Some Alternative Actuarial Pricing Methods: Application to ... · Some alternative actuarial pricing methods… 3 tion premiums are obtained. At the same time this example justifies

Reports on Economics and Finance, Vol. 2, 2016, no. 1, 1 - 35

HIKARI Ltd, www.m-hikari.com

http://dx.doi.org/10.12988/ref.2016.51114

Some Alternative Actuarial Pricing Methods:

Application to Reinsurance and Experience Rating

Werner Hürlimann

Swiss Mathematical Society, University of Fribourg

CH-1700 Fribourg, Switzerland

Copyright © 2015 Werner Hürlimann. This article is distributed under the Creative Commons

Attribution License, which permits unrestricted use, distribution, and reproduction in any medium,

provided the original work is properly cited.

Abstract

The present work concerns reinsurance rate-making in a distribution-free

environment and fair premium calculation principles. Applications to the design

of perfectly hedged experience rating contracts in a risk-exchange and reinsurance

environment are discussed. Special emphasis is put on distribution-free and

immunization aspects as well as on long-term optimal and competitive strategies

in the sense of strategic financial management.

Keywords: additive premium principle, fair premium, generalized variance

principle, utility theory, CAPM, extended mean-variance analysis, perfect hedge

1. Introduction

The present work offers an overview and synthesis around several studies of the

author. The main topics of theoretical interest are reinsurance rate-making in a

distribution-free environment and fair premium calculation principles.

Applications to the design of perfectly hedged experience rating contracts in a

risk-exchange and reinsurance environment are discussed. Special emphasis is put

on distribution-free and immunization aspects as well as on long-term optimal and

competitive strategies. A more detailed outline of the content follows.

Section 2 is devoted to a special case of the following more general problem in

pricing theory. Given is a risk that can be decomposed in a finite number of

splitting risk components. If all involved risk premiums are calculated according

2 Werner Hürlimann

to some premium calculation principle, what are adequate premium principles,

which besides other desirable properties satisfy the superadditive property for

splitting risk components? In situations where arbitrage opportunities should be

avoided, as in reinsurance markets with trading possibilities, the stronger additive

property is required. Examples, which illustrate the use of a (strict) superadditive

property are found in Hürlimann [19], Section 1, Hürlimann [20], Section 3, and

here later in Section 5. An overview about known results in the additive case is

Aase [1]. In the present work, only the special case of two splitting risk

components is touched upon. Despite this restriction, the potential practical

applications are numerous. It is shown that the generalized variance principle by

Borch [4], obtained from an insurance version of the Capital Asset Pricing Model

(CAPM), can be justified on the basis of four alternative mathematical, economics

and actuarial arguments. In particular Proposition 2.1 links the economics form of

Borch's principle with the class of biatomic risks.

Section 3 is a synthesis of the main ideas presented in Hürlimann [19] to [22].

A set of feasible reinsurance contracts with a fixed maximum deductible is

considered. This reinsurance structure is part of a more general structure, which

minimizes the square loss risk of the insurer by offering to the insured a claims

dependent bonus under the help of risk-exchanges, in particular reinsurance

treaties. For the considered set, the minimum square loss risk vanishes and defines

perfectly hedged experience rating contracts in a reinsurance environment, which

are mathematically characterized in Proposition 3.1. The needed fair premium of

such a contract equals the sum of the expected claims, the expected bonus

payment and the loading for reinsurance. Under a fair exchange of the risk profit

loading between the insurer and the reinsurer, fair premiums are characterized in

Proposition 3.2 by a property of complete immunization. Alternative

interpretations are proposed in the Remarks 3.1 and a fair stop-loss premium

rating model is proposed in Example 3.2.

The results of Section 2 are applied in Section 4 to the construction of a

distribution-free premium rating system for the class of perfectly hedged

experience rating contracts. It satisfies a criterion of (relative) safeness and may

be viewed as a biatomic approximation to Borch's additive CAPM premium

principle (see Theorem 4.1). With the help of the immunization argument of

Proposition 3.2, this distribution-free premium methodology is then applied to

develop explicit examples for a stop-loss contract in environments of positive

risks (Section 4.1) respectively arbitrary risks (Section 4.2). In Section 4.1 a new

insurance economics interpretation of the "Karlsruhe" pricing principle proposed

by Heilmann [12] is obtained (see also Hürlimann [26]). The link with utility

theory allows for an interesting interpretation of the expected value principle in

Example 4.2. The assumption of a quadratic utility function with saturation leads

in Example 4.3 to a non-trivial generalized variance principle of Borch's type. In

Section 4.2 the list of practical applications is continued for the class of arbitrary

risks. In Example 4.4 the insurance market based distribution-free stop-loss

premium formula first proposed in Hürlimann [16] finds a rigorous model theoretical justification. Example 4.5 shows how distribution-free fair standard devia-

Some alternative actuarial pricing methods… 3

tion premiums are obtained. At the same time this example justifies the most

competitive choice, which can be made in Example 4.4. A discussion of the

"Karlsruhe" principle in a distribution-free environment of arbitrary risks is given

in Example 4.6.

In Section 5 it is shown how perfectly hedged experience rating strategies,

which are long-term optimal and competitive in the insurance market, can be

defined. The corresponding possible minimum and maximum premiums are

determined in Proposition 5.1 and its Corollary 5.1. Moreover, formula (5.11)

shows that it is possible to obtain optimal strategies with fair variance premiums

that do not depend upon the unknown risk loading factor. Two common situations

illustrate the use of the method. In Section 5.1 an optimal long-term stop-loss

pricing strategy is displayed. Applied to financial risks, an extended version of the

classical mean-variance approach to portfolio theory by Markowitz, considered in

Hürlimann [15], [24], finds herewith a rigorous justification. Finally, it is shown

in Section 5.2 that an optimal long-term and competitive perfectly hedged

experience rating strategy based on a linear combination of proportional and stop-

loss reinsurance does not exist in case of a non-vanishing proportional reinsurance

payment. The proof of this result is based on the inequalities of Kremer [28] and

Schmitter [30], which are reviewed in the Appendix.

2. Splitting risk premiums and biatomic risks

Consider the following rate-making problem encountered in the economic theory

of insurance under uncertainty. Given a risk X with risk premium P=H[X]

calculated according to some pricing calculation principle H[], and given a

splitting of the risk in smaller parts Xi with splitting risk premiums Pi=H[Xi],

i=1,...,n, such that X=X1+...+Xn, what are appropriate pricing principles, which

satisfy the following superadditive property

.11 ni i

ni i PXHXHP (2.1)

In cases where the strict inequality holds, it is more effective to insure the splitting

parts Xi separately. For some examples consult Hürlimann [19], Section 1, [20],

Section 3, as well as the later Section 5. In particular, to avoid arbitrage

opportunities, one requires the equality sign for both independent and dependent

risk components Xi. An overview of known results in the latter case is offered by

Aase [1].

In the present Section we restrict our attention to the additive property in the

special case n=2 that will be needed later. Emphasis is put on additive pricing

principles, whose general forms are determined by the values they take on the set

D2[a,b]:=D2([a,b];,) of biatomic risks with given mean and standard

deviation defined on the interval [a,b], a, b R.

4 Werner Hürlimann

Suppose a risk X is split into two transformed components Y=f(X), Z=g(X)

such that Y+Z=X. The problem consists to construct pricing principles H[] that

satisfy the additive property

H[X] = H[Y] + H[Z]. (2.2)

Applying the functional representation theorem of Riesz [29], one knows that

there exists some random variable W such that the following relations hold:

H[X] = E[XW] = E[X]E[W] + Cov[X,W]

H[Y] = E[YW] = E[Y]E[W] + Cov[Y,W] (2.3)

H[Z] = E[ZW] = E[Z]E[W] + Cov[Z,W]

Example 2.1.

