some alternative actuarial pricing methods: application to ... · some alternative actuarial...
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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
Some alternative actuarial pricing methods… 5
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
Some alternative actuarial pricing methods… 7
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
Some alternative actuarial pricing methods… 9
(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.
Some alternative actuarial pricing methods… 11
(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)
12 Werner Hürlimann
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
Some alternative actuarial pricing methods… 13
)( *
*
* 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)
14 Werner Hürlimann
)()()(
)(*
*** 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( ) , , ( )*
Some alternative actuarial pricing methods… 15
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)
16 Werner Hürlimann
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
Some alternative actuarial pricing methods… 17
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
18 Werner Hürlimann
)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
.
Some alternative actuarial pricing methods… 19
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
20 Werner Hürlimann
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:
Some alternative actuarial pricing methods… 21
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
22 Werner Hürlimann
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:
Some alternative actuarial pricing methods… 23
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
24 Werner Hürlimann
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.
Some alternative actuarial pricing methods… 25
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
26 Werner Hürlimann
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
Some alternative actuarial pricing methods… 27
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
28 Werner Hürlimann
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.
Some alternative actuarial pricing methods… 29
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
30 Werner Hürlimann
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)
Some alternative actuarial pricing methods… 31
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)
32 Werner Hürlimann
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]).
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Received: December 9, 2015; Published: January 19, 2016