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Reinsurance: Optimization andanalysis
MMA713 Actuarial Mathematics
Polite Mpofu Esther Torres Victor Lopez LopezShedrack Lutembeka Bo Yuan Benas Bacanskas
22nd May 2015
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1 Abstract
The purpose of this seminar is to consider optimal reinsurance from an in-surer’s point of view by the application of utility theory in reinsurance, andillustrate two different approaches to the problem. We will show how util-ity theory can be applied to determine the optimal retention level underboth proportional and excess of loss reinsurance. Furthermore, we have ana-lyzed the behavior of the results depending on different parameters using themathematical software R.
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Contents
1 Abstract 2
2 Introduction 4
3 Reinsurance 53.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.2 History of Reinsurance . . . . . . . . . . . . . . . . . . . . . . 63.3 Types of Reinsurance Treaties . . . . . . . . . . . . . . . . . . 83.4 Types of Treaty/Obligatory Reinsurance . . . . . . . . . . . . 8
3.4.1 Proportional Reinsurance . . . . . . . . . . . . . . . . 83.4.2 Non Proportional Reinsurance . . . . . . . . . . . . . . 10
4 Application of Utility Theory in Insurance 124.1 Utility Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.2 Utility Functions . . . . . . . . . . . . . . . . . . . . . . . . . 134.3 Exponential utility function . . . . . . . . . . . . . . . . . . . 14
5 Proportional reinsurance 145.1 Utility optimization . . . . . . . . . . . . . . . . . . . . . . . . 145.2 Simulation and analysis . . . . . . . . . . . . . . . . . . . . . 165.3 a as a function of A . . . . . . . . . . . . . . . . . . . . . . . . 165.4 a as a function of β . . . . . . . . . . . . . . . . . . . . . . . . 18
6 Excess of loss reinsurance 206.1 Utility optimization . . . . . . . . . . . . . . . . . . . . . . . . 206.2 Simulation and analysis . . . . . . . . . . . . . . . . . . . . . 23
7 Conclusion 27
8 Appendix 28
9 Reference 31
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2 Introduction
In early periods the term reinsurance was used to refer to different types ofagreement from what we have today, there was no uniform meaning. An in-surer took over the obligation accepted by another insurer who either wishedto withdraw from the business or had died or gone bankrupt. Today rein-surance is refered to as the transfer of part of the risks that a direct insurerassumes by way of insurance contract on behalf of an insured, to a secondinsurance carrier, the reinsurer, who has no direct contractual relationshipwith the insured. It is a form of risk sharing between two parties rather thantransfer of total risk to one party.
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3 Reinsurance
3.1 Definitions
Reinsurance is an insurance on insurance or an insurance for the insurers.It is a contractual agreement between an insurer (cedent) and a reinsurerwhereby the reinsurer indemnifies part of the losses incurred on the insurer.The figure below illustrates this concept.
Figure 1: see reference 4
In a reinsurance contract, one party (the reinsurer) for a certain premiumagrees to indemnify another party (called the reinsured, the first-line insureror also the cedant) for specified parts of its underwritten insurance risk. Thereinsurance treaty thus becomes a blank cheque which the reinsurer grantsthe cedant in order to provide it with reinsurance capacity during the periodof the treaty. This treaty is also known as blind treaty because the cedantknows the risks ceded to the treaty before the reinsurer does. The reinsureris informed of the treaty development by means of quarterly or, half-yearlyaccounts.
There are different motivations for an insurance company to buy reinsur-ance, these include:
• To reduce the probability of suffering losses that it cannot easily copewith, in particular the appearance of excessively large or unusuallymany claims from the underwritten policies.
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• To increase the underwriting capacity (which is particularly but notexclusively relevant for smaller insurance companies) and can be in-terpreted as a (virtual) increase of the solvency capital of the cedingcompany.
• To smooth out the financial results of an insurance company, makingthem more predictable by absorbing larger losses. This enables easierbusiness planning and financial projections.
• To gain arbitrage in purchasing reinsurance coverage at a lower ratethan they charge the insured for the underlying risk, whatever theclass of insurance
• To gain the expertise of a reinsurer, or the reinsurer’s ability to set anappropriate premium, in regard to a specific (specialised) risk.