In a simple "linear world" such that

W = + (X - E[X]), (2.4)

one sees that

H[X] = E[X] + Var[X] (2.5)

is the generalized variance principle obtained by Borch [5], [6], from the CAPM.

Now, one has

H[Y] = E[Y] + Cov[X,Y],

H[Z] = E[Z] + Cov[X,Z], (2.6)

and eliminating using (2.5) one gets Borch’s formula

).(,

),(,

XEXHXVar

ZXCovZEZH

XEXHXVar

YXCovYEYH

(2.7)

To interpret this simple example from the point of view of an economic theory

under uncertainty, one can assume that W=u'(X), where u'(x) is a marginal utility

function of some representative insurer (see Theorem 1 in Aase [1] and its

consequence (2.7)). Disregarding the needed technical assumptions for the

existence of a competitive equilibrium implying W=u'(X), our utility functions are

supposed to satisfy the property u'(x)0, u''(x)0, which includes in particular the

possibility of a risk-neutral representative insurer with linear utility function. In


this setting the "linear world" of Example 2.1 follows by assuming a quadratic

utility function.

Besides the above and Borch [4], let us give a third alternative justification,

which is based on a criterion of "safeness" or prudent pricing. If one considers

only positive risks X0, it follows from the assumption W=u'(X)0 that

H X E X E W X W Var X Var W ( , ) , (2.8)

(X,W) the correlation coefficient between X and W,

is always on the safe side provided (X,W)=1, which implies a linear

transformation W = + (X - E[X]), with =E[W], >0. Therefore Borch's

principle (2.7) is also the safest possible choice in an economic world under

uncertainty for which W=u'(X).

A fourth derivation of the formulas (2.7) follows by considering biatomic risks

as suggested in Hürlimann [16], [19]. Let X D2[a,b] has support {x1,x2} and

probabilities {p1,p2} such that

p

x

x xp

x

x xx x

a b a b

1

2

2 1

2

1

2 1

2

1 2

20

, , ( )( ),

, ( )( ).

(2.9)

Let {w1=u'(x1),w2=u'(x2)} be the support of the transformed random variable

W=u'(X). A calculation shows that

Cov X W

Var X

w w

x x

,,

2 1

2 1

(2.10)

and similar formulas for the risk components Y=f(X), Z=g(X). This observation

implies immediately the following result.

Proposition 2.1. In the class D2 [a,b] of biatomic risks, an additive premium

principle of the form (2.3) with W=u'(X) is necessary of the Borch's form (2.5),

(2.7) with parameters

E W p u x p u x

Cov X W

Var X

u x u x

x x

1 1 2 2

2 1

2 1

' ( ) ' ( ),

, ' ( ) ' ( ). (2.11)

As a particular feature of this model, one gets a simple link with utility theory, which will be exploited later in Section 4.1. Moreover in an economic world under

6 Werner Hürlimann

uncertainty for which W=u'(X) holds, an additive pricing principle characterized

by the values it takes on biatomic risks is necessarily of the CAPM form (2.5),

(2.7).

3. Fair premiums and perfectly hedged experience rating

The present Section offers a synthesis of the main ideas contained in the original

papers Hürlimann [19] to [22] on this topic. Notations are as in Section 2. In

reinsurance theory, one often restricts the set of transformations f(x), g(x) to those

compensation functions for which neither the cedant nor the reinsurer will benefit

in case the claim amount increases. In this situation, one assumes that f(x), g(x)

are non-decreasing functions such that f(x), g(x) x and f(x)+g(x)=x. This means

that feasible reinsurance contracts are described by the class of comonotonic

random variables

Com(X) = {(Y,Z) : Y=f(X), Z=g(X) are comonotonic such that Y+Z=X}. (3.1)

In this setting, the random variable Z describes the reinsurance payment and Y

denotes the retained amount of the direct insurer. One says that a feasible

reinsurance contract has a maximum deductible d if the following number exists

and is finite

.)(sup

xfdRx

(3.2)

Examples 3.1.

(i) A stop-loss contract Z=(X-d)+ has (maximum) deductible d.

(ii) A linear combination of proportional and stop-loss reinsurance of the form

Z=(1-r)X+r(X-T)+ has a maximum deductible d=rT. For a detailed study of this

contract see Hürlimann [22].

(iii) A linear combination of stop-loss contracts in layers Z=r(X-L)++(1-r)(X-

M)+, M>L, has a maximum deductible d=rL+(1-r)M.

(iv) Consider a compound Poisson risk

X U i

i

N

1

, (3.3)

where N is a Poisson random variable, and the Ui's are independent and

identically distributed random variables, which are independent from N. Then the

reinsurance payment


Z b N c U bi

i

N

( ) ( )1

(3.4)

defines a feasible reinsurance contract with maximum deductible d=bc, which

might be attractive in the framework of the classical model of risk theory.

The set of feasible reinsurance contracts with maximum deductible d is denoted

by

.)(sup)(),(:)(),((

xfdandXComZYXgZXfYSRx

d (3.5)

For (Y,Z)Sd the function defined by

d(x) = d - f(x) = d + g(x) – x (3.6)

is always non-negative and defines a transformed random variable D=d(X) such

that with probability one

d + Z = X + D. (3.7)

Setting xxdxf

d

)(

inf one has for all xxd

d(x) = 0, g(x) = x - d. (3.8)

It has been shown in Hürlimann [19], [22], that the set Sd defines a

convenient reinsurance substructure of a more general structure, which minimizes

the square loss of the insurer’s risk by offering to the insured a claims dependent

bonus under the help of risk-exchanges, in particular reinsurance treaties. This

general structure parameterizes the set of experience rating contracts in a risk-

exchange environment. In general, an experience rating contract with premium P

offers besides claims payment X a bonus D=D[X]0, which usually is paid out

in case the risk profit P-X is positive. In this situation, the liability of the insurer

is X+D. To reduce the financial risk of a loss X+D>P, suppose the insurer splits

the liability in two smaller parts, say X+D=(Y+D)+Z, where Z is some risk-

exchange. Then, the needed premium P=P[X+D] is the sum of the net retained

premium PN=PN[Y+D] of the insurer plus the price paid for the risk-exchange,

that is one has P=PN+H[Z], where H[] is the pricing principle applied to the

risk-exchange. An important problem consists to design appropriate pairs (Z,D)

satisfying some desirable properties. To limit the insurer's risk, one minimizes the

expected square difference between assets and liabilities, that is one considers the

optimization problem

8 Werner Hürlimann

R = E[(PN - Y - D)2] = min. (3.9)

over an appropriate space of pairs (Y,D). Taking into account the relation

R = (PN - E[Y+D])2 + Var[Y] + Var[D] + 2Cov[Y,D], (3.10)

one sees that a minimizing solution necessarily satisfies the conditions

PN = E[Y+D], (3.11)

Cov[Y,D] = -Var[D], or (3.12a)

Cov[Y,D] = -Var[Y]. (3.12b)

In the case (3.12a) one has Rmin=Var[Y]{1 - (Y,D)2}, where (Y,D) is the

correlation coefficient between Y and D, and in case (3.12b) one has

Rmin=Var[D]{1 - (Y,D)2}. In particular, the insurer's risk is completely

eliminated, a so-called perfect hedge with Rmin=0, for all pairs (Y,D) such that

Cov[Y,D]= - Var[D]= - Var[Y]. The condition Cov[Y,D]= - Var[D] means that

the systematic risk of the retained amount relative to the bonus equals - 1. The

formula (3.11) says that the fair net premium after reinsurance equals the expected

costs of the ceding company and finally the risk quantity Rmin is an intrinsic risk

measure, on which any adjustment of the fair net premium by a suitable security

loading should be based. In the perfect hedge situation Rmin=0 the needed fair

premium P=PN+H[Z], where PN=E[Y+D], can be decomposed into three

components as follows:

P = E[X] + E[D] + (H[Z] - E[Z]). (3.13)

fair premium expected expected loading in

claims bonus risk-exchange price

Besides the expected costs for claims and bonus payments only the loading of the

risk-exchange price has to be paid for a perfectly hedged experience rating

contract in a risk-exchange environment. This feature is similar to the "Dutch

property" of the Dutch premium principle (see Van Heerwaarden and Kaas [11]).

