Auni ed approach to generate risk measures

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1 Auni ed approach to generate risk measures Marc J. Goovaerts a;b, Rob Kaas b, Jan Dhaene a;b, Qihe Tang b a CRIS, Catholic University of Leuven b Department of Quantitative Economics, University of Amsterdam March 13, 2003 Abstract The paper derives many existing risk measures and premium principles by minimizing a Markov bound for the tail probability. Our approach involves two exogenous functions v(s) and Á(S; ¼) and another exogenous parameter 1. Minimizing a general Markov bound leads to the following unifying equation: E[Á(S;¼)] = E[v(S)]: For any random variable, the risk measure ¼ is the solution to the unifying equation. By varying the functions Á and v, the paperderives the mean value principle, the zeroutility premium principle, the Swiss premium principle, Tail VaR, Yaari's dual theory of risk, mixture of Esscher principles and more. The paper also discusses combining two risks with super-additive properties and sub-additive properties. In addition, we recall some of the important characterization theorems of these risk measures. Keywords: Insurance premium principle, Risk measure, Markov inequality 1 Introduction In the economic and actuarial nancial literature the concept of insurance premium principles (risk measures) has been studied from di erent angles. An insurance premiumprinciple is a mapping from the set of risks to the reals, cf. e.g. Gerber (1979). The reason to study insurance premium principles is the well-known fact in the actuarial eld that if the premium income equals the expectation of the claim size or less, ruin is certain. In order to keep the ruin probability restricted one considers a risk characteristic or a risk measure for calculating premiums that includes a safety loading. Thisconcept is essential for the economics of actuarial evaluations. Several types of insurancepremium principleshave been studied and characterized by means of axiomsasin Goovaertset al. (1984). On the other hand, desirablepropertiesfor premiums relevant from an economic point ofview havebeen considered. An insurancepremiumprincipleisoften considered asthe \price"ofa risk (or 1

2 of a tail riskin reinsurance), as the valueof a stochastic reserve, or asan indication of the maximal probableloss. This givestherelation to ordering of risksthat isrecentlydeveloped in the actuarial literature. In Artzner (1999), see also Artzner et al. (1999), a risk measure is also de ned as a mapping from the set of r.v.'s to the reals. It could be argued that a risk measure is a broader concept than an insurance premium calculation principle. Indeed, for a risk, the probability½() = Pr[> 1:10E[]] is a risk measure, but this is not a premiumcalculation principle becausetacitlyit is assumed that premiums are expressed in monetary units. However, assuming homogeneity for a risk measure, hence ½(a) = a½() for all reala>0and all risks, implies that½() allows changing the monetary units. On the other hand, because the parameters appearing in the insurance premiumprinciplesmay depend on monetaryunits, theclass of insurancepremiumscontains therisk measuresthat arehomogeneousasa special case. In addition, let beariskvariablewith niteexpectation and let u be an initial surplus. De ning a transformed random variable describing risk as Y = + ( E[])2 u also allowsriskmeasuresto depend on other monetary quantities. Consequently it is di±cult togiveadistinction between insurancepremiumprinciplesand homogeneousriskmeasures. Di erent setsof axioms lead to di erent risk measures. Thechoiceof the relevant axiomsof coursedepends on the economics of thesituationswhere it is used for. Desirableproperties might be di erent for actual calculation of premiums, for reinsurance premiums, or for allocation, and so on. In this paper we present a uni ed approach to some important classes of premium principlesaswell as riskmeasures, based on the Markovinequalityfor tail probabilities. We prove that most well-known insurancepremiumprinciples can be derived in this way. In addition, we will refer to someof the important characterization theorems of these risk measures. Basicmaterial on utilitytheory and insurancegoesbackto Borch (1968, 1974), using the utility concept of von Neumann and Morgenstern (1944). Thefoundation of premium principleswaslaid bybäuhlmann (1970) who introduced thezero-utilitypremium, Gerber (1979) and comprehensivelybygoovaertset al. (1984). Theutilityconcept, themeanvaluepremiumprincipleaswell astheexpected valueprinciplecan bededuced fromcertain axioms. An early source is Hardy et al. (1952). The Swiss premiumcalculation principle was introduced by Gerber (1974) and De Vijlder and Goovaerts (1979). A multiplicative equivalent of the utility framework has led to theorlicz principle as introduced by Haezendonck and Goovaerts (1982). A characterization for additive premiums has been introduced by Gerber and Goovaerts (1981), and led to the so-called mixture of Esscher premium principles. Morerecently, Wang (1996) introduced in theactuarial literaturethe distortion 2

