chap-4.pdf

0 INTRODUCTION 1

Chapter 4Reinsurance

0 Introduction

• A reinsurance policy

• For the purposes of this chapter, it will be assumedthat the reinsurance contract is one of two verysimple types:

– individual excess of loss reinsurance

– proportional reinsurance

• Reinsurance can be classified into two forms:

– proportional

– non-proportional

0 INTRODUCTION 2

0.1 Proportional reinsurance

• Under a proportional reinsurance arrangement,the original insurance company and the reinsurershare the cost of all claims for each risk. The directwriter must pay a premium to effect this reinsur-ance. This premium may, or may not, be calcu-lated on the same basis as the premiums chargedby the insurer.

• Proportional reinsurance operates in two forms:

1. With quota share reinsurance, the proportionsare the same for all risks.

2. With surplus reinsurance, the proportions canvary from one risk to the next.

0 INTRODUCTION 3

0.2 Non-proportional reinsurance

• Under a non-proportional reinsurance ar-rangement, the direct writer pays a fixed premiumto the reinsurer. The reinsurer will only be re-quired to make payments where part of the claimamount falls in a particular reinsurance layer.

• The layer will be defined by a lower limit, and anupper limit (or ”infinity” if the cover is unlimited).

0 INTRODUCTION 4

• Two forms of non-proportional reinsurance:

1. With individual excess of loss(XOL) reinsur-ance, the reinsurer will be required to make apayment when the claim amount for any indi-vidual claim exceeds a specified excess point orretention.

2. With stop loss reinsurance, the reinsurer willbe required to make payments if the total claimamount for a specified group of policies exceedsa specified amount (usually expressed as a per-centage of the gross premium).

– For example, the reinsurer might. agree topay 90% of the excess when the total claimamount for all motor policies exceeds 105%of the total gross premium, with no upperlimit.

• See Figures on page 3.

1 REINSURANCE ARRANGEMENTS 5

1 Reinsurance arrangements

• The actual claim amount that the direct insurerends up paying after allowing for payments un-der the reinsurance arrangements is called the netclaim amount.

• The actual premium that the direct writer gets tokeep after making any payments for reinsurance iscalled the net premium.

• The original amounts without adjustment for rein-surance are referred to as the gross claim amountand the gross premium.


• We can calculate the moments of the net claimamount paid by a direct insurer or the amountpaid by the reinsurer by considering the net amountpaid for different sizes of the gross claim.

• Example on pages 4-5.

• We will consider the statistical calculations relat-ing to reinsurance arrangements in the followingsubsections.


1.1 Excess of loss reinsurance - the insurer

• In excess of loss reinsurance, the insurer will payany claim in full up to an amount M , the retentionlevel; any amount above M will be borne by thereinsurer.

• The excess of loss reinsurance arrangement can bewritten in the following way:

Y = X if X ≤ M

Y = M if X > M

i.e.

Y = min(X,M)

Where X denotes the claim amount and Y denotesthe amount paid by the insurer.

• The reinsurer pays the amount Z = X − Y . i.e.Z = max(X −M, 0).


• If only a layer between M and 2M is reinsured,then the amount paid by the insurer is

Y = X if X ≤ M

Y = M if M < X ≤ 2M

Y = X −M if X > 2M


• The insurer’s liability is affected in two obviousways by reinsurance:

1. the mean amount paid is reduced;

2. the variance of the amount paid is reduced.

• Both the mean and the reduction in the mean ofthe amount paid by the insurer under excess ofloss reinsurance can be obtained.


• Observe that the mean amount paid by the insurerwithout reinsurance is

E(X) =

∫ ∞

0

xf (x)dx

where f (x) is the PDF of the claim amount X .

• With a retention level of M the mean becomes

E(Y ) =

∫ M

0

xf (x)dx + M P(X > M)

• The main difficulty in dealing with excess of lossreinsurance is that the integral is incomplete. (Thesame difficulty will arise when considering a policyexcess.)


• With excess of loss reinsurance there is a methodthat converts the incomplete integral to a completeintegral.