In the perfect hedge situation, one has PN+Z=X+D with probability one. In this

special case, the design of pairs (Z,D) such that the fair premium satisfies some

desirable properties has been studied in Hürlimann [21]. Here, we include

examples of risk-exchanges, which are not reinsurance contracts in the sense that


(Y,Z)Com(X). Restricting the space of random pairs (Y,Z) to Com(X), the

following characterization of the set Sd has been obtained in Hürlimann [19].

Proposition 3.1. Suppose (Y,Z)Com(X) and D=D[X]0 define an experience

rating contract in a reinsurance environment. Assume the set {x R: D[X=x]=0}

is non-empty. Then the following conditions are equivalent:

One has PN=d= )(sup xfRx

and Rmin=E[(PN - Y - D)2]=0. (C1)

One has Cov[Y,D]= - Var[D]= - Var[Y]. (C2)

(Y,Z)Sd defines a perfectly hedged experience rating contract with maximum

deductible d and bonus payment D=d - Y. (C3)

In the following, we restrict our attention to the set Sd of perfectly hedged

experience rating contracts in a reinsurance environment. Since the fair premium

(3.13) of such a contract depends upon the pricing principle H[] applied by the

reinsurer and generally not known with certainty, premium calculation simplifies

provided conditions, which characterize fair premiums, can be derived. As a

reasonable compromise for decision, let us adopt the following fair premium

condition. Given any premium H[X]=+ with loading >0, suppose that the

insurer and the reinsurer exchange the expected profit in an economic fair manner

such that the loading goes half and half to the ceding company and the reinsurer.

Then, one has the relations

H[Y] = E[Y] + 12, H[Z] = E[Z] + 1

2. (3.14)

In case (Y,Z)Sd the guaranteed bonus payment D belongs to the insured. The

insurer's net outcome after payment of the bonus equals

(H[Y] - Y) - D = H[Y] - d, (3.15)

which will be immunized with probability one provided H[Y]d. In fact the

following result characterizes fair premiums in the defined sense.

Proposition 3.2. Let H[] be a pricing principle and (Y,Z)Sd a perfectly

hedged experience rating contract with fair premium P=E[X+D]+(H[Z]-E[Z]),

and suppose the fair premium condition (3.14) holds. Then H[] is a fair pricing

principle such that P=H[X] if, and only if, the insurer's balance is completely

immunized, that is H[Y]=d.

10 Werner Hürlimann

Proof. Let us first show that the condition H[Y]=d is sufficient. By assumption

and (3.14) one gets 12=d - E[Y]=E[D], where the last equality holds because

(Y,Z)Sd. Rearranging terms using (3.14) one has

H[X] = + 12 + 1

2 = + E[D] + (H[Z] - E[Z]) = P, (3.15)

which identifies H[X] with the fair premium. To show that the condition is

necessary one proceeds as follows. Look at the balance of the insurer. His income

consists of the fair premium P plus the reinsurance payment Z while his outcome

consists of the claims payment X, the guaranteed bonus D and the reinsurance

premium H[Z]. By construction of the fair premium, his net outcome, that is the

difference between income and outcome, vanishes. By (3.15) this means that

H[Y]=d, as desired. ◊

Remarks 3.1.

(i) The necessary condition in Proposition 3.2 is implicitely contained in the

argument following formula (4.15) in Hürlimann [19], while the sufficient

condition has been formulated in a special case in Hürlimann [20], Section 4.3.

(ii) As shown in Hürlimann [20], Section 4.1, the fair premium condition (3.14)

approximately holds in a "distribution-free" sense in the stop-loss case Z=(X-d)+.

Let X be an arbitrary risk, defined on the whole real line, with finite mean

and standard deviation , and let H E Var be the standard

deviation pricing principle. Consider the distribution-free standard deviation

premiums defined by H*[Y]=H[Y*], H*[Z]=H[Z*], where the risk X has been

replaced by a risk X* with Bowers' distribution function

),,(),)(

1(2

1)(

22

*

x

x

xxF

(3.16)

first considered in Hürlimann [18]. Furthermore set H*[X]=+=H[X] (since

the variance of X* is infinite H[X*] is actually not defined, but this does not

disturb the results). Since E[Z*] coincides with the best stop-loss upper bound by

Bowers [6], this choice defines safe stop-loss premiums uniformly for all

deductibles d. Moreover, from the property Var[Y*]=Var[Z*]= 14

2 for all d,

one sees that H*[Y]=H[Y*]=E[Y*]+ 12 and H*[Z]=H[Z*]=E[Z*]+ 1

2 ,

which defines a "distribution-free" weak fair premium condition of the form

(3.14). In particular, the strong condition (3.14) holds true in case X equals X* in

distribution. The above situation will be further discussed in Example 4.5.


(iii) The condition (3.14) has the following alternative interpretations. Given

P=H[X] is a fair premium, then the redistribution of the risk profit loading

(H[X]-E[X]) is fair in the long run, or mean fair, for all three contractual

members in the insurance agreement (consult [25] for a precise mathematical

concept). Since the insurer's risk has been eliminated (perfect hedge condition), no

risk profit loading goes to the insurer. The expected risk profit goes half and half

to the reinsurer (needed risk profit loading to absorb reinsurance payment

fluctuations) and to the insured (in form of the bonus of expect amount equal to

half the risk profit loading). In the situation, where the insurer and the reinsurer

operations are exercised by the same insurance group, one can say that half of the

risk profit loading belongs to the insured and half to the shareholders of the group,

which provide the economic capital.

Example 3.2: a fair stop-loss premium model

Let X be a positive risk and let Z=(X-d)+ define a perfectly hedged experience

rating stop-loss contract with bonus D=(d-X)+. Suppose insurance premiums are

set according to the standard deviation pricing principle P=H[X]=+ and that

the fair premium condition (3.14) holds. Denote by (d)=E[(X-d)+] the net stop-

loss premium calculated according to a realistic claims model and let ( )d =E[(d-

X)+] the "conjugate" stop-loss premium representing the expected bonus of the

experience rating contract. By (3.14) one has

H Y d ( ) 12

, (3.17)

H Z d ( ) 12

. (3.18)

If the immunization condition H[Y]=d of Proposition 3.2 is imposed, that is the

risk loading satisfies the equality

12 ( )d , (3.19)

then the standard deviation premium P=+ is necessarily equal to the fair

premium

P d H Z d d d d ( ) ( ( )) ( ) ( ) . (3.20)

Observe that if the market price P is known, that is is known, then d is

uniquely determined by (3.19). Alternatively, if stop-loss market prices H[Z] are

known, then d is uniquely determined by the equivalent implicit equation

( ) ( ) ( )d d H X d . (3.21)


Further, in an uncertain insurance economy, market prices are not known with

certainty. In this situation, we use the distribution-free and parameter-free

premium P=(1+k2), where k is the coefficient of variation, as justified later in

Section 4.1. Then, one has necessarily =k and the "optimal fair" deductible d

is unique solution of the equation

12

2k d ( ). (3.22)

4. Distribution-free CAPM fair premiums

In this Section, the CAPM splitting premium formulas (2.5), (2.7) and Proposition

2.1 are used to construct (relatively) safe distribution-free reinsurance premiums

and fair premiums for the class Sd of perfectly hedged experience rating contracts.

The method follows closely Hürlimann [19], Section 5. Notations are the same as

in the preceding Sections.

Consider the biatomic risk X* such that the reinsurance payment Z*=g(X*) has

a maximum expected value over all biatomic risks:

)(max,

*

2

XgEZEbaDX

. (4.1)

Standard real analysis leads to the following result (proof in Hürlimann [19],

Lemma 5.1).