3 functionsinto theframework of riskmeasures, using Yaari's (1987) dual theory of choice under risk. This approach also can be introduced in an axiomatic way. Artzner (1999) restricted theclass of Orliczpremiumprinciplesbyadding therequirement of translation invariance to its axioms, weakened by Jarrow (2002). This has mathematical consequences that are sometimes contrary to practical insurance applications. In the 1980's the practical signi cance of thebasicaxiomshasbeen discussed; seegoovaertset al. (1984). On thesamegroundsartzner (1999) providesan argumentation for selecting a set of desirable axioms. In Goovaerts et al. (2003) it is argued that there are no sets of axioms generally valid for all typesofriskysituations. Thereisadi erencein desirablepropertieswhen one considers a risk measure for allocation of capital, a risk measure for regulating purposes or a risk measure for premiums. There is a parallel with mathematical statistics, where characteristicsofdistributions mayhavequitedi erent meaningsand uses, likee.g. the mean to measurecentral tendency, thevariancetomeasurespread, theskewnessto re ect asymmetryand thepeakedness tomeasurethe thicknessof thetails. In an actuarial context, risk measuresmight havedi erent properties than in other economiccontexts. For instance, if we cannot assume that there are two di erent reinsurers willing to cover both halves of a risk separately, therisk measure (premium) for the entirerisk should be larger than twicetheoriginal riskmeasure. This paper aims to introducemany di erent riskmeasures (premiumprinciples) now available, each with their desirable properties, within a uni ed framework based on the Markovinequality. Togivean ideahowthisisachieved, wegiveasimpleillustration. Example 1.1. The exponential premium is derived as the solution to the utility equilibrium equation E[ e (w ) ] = E[ e (w ¼) ]; (1.1) wherew is the initial capital andu(x) = e x is the utility attached to wealth levelx. This is equivalent to hence we get the explicit solution E[e (¼ ) ] = 1; (1.2) ¼ = 1 loge[e ]: (1.3) TakingY =e andy =e ¼ and applying Pr[Y >y] 1 E[Y] (Markov inequality), we get y the following inequality for the survival probabilities with: Pr[>¼] e ¼E[e 1 ]: (1.4) For this Markov bound to be non-trivial, the r.h.s. of (1.4) must be at most 1. It equals 1 when¼is equal to the exponential( ) premium with. This procedure leads to an equation which gives the premium for from a Markov bound. 3

4 Two more things must be noted. First, for xed¼, we write the bound in (1.4) as f( ) = E[e ( ¼) ]. Sincef 00 ( ) = E[( ¼) 2e ( ¼) ]>0the functionf( ) is convex in some relevant -region. The risk aversion 0 for which this bound f( ) is minimal hasf 0 ( 0) = 0, hence¼=e[e 0 ]=E[e 0 ], which is the Esscher premium for with parameter 0. This way, also the Esscher premium has been linked to a Markov bound. The r.h.s. of (1.4) equals 1 for = 0 as well, and is less than or equal to 1 for in the interval [0; 1], where 1 is the risk aversion for which the exponential premium equals¼. Second, if ¼ = 1 loge[e ] holds, we have the following exponential upper bound for tail probabilities: for anyk> 0, Pr[>¼+k] e (¼+k)E[e 1 ] =e k : (1.5) Using variations of themarkov bound above, the various equations that generate various premiumprinciples (or riskmeasures) can bederived. Section 2 presentsa method to do this, Section 3 applies this method to many such principles, and discusses their axiomatic foundations as well as some other properties; Section 4 concludes. 2 GeneratingMarkovianrisk measures Throughout this paper, wedenotethe cumulative distribution function (cdf) of a random variables byf S. For any non-negative and non-decreasing functionv(s) satisfying we de ne an associated r.v.s having a cdf with di erential It iseasyto provethat E[v(S)]<+1; (2.1) df S (s) = v(s)df S(s) ; <s<+1: (2.2) E[v(S)] Pr[S>¼] Pr[S >¼]; <¼<+1: (2.3) For anylebesguemeasurablebivariatefunction Á( ; ) satisfying we have the following inequalities: Then it follows from (2.3) that Á(s;¼) I (s>¼) ; (2.4) Pr[S >¼] = E I (S >¼) E[Á(S ;¼)]: (2.5) Pr[S>¼] E[Á(S;¼)v(S)] : [GMI] E[v(S)] 4