E(Y ) =

∫ ∞

0

xf (x)dx−∫ ∞

M

xf (x)dx

+M

∫ ∞

M

f (x)dx

= E(X)−∫ ∞

M

(x−M)f (x)dx

= E(X)−∫ ∞

0

zf (z + M)dz

by setting z = x−M.


• If X ∼ Pareto(α, λ), and the retention level isM , then

E(Y ) =λ

α− 1−

∫ ∞

0

zαλα

(λ + z + M)α+1dz

=λ

α− 1−

(λ

λ + M

)α ∫ ∞

0

zα(λ + α)α

(λ + z + M)α+1dz

=λ

α− 1−

(λ

λ + M

)αλ + M

α− 1


• The usefulness of the formula given above doesdepend on being able to carry out the integrationof zf (z +M) . For cases where this integral is noteasy to calculate, the formula has limited use.

• If X ∼ Gamma(α, λ), and the retention level isM , then

E(Y ) =α

λ−

∫ ∞

0

z1

Γ(α)λα(z + M)α−1e−λ(z+M)dz

This is not going to be easy to evaluate.


1.2 Excess of loss reinsurance - the reinsurer

• The reinsurer may have a record only of claimsthat are greater than M . If a claim is for lessthan M the reinsurer may not even know a claimhas occurred. The reinsurer thus has the prob-lem of estimating the underlying claims distribu-tion when only those claims greater than M areobserved.

• The statistical terminology is to say that the rein-surer observes claims from a truncated distribu-tion.


• In this case the values observed by the reinsurerrelate to a conditional distribution, since the num-bers are conditional on the original claim amountexceeding the retention limit.

• We write the random variable for the amount paidby the reinsurer asZ = X −M |X > M .


• Suppose that the underlying claim amounts havePDF f (x) and CDF F (x). Suppose that the rein-surer is only informed of claims greater than theretention M and has a record of z = x − M .What is the PDF g(z) of the amount, z, paid bythe reinsurer?


• The argument goes as follows:

P(Z < z) = P(X < z + M |X > M)

=

∫ z+M

M

f (x)

1− F (M)dx

=F (z + M)− F (M)

1− F (M)

the PDF of the reinsurer’s claims is

g(z) =f (z + M)

1− F (M)z > 0.

• NOTE: This is just the original PDF applied to thegross amount z + M , divided by the probabilitythat the claim exceeds M .


• It is also possible that the reinsurer is aware of allthe claims that are made under the original poli-cies. In this case the distribution of the reinsurer’soutgo may include those claims on which it doesnot in fact pay anything.

• In this case we are looking at the unconditionaldistribution of the reinsurer’s outgo Z, where:

Z = 0 if X < M

Z = X −M if X ≥ M

• The calculation of the mean and variance of thereinsurer’s outgo in this case will be similar to thecorresponding calculations for the insurer.


• If we now use W to denote the amount payable bythe reinsurer on claims in which it is involved, i.e.

W = Z|Z > 0

= X −M |X > M

then

g(w) =fZ(w)

1− FZ(0)

E(W ) =E(Z)

P(Z > 0)=

E(Z)

P(X > M)

• Example on page 9.


1.3 Proportional reinsurance

• In proportional reinsurance the insurer pays a fixedproportion of the claim, whatever the size of theclaim. Using the same notation as above, the pro-portional reinsurance arrangement can be writtenas follows: if the claim is for an amount X thenthe company will pay Y where

Y = αX 0 < α < 1.

• The parameter α is known as the retention level.


• As the amount paid by the insurer on a claim Xis Y = αX and the amount paid by the reinsureris Z = (1−α)X , the distribution of both of theseamounts can be found by a simple change of vari-able.

fY (y) =1

αfX

(y

α

)fZ(z) =

1

1− αfX

(z

1− α

)


• If X represents the size of an individual claim, theretention level α is 0.8, then the amount paid bythe insurer is Y = 0.8X , and E(Y ) = 0.8 E(X),Var(Y ) = 0.64 Var(X). The amount paid by thereinsurer is Z = 0.2X , and E(Z) = 0.2 E(X),Var(Z) = 0.04 Var(X).