Lemma 4.1. Let Z=g(X) be the transform of XD2[a,b]. Then, the maximizing

biatomic risk X*D2[a,b] solving the optimization problem (4.1) with support

{x,x*}, where xx

*

2

defines an involution on [a,b], satisfies one of the

following conditions:

(i) x, x*(a,b) is solution of the equation

g x g x

x xg x g x

( ) ( )( ' ( ) ' ( ))

*

*

*

1

2 (4.2)

(ii) x=a, x*=a*

(iii) x=b*, x*=b**=b

Example 4.1.

Let Z=g(X)=(X-d)+ be a stop-loss claim. Then the maximum (4.1) equals


)( *

*

* dxxx

xZE

, (4.3)

and is attained as follows :

Case 1: a d a a x a 12( ),*

Case 2: 12

12

2 2( ) ( ), ( )* *a a d b b x d d

Case 3: 12( ) ,* *b b d b x b

In this special case (4.3) leads to a rigorous safe net stop-loss premium. Indeed, it

is actually the best upper bound over all risks defined on [a,b] with given mean

and variance (e.g. DeVylder and Goovaerts [7], Goovaerts et al. [8], p. 316,

Jansen et al. [217], Goovaerts et al. [9]). The limiting Case 2 when a - , b

is due to Bowers [6].

Typical often encountered situations include positive risks (limiting Case a=0,

b solved by Case 1 and Case 2) and arbitrary risks (limiting Case a - ,

b solved by Case 2). Non-life insurance concerns mainly positive risks.

Applications, which require the study of arbitrary risks, include financial risks

(rate of return, asset and liability management, portfolio theory, etc.) and some of

the life-insurance risks (e.g. mixed portfolios of whole life insurances and life

annuities).

A distribution-free pricing system for the class of contracts Sd is now obtained

as follows. Let X be a risk defined on [a,b] with finite mean and variance. In a

distribution-free world any such risk will be associated the same insurance

premium P=H[X], where H[] is a pricing principle depending stochastically only

on the mean and variance (e.g. variance principle, standard deviation principle,

etc.). Consider the biatomic risk X*={x,x*} that is solution of the optimization

problem (4.1) described in Lemma 4.1, and let Y*=f(X*), Z*=g(X*) be the

biatomic splitting risk components such that Y*+Z*=X*. Since X* has the same

mean and variance as X, its associated insurance premium equals H[X*]=H[X]=P.

On the other hand, consider Borch's additive premium principle defined by (2.5),

(2.7) and denoted by B[]. Assume that the compatibility condition

B[X*]=B[X]= 2=P holds and that 1, 0. Applying Proposition 2.1,

one gets the following CAPM splitting risk premiums

)()()(

)(

)(,

*

*

*

*

****

xPxx

xgxgxg

XEPXVar

ZXCovZEZB

(4.4)


)()()(

)(*

*** xP

xx

xfxfxfZBPYB

(4.5)

The CAPM based distribution-free pricing system H*[] obtained by setting

H*[X]:=B[X*], H*[Y]:=B[Y*], H*[Z]:=B[Z*], satisfies the desired splitting

property P=H*[X]=H*[Y]+H*[Z]. It may be viewed as a biatomic approximation

to Borch's additive CAPM premium principle, which satisfies the splitting

property P=B[X]=B[Y]+B[Z]. Furthermore the maximum (4.1), denoted by

*[Z]:=E[Z*] equals

)()()(

)(*

** x

xx

xgxgxgZ

. (4.6)

Using that P= 2 , 1, 0, x, it follows that

Zxxx

xgxgxgZBZH *2

*

*** ))((

)()()(

. (4.7)

A generalized version of Theorem 5.1 in Hürlimann [19] has been shown.

Theorem 4.1. Given is a risk X defined on [a, b] with finite mean and variance.

Let (Y,Z)=(f(X),g(X))Sd be a perfectly hedged experience rating contract with

maximum deductible d and bonus D=d-Y. Then the CAPM based distribution-

free pricing system H*[X]=B[X*], H*[Y]=B[Y*], H*[Z]=B[Z*], defined by

(4.4), (4.5), satisfies the splitting property P=H*[X]=H*[Y]+H*[Z] as well as the

criterion of (relative) safeness

)(max,

**

2

XgEZZHbaDX

. (4.8)

To illustrate let us develop some explicit results in case Z=(X-d)+ is a perfectly

hedged experience rating stop-loss contract with bonus D=(d-X)+ for the two

typical situations of positive risks and arbitrary risks. Results for a general risk

defined on [a,b] and other reinsurance structures can be obtained similarly.

4.1. Positive risks.

Consider the limiting case a=0, b of Example 4.1. The coefficient of

variation is denoted by k. Two cases must be distinguished:

Case 1: 0 1 0 112

2 2 d k x x k( ) , , ( )*


One has the parameter-free pricing formulas:

* ( )

Zk d

k

1

1

2

2, (4.9)

H Zk

dP* ( )

1

1

1 2 , (4.10)

H Yk

dP*

1

1 2 . (4.11)

Case 2: d k x d d x d d 12

2 2 2 2 21( ) , ( ) , ( )*

* ( ( ) ( ))Z d d 1

2

2 2 , (4.12)

H Z P x* ( ) 1

2 , (4.13)

H Y P x* ( ) 1

2 . (4.14)

Let us apply the immunization argument of Section 3. In Case 1 and for d>0, the

immunization condition H*[Y]=d implies that P=(1+k2), which is thus shown

to be a distribution-free and parameter-free fair premium by Proposition 3.2. By

continuity the same holds true in the limiting case as the deductible d goes to zero

(see also Hürlimann [19], Proposition 4.1). This argument provides a new

insurance economics interpretation of the "Karlsruhe" pricing principle introduced

by Heilmann [12] (see also Hürlimann [26]). In Case 2 one has x>0 and the

condition H*[Y]=d is equivalent to the relation

xPd /)2( , (4.15)

which restricts the possible deductible choices. Since P= 2 is assumed,

(4.15) is also equivalent to the relation

2 2 d x( ) . (4.16)

In view of Proposition 2.1, interesting interpretations in terms of utility theory are

possible. By (2.11) the following relations hold:

)(')(' *

**

*

xuxx

xxu

xx

x

, (4.17)


P u x u x

x x2

' ( ) ' ( )*

*. (4.18)

Example 4.2: linear utility

If the representative insurer is risk-neutral with utility function u(x)=ax+b, a>0,

then one has =a1, =0, which implies that P=H[X]=aE[X] is an expected

value principle. In Case 1 all deductibles are feasible while in Case 2 only those

d's satisfying (4.15) with P=a are feasible. After calculation one finds that the

deductible must satisfy the condition

d P a k 12

12

12

21 ( ) , (4.19)

which is feasible only in case a>(1+k2). If P=(1+k2) no deductible is feasible

in Case 2.

Example 4.3: quadratic utility with saturation

Let the representative insurer has marginal utility

bx

bxb

x

xu

,0

,1)(' (4.20)

In Case 2 one sees that (4.17), (4.18) are equivalent to

11

b b, . (4.21)

Feasible deductibles are solutions of the equation (4.15), that is

2d P

x

b

b

. (4.22)

In case P=(1+k2) one has necessarily

2

21

k

kb , (4.23)

which implies the parameter values


01

1,1

1

212

2

2

2

k

k

k

k

. (4.24)

After calculation one obtains two solutions to (4.22):

d kk

k

dk

k

1

22

2

2

4

2

1

21

1

1

21

1

( ) ,

( ) .

(4.25)

Only the first solution satisfies the required inequality d> 12

21( ) k of Case 2.

In particular, a non-trivial pricing principle P=H[X]= 2 with >1, >0,

finds herewith an actuarial application. Note that this example does not exist if the

normalization =1 is made as in Aase [1] and Hürlimann [19].

Remark 4.1. Similar results can be derived for other utility functions of common

use, e.g. exponential utility and power utility functions.