5 Thisis a generalized version of the Markovinequality, which hass 0 with probability1 and¼ 0,Á(s;¼) =s=¼ andv(s) 1. Therefore, we denote it by the acronym [GMI]. Similar discussions can be found in Runnenburg and Goovaerts (1985), where the functions v( ) andá( ; ) are speci ed asv( ) 1; andá(s;¼) =f(s)=f(¼) respectively, for some non-negativeand non-decreasingfunction f( ). For the inequality [GMI] to make sense, the bivariate functioná(s;¼) given in (2.4) and the r.v.s should satisfy E[Á(S; ¼)v(S)] < +1 (2.6) for all relevant¼. Note that if (2.6) holds for some <¼< +1 then (2.1) does as well. By assuming (2.6), it is clear that thefamilyof r.v.'s S considered in theinequality [GMI] is restricted, in the sense that the right tail ofs can not be arbitrarily heavy. For the given functionsá( ; ) and v( ) asabove, weintroducebelowa family of all admissibler.v.'swhich satisfy (2.6): S Á;v = fs : E[Á(S;¼)v(S)]<+1 for all large¼g: (2.7) Sometimes we are interested in the case that there exists a minimal value¼ ( ) M [GMI] gives a bound Pr[S>¼ M ] E[Á(S;¼ M)v(S)] E[v(S)] such that 1: (2.8) Note that both (2.8) produces an upper bound for the -quantileq (S) ofs. For each 0 1, the restriction (2.4) oná( ; ) allows us further to introduce a subfamily of S Á;v as follows: ½ S Á;v; = S : E[Á(S;¼)v(S)] ¾ for all large¼ : (2.9) E[v(S)] If in (2.4) the functioná(s;¼) is strictly smaller than 1 for at least one point (s;¼), then it is not di±cult to prove that there are some values of 0 <1such that the subfamilies S Á;v; are not empty. We also note that S Á;v; increases in 0. Hereafter, for a real functionf( ) de ned on an intervaldand a constantbin the range of the functionf( ), we write an equationf(¼) =b with understanding that its root is the minimal value of¼satisfyingf(¼) band maxff(x)jx 2 (¼ ";¼ +") \Dg bfor any ">0: With this convention, the minimal value¼ ( ) M such that the second inequality in (2.8) holdsissimplythesolution of theequation When =1, we call E[Á(S;¼ M )v(s)] E[v(S)] E[Á(S;¼ M )v(s)] E[v(S)] 5 = : [UE ] = 1 [UE]

6 the unifying equation, or [UE] in acronym. This equation will act as the unifying form to generate many well-known risk measures. Theequation [UE] givestheminimal percentile for which the upper bound for the tail probability ofs still makes sense. It will turn out that theseminimal percentilescorrespond toseveral well-known premiumprinciples (risk measures). It is clear that the solution of the equation [UE] is not smaller than the minimal value of the r.v.s. De nition 2.1. LetS be an admissible r.v. from the family S Á;v; for some 0 1, whereá=á( ; ) andv=v( ) are two given measurable functions withásatisfying (2.4) andvnon-negative and non-decreasing. The solution¼ ( ) M of the equation [UE ] is called a Markovian risk measure of the r.v.s at level. Remark 2.1. About the actuarial meaning of the ingredientsá( ; ),v( ) and in De nition 2.1 we remark that represents a con dence bound, which in practical situations is determined by the regulator or the management of an insurance company. In principle the actuarial risk measures considered are intended to be approximations (on the safe side) for the VaR of order, and to have some desirable actuarial properties such as additivity, subadditivity, or superadditivity, according to actuarial applications for calculating solvency margins, for RBC calculations, as well as for the top-down approach of premiums calculations. The functionsá( ; ) andv( ) are introduced to derive bounds for the VaR, so that these bounds have some desirable properties for applications. In addition, because a risk measure provides an upper bound for the VaR, it might be interesting to determine the minimal value of the risk measure attached by the di erent choices ofá( ; ) andv( ). Remark 2.2. Clearly, given the ingredientsá( ; ),v( ) and, the Markovian risk measure ¼ ( ) M (S) involves only the distribution of the admissible r.v. S. A Markovian risk measure provides an upper bound for the VaR at the same level. By selecting appropriate functionsá the Markovian risk measures can re ect desirable properties when adding r.v.'s in addition to their dependence structure. Remark 2.3. Let 1 and 2 be two admissible r.v.'s, with Markovian risk measures ¼ ( ) M ( 1) and¼ ( ) M ( 2). Then we have Pr h 1 >¼ ( ) M ( 1) i h i ; Pr 2 >¼ ( ) M ( 2) : (2.10) We can obtain from the equation [UE ] that h i Pr >¼ ( ) M ( 1)+¼ ( ) M ( 2) (2.11) ² in case 1 and 2 are independent when the risk measure¼ ( ) M sums of independent risks; involved is subadditive for 6