3.2 History of Reinsurance
Insurance and Reinsurance basically originated in maritime trade, the rein-surance institution grew up mainly in relation to marine transport. DuringGreek and Roman times, a type of marine insurance known as bottomryloans existed and was governed by Roman law. This continued up to theMiddle Ages and constituted a fledgling insurance system. These types ofcontracts were used to finance the purchase of commodities that were goingto be transported by sea so that, if the cargo reached its destination safe andsound, the person financing the voyage received the amount of the loan plussubstantial interest.
The first known reinsurance contract (written in Latin) was affected inGenoa in July 1370. It concerned a cargo that was to be carried by sea fromCadiz (in Spain) to Sluis (in Flanders) and was insured. However because ofthe dangerous nature of the voyage, the insurer decided to transfer most ofthe risk to a second insurer, who accepted it. The owner of the cargo didnthave any contractual relationship with the reinsurer. This contract had 2interesting aspects in relation to reinsurance:
• Only the last part of the route was reinsured (not form Genoa to Cadiz,but from Cadiz to Flanders) due to its particular risk.
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• The most substantial risk was transferred, a form in general use today.
According to records, the first insurer to have used printed insurancepolicies was based in Seville in 1552. Those early insurers needed to dilutethe risks accepted by adopting a system of coinsurance: The largest riskswere shared amongst themselves according to different, previously agreed,percentages. The sums insured per policy grew rapidly, leading the numberof coinsurers per risk to grow too, as each one didnt want to have significantshares.
This situation led to these first insurers into cumbersome administrativesituations and even, in some cases to a lack of capacity. This led to develop-ment of pure reinsurance, becoming thoroughly accepted in the 17th century.These grew stronger over time, and even more with industrial developmentand the appearance of large accumulations of sums insured per risk.
Examples of early reinsurances
• First known fire reinsurance contract dates back to 1813 between EagleFire Insurance Company, of New York, and Union Insurance.
• Fire reinsurance contract between National and Imperial Fire, 1824
• Hail reinsurance contract dates back to 1854 in Trieste between Mag-deburger and Riunione Adriatica.
• First life reinsurance contract were effected in England in 1844 andon the Continent in 1858 between Schweizerische Rentenanstalt andFrankfurter Ruck.
• First accident reinsurance was taken out by two Scottish companies in1888.
Today the two largest reinsurance companies (in terms of volume of busi-ness) have their head offices in Germany and Switzerland. They are followedby the North American reinsurers.
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3.3 Types of Reinsurance Treaties
There are two types of reinsurance treaties; these are facultative reinsuranceand treaty/obligatory or automatic reinsurance.
Where the subject-matter of the treaty is a policy or an individual risk,this is known as facultative reinsurance; here the direct insurer chooses freelywhich particular, individual risks he wants to offer to a reinsurer. Likewisethe reinsurers, for their part, are at liberty to accept or reject the offer.
With obligatory reinsurance, a treaty is arranged for the reinsurance of aspecific portfolio, with risks being ceded automatically for the entire portfoliowithin the terms of the treaty. Cedents do not have to decide whether ornot to cede each individual risk but undertake to cede the entire portifolio.Nor do the reinsurers have to go through an individual acceptance procedure,since they are contractually bond to accept the entire portfolio.
3.4 Types of Treaty/Obligatory Reinsurance
Let X(t) =∑N(t)
i=1 Xi denote the aggregate claim amount in an insuranceportfolio up to time t, where the counting process Nt specifies the numberof claims up to time t and the random variable Xi denote the claim sizes.In every reinsurance treaty, this amount is split into X(t) = D(t) + R(t),Where D(t) is the amount that the insurance company retains for itself (thedeductible) and R(t) is the reinsured amount to be covered by the reinsurancecompany.
3.4.1 Proportional Reinsurance
Reinsurance treaty is called proportional when the cession to reinsurance ismade proportionately to the characteristics of the policy. The cedant andthe reinsurer participate proportionally in the insured risk. This proportion-ality means that premiums, claims and expenses will be apportioned in theproportion established in the treaty.
The types of proportional reinsurance are Quota Share, Surplus, Com-bined (Quota Share and Surplus) and Mixed (Facultative Obligatory and
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Obligatory Facultative). Some of these are discussed below.