4.2. Arbitrary risks.

In the limiting case a - , b of Example 4.1, only Case 2 occurs, for

which the formulas (4.12) to (4.14) hold.

Example 4.4: An insurance market distribution-free stop-loss premium formula

Invoking the immunization condition H*[Y]=d in the special case d=0, for which

the whole premium and claim goes to the reinsurer (respectively the insurer acts

itself as reinsurer), one sees that the following equations must hold:

P H Z P x * ( )1

2 . (4.26)

It follows that

P k 1 2 . (4.27)

Solving for and inserting the result in the right-hand side of (4.26), one gets

the distribution-free stop-loss premium


)2()(1

**

2

*

dk

PZH , (4.28)

where )()()( 22

21* ddd denotes Bowers' best stop-loss upper

bound. Since 1 by assumption, one recovers the main result from Hürlimann

[16], which finds herewith a rigorous theoretical justification. Furthermore, the

arbitrage-free stop-loss premium rate H*[Z]/P is a distribution-free premium rate

depending only on , and d. The problem of the partition of the risk profit

loading between insurer and reinsurer (see Amsler [3]) is herewith solved in a

natural way (the stop-loss premium is coupled with the market risk premium) and

the inequality H*[Z]*(d) guarantees safeness for the reinsurer.

In the table below, we compare the distribution-free stop-loss premium with

the net stop-loss premium obtained from the Erlang approximation of the

probability density function. To get an idea of the probability of occurrence of a

stop-loss claim, the values Pr(X>d) for the Erlang approximation are displayed.

The required formulas are

Nx

xxxg

,)exp(

!);,( (probability density) (4.29)

G x g k xk

( , ; ) ( , ; )

11

1

(cumulative distribution) (4.30)

);,();,(1)()( dgd

dGddXEd

. (4.31)

In the numerical example, the parameters are

,

2

.


Table 4.1: distribution-free stop-loss premiums vs. other approximations

Parameters

a) =100 =5 b) =100 =10 c) =100 =20 d) =100 =25

case) d 100H*[Z]/P *(d) (d) Pr(X>d) in %

a) 100

110

120

150

200

2.56

0.65

0.37

0.19

0.12

2.50

0.59

0.31

0.12

0.06

1.99

0.052

0.00011

< 10-6

< 10-6

49.34

2.54

0.0079

< 10-5

< 10-5

b) 100

110

120

150

200

5.22

2.31

1.42

0.74

0.50

5.00

2.07

1.18

0.50

0.25

3.99

0.91

0.12

0.000016

< 10-6

48.67

15.83

2.79

0.00059

< 10-5

c) 100

110

120

150

200

10.78

7.03

5.03

2.86

1.94

10.00

6.18

4.14

1.93

0.99

7.95

4.16

1.97

0.12

0.00025

47.34

29.10

15.72

1.26

0.0035

d) 100

110

120

150

200

13.62

9.70

7.32

4.36

2,99

12.50

8.46

6.01

2.95

1.54

9.92

6.01

3.44

0.47

0.0072

46.67

31.91

20.21

3.44

0.066

Remark 4.2. Since the formula (4.28) is now established rigorously, all the

implications made in Hürlimann [16], Sections 4 to 7, are valid when working in a

distribution-free environment of arbitrary risks. It remains to be checked if and

under which conditions the same or similar conclusions may be true or

approximately true in a distribution-free environment of positive risks. An

illustration of this phenomenon follows in Example 4.6.

Example 4.5: a distribution-free fair standard deviation pricing principle

Given is the situation described in (ii) of Remark 3.1, which also concerns an

environment of arbitrary risks. However, observe that the definition of H*[]

differs from that of Theorem 4.1. A calculation shows that the condition H*[Y]=d

holds if, and only if, one has


21

2

1d . (4.32)

In the special case d=0, the standard deviation loading equals

k

k1 1 2, (4.33)

which yields the distribution-free and parameter-free fair standard deviation

pricing principle

P H X k 1 2 , (4.34)

first observed in Hürlimann [20], Section 4.1. In a distribution-free environment

of arbitrary risks, (4.34) is always less than the "Karlsruhe" price P=(1+k2)

obtained in a distribution-free environment of positive risks. Note that (4.34)

justifies the most competitive and unique choice =1, which can be made in

practical applications of Example 4.4.

Example 4.6: the "Karlsruhe" principle in a distribution-free environment

Consider the modelling situation of Hürlimann [20], Section 4.2. Suppose the

cedant operates according to the distribution-free standard deviation principle

H*[Y] as in Example 4.5. On the other hand, suppose the reinsurer sets premiums

following the principle (4.28):

)2()(1

**

2

d

k

PZH R . (4.35)

The condition P k 1 2 is equivalent to the inequality

1 1 2 k k . (4.36)

As in Example 4.5, one has

H Y d* * ( ) 12

. (4.37)

Since P=H*[Y]+HR[Z] the following equation must hold:


H Z dR * ( ) 12

. (4.38)

If the cedant wants to guarantee exactly the surplus D=(d-X)+ (complete

immunization argument), then (4.32) holds. Solving simultaneously the pair of

equations (4.32), (4.37) in the unknowns , d, yields after algebraic calculation

two feasible solutions, namely

Solution 1: d=0,

k

k1 1 2

Solution 2: d k k 12

21( ) ,

Although the insurer and the reinsurer operate according to different pricing

principles, the above model specification contains as special case the pricing

principle (4.34) of Example 4.5. Again the lower bound in (4.36) is attained. The

second solution identifies the premium with the "Karlsruhe" premium, giving to it

an arbitrary risk based interpretation. Solution 1 guarantees a bonus D=(- X)+

while solution 2 guarantees

XkD )1( 2

21 . As seen in Section 4.1, a

market premium P=(1+k2) suffices to guarantee a bonus D=(d - X)+ for all

0 112

2 d k( ) in a distribution-free environment of positive risks. In

particular, for the same premium P=(1+k2), the bonus

XkD )1( 2

21

can be guaranteed independently of whether X is a positive or an arbitrary risk

(cf. Remark 4.2).

5. Optimal long-term perfectly hedged experience rating

We show how perfectly hedged experience rating strategies, which are "optimal"

in the long-run and in a competitive environment, can be obtained for the class Sd

from Section 3. Optimality of a given pair (Y,Z)Sd is considered with respect to

the following two relevant properties:

(P1) Acceptable for the cedant are only contracts, which are mean self-financing.

(P2) The insurance premium of the contract should be competitive

To motivate the first property, consider the perfect hedge relation d+Z=X+D. A

time dependent mean self-financing dynamic strategy for the cedant can be

formulated as follows. At the beginning of the first period, the cedant puts aside

the maximum deductible d and pays the reinsurance premium E[Z], which in

general is adjusted by some loading. At the end of the first period, the reinsurance

payment Z together with the maximum deductible permits to pay the claims and

there remains the guaranteed bonus, which can be used to finance the maximum

deductible and the reinsurance premium of the next period, and so on. To be mean


self-financing, the expected bonus payment must at least be equal to the expected

reinsurance payment, that is (see also Hürlimann [25])

E[D] - E[Z] = E[d - X] = d - 0. (5.1)

Therefore, the maximum deductible should be greater or equal to the mean

amount of claims. Suppose premiums are set according to the variance principle.