7 ² in case 1 and 2 are comonotonic when the risk measure¼ ( ) M involved is subadditive for sums of comonotonic risks; ² for any 1 and 2 ; regardless of their dependence structure, in case a subadditive risk measure¼ ( ) M is applied. 3 SomeMarkovian risk measures In what follows we will provide a list of some important insurance premium principles (or risk measures) and show how they can be derived from the equation [UE]. We will also list a set of basicunderlying axioms. In practice, for di erent situations di erent sets of axioms are needed. 3.1 Themeanvalueprinciple Themean value principle has been characterized by Hardy et al. (1952); see also Goovaerts et al. (1984), Chapter 2.8, in the framework of insurance premiums. De nition 3.1. Let S be a risk variable. For a given non-decreasing and non-negative functionf( ) such that E[f(S)] converges, the mean value risk measure¼=¼ f is the root of the equationf(¼) = E[f(S)]. Clearly, wecan obtain themean valueriskmeasure bychoosing in the equation [UE] the functionsá(s;¼) =f(s)=f(¼) andv( ) 1: As veri ed in Goovaerts et al. (1984), p. 57{61, this principlecan be characterized by thefollowingaxioms (necessary and su±cient conditions): A1.1.¼(c) =c for any degenerate riskc; A1.2. Pr[ Y] = 1 =)¼() ¼(Y); A1.3. If¼() =¼( 0 ),Y is a r.v. andi is a Bernoulli variable independent of the vector f; 0 ;Y g, then¼(i +(1 I)Y) =¼(I 0 +(1 I)Y): Remark 3.1. This last axiom can be expressed in terms of distribution functions by assuming that mixingf Y withf or withf 0 leads to the same risk measure, as long as the mixing weights are the same. Remark 3.2. Under the condition that E[f(S)] converges one obtains as an upper bound for the survival probability Pr[S>¼+u] E[f(S)] f(¼ +u) = f(¼) f(¼ +u) : (3.1) 7

8 Speci cally, when E e S < 1 for some >0 one obtainse u as an upper bound for the probability Pr[S>¼+u], see the example in Section 1. In case¼ is the root of E[f(S)] = f(¼), by the inequality [GMI] one gets Pr[S>¼ ]. 3.2 The zero-utility premium principle The zero-utility premium principle was introduced by BÄuhlmann (1970). De nition 3.2. Let u( ) be anon-decreasingutility function. The zero-utility premium ¼(S) is the solution ofu(0) = E[u(¼ S)]. We assume that either the risk or the utility function is bounded from above. Because u( ) and u( ) + c de ne the same ordering in expected utility, the utility is determined such that u(x)! 0 as x! +1. To obtain the zero-utilitypremium principle, one chooses in the equation [UE] the functionsá(s;¼) =u(¼ s)=u(0) andv( ) 1. In order to relate the utility to the VaR one should proceed as follows. By the inequality [GMI], we get u(¼ S) Pr[S>¼ ] E = ; (3.2) u(0) where¼ is the solution of the equation E[u(¼ S)] = u(0). The result obtained here requires that the utility function u( ) is bounded from below. However, this restriction can be weakened by considering limits for translated utility functions. Let the symbol ¹ eu represent the weak order with respect to the zero-utility premium principle, that is, ¹ eu Y means that is preferable toy. We write» eu Y if both ¹ eu Y andy ¹ eu. It is well-known that the preferences of a decision maker between risks can bedescribed by means of comparing expected utility as a measure of the risk if they ful ll the following ve axioms which are due to von Neumann and Morgenstern (1944) (combining Denuit et al. (1999) and Wangand Young (1998)): A2.1. IfF =F Y then» eu Y; A2.2. The order ¹ eu is re exive, transitive and complete; A2.3. If n ¹ eu Y andf n!f then ¹ eu Y; A2.4. IfF F Y then ¹ eu Y; A2.5. If ¹ eu Y and if the distribution functions of 0 p andy 0 p are given byf 0 p (x) = pf (x)+(1 p)f Z (x) andf Y 0 p (x) =pf Y (x)+(1 p)f Z (x); wheref Z is an arbitrary distribution function, then 0 p ¹ euy 0 p for anyp 2 [0;1]. From these axioms, the existence of a utility functionu( ) can be proven, with the property that¹ eu Y if and only if E[u( )] E[u( Y)]. 8