Quota-Share ReinsuranceIn this case a reinsurer participates with a fixed share in all risks accep-
ted by the cedant in the class of business or type covered by the treaty. Thecedant in turn retains part of each risk. The cedants and reinsurers parti-cipations are fixed based on a certain percentage of each risk written by thecedant. This treaty is suitable when the aim is to share all the risks in aportfolio. It is mostly preferred by the newly formed insurance companies,since they do not have prior statistics and experience and they require thefinance of reinsurers during the early years.
In the Quota-Share (QS) treaty, one has:
R(t) = aX(t), D(t) = (1− a)X(t) (1)
For some proportionality factor 0 < a < 1. In this case it is straight-forward to calculate the distribution of R(t) and D(t) , once the distributionof X(t) is known.
Surplus ReinsuranceSurplus reinsurance is also known as reinsurance of amounts because the
cedants retention is fixed as an amount of each risk or policy. This amountmay vary depending on the different types of risk forming a portfolio, thusallowing the cedant to retain a greater or lesser proportion of risks. It isa proportional treaty because both parties participate in the premiums andclaims in the same percentage as they have retained and accepted, respect-ively. Surplus treaties are quite popular in practice, in particular in fire,marine and storm insurance.
Whereas in a QS treaty also small claims are reinsured, a surplus treatyonly reinsures claims whose corresponding insured sum exceeds some reten-tion M. In the latter case the first line insurer deducts a proportion and shiftsthe remaining part to the reinsurer. Consequently,
R(t) =
N(t)∑i=1
(1− M
Qi
)Xi1Qi>M (2)
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D(t) =
N(t)∑i=1
XiI(Qi 6M) + (MXi
Qi
)1Qi>M (3)
Combined Reinsurance (Quota Share/Surplus)Combined treaties favor portfolio growth and the cedants possibilities of
writing business, thereby creating a larger and increasing premium volumewhich, in turn, will enable the cedant to increase retentions in the future andtake over an ever greater portion of the Quota Share. This is sometimes usedin companies or classes of business that are starting up because it allows thereinsurer to participate in the cedants net retention, thus compensating forthe initial imbalance in the Surplus treaty.
3.4.2 Non Proportional Reinsurance
Unlike proportional insurance, in non proportional reinsurance, the alloca-tion of liabilities between the cedant and the reinsurer is based on claims andnot on the sum insured. The reinsurer accordingly undertakes to indemnifythe cedant when the amount of claims exceeds a previously agreed amount(the deductible) and up to a maximum limit (the limit cover).
The types of proportional reinsurance are Per-risk XL cover, Per eventcover (Catastrophe XL), and Stop-Loss Cover and Large Claim Reinsurance,as discussed below.
Excess-of-Loss ReinsuranceIn an Excess-of-Loss (XL) treaty, for each individual claim the excess oversome retention M is paid by the reinsurer:
R(t) =i=1∑N(t)
(Xi −M)+, D(t) =i=1∑N(t)
min(Xi,M) (4)
Excess-of-Loss (XL) treaty can either be Per-risk XL cover or Per eventcover (Catastrophe XL).
Per- risk XL cover
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This type of treaty protects the insurer against claims exceeding theamount that it has decided to retain for own account on a given risk. Thistype of cover generally replaces part of the surplus reinsurance programme.
Per event cover (Catastrophe XL)
This treaty offers the insurer protection against accumulations which arisewhen numerous claims are caused by the same event (storm, earthquake) af-fecting many policies, it generally covers the retention against catastropherisks. It is used in all classes of business in which there is the possibility ofaccumulations: Fire, Marine, Personal Accident, Glass, etc.
Stop-Loss Reinsurance
The purpose of this is to protect the companys annual results in a classof business against a negative deviation due to a high incidence of claims,either in number or size.The Stop-Loss (SL) treaty acts on the aggregate claim amount of the port-folio:
R(t) = {N(t)∑i=1
Xi − C}+
, D(t) = min(X(t), C) (5)
where C is some retention. In general there is again an additional upperlimit on R(t).
In practice, often combinations of the above reinsurance forms are em-ployed (together with upper limits on the total reinsurance cover).
Large Claim Reinsurance
Large claim reinsurance treaties are rarely applied in practice.