Taking into account the bonus, which belongs to the cedant (resp. the shareholders

and/or the insureds), the needed periodic random payment of the cedant equals

P Y Z d d E Z Var Z D E Z Var Z Y, ; . (5.2)

The needed variance premium (different from the fair premium of Section 3) is

P E P Y Z d Var P Y Z d R Y ZX , ; , ; , , (5.3)

where the risk quantity

R Y Z Var Y Var Z Var X Cov Y ZX , , 2 (5.4)

is called "total splitting risk" (as measured by the variance) of the contract. By

Chebychev's inequality one has Cov[Y,Z]0 with equality sign if, and only if, Y

or Z is a constant (e.g. Hardy, Littlewood and Polya [10], no. 43, or alternatively

note that the pair (X,X) is positively quadrant dependent, which implies

Cov[f(X),g(X)]0 for all non-decreasing functions f(x), g(x)). It follows that the

needed variance premium is strictly less than the variance premium 2 of

the original risk X. The part of the risk stemming from the dependence between

the retained amount Y and the reinsurance payment Z has been eliminated by the

perfectly hedged experience rating contract (variance reduction through

diversification). One should note that the premium formula (5.3) defines a new

"reinsurance based" pricing principle.

To satisfy also property (P2), the premium (5.3) must be minimized. Therefore

"optimal" experience rating strategies (Y,Z)Sd are obtained as solutions of the

constrained optimization problem

RX[Y,Z] = min. under the constraint d. (5.5)

Equivalently by (5.4) one has

Cov[Y,Z] = max. under the constraint d. (5.6)

Let us determine the possible minimum and maximum premiums.

Proposition 5.1. For all (Y,Z)Sd the following bounds hold:


P P R Y Z PXmin max, 12

2 2. (5.7)

Proof. The upper bound follows from the mentioned inequality by Chebychev.

The lower bound is a Corollary to Proposition 1.1 and Proposition 1.3 in

Hürlimann [19]. In Proposition 1.1, it is shown that the minimum of RX[Y,Z]

over all transformed random variables Y, Z is attained by a linear transformation

(application of the Cauchy-Schwarz inequality). In Proposition 1.3, this lower

bound is also attained for a stop-loss experience rating contract Z=(X-d)+ for a

biatomic risk, which maximizes the net stop-loss premium.

In a distribution-free environment with positive risks, one can assume that

Pmax=(1+k2) as justified in Section 4.1. In this case one has necessarily

1.

Corollary 5.1. Let X0 be a positive risk and let (Y,Z)Sd. If Pmax=(1+k2),

then the following bounds hold:

P k PR Y Z

P kX

min max( ) (,

) ( ) 1 1 112

2

2

2

. (5.8)

Remark 5.1. The above minimum and maximum premium bounds are

reminiscent of the experience rating method introduced by Ammeter [2] and

which, as our starting point, has been given different equivalent characterizations

(see Hürlimann [14]).

On the other hand, if one requires additionally that the premium (5.3) should be a

fair premium in the sense of Section 3, then one must have

P E D Var Z . (5.9)

Now, both premiums can be equal only if the following relation holds:

Var[Y] = E[D]. (5.10)

Therefore "optimal" strategies (Y,Z)Sd with fair variance premiums of the

form

PR Y Z

Var YE D

X

, (5.11)

solve the constrained minimization problem


R Y Z

Var YE D

X , = min. under the constraint d. (5.12)

Two illustrate the many applications of the introduced methodology, let us search

for "optimal" perfectly hedged experience rating contracts in two common

situations.

5.1. An optimal long-term stop-loss pricing strategy.

Let us reproduce the results sketched in Hürlimann [20], Section 3. For Z=(X-d)+

use the "conjugate" notations (d)=E[Z], ( )d E D , F(x)=Pr(Xx),

F x F x( ) ( ) 1 , 2 ( ) ( )d Var X d , 2 ( ) ( )d Var X X d . The

total stop-loss risk equals the univariate function R d d d( ) ( ) ( ) 2 2 . A

minimum under the constraint d satisfies the necessary condition

R d F d d F d d d'( ) ( ) ( ) ( ) ( ) , 0 . (5.13)

Using that ( ) ( ) , ( ) ( )d F d E X d X d d F d E d X X d , one

obtains the equivalent conditional expected equation

E X d X d E d X X d d , . (5.14)

In order that a stop-loss deductible is optimal, it is necessary that the conditional

expected amount of stop-loss claims equals the conditional expected bonus.

Rearrangement shows that alternatively the following fixed-point equation must

hold:

d E X X d E X X d 12( ). (5.15)

In case a fixed-point d has been found, this will be a guaranteed local

minimum provided

0)()()()()(2)('' dddfdFdFdR . (5.16)

In modern analysis general conditions under which a fixed-point equation like

(5.15) has a solution are well-known. However, in our context there is an

elementary proof, which shows that the above necessary condition is, in many

cases of practical importance, fulfilled.

Proposition 5.2. Let F(x) and (x) be continuous real functions. If F( ) 12

,

then there exists at least one d [,) such that R'(d)=0.


Proof. First of all one has from (5.13) and by assumption R'()=(1-2F())()0.

By continuity it remains to check that there exists d such that g(d):=R'(d)0.

Elementary calculation shows that

)(1)(2)()()( ddFdFddg .

With the inequality of Bowers [6]

,)()()( 22

21 ddd

one has for all d :

22

2122

21 )()()()()()()( ddFdddFdFddg

For a fixed d0 such that F d( )012

, it follows that for all

1)(4,max

2

0

0

dFdd

,

one has

g d d F d d( ) ( ) ( ) ( ) 012

2 2 0,

as was to be shown. ◊

In contrast to the above result, the uniqueness of a solution to the fixed-point

equation (5.15) cannot in general be guaranteed without further assumptions. An

interesting useful example illustrates this fact.

Example 5.1: the distribution of Bowers

In Hürlimann [18] the following distribution function has been considered:

).,(,)(

12

1)(

22

x

x

xxF

(5.17)

Integrating the differential equation '(x)= - (1-F(x)), one sees that the associated

net stop-loss premium

)()()( 22

21 ddd (5.18)

coincides with the best upper bound of Bowers given above. One checks the

"uniform invariant conjugate" property


F d d F d d( ) ( ) ( ) ( ) for all d, (5.19)

which in particular shows that R'(d)=0 for all d. From (5.7) and (A.1), (A.2)

in the Appendix, one borrows the inequality

2222

2

1),(

)(

)()(

)(

)(max)()()( d

dF

dFd

dF

dFdddR . (5.20)

But 2 2 14

2( ) ( )d d uniformly for all d, which shows in particular that the

minimum is uniformly attained for all d. We have shown that the distribution of

Bowers satisfies the following two extremal properties. Independently of the

deductible it maximizes the net stop-loss premium and minimizes the total stop-

loss risk. The simultaneous use of this distribution as "risk valuation function" by

the cedant and the reinsurer can thus be mathematically justified through optimal

properties. Our first application in this direction is given in Hürlimann [18].

Similarly, the "optimal" choice d=, which played up to now only a guiding

role in an extended version of the classical mean-variance approach to portfolio

theory by Markowitz (see Hürlimann [15], [24]), is justified below using the

notion of "total stop-loss risk".

Example 5.2: the normal distribution

Let N(x) be the standard normal distribution, (x)=N'(x) the normal density,

and assume that F(d)=N(z) with z=(d-)/. Then, one has R'()=0 and

2

1)()()()()(2)('' NNR . (5.21)

If < /2 the retention level d= is a local minimum of the total stop-loss risk

function. It is important to observe that the required technical condition about the

standard deviation or similar volatility is almost always fulfilled in applications.

Example 5.3: the distribution of a whole life insurance portfolio

Computations with the exact distribution have shown that an optimal deductible

lies above but often close to the mean.

5.2. On the linear combination of proportional and stop-loss reinsurance.

The perfectly hedged experience rating contract defined by Z=(1-r)X+r(X-s)+,

(r,s)(0,1]x[0,), with d=rs the maximum deductible and D=r(s-X)+ the bonus,

has been studied in Hürlimann [22]. In particular, its fair premium, when the


reinsurer uses the variance principle, has been determined and shown to be

bounded by the variance premium 2 in case r(0,1] and 2 1( )s .

Properties of the minimum fair premium with respect to the proportional retention

level r have also been discussed.