9 3.3 The Swiss premium calculation principle The Swiss premium principle was introduced by Gerber (1974) to put the mean value principleand thezero-utilityprinciplein auni ed framework. De nition 3.3. Letw( ) be a non-negative and non-decreasing function on R and 0 z 1 be a parameter. Then the Swiss premium principle ¼ = ¼(S) is the root of the equation E[w(S z¼)] =w((1 z)¼): (3.3) This equation is the special case of [UE] withá(s;¼) =w(s z¼)=w((1 z)¼) and v( ) 1. It is clear thatz= 0 provides us with the mean value premium, whilez= 1 gives the zero-utility premium. Recall that by the inequality [GMI], the root¼ of the equation E[w(S z¼)] = w((1 z)¼) determines an upper bound for the VaR. Remark 3.3. Because one still may choosew( ), it can be arranged to have supplementary properties for the risk measures. Indeed if we assume thatw( ) is convex, we have cx Y =)¼() ¼(Y): (3.4) See for instance Dhaene et al. (2002a, b) for the de nition of cx (convex order). For two random pairs (S 1 ;S 2 ) and ( S e 1 ; S e 2 ) with the same marginal distributions, we call (S 1 ;S 2 ) more related than ( S e 1 ; S e 2 ) if the probability Pr[S 1 x;s 2 y] thats 1 ands 2 are both small is larger than fors e 1 ands e 2, for allxandy; see e.g. Kaas et al. (2001), Chapter In this case one gets from (3.4) ¼( S e 1 + S e 2 ) ¼(S 1 +S 2 ): (3.5) The risk measure of the sum of a pair of r.v.'s with the same marginal distributions depends on the dependence structure, and in this case increases with the degree of dependence between the terms of the sum. Remark 3.4. Gerber (1974) proves the following characterization: Let w( ) be strictly increasing and continuous, then the Swiss premiumcalculation principle generated by w( ) is additive for independent risks if and only ifw( ) is exponential or linear. 3.4 The Orlicz premium principle TheOrliczprinciplewasintroduced byhaezendonckand Goovaerts (1982) asamultiplicativeequivalent ofthezero-utilityprinciple. Tointroducethispremiumprinciple, theyused the concept of a Young functionã, which is a mapping from R + 0 into R+ 0 that can be written as an integral of the form Z x Ã(x) = f(t)dt; x 0; (3.6) 0 9

10 wheref is a left-continuous, increasing function on R + 0 withf(0) = 0 and lim x!+1f(x) = +1. It is seen that a Young functionãis absolutely continuous, convex and strictly increasing, and hasã 0 (0) = 0. We say thatãis normalized ifã(1) = 1. De nition 3.4. Let à be anormalized Youngfunction. The root oftheequation E[Ã(S=¼)] = 1 (3.7) is called the Orlicz premium principle of the risks. The uni ed approach follows from the equation [UE] withá(s;¼) replaced byã(s=¼) and v(s) 1. TheOrliczpremiumsatis esthefollowingproperties: A4.1. Pr[ Y] = 1 =)¼() ¼(Y); A4.2.¼() = 1 when 1; A4.3.¼(a) =a¼() fora>0 and any risk; A4.4.¼( +Y) ¼()+¼(Y): Remark 3.5. A4.3 above says that the Orlicz premium principle is positively homogenous. In the literature, positive homogeneity is often confused with currency independence. As an example, we look at the standard deviation principle¼ 1 () = E[] + ¾[] and the variance principle¼ 2 () = E[]+ Var[], where and are two positive constants, is dimension-free but the dimension of 1= is money. Clearly¼ 1 () is positive homogenous but¼ 2 () is not. But it stands to reason that when applying a premium principle, if the currency is changed, so should all constants having dimension money. So going from BFr to Euro, where 1 Euro ¼ 40 BFr, the value of in¼ 2 () should be adjusted by the same factor. In this way both¼ 1 () and¼ 2 () are independent of the monetary unit. Remark 3.6. These properties remain exactly the same for risks that may also be negative, such as they are used in the de nition of coherent risk measures by Artzner (1999). Indeed if¼() = and one extends these properties to r.v.'s supported on the whole line R, then ¼( +a a) ¼( +a) a: (3.8) Hence¼( +a) ¼()+a and consequently¼( +a) =¼()+a. Theinterested reader isreferred to Haezendonckand Goovaerts (1982). If in addition translation invariance isimposed for non-negativerisks, it turnsout that the only coherent riskmeasurefor non-negativeriskswithin theclassoforliczprinciplesisan expectation ¼() = E[]. Remark3.7. The Orlicz principle can also be generalized to cope with VaR. Actually, from the inequality [GMI], the solution¼ of the equation E[Ã(S=¼)] = gives Pr[S>¼ ]. 10

