Some Alternative Actuarial Pricing Methods: Application to Reinsurance and Experience Rating

Transcription

1 Reports on Economics an Finance, Vol., 06, no., - 35 HIKARI Lt, Some Alternative Actuarial Pricing Methos: Application to Reinsurance an Experience Rating Werner Hürlimann Swiss Mathematical Society, University of Fribourg CH-700 Fribourg, Switzerlan Copyright 05 Werner Hürlimann. This article is istribute uner the Creative Commons Attribution License, which permits unrestricte use, istribution, an reprouction in any meium, provie the original work is properly cite. Abstract The present work concerns reinsurance rate-making in a istribution-free environment an fair premium calculation principles. Applications to the esign of perfectly hege experience rating contracts in a risk-exchange an reinsurance environment are iscusse. Special emphasis is put on istribution-free an immunization aspects as well as on long-term optimal an competitive strategies in the sense of strategic financial management. Keywors: aitive premium principle, fair premium, generalize variance principle, utility theory, CAPM, extene mean-variance analysis, perfect hege. Introuction The present work offers an overview an synthesis aroun several stuies of the author. The main topics of theoretical interest are reinsurance rate-making in a istribution-free environment an fair premium calculation principles. Applications to the esign of perfectly hege experience rating contracts in a risk-exchange an reinsurance environment are iscusse. Special emphasis is put on istribution-free an immunization aspects as well as on long-term optimal an competitive strategies. A more etaile outline of the content follows. Section is evote to a special case of the following more general problem in pricing theory. Given is a risk that can be ecompose in a finite number of splitting risk components. If all involve risk premiums are calculate accoring

2 Werner Hürlimann to some premium calculation principle, what are aequate premium principles, which besies other esirable properties satisfy the superaitive property for splitting risk components? In situations where arbitrage opportunities shoul be avoie, as in reinsurance markets with traing possibilities, the stronger aitive property is require. Examples, which illustrate the use of a strict superaitive property are foun in Hürlimann [9], Section, Hürlimann [0], Section 3, an here later in Section 5. An overview about known results in the aitive case is Aase []. In the present work, only the special case of two splitting risk components is touche upon. Despite this restriction, the potential practical applications are numerous. It is shown that the generalize variance principle by Borch [4], obtaine from an insurance version of the Capital Asset Pricing Moel CAPM, can be justifie on the basis of four alternative mathematical, economics an actuarial arguments. In particular Proposition. links the economics form of Borch's principle with the class of biatomic risks. Section 3 is a synthesis of the main ieas presente in Hürlimann [9] to []. A set of feasible reinsurance contracts with a fixe maximum euctible is consiere. This reinsurance structure is part of a more general structure, which minimizes the square loss risk of the insurer by offering to the insure a claims epenent bonus uner the help of risk-exchanges, in particular reinsurance treaties. For the consiere set, the minimum square loss risk vanishes an efines perfectly hege experience rating contracts in a reinsurance environment, which are mathematically characterize in Proposition 3.. The neee fair premium of such a contract equals the sum of the expecte claims, the expecte bonus payment an the loaing for reinsurance. Uner a fair exchange of the risk profit loaing between the insurer an the reinsurer, fair premiums are characterize in Proposition 3. by a property of complete immunization. Alternative interpretations are propose in the Remarks 3. an a fair stop-loss premium rating moel is propose in Example 3.. The results of Section are applie in Section 4 to the construction of a istribution-free premium rating system for the class of perfectly hege experience rating contracts. It satisfies a criterion of relative safeness an may be viewe as a biatomic approximation to Borch's aitive CAPM premium principle see Theorem 4.. With the help of the immunization argument of Proposition 3., this istribution-free premium methoology is then applie to evelop explicit examples for a stop-loss contract in environments of positive risks Section 4. respectively arbitrary risks Section 4.. In Section 4. a new insurance economics interpretation of the "Karlsruhe" pricing principle propose by Heilmann [] is obtaine see also Hürlimann [6]. The link with utility theory allows for an interesting interpretation of the expecte value principle in Example 4.. The assumption of a quaratic utility function with saturation leas in Example 4.3 to a non-trivial generalize variance principle of Borch's type. In Section 4. the list of practical applications is continue for the class of arbitrary risks. In Example 4.4 the insurance market base istribution-free stop-loss premium formula first propose in Hürlimann [6] fins a rigorous moel theoretical justification. Example 4.5 shows how istribution-free fair stanar evia-