Let X1, X2, ..., XN(t) denote the order statistics of the claims ordered ac-cording to their size. The largest claims reinsurance is defined through
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R(t) =r∑i=1
XN(t)−i+1, D(t) =
N(t)−r∑i=1
Xi (6)
i.e. the insurer covers the r largest claims of the portfolio that have oc-curred up to time t(usually 1 year), Another variant called ECOMOR wasintroduced by Thepaut [27] with a reinsured amount of the form
R(t) = {N(t)∑i=1
Xi −XN(t)−r}+
(7)
In this case the reinsurer covers only that part of the r largest claimsthat overshoots the random retention. Such a contract gives the reinsurerprotection against claim inflation, since as the claim amounts increase, theretention will also increase.
4 Application of Utility Theory in Insurance
4.1 Utility Theory
The notion of utility goes back to Daniel Bernoulli (1738). Because thevalue of money does not solve the paradox of St. Petersburg, he proposedthe moral value of money as a standard of judgment. According to Borch(1974, p.26). Several mathematicians discussed the Bernoulli principle in thefollowing century, and its relevance to insurance systems seemed to be gener-ally recognized. However during the next hundred years Bernoulli principlewas almost completely forgotten, by actuaries and economists alike.
Utility theory came to life again in the middle of this century. This wasabove all the merit of von Neumann and Morgenstern (1947), who arguedthat the existence of a utility function could be derived from a set of axiomsgoverning a preference ordering. Borch showed how utility theory could beused to formulate and solve some problems that are relevant to insurance.
Since then many studies have shown that the expected utility decisioncriterion provides useful insights for certain insurance problems, such as un-derwriting, reinsurance and portfolio optimization problems. In this seminar
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we are going to focus on reinsurance, specifically, we will illustrate how utilitytheory can be applied to determine the optimal retention level under bothproportional and non proportional reinsurance.
4.2 Utility Functions
Often it is not appropriate to measure the usefulness of money on the mon-etary scale. To explain certain phenomena, the usefulness of money must bemeasured on a new scale. Thus, the usefulness of x is u(x), the utility (ormoral value) of x.
A utility function, u(x), can be described as a function which measuresthe value, or utility, that an individual (or institution) attaches to the mon-etary amount x. We suppose that a utility function u(x) has the followingtwo basic properties:
• u(x) is an increasing function of x, expressed as u′(x) > 0.
• u(x) is a concave function of x, expressed as u′′(x) > 0.
The first property amounts to the evident requirement that more is bet-ter i.e. an individual whose utility function is u prefers amount y to amountz provided that y > z. An individual whose utility function satisfies thiscondition is said to be risk averse.
The second property states that as the individuals wealth increases, theindividual places less value on a fixed increase in wealth. For example, anincrease in wealth of 1000 is worth less to the individual if the individu-als wealth is 2,000,000 compared to the case when the individuals wealth is1,000,000.
Utility functions can take various forms including; exponential form,quadratic form, logarithmic form or fractional power form. However in thenext section we are going to cover the exponential utility function only be-cause our assumption in this seminar is that an insurer makes decisions onthe basis of the exponential utility function.
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4.3 Exponential utility function
An exponential utility function takes a form u(X) = −exp{−βX}, whereβ > 0.Under this utility function, the decisions do not depend on the individualswealth W. To illustrate this, consider the case of an individual with wealthW who has a choice between n courses of action.
Suppose that the i th course of action will result in random wealth ofW + Xi, for i = 1, 2, .., n. Then under the expected utility criterion, theindividual would calculate E[u(W +Xi)] for i = 1, 2, ..., n, and would choosecourse of action j if and only if
E[u(W +Xj)] > E[u(W +Xi)] (8)
for i = 1, 2, .., n, andi 6= j. Inserting for u in the equation above, thiscondition becomes
−E[exp{−β(W +Xj)}] > −E[exp{−β(W +Xi)}] (9)
Or equivalently,
E[exp{−βXj}] < E[exp{−βXi}] (10)
The maximm premium, P that an individual with utility function wouldbe prepared to pay for insurance against a random loss, X, is
P = β−1logMx, (β) (11)
5 Proportional reinsurance
5.1 Utility optimization
Throughout this section we assume that an insurer makes decisions on thebasis of the exponential utility function u(x) = −exp{−βx} where β > 0.We consider a (reinsured) risk over a oneyear period, so that the insurerswealth at the end of the year is W + P − PR − SI .