11 3.5 More general risk measures derived from Markov bounds For this section, we con ne to risks with thesame mean. We consider more general risk measures derived from Markov bounds, applied to sums of pairs of r.v.'s, which may or may not be independent. The generalization consists in the fact that we consider the dependence structure to someextent in the risk premium, letting the premium for the sum +Y depend both on the distribution of the sum+y and on the distribution of the sum c +Y c of the comonotonic (maximally dependent) copies of the r.v.'s and Y. Because of this, we would rather denote the premium for the sum +Y by¼(;y) rather than by¼( +Y). When ther.v.'s and Y are comonotonic, however, there is no di erence in understanding between the two symbols¼(;y) and¼( +Y). Taking ¼() simply equal to ¼(; 0), we consider the following properties: A5.1.¼(a) =a¼() for anya>0; A5.2.¼( +b) =¼()+b for anyb 2 R; A5.3 a.¼(;y) ¼()+¼(Y); A5.3 b.¼(;y) ¼()+¼(Y); A5.3 c.¼(;y) =¼()+¼(Y): Remark 3.8. A5:3 a describes the subadditivity property, which is realistic only in case diversi cation of risks is possible. However, this is rarely the case in insurance. Subadditivity gives rise to easy mathematics because distance functions can be used. The superadditivity property for a risk measure (that is not Artzner coherent) is redundant in the following practical situation of capital allocation or solvency assessment. Suppose that two companies with risks 1 and 2 merge and form a company with risk Letd 1,d 2 andddenote the allocated capitals or solvency margins. Then, withprobability 1, ( d 1 d 2 ) + ( 1 d 1 ) + +( 2 d 2 ) + : (3.9) This inequality expresses the fact that, with probability 1, the residual risk of the merged company is smaller than the risk of the split company. In cased d 1 +d 2, one gets, also with probability 1, ( d) + ( 1 d 1 ) + +( 2 d 2 ) + : (3.10) Hence in case one calculates the capitalsd 1,d 2 anddby means of a risk measure it should be superadditive (or additive) to describe the economics in the right way. Subadditivity is only based on the idea that it is easier to convince the shareholders of a conglomerate in failure to provide additional capital than the shareholders of some of the subsidiaries. Recent cases indicate that for companies in a nancial distress situation splitting is the only way out. 11

12 Remark 3.9. It should also be noted that subadditivity cannot be used as an argument for a merger of companies to be e±cient. The preservation (3.9) of the inequality of risks with probability one expresses this fact; indeed ½(( d 1 d 2 ) + ) ½(( 1 d 1 ) + +( 2 d 2 ) + ) (3.11) expresses the e±ciency of a merger. It has nothing to do with the subadditivity. A capital d<d 1 +d 2, for instance derived by a subadditive risk measure, can only be considered if the dependence structure allows it. For instance ifdis determined as the minimal root of the equation ½(( d) + ) ½(( 1 d 1 ) + +( 2 d 2 ) + ); (3.12) d obviously depends on the dependence between 1 and 2. Note that in ( 1 d 1 ) + +( 2 d 2 ) +, the two terms are dependent. Taking this dependence into account, the risk measure providing the capitalsd,d 1 andd 2 will not always be subadditive, nor always superadditive, but may instead exhibit behavior similar to the VaR, see Embrechts et al. (2002). Letà beanincreasingandconvexnon-negativefunctionon Rsatisfyinglim x!+1 Ã(x) = +1. For xed 0<p<1, we get, by choosingv( ) = 1, the equality à Á(;Y;¼) = 1 +Y F Ã(1) à c +Y c(p)! + ¼ F c +Y c(p) (3.13) and bysolving [UE] for ¼, thefollowingriskmeasurefor thesumoftwo r.v.'s: " à +Y F E à c +Y c(p)!# + ¼(;Y) F c +Y c(p) =Ã(1) (3.14) for some parameter 0<p<1. Hereafter, thepth quantile of a r.v. with d.f. F is, as usual, de ned by F (p) = inf fx 2 RjF (x) pg; p 2 [0;1]: (3.15) It is easily seen that there exists a unique constanta(p)>0 such that " à +Y F E à c +Y c(p)!# + =Ã(1): (3.16) a(p) Thus¼(;Y) =F c +Yc(p)+a(p). Especially, lettingy be degenerate at 0 we get¼()> F (p): 12

13 Now we check that A5:1, A5:2 and A5:3 a are satis ed by¼ (subadditive case). In fact, the proofs for the rst two axioms are trivial. As for A5:3 a ; we derive " Ã +Y F E Ã c +Y c(p)!# + ¼()+¼(Y) F c +Y " Ã c(p) F E Ã (p) + Y F + Y (p)!# + ¼()+¼(Y) F (p) F Y µ (p) ¼() F = E Ã (p) This proves A5:3 a : ¼()+¼(Y) F (p) F Y (p) F (p) ¼() F (p) ¼(Y) F Y + (p) ¼()+¼(Y) F (p) F Y (p) Y FY (p) ¼(Y) F Y (p) ¼() F E (p) µ F ¼()+¼(Y) F Ã (p) (p) F Y (p) ¼() F (p) ¼(Y) F Y +E (p) µ Y F Y ¼()+¼(Y) F Ã (p) (p) F Y (p) ¼(Y) F Y (p) =Ã(1) = E Ã µ +Y F c +Y c(p) ¼(;Y) F c +Y c(p) : (3.17) Remark If the functionã( ) above is restricted to satisfyã(1) =Ã 0 (1), then it can be proven that the risk measure h i ¼ l () =F (p)+e F (p) + (3.18) gives the lowest generalized Orlicz measure. In fact, sinceã is convex on R and satis es Ã(1) =Ã 0 (1), we have Ã((x) + ) Ã(1) (x) + for anyx 2 R: (3.19) Let¼() be a generalized Orlicz risk measure of the risk, that is,¼() is the solution of the equation E " Ã Ã F (p)!# + ¼() F (p) =Ã(1): (3.20) By (3.19) and recalling that¼()>f (p), we have " Ã F Ã(1) = E Ã (p)!# " + F ¼() F (p) Ã(1) E (p) # + ¼() F (p) ; (3.21) which implies that h i ¼() F (p)+e F (p) =¼ + l (): (3.22) 13