3 Some alternative actuarial pricing methos 3 tion premiums are obtaine. At the same time this example justifies the most competitive choice, which can be mae in Example 4.4. A iscussion of the "Karlsruhe" principle in a istribution-free environment of arbitrary risks is given in Example 4.6. In Section 5 it is shown how perfectly hege experience rating strategies, which are long-term optimal an competitive in the insurance market, can be efine. The corresponing possible minimum an maximum premiums are etermine in Proposition 5. an its Corollary 5.. Moreover, formula 5. shows that it is possible to obtain optimal strategies with fair variance premiums that o not epen upon the unknown risk loaing factor. Two common situations illustrate the use of the metho. In Section 5. an optimal long-term stop-loss pricing strategy is isplaye. Applie to financial risks, an extene version of the classical mean-variance approach to portfolio theory by Markowitz, consiere in Hürlimann [5], [4], fins herewith a rigorous justification. Finally, it is shown in Section 5. that an optimal long-term an competitive perfectly hege experience rating strategy base on a linear combination of proportional an stoploss reinsurance oes not exist in case of a non-vanishing proportional reinsurance payment. The proof of this result is base on the inequalities of Kremer [8] an Schmitter [30], which are reviewe in the Appenix.. Splitting risk premiums an biatomic risks Consier the following rate-making problem encountere in the economic theory of insurance uner uncertainty. Given a risk X with risk premium P=H[X] calculate accoring to some pricing calculation principle H[], an given a splitting of the risk in smaller parts X i with splitting risk premiums P i =H[X i ], i=,...,n, such that X=X +...+X n, what are appropriate pricing principles, which satisfy the following superaitive property n n X HX. P H. i i i Pi In cases where the strict inequality hols, it is more effective to insure the splitting parts X i separately. For some examples consult Hürlimann [9], Section, [0], Section 3, as well as the later Section 5. In particular, to avoi arbitrage opportunities, one requires the equality sign for both inepenent an epenent risk components X i. An overview of known results in the latter case is offere by Aase []. In the present Section we restrict our attention to the aitive property in the special case n= that will be neee later. Emphasis is put on aitive pricing principles, whose general forms are etermine by the values they take on the set D [a,b]:=d [a,b];, of biatomic risks with given mean an stanar eviation efine on the interval [a,b], a, b R.

4 4 Werner Hürlimann Suppose a risk X is split into two transforme components Y=fX, Z=gX such that Y+Z=X. The problem consists to construct pricing principles H[] that satisfy the aitive property H[X] = H[Y] + H[Z].. Applying the functional representation theorem of Riesz [9], one knows that there exists some ranom variable W such that the following relations hol: H[X] = E[XW] = E[X]E[W] + Cov[X,W] H[Y] = E[YW] = E[Y]E[W] + Cov[Y,W].3 H[Z] = E[ZW] = E[Z]E[W] + Cov[Z,W] Example.. In a simple "linear worl" such that one sees that W = + X - E[X],.4 H[X] = E[X] + Var[X].5 is the generalize variance principle obtaine by Borch [5], [6], from the CAPM. Now, one has H[Y] = E[Y] + Cov[X,Y], H[Z] = E[Z] + Cov[X,Z],.6 an eliminating using.5 one gets Borch s formula H H Y Z E Z X X Cov, EY Var X Cov, Var X Y Z H H X EX, X EX..7 To interpret this simple example from the point of view of an economic theory uner uncertainty, one can assume that W=u'X, where u'x is a marginal utility function of some representative insurer see Theorem in Aase [] an its consequence.7. Disregaring the neee technical assumptions for the existence of a competitive equilibrium implying W=u'X, our utility functions are suppose to satisfy the property u'x0, u''x0, which inclues in particular the possibility of a risk-neutral representative insurer with linear utility function. In