2 PARTICULAR DISTRIBUTIONS 23

2 Particular distributions

• There are a number of useful integral formulaethat simplify reinsurance calculations when work-ing with particular distributions.

2.1 Lognormal distribution

• Property of the lognormal distribution (SEE page12.)

2.2 Normal distribution

• Property of the normal distribution (SEE pages17-18.)

3 INFLATION 24

3 Inflation

• The examples so far have assumed that claim dis-tributions remain constant over time. In fact claimsare likely to increase because of inflation, at leastin the longer term.

• We need to adjust our claim distributions to allowfor inflation.

• In this section we will look at how claims inflationaffects reinsurance arrangements.

3 INFLATION 25

• Suppose that the claims X are inflated by a fac-tor of k but the retention M remains fixed. Theamount claimed is kX , and the amount paid bythe insurer, Y , is

Y = kX if kX ≤ M

Y = M if kX > M.

The mean amount paid by the insurer is

E(Y ) =

∫ M/k

0

kxf (x)dx + M P(X > M/k)

3 INFLATION 26

which can be written as

E(Y ) =

∫ ∞

0

kxf (x)dx−∫ ∞

M/k

kxf (x)dx

+M

∫ ∞

M/k

f (x)dx

= k E(X)− k

∫ ∞

M/k

(x−M/k)f (x)dx

= k

(E(X)−

∫ ∞

0

yf (y + M/k)dy

).(

y ≡ x− M

k

)

3 INFLATION 27

• One important general point that can be made isthat the new mean claim amount is NOT k timesthe mean claim amount without inflation.

• Of course if the retention limit increases by a factorof k as well, both mean claim amounts (for theinsurer and reinsurer) will increase by the samefactor.

• When examining the details of a reinsurance ar-rangement in real life it is very important to checkwhether the retention limits are fixed or linked toan agreed inflation index.

• Example on pages 21-23.

4 ESTIMATION 28

4 Estimation

• Consider the problem of estimation in the presenceof excess of loss reinsurance. Suppose that theclaims record shows only the net claims paid bythe insurer.

• A typical claims record might be

x1, x2, M, x3, M, x4, x5, . . .

and an estimate of the underlying gross claims dis-tribution is required.

4 ESTIMATION 29

• The statistical terminology for a sample of thisform is censored. In general, a censored sampleoccurs when some values are recorded exactly andthe remaining values are known only to exceed aparticular value.

• The method of moments is not available since eventhe mean claim amount cannot be computed.

• It may be possible to use the method of percentileswithout alteration; this would happen if the reten-tion level M is high and only the higher samplepercentiles were affected by the (few) reinsuranceclaims.

4 ESTIMATION 30

• Maximum likelihood can be applied to censoredsamples.

• The likelihood is made up of two parts. The com-plete likelihood is

L(θ) =

n∏1

f (xi; θ)× [1− F (M ; θ)]m

• Example on page 25.

5 POLICY EXCESS 31

5 Policy excess

• Insurance policies with an excess are common inmotor insurance and many other kinds of propertyand accident insurance.

• Under this kind of policy, the insured agrees tocarry the full burden of the loss up to a limit,L, called the excess. If the loss is an amount X ,greater than L, then the policyholder will claimonly X − L. If Y is the amount actually paid bythe insurer, then

Y = 0 if X ≤ L

Y = X − L if X > L.

5 POLICY EXCESS 32

• Clearly, the premium due on any policy with anexcess will be less than that on a policy withoutan excess.

• The position of the insurer for a policy with anexcess is exactly the same as that of the reinsurerunder excess of loss reinsurance. The position ofthe policyholder as far as losses are concerned isexactly the same as that of an insurer with anexcess of loss reinsurance contract.

5 POLICY EXCESS 33

• In practice, expenses form a significant part of theinsurance cost. So the presence of an excess mightnot affect the premium as much as might be ex-pected.

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