14 Remark Now we consider the risk measure " Ã F E Ã (p)!# + ¼() F (p) = 1 p (3.23) for some parameter 0<p<1. Similarly as in Remark 3.12, if the functionã( ) is restricted to satisfyã(1) =Ã 0 (1), we obtain the lowest risk measure as ¼() =F (p)+ 1 h i 1 p E F (p) = E >F + (p) : (3.24) Remark Another choice is to consider the root of the equation µ 1 j E[]j E Ã(1) Ã = ; (3.25) ¼() E[] de ning, in general terms, a risk measure for the deviation from the expectation. As a special case whenã(t) t 2 I (t 0) one gets ¼() = E[]+ ¾[] p : (3.26) Note that both (3.24) and (3.26) produce an upper bound for the -quantileq () of. Remark For a risk variable, one could consider a risk measure¼ c () which is additive, and de ne another risk measure½(), where the deviationa() =½() ¼ c () is determined by E Ã µ j ¼c ()j =Ã(1): (3.27) a() Here the role of¼ c () is to measure central tendency whilea() measures the deviation of the risk variable from¼ c (). If¼ c () is positively homogenous, translation invariant and additive, then½() is positively homogenous and translation invariant. The measure ½() may be subadditive or superadditive, depending on the convexity or concavity of the functionã( ). 3.6 Yaari's dual theory of choice under risk Yaari (1987) introduced the dual theory of choice under risk. It was used by Wang (1996), whointroduced distortion functionsin theactuarial literature. A distortion function is de ned as a non-decreasing functiong: [0;1]! [0;1] such thatg(0) = 0 andg(1) = 1. De nition 3.5. LetS be a non-negative r.v. with d.f.f S, andg( ) be a distortion function de ned as above. The distortion riskmeasure associated with the distortion function g is de ned by ¼ = Z +1 0 g(1 F S (x))dx: (3.28) 14

15 Choosing the functioná( ; ) in the equation [UE] such thatá(s;¼) =s=¼ and using the left-hand derivativeg 0 (1 F S (s)) instead ofv(s), using integration by parts we get the desired unifying approach. The choiceof v( ), which at rst glance may look arti cial, is very natural if one wants to have E[v(S)] = 1. This risk measurecan be characterized by the following axioms: A6.1. Pr[ Y] = 1 =)¼() ¼(Y); A6.2. If risks andy are comonotonic then¼( +Y) =¼()+¼(Y); A6.3.¼(1) = 1. Remark It is clear that this principle results in large upper bounds because g 0 (1 F ()) Pr[ ¼+u] E = ¼ ¼ +u ¼ +u : (3.29) It is also clear that the set of risks for which¼is nite contains all risks with nite mean. 3.7 Mixtures of Esscher principles Themixture of Esscher principles was introduced by Gerber and Goovaerts (1981). It is de ned as follows: De nition 3.6. For a bounded r.v.s, we say a principle¼=¼(s) is a mixture of Esscher principles if it is of the form ¼ F (S) =F()Á()+ Z +1 Á(t)dF(t)+(1 F(+1))Á(+1); (3.30) wheref is a non-decreasing function satisfying 0 F(t) 1 andáis of the form Á(t) =Á S (t) = d dt loge e ts ; t 2 R: (3.31) Actually we can regardf as a possibly defective cdf with mass at both and +1. Since the variables is bounded,á() = min[s] andá(+1) = max[s]. In addition,á S (t) is the Esscher premium ofs with parametert 2 R. In thespecial case where thefunction F iszerooutsidethe interval [0; 1]; themixture of Esscher principlesis a mixtureof premiums with a non-negativesafetyloading coe±cient. We show that in this case the mixture of Esscher premiums can also be derived from the Markov inequality. Actually, Z +1 Z ¼ F (S) = Á(t)dF(t)+(1 F(+1))Á(+1) = 0 15 [0;+1] Á(t)dF(t): (3.32)