5 Some alternative actuarial pricing methos 5 this setting the "linear worl" of Example. follows by assuming a quaratic utility function. Besies the above an Borch [4], let us give a thir alternative justification, which is base on a criterion of "safeness" or pruent pricing. If one consiers only positive risks X0, it follows from the assumption W=u'X0 that H X E X E W X, W Var X Var W,.8 X,W the correlation coefficient between X an W, is always on the safe sie provie X,W=, which implies a linear transformation W = + X - E[X], with =E[W], >0. Therefore Borch's principle.7 is also the safest possible choice in an economic worl uner uncertainty for which W=u'X. A fourth erivation of the formulas.7 follows by consiering biatomic risks as suggeste in Hürlimann [6], [9]. Let X D [a,b] has support {x,x } an probabilities {p,p } such that p x x x x, p, x x, x x a b, 0 a b..9 Let {w =u'x,w =u'x } be the support of the transforme ranom variable W=u'X. A calculation shows that Cov X, W Var X w x w x,.0 an similar formulas for the risk components Y=fX, Z=gX. This observation implies immeiately the following result. Proposition.. In the class D [a,b] of biatomic risks, an aitive premium principle of the form.3 with W=u'X is necessary of the Borch's form.5,.7 with parameters E W p u' x p u' x, Cov X, W u' x u' x.. Var X x x As a particular feature of this moel, one gets a simple link with utility theory, which will be exploite later in Section 4.. Moreover in an economic worl uner

6 6 Werner Hürlimann uncertainty for which W=u'X hols, an aitive pricing principle characterize by the values it takes on biatomic risks is necessarily of the CAPM form.5, Fair premiums an perfectly hege experience rating The present Section offers a synthesis of the main ieas containe in the original papers Hürlimann [9] to [] on this topic. Notations are as in Section. In reinsurance theory, one often restricts the set of transformations fx, gx to those compensation functions for which neither the ceant nor the reinsurer will benefit in case the claim amount increases. In this situation, one assumes that fx, gx are non-ecreasing functions such that fx, gx x an fx+gx=x. This means that feasible reinsurance contracts are escribe by the class of comonotonic ranom variables ComX = {Y,Z : Y=fX, Z=gX are comonotonic such that Y+Z=X}. 3. In this setting, the ranom variable Z escribes the reinsurance payment an Y enotes the retaine amount of the irect insurer. One says that a feasible reinsurance contract has a maximum euctible if the following number exists an is finite sup xr f x. 3. Examples 3.. i A stop-loss contract Z=X- + has maximum euctible. ii A linear combination of proportional an stop-loss reinsurance of the form Z=-rX+rX-T + has a maximum euctible =rt. For a etaile stuy of this contract see Hürlimann []. iii A linear combination of stop-loss contracts in layers Z=rX-L + +-rx- M +, M>L, has a maximum euctible =rl+-rm. iv Consier a compoun Poisson risk X N i U i, 3.3 where N is a Poisson ranom variable, an the U i 's are inepenent an ientically istribute ranom variables, which are inepenent from N. Then the reinsurance payment

7 Some alternative actuarial pricing methos 7 Z b N c U b N i i 3.4 efines a feasible reinsurance contract with maximum euctible =bc, which might be attractive in the framework of the classical moel of risk theory. The set of feasible reinsurance contracts with maximum euctible is enote by S, :, sup Y f X Z g X Y Z Com X an f x. 3.5 xr For Y,ZS the function efine by x = - fx = + gx x 3.6 is always non-negative an efines a transforme ranom variable D=X such that with probability one + Z = X + D. 3.7 Setting x f x x inf one has for all xx x = 0, gx = x It has been shown in Hürlimann [9], [], that the set S efines a convenient reinsurance substructure of a more general structure, which minimizes the square loss of the insurer s risk by offering to the insure a claims epenent bonus uner the help of risk-exchanges, in particular reinsurance treaties. This general structure parameterizes the set of experience rating contracts in a riskexchange environment. In general, an experience rating contract with premium P offers besies claims payment X a bonus D=D[X]0, which usually is pai out in case the risk profit P-X is positive. In this situation, the liability of the insurer is X+D. To reuce 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 riskexchange. Then, the neee premium P=P[X+D] is the sum of the net retaine premium P N =P N [Y+D] of the insurer plus the price pai for the risk-exchange, that is one has P=P N +H[Z], where H[] is the pricing principle applie to the risk-exchange. An important problem consists to esign appropriate pairs Z,D satisfying some esirable properties. To limit the insurer's risk, one minimizes the expecte square ifference between assets an liabilities, that is one consiers the optimization problem