• W: the insurers wealth at the start of the year
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• P: the premium the insurer receives to cover the risk
• PR: the amount of the reinsurance premium
• SI : the amount of claims paid by the insurer net of reinsurance
Our objective is to find the retention level that maximises the insurersexpected utility of year end wealth. As neither W nor P depends on theretention level, and as we are applying an exponential utility function, ourobjective is to maximise
−exp{βPR}E[exp{βSI}] (12)
Let us assume that the insurer effects proportional reinsurance and paysproportion a of each claim, and that the reinsurance premium is calculatedby the exponential principle with parameter A.
The reinsurance premium is:
PR =λ
A(
∫ ∞0
e(1−a)Axf(x)dx− 1) (13)
Similarly, as SI has a compound Poisson distribution with Poisson para-meter λ and with individual claim amounts distributed as aX,
E[exp(βS1)] = exp{λ(
∫ ∞0
eaβxf(x)dx− 1)} (14)
−exp(βPR)E[exp(βS1)] = −exp{λβA
(
∫ ∞0
e(1−a)Axf(x)dx−1)+λ(
∫ ∞0
eaβxf(x)dx−1)}
(15)Finding a to maximise this expression is the same as finding a to minimise
h(a) where
h(a) =λβ
A
∫ ∞0
e(1−a)Axf(x)dx+ λ
∫ ∞0
eaβxf(x)dx
= λ
∫ ∞0
(A−1βe(1−a)Ax + eaβx)f(x)dx
(16)
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Differentiation gives
d
dah(a) = λ
∫ ∞0
(−xβe(1−a)Ax + βxeaβx)f(x)dx (17)
and this equals 0 when
a =A
A+ β(18)
5.2 Simulation and analysis
5.3 a as a function of A
Firstly, we are going to calculate the first derivative to check if the functionis an increasing or decreasing function of A.
• a: The proportion of each claim to be payed
• A: The reinsurer’s coefficient of risk aversion
∂a
∂A=
β
(A+ β)2> 0 (19)
forβ > 0Hence, it follows that it is an increasing function of A. This result can
also be seen in the next graph. We have used a fixed a value of the riskaversion such as β= 0.01.
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Secondly, we are going to see how the payment of each claim as a functionof A behaves for different values of the risk aversion parameter. The resultsare shown in the following plot.
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Those are the values of β that we have use. As the value of β increases,the value of the payment decreases. On the other hand, the payment increasealong with the coefficient A.
5.4 a as a function of β
As in the first part, we begin by find out the first derivative of this newfunction.
∂a
∂β=
−A(A+ β)2
< 0 (20)
for A > 0Thus we find out this function is a decreasing function. And we can ob-
serve this behavior in the next plot. Let the reinsurer’s coefficient of risk
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aversion be 2%.
As β increase, the payment of claim decrease. And then, we are going tostudy the behavior by fixing different values of the coefficient A as following.
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As we can see, as we increase the value of the coefficient A, the paymentalso increases. On the other hand, the payment will decrease if β decrease.
In conclusion, a small value of the parameter of the utility function and ahigh value of the coefficient A will maximize the payment a. By maximizinga, the insurer’s expected utility of year end wealth will also be maximized.
6 Excess of loss reinsurance
6.1 Utility optimization
The excess of loss model is based on the following assumptions:
• The insurer makes decisions under the utility function: u(x) = − exp{−βx}.
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• The wealth of the insurer after the end of the year is W +P −PR−SIwhere W is the wealth at the beginning of the year, P is the premiumreceived to cover the risk, PR is the reinsurance premium and SI is theamount of claims paid by the insurer net of reinsurance.
• The aggregate claims from the risk before the reinsurance follows acompound Poisson distribution with λ parameter and the continuousindividual claims amount follow a distribution F with F (0) = 0.
As neither W nor P depend on the retention level, and as we are applyingan exponential utility function, our objective is to get the retention level thatmaximizes the insurer’s expected utility. Therefore, we will need to maximize
− exp{βPR}E[exp{βSI}] (21)
The value of the reinsurance premium is
PR = (1 + θ)λ
∫ ∞M
(x−M)f(x)dx (22)
where θ is the loading.