We show that the optimization problem (5.6) has no extremal solution (except

perhaps for the degenerate biatomic distribution of Subcase 2 below) in the inner

of the domain { (r,s)(0,1]x[0,) : d=rs }. Therefore the optimal contract lies

necessarily on the boundary of this domain and is thus the stop-loss contract r=1,

d=s, studied in Section 5.1.

For the considered contract one has

)()1()()()12()1(,:),( 22 srssrrrZYCovsrC . (5.22)

Using the derivatives ' ( ) ( ), '( ) ( ), ( )' ( ) ( ),s F s s F s s F s s 2 2

2 2( )' ( ) ( )s F s s , one gets the partial derivatives

C s s r s s sr ( ) ( ) ( )( ( ) ( ) ( ))2 1 22 2 , (5.23)

)()()12()()( ssFrssFrCs . (5.24)

To show that a stationary point (r,s) satisfying Cr=Cs=0 does not exist for r

(0,1) let us distinguish between two cases :

Case 1: 2 2 22( ) ( ) ( )( )

( )( )s s s

F s

F ss

In this situation, the inequality of Kremer [28] (simple probabilistic proof in the

Appendix) is strict. In particular the inequality 2 2 2( ) ( ) ( )s s s of

Hürlimann [17] is strict and F s s( ) ( ) 0 (F s s( ) ( ) 0 corresponds to Case 1

and Case 2 in the proof of the inequality of Schmitter [30] in the Appendix, that is

one has also ( ) ( )s s 0 and F s s( ) ( ) 0). The system of equations Cr=Cs=0

is equivalent to the system

)()(

)()(1

2

1

ssF

ssFr

, (5.25)

F s s

F s s

s s

s s s

( ) ( )

( ) ( )

( ) ( )

( ) ( ) ( )

2 22. (5.26)

Solving for 2 ( )s in (5.26) and using that ( ) ( )s s s , one gets


2 2 22( ) ( ) ( )( )

( )( )s s s

F s

F ss , (5.27)

which contradicts the assumption made.

Case 2: 2 2 22( ) ( ) ( )( )

( )( )s s s

F s

F ss

This means that the upper bound of the inequality of Kremer [28] is attained. By

the inequality of Schmitter [30], proved in the Appendix, the upper bound is

attained in the space of all risks with given mean , variance 2 and net stop-loss

premium (s), by a biatomic distribution and equals

)()(4)()(22

1)(max 222 sssss . (5.28)

Subcase 1: ( )s s 0 (Case 1 in Appendix)

One has ( ) , ( ) , , ( )( )s s C C rF s sr s 0 0 02 2 . The solution

s= implies, using the restriction d=rs, that r=1.

Subcase 2: ( )s s 0 (Case 2 in Appendix)

One has ( ) , ( ) , , ( ) ( )( )s s C C r r F s sr s 0 0 2 1 02 2 for

a degenerate limiting diatomic distribution for which F s( ) 0.

Subcase 3 : ( ) ( )s s 0

As observed in Case 1 this implies F s s( ) ( ) 0 . Set

zF s s

F s s s s

( ) ( )

( ) ( ),

( ) ( )

2

4. (5.29)

With (5.28) one sees that if the upper bound of Kremer [28] is attained, one must

have the relation (multiply with 1)()(4

ss ) :

)1(21

21

41

21 z . (5.30)

Solving for z one gets

z 2 1 2 1 ( ) , (5.31)

where 1 by the inequality of Bowers [6]. Now by (5.24) one should have

rz

12

11 1( ) , hence z1. By (5.31) this is only possible if =1, hence z=1 and

r=1.


Appendix: Dual inequalities of Kremer [28] and the inequality of Schmitter [30]

First, a simple probabilistic derivation of a slightly generalized version of the

inequality by Kremer [28] on the stop-loss variance is presented. Based on this

result, it is shown that among all risks with a given mean, variance and net stop-

loss premium, a biatomic distribution maximizes the stop-loss variance, a result

due to Schmitter [30].

We use the following "conjugate" notations. For a risk X with finite mean

and variance 2, let F(x)=Pr(Xx) the corresponding distribution function,

F x F x( ) ( ) 1 the survival function, (d)=E[(X-d)+] the net stop-loss

premium, (d)=E[(d-X)+]=d-+(d) the "conjugate" net stop-loss premium,

2(d)=Var[(X-d)+] the stop-loss variance and 2 ( ) ( )d Var X X d the

variance of the retained claims. The "conjugate" attribute has two motivations.

First in the language of "life contingencies" the quantities F(d) and

( ) ( )d F d E X d X d are associated to the "mortality" of the risk X at

"age" d, while the quantities F d( ) and ( ) ( )d F d E d X X d are given

the corresponding "survival" metaphoric interpretations. Second, there is a

"conjugate" property relating the inequalities for 2 ( )d and 2 ( )d , which

render them easy to remind of. To pass from one inequality to the other, it suffices

to take "conjugate" quantities with the convention that

2 2 , ( ) ( ), ( ) ( )F d F d d d .

Proposition A.1. (Kremer [28]) Given is a risk X with known , 2, (d), and

F(d). Then, in the "conjugate" notation, the following inequalities hold:

F d

F dd d d d

F d

F dd

( )

( )( ) ( ) ( ) ( )

( )

( )( ) 2 2 2 22 , (A.1)

F d

F dd d d d

F d

F dd

( )

( )( ) ( ) ( ) ( )

( )

( )( ) 2 2 2 22 . (A.2)

Proof. Conditioning on the event {X>d} one has

.)(

)()(

)(

)()()(

2

2

2222

dF

ddXdXVardF

dXdXEdXdXVardF

dXdXEdFdXEdd

(A.3)

Rearranging, one obtains the relation


2 2( ) ( )( )

( )( )d F d Var X d X d

F d

F dd , (A.4)

which implies the lower bound in (A.1). On the other hand, combining the relation

2 2 2 2 2 ( ) ( ) ( ) ( ) ( ) ( )d E X d F d E X d X d F d E X d X d

(A.5)

with (A.3), one shows similarly that

2 2 22( ) ( ) ( )( )

( )( ) ( )d d d

F d

F dd F d Var X d X d , (A.6)

which implies the upper bound in (A.1). The inequalities (A.2) follow from (A.1)

and the relation 2 2 2 2( ) ( ) ( ) ( )d d d d .

Remark A.1. In case the probability F(d) is not known, one has the simpler

"self-conjugate" upper bounds (see Hürlimann [17], Hesselager [13]):

2 2 2( ) ( ) ( ).d d d (A.7)

2 2 2( ) ( ) ( ).d d d (A.8)

The following result is originally due to Schmitter [30]. We obtain a slightly more

general version based on a simpler and more rigorous proof.

Proposition A.2. The maximal stop-loss variance to the deductible d for a risk X

with mean , variance 2 and net stop-loss premium (d), is given by

,)()(4)()(22

1)(max 222 ddddd (A.9)

and is attained by a biatomic random variable Z with support {a,b} such that

,

,

1

1

p

p

p

p

b

a

(A.10)

where the probability p=Pr(Z=a) satisfies the relation

p

p

d d d d

d1

1

2

2 42 2

2

( ) ( ) ( ) ( )

( ) (A.11)


Proof. It suffices to consider distributions which, for given p=F(d), maximize

2(d), that is which satisfy equality in the upper bound (A.1). A maximum is

obtained if we determine p so that the right hand side of (A.1) is maximal. Since

the derivative of that right hand side with respect to p is positive, this expression

is a monotone function. Thus the probability p must be as great as possible. On

the other hand, the variation of p is restricted by the lower bound constraint in

(A.1):

22 )(1

)( dp

pd

, (A.12)

where equality holds for biatomic distributions with given and 2, which are

always of the type (A.10). Moreover the right hand side of (A.12) is maximum in

case p is greatest possible. Provided equality holds in (A.12) and inserting this

relation into the upper bound (A.1) with equality sign, it follows that the

maximizing probability p is the greatest solution of the quadratic equation

22)1()()1()( ppdpdp . (A.13)

But, this equation is equivalent to the quadratic equation for p

p1:

0)(1

)()(2

12

1)( 22

2

2

d

p

pdd

p

pd , (A.14)

whose greatest solution is (A.11). Inserting this value into (A.12) with equality

sign one gets the maximum (A.9). It remains to show that the maximum is

attained by a biatomic distribution of the type (A.10), with p from (A.11), such

that the quantities , 2, (d) are the given ones. Using the well-known

properties of (A.10), it remains only to check that E[(Z-d)+]=(d) is the given net

stop-loss premium. We distinguish between three cases.