8 8 Werner Hürlimann R = E[P N - Y - D ] = min. 3.9 over an appropriate space of pairs Y,D. Taking into account the relation R = P N - E[Y+D] + Var[Y] + Var[D] + Cov[Y,D], 3.0 one sees that a minimizing solution necessarily satisfies the conitions P N = E[Y+D], 3. Cov[Y,D] = -Var[D], or Cov[Y,D] = -Var[Y]. 3.a 3.b In the case 3.a one has R min =Var[Y]{ - Y,D }, where Y,D is the correlation coefficient between Y an D, an in case 3.b one has R min =Var[D]{ - Y,D }. In particular, the insurer's risk is completely eliminate, a so-calle perfect hege with R min =0, for all pairs Y,D such that Cov[Y,D]= - Var[D]= - Var[Y]. The conition Cov[Y,D]= - Var[D] means that the systematic risk of the retaine amount relative to the bonus equals -. The formula 3. says that the fair net premium after reinsurance equals the expecte costs of the ceing company an finally the risk quantity R min is an intrinsic risk measure, on which any ajustment of the fair net premium by a suitable security loaing shoul be base. In the perfect hege situation R min =0 the neee fair premium P=P N +H[Z], where P N =E[Y+D], can be ecompose into three components as follows: P = E[X] + E[D] + H[Z] - E[Z]. 3.3 fair premium expecte expecte loaing in claims bonus risk-exchange price Besies the expecte costs for claims an bonus payments only the loaing of the risk-exchange price has to be pai for a perfectly hege experience rating contract in a risk-exchange environment. This feature is similar to the "Dutch property" of the Dutch premium principle see Van Heerwaaren an Kaas []. In the perfect hege situation, one has P N +Z=X+D with probability one. In this special case, the esign of pairs Z,D such that the fair premium satisfies some esirable properties has been stuie in Hürlimann []. Here, we inclue examples of risk-exchanges, which are not reinsurance contracts in the sense that

9 Some alternative actuarial pricing methos 9 Y,ZComX. Restricting the space of ranom pairs Y,Z to ComX, the following characterization of the set S has been obtaine in Hürlimann [9]. Proposition 3.. Suppose Y,ZComX an D=D[X]0 efine an experience rating contract in a reinsurance environment. Assume the set {x R: D[X=x]=0} is non-empty. Then the following conitions are equivalent: One has P N == sup f x xr an R min =E[P N - Y - D ]=0. C One has Cov[Y,D]= - Var[D]= - Var[Y]. C Y,ZS efines a perfectly hege experience rating contract with maximum euctible an bonus payment D= - Y. C3 In the following, we restrict our attention to the set S of perfectly hege experience rating contracts in a reinsurance environment. Since the fair premium 3.3 of such a contract epens upon the pricing principle H[] applie by the reinsurer an generally not known with certainty, premium calculation simplifies provie conitions, which characterize fair premiums, can be erive. As a reasonable compromise for ecision, let us aopt the following fair premium conition. Given any premium H[X]=+ with loaing >0, suppose that the insurer an the reinsurer exchange the expecte profit in an economic fair manner such that the loaing goes half an half to the ceing company an the reinsurer. Then, one has the relations H[Y] = E[Y] +, H[Z] = E[Z] In case Y,ZS the guarantee bonus payment D belongs to the insure. The insurer's net outcome after payment of the bonus equals H[Y] - Y - D = H[Y] -, 3.5 which will be immunize with probability one provie H[Y]. In fact the following result characterizes fair premiums in the efine sense. Proposition 3.. Let H[] be a pricing principle an Y,ZS a perfectly hege experience rating contract with fair premium P=E[X+D]+H[Z]-E[Z], an suppose the fair premium conition 3.4 hols. Then H[] is a fair pricing principle such that P=H[X] if, an only if, the insurer's balance is completely immunize, that is H[Y]=.

10 0 Werner Hürlimann Proof. Let us first show that the conition H[Y]= is sufficient. By assumption an 3.4 one gets = - E[Y]=E[D], where the last equality hols because Y,ZS. Rearranging terms using 3.4 one has H[X] = + + = + E[D] + H[Z] - E[Z] = P, 3.5 which ientifies H[X] with the fair premium. To show that the conition is necessary one procees 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 guarantee bonus D an the reinsurance premium H[Z]. By construction of the fair premium, his net outcome, that is the ifference between income an outcome, vanishes. By 3.5 this means that H[Y]=, as