The claims follow a compound Poisson distribution with λ parameter.Thus, we will have:
E[exp{βSI}] = exp{λ(
∫ M
0
eβxf(x)dx+ eβM(1− F (M)− 1)} (23)
Hence,
− exp{βPR}E[exp{βSI}] = − exp{(1 + θ)λβ
∫ ∞M
(x−M)f(x)dx}
exp{λ(
∫ M
0
eβxf(x)dx+ eβM(1− F (M)− 1)}(24)
so this is the equation we need to maximize in order to get the optimalretention level. We can do it easier if we find the minimum of the total
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exponent since we can put the two exponents together. The result will be anegative exponential function with exponent g(M).
g(M) = (1+θ)λβ
∫ ∞M
(x−M)f(x)dx+λ(
∫ M
0
eβxf(x)dx+eβM(1−F (M)−1)
(25)Therefore, to find its min we need to differentiate g(M), getting
d
dM= −(1 + θ)λβ
∫ ∞M
f(x)dx+ λβeβM(1− F (M))
= λβ(1− F (M))(eβM − 1− θ)
Now, for it to be equal to 0 we have
M =1
βlog(1 + θ) (26)
Furthermore, the second difference will be
d2
dM2= −λβf(M) + λβ(eβM − 1− θ) + λβ2(1− F (M))eβM
And substituting the value of M gotten above, we have that the secondderivative is positive. Therefore, it minimizes g(M) and we can affirm thatM = 1
βlog(1+θ) maximizes the insurer’s expected utility of year-end wealth.
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6.2 Simulation and analysis
Once we get the function that maximizes the insurers expected utility ofyear-end wealth, we can see that it does not depend on the individual claimamount distribution, but on the parameter of the insurers utility function(β) and on the parameter of the reinsurers premium calculation principle (θ).
In the next step we are going to do some simulations to observe howthe optimal retention level behaves if we consider it as function of these twoparameters.Note: The parameter θ is shown as a percentage (0 < θ < 1) and β > 0.
M as function of θ
Firstly, we are going to calculate the first derivative to check if the func-tion is an increasing or decreasing of θ.
∂M
∂θ=
1
β(1 + θ)> 0 (27)
for β > 0 and 0 < θ < 1
Hence, it follows that it is an increasing function of θ. This result can alsobe seen in the next graph. We have used a fixed a value of the risk aversionsuch as β = 0.01.
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Secondly, we are going to see how the retention level as function of θbehaves for different values of the risk aversion parameter. The results areshown in the following plot.
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Those are the values of β that we have use. As the value of β increases,the value of the retention level decreases. The higher the theta (reinsurer’srisk loading) the higher the retention level for the insurer.
The insurer the wants to maximise returns so a high value of theta meanshigher reinsurance premiums which pushes the insurer to cover more risk onhis own getting a bigger value of retention level.
M as function of β
In this part, we are going to analyze the function that maximizes theinsurers expected utility and we first calculate the first derivative.
∂M
∂β=−log(1 + β)
β2< 0 (28)
for β > 0 and 0 < θ < 1
The retention level as function of β is decreasing, and we can observe thisbehavior in the next plot. Let the loading factor be the 20%.
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As β increases, the retention level decreases. And then, we are going tostudy the behavior by fixing different values of the loading factor (10, 20, 30and 40%).
As we can see, as we increase the value of the loading factor, the retentionlevel increases. The higher the beta (insurer’s risk aversion) the more theinsurer will want to avoid risk and therefore, the lower the retention levelwill be, this will lead to the insurer paying higher premiums.
In conclusion, a small value of the parameter of the utility function anda high value of the loading factor will maximize the retention level M.
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7 Conclusion
Results for the application of utility theory on proportional and excess ofloss reinsurance show that the optimal retention level is independent of theindividual claim amount distribution, and depends only on the parameter ofthe insurer’s utility function and the parameter of the reinsurance premiumprinciple. Although these results which are based single period analysis areintuitively appealing, they also have limitation. Since the analysis is basedon an exponential utility function, the premium that the insurer receives tocover the risk doesn’t affect the decision.
