Economics, 30, 2002,37-50
Centeno, M. L. On Combining Quota-share and Excess of loss, Astin Bulletin, 15, 1985, 49-63
Daykin ,C.D., Pentikaïnen, T.& Pesonen, M.:
Practical Risk Theory for Actuaries, Chapman & Hall, 1994
Embrechts, P., KlüppelbergC. & Mikosch, T.
Modelling Extremal Events for Insurance and Finance, Springer, Berlin,1997
Gerathewohl, K. Reinsurance Principles and Practice, Volume I, VerlagVersicherungswirtschaft e. V., Karlsruhe, 1980
Gerathewohl, K. Reinsurance Principles and Practice, Volume II, VerlagVersicherungswirtschaft e. V., Karlsruhe, 1982
Goovaerts, M.J., De Vylder,F., Haezendonck, J.
Insurance Premiums, Theory and Applications, North-Holland, 1983
Goovaerts, M.J., Kaas, R.,Van heerwaarden, A.E.,Bauwelinckx, T.
Effective Actuarial Methods, North-Holland, Amsterdam, 1993
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Goovaerts, M.J., Kaas, R.,Van Heerwaarden, A.E.,Bauwelinckx, T.
Ordening of Actuarial Risks, Caire Education Series 1, 1994
Kaas, R., Goovaerts, M.J. Inleiding Risicotheorie, 1993Kaluszka, M Optimal reinsurance under mean-variance premium principles, Insurance:
Mathematics and Economics, 28, 2001, 61- 67Kiln, R. Reinsurance in Practice, Witherby & Co., London, 1982Klugman, S. A., Panjer, H.,Willmot, G. E.
Loss Models, From Data to Decisions, John Wiley & Sons, New York, 1998
Schmitter, H. Setting optimal reinsurance retentions, Swiss Reinsurance Company, Zurich,2001
Straub, E. Non-life Insurance Mathematics, Springer Verlag, 1988