16 It can be shown that the mixture of Esscher principles is translation invariant. Hence in what follows, we simplyassume, without lossof generality, that min[s] 0 because otherwisea translation ons can be used. We notice that, for anyt 2 [0;+1], Á(t) = E Se ts E[S]: (3.33) E[e ts ] The inequality (3.33) can, for instance, be deduced from the fact that the variabless and e ts are comonotonic, hence positively correlated. Since we have assumed that min[s] 0, now we choose in [GMI] the functionsv( ) 1 andá(s;¼) =s=¼, then we obtain that Pr[S>¼] E[Á(S;¼)] 1 ¼ E[S] 1 Z Á(t)dF(t); (3.34) ¼ [0;+1] where the last step in (3.34) is due to the inequality (3.33) and the fact thatf([0;+1]) = 1. Letting the r.h.s. of (3.34) be equal to 1, we immediately obtain (3.32). We now verify another result: the tail probability Pr[S>¼+u] decreases exponentially fast in u2[0; +1). The proof is not di±cult. Actually, since therisk variable S is bounded, it holds for any >0 that Pr[S>¼+u] expf (¼ +u)g E[exp f Sg]: (3.35) Hence, in order to get the announced result, it su±ces to prove that, for some >0, ½ Z ¾ E[exp f Sg] expf ¼g = exp Á(t)dF(t) ; or equivalently to prove that, for some >0, Z loge[exp f Sg] [0;+1] [0;+1] Á(t)dF(t): (3.36) In thetrivial casewheretherisks isdegenerate, bothsidesof (3.36) areequal for any >0. IfF is not degenerate, the Esscher premiumá(t) is strictly increasing int2[0;+1], and we can nd some 0 > 0 such that Z Á( ) [0;+1] Á(t)dF(t) (3.37) holds for any 2 [0; 0 ]. Thus in any case we obtain that (3.36) holds for any 2 [0; 0 ]. We summarize: Remark For the mixture of Esscher premiums¼de ned above, iff is concentrated on [0;+1], then Pr[S>¼+u] expf 0 ug (3.38) holds for anyu 0, where the constant 0 > 0 is the solution of the equation Z Á( ) = Á(t)dF(t): (3.39) [0;+1] 16

17 Themixture ofesscher premiums ischaracterized bythefollowingaxioms; seegerber and Goovaerts (1981): A7.1.Á 1 (t) Á 2 (t) 8t 2 R =)¼ F ( 1 ) ¼ F ( 2 ); A7.2. It holds for any two independent risks 1 and 2 that ¼ F ( ) =¼ F ( 1 )+¼ F ( 2 ): (3.40) Hence this risk measure is additive for independent risks. When the function F in (3.32) isde ned on theinterval [0; 1], thepremiumcontainsa positivesafetyloading. 4 Conclusions Thispaper showshowmanyof theusual premium calculation principles (or riskmeasures) can bededuced fromageneralized Markovinequality. All riskmeasuresprovideinformation concerning the VaR, as well as the asymptotic behavior of Pr[S>¼+u]. Therefore, the e ect ofusing ariskmeasureand requiringadditional propertiesis equivalent to makinga selection of admissible risks. Noticethat when using arisk measure, additional requirements are usually needed about convergence of certain integrals. In this way, the set of admissible risks isrestricted, e.g. theone having nite mean, nitevariance, nitemoment generating function and so on. References [1] Artzner, Ph., Application of coherent risk measures to capital requirements in insurance. North American Actuarial Journal, 3, no. 2, 11{25. [2] Artzner, P., Delbaen, F., Eber, J.-M., Heath, D., Coherent measures of risk. Mathematical Finance, 9, 203{228. [3] Borch, K., The economics ofuncertainty. Princeton UniversityPress, Princeton. [4] Borch, K., The mathematical theory of insurance. Lexington Books, Toronto. [5] BÄuhlmann, H., Mathematical Methods in Risk Theory. Springer-Verlag, Berlin. [6] Denuit, M., Dhaene, J., Van Wouwe, M., The economics of insurance: a review and some recent developments. Mitt. Schweiz. Aktuarver., no. 2, 137{175. [7] DeVijlder, F., Goovaerts, M. J., An invariancepropertyof theswisspremium calculation principle. Mitt. Verein. Schweiz. Versicherungsmath., 79, no. 2, 105{

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19 [24] Yaari, M. E., The dual theory of choice under risk. Econometrica, 55, no. 1, 95{115. Marc Goovaerts Center for Risk and Insurance Studies (CRIS) KatholiekeUniversiteit Leuven B Leuven, BELGIUM Marc.Goovaerts@econ.kuleuven.ac.be Rob Kaas Department of Quantitative Economics University of Amsterdam Roetersstraat 11, 1018 WB Amsterdam, THE NETHERLANDS r.kaas@uva.nl Jan Dhaene Center for Risk and Insurance Studies (CRIS) KatholiekeUniversiteit Leuven B Leuven, BELGIUM Jan.Dhaene@econ.kuleuven.ac.be Qihe Tang Department of Quantitative Economics University of Amsterdam Roetersstraat 11, 1018 WB Amsterdam, THE NETHERLANDS q.tang@uva.nl 19