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8 Appendix
#####SIMULATION
########## Proportional reinsurance
## A as a function of Beta
beta<-seq(0.05,1, 0.001)
MM1<-(0.02)/((0.02+beta))
plot(beta,MM1, type="l", ylab="The proportion of each claim to be payed
, a", xlab="Risk aversion, Beta", main="a as a function of Beta for A=0.02")
M1<-(0.01)/((0.01+beta))
M2<-(0.02)/((0.02+beta))
M3<-(0.03)/((0.03+beta))
M4<-(0.04)/((0.04+beta))
plot(beta,M1, type="l", ylab="The proportion of each claim to be payed,a",
xlab="Risk aversion, Beta", main="a as a function of Beta")
lines(beta,M2,type="l", col="red")
lines(beta,M3,type="l", col="blue")
lines(beta,M4,type="l", col="green")
legend("topright",legend=c("A=1% ","A=2% ","A=3% ","A=4% "),
lty=c(1,1,1,1),col=c("black","red", "blue", "green"))
##A as a function of a
A<-seq(0.1, 1, 0.005)
a<-A/(A+0.01)
plot(A,a, type="l", ylab="The proportion of each claim to be payed, a",
xlab="The reinsurer’s coefficient of risk aversion , A",
main="a as a function of A for Beta=0.01")
M5<-A/(A+0.003)
M6<-A/(A+0.004)
M7<-A/(A+0.005)
a<-A/(A+0.01)
plot(A,M5, type="l", ylab="The proportion of each claim to be payed, a",
xlab="The reinsurer’s coefficient of risk aversion , A",
main="a as a function of A")
lines(A,M6,type="l", col="red")
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lines(A,M7,type="l", col="blue")
lines(A,a,type="l", col="green")
legend("bottomright",legend=c("Beta=0.003 ","Beta=0.004 ","Beta=0.005 ",
"Beta=0.01 "),lty=c(1,1,1,1),col=c("black","red", "blue", "green"))
##########Excess of loss reinsurance
##M as a function of Beta
beta<-seq(0.05,1, 0.001)
MM1<-1/beta* log(1+0.2)
plot(beta,MM1, type="l", ylab="Retention level, M", xlab="Risk aversion, Beta",
main="Retention level M as function of Beta for Theta=0.2")
M1<-1/beta* log(1+0.1)
M2<-1/beta* log(1+0.2)
M3<-1/beta* log(1+0.3)
M4<-1/beta* log(1+0.4)
plot(beta,M1, type="l", ylab="Retention level, M", xlab="Risk aversion, Beta",
main="Retention level M as function of Beta")
lines(beta,M2,type="l", col="red")
lines(beta,M3,type="l", col="blue")
lines(beta,M4,type="l", col="green")
plot (c(0,10),c(0,10), col="white")
legend("topright",legend=c("Theta=10% ","Theta=20% ","Theta=30% ","Theta=40% "),
lty=c(1,1,1,1),col=c("black","red", "blue", "green"))
##M as a function of Theta
tita<-seq(0.01, 1, 0.05)
MM<-(1/0.01)* log(1+tita)
plot(tita,MM, type="l", ylab="Retention level, M", xlab="Loading factor, Theta",
main="Retention level M as function of Theta for Beta=0.01")
M5<-(1/0.003)* log(1+tita)
M6<-(1/0.004)* log(1+tita)
M7<-(1/0.005)* log(1+tita)
M8<-(1/0.01)* log(1+tita)
plot(tita,M5, type="l", ylab="Retention level, M", xlab="Loading factor, Theta",
main="Retention level M as function of Theta")
lines(tita,M6,type="l", col="red")
lines(tita,M7,type="l", col="blue")
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lines(tita,M8,type="l", col="green")
plot (c(0,10),c(0,10), col="white")
legend("topright",legend=c("Beta=0.003 ","Beta=0.004 ","Beta=0.005 ","Beta=0.01 "),
lty=c(1,1,1,1),col=c("black","red", "blue", "green"))
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9 Reference
1 Albrecher, H. 2010. Reinsurance. Encyclopedia of Quantitative Finance.
2 FUNDACION MAPFRE, Institute of Insurance Sciences. 2013. An Intro-duction to Reinsurance,: FUNDACIN MAPFRE, Institute of InsuranceSciences, Madrid
3 Gerber, H. U and Pafum, G (1998) Utility Functions: From Risk Theoryto Finance. North American Actuarial Journal 2, No. 3, 74-100
4 Swiss Re 1997, Proportional and non proportional reinsurance, Swiss Republishing, 31
5 Dickson, David C. M. (2005), Insurance Risk and Ruin, Cambridge
6 ”http://www.r-project.org/”
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