Case 1: (d)=-d0

One has (d)=0 and thus max{2(d)}=2. The appropriate biatomic distribution

of the type (A.10), (A.11) is

a d

bd

p Z ad

2

2

2 2Pr( )

( )

(A.15)


Case 2: (d)=d-0

One has (d)=0 and thus max{2(d)}=2. The maximizing biatomic distribution

of the type (A.10), (A.11) is the limiting biatomic distribution obtained setting p

1, a, b.

Case 3: (d)-d

It suffices to consider the case ad<b. Otherwise, one has a<bd with E[(Z-

d)+]=0, leading to the Case 2, or d<a<b with E[(Z-d)+]=-d, which is Case 1.

One has to satisfy

E Z d d p p p d( ) ( )( ) ( ) ( ), 1 1 (A.16)

which is seen to be equivalent to

p p p d p d( ) ( ) ( ) ( ).1 1 (A.17)

Taking squares one sees that p must be solution of (A.13).

Remarks A.2. Schmitter [30] does not provide (A.11) and verify that the

maximizing distribution has as net stop-loss premium the given (d). Case 1

reveals a statement similar to that of Proposition A.2 for the weaker inequality

(A.7). Provided d, the equality sign in (A.7) is attained by the biatomic

distribution (A.15) and gives also the maximum of the stop-loss variance, namely

max{2(d)}=2. This fact finds an interesting application (see Hürlimann [22]).

References

[1 K.K. Aase, Equilibrum in a reinsurance syndicate; existence, uniqueness and

characterization, ASTIN Bulletin, 23 (1993), 185-211.

http://dx.doi.org/10.2143/ast.23.2.2005091

[2] H. Ammeter, Risikotheoretische Grundlagen der Erfahrungstarifierung,

Bulletin of the Swiss Association of Actuaries, (1961), 183-217.

[3] M.-H. Amsler, Sur les marges de sécurité des primes "stop-loss", Bulletin of

the Swiss Association of Actuaries, (1986), 77-87.

[4] K. Borch, Additive insurance premiums: a note, The Journal of Finance, 37

(1982), 1295-1298, Reprinted in Borch, (1990), 192-197.

http://dx.doi.org/10.2307/2327851


[5] K. Borch, Advanced Textbooks in Economics, Chapter in Economics of

Insurance, Elsevier Science Publishers, Amsterdam, 29, 1990.

http://dx.doi.org/10.1016/b978-0-444-87344-6.50001-4

[6] N.L. Bowers, An upper bound for the net stop-loss premium, Transactions of

the Society of Actuaries, XIX (1969), 211-216.

[7] F. DeVylder and M.J. Goovaerts, Upper and lower bounds on stop-loss

premiums in case of known expectation and variance of the risk variable,

Bulletin of the Swiss Association of Actuaries, (1982), 149-164.

[8] M.J. Goovaerts, F. DeVylder and J. Haezendonck, Insurance Premiums,

North-Holland, 1984.

[9] M.J. Goovaerts, R. Kaas, A.E. Van Heerwaarden and T. Bauwelinckx,

Effective actuarial methods, Insurance Series 3, North-Holland, Amsterdam,

1990.

[10] G. Hardy, J.E. Littlewood and G. Polya, Inequalities, 8th Reprint of 2nd ed.,

Cambridge University Press, 1989.

[11] A.E. Van Heerwaarden and R. Kaas, The Dutch premium principle,

Insurance: Mathematics and Economics, 11 (1992), 129-133.

http://dx.doi.org/10.1016/0167-6687(92)90049-h

[12] W.-R. Heilmann, Fundamentals of Risk Theory, Verlag Versicherungs-

wirtschaft, Karlsruhe, 1988.

[13] O. Hesselager, An improved elementary upper bound for the variance of a

stop-loss risk: a comment on the article by W. Hürlimann, Bulletin of the

Swiss Association of Actuaries, (1993), 277-278.

[14] W. Hürlimann, On algebraic equivalence of tariffing systems, Insurance:

Mathematics and Economics, 7 (1988), 35-37.

http://dx.doi.org/10.1016/0167-6687(88)90093-5

[15] W. Hürlimann, Absicherung des Anlagerisikos, Diskontierung der Passiven

und Portfoliotheorie, Bulletin of the Swiss Association of Actuaries, (1991),

217-250.

[16] W. Hürlimann, An insurance market based distribution-free stop-loss

premium principle, Proceedings 24th Int. ASTIN Coll., Cambridge, 1993.


[17] W. Hürlimann, An elementary upper bound for the variance of a stop-loss

risk, Bulletin of the Swiss Association of Actuaries, (1993), 97-99.

[18] W. Hürlimann, Solvabilité et réassurance, Bulletin of the Swiss Association of

Actuaries, (1993), 229-249.

[19] W. Hürlimann, Splitting risk and premium calculation, Bulletin of the Swiss

Association of Actuaries, (1994), 167-197.

[20] W. Hürlimann, From the inequalities of Bowers, Kremer and Schmitter to the

total stop-loss risk, Proceedings 25th Int. ASTIN Coll., Cannes, 1994.

[21] W. Hürlimann, A note on experience rating, reinsurance and premium

principles, Insurance: Mathematics and Economics, 14 (1994), 197-204.

[22] W. Hürlimann, Experience rating and reinsurance, Proceedings 25th Int.

ASTIN Coll., Cannes, 1994.

[23] W. Hürlimann, A distribution-free all-finance binomial valuation model,

Trans. 25th Int. Congress of Actuaries, Brussels, 3 (1995), 365-382.

[24] W. Hürlimann, Mean-variance portfolio selection under portfolio insurance,

In: P. Albrecht (Ed.), Aktuarielle Ansätze für Finanz-Risiken, Proceedings

6th Int. AFIR Colloquium Nürnberg, Verlag Versicherungswirtschaft,

Karlsruhe, 1, 1996, 347-374.

[25] W. Hürlimann, Non-optimality of a linear combination of proportional and

non-proportional reinsurance, Insurance: Mathematics Economics, 24

(1999), no. 3, 219-227. http://dx.doi.org/10.1016/s0167-6687(98)00054-7

[26] W. Hürlimann, Is the Karlsruhe premium a fair value premium? Blätter

DGVFM, 26 (2004), no. 4, 701-708.

http://dx.doi.org/10.1007/bf02808974

[27] K. Jansen, J. Haezendonck and M.J. Goovaerts, Upper bounds on stop-loss

premiums in case of known moments up to the fourth order, Insurance:

Mathematics and Economics, 5 (1986), 315-334.

http://dx.doi.org/10.1016/0167-6687(86)90027-2

[28] E. Kremer, An elementary upper bound for the loading of a stop-loss cover,

Scandinavian Actuarial Journal, 1990 (1990), 105-108.

http://dx.doi.org/10.1080/03461238.1990.10413875

[29] F. Riesz, Sur les opérations fonctionelles linéaires, Comptes Rendus de

l'Académie des Sciences de Paris, (1909), 974-976.


[30] H. Schmitter, An upper limit for the stop-loss variance, Preprint, July 7,

1993.

Received: December 9, 2015; Published: January 19, 2016

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Some Alternative Actuarial Pricing Methods: Application to ... · Some alternative actuarial pricing methods… 3 tion premiums are obtained. At the same time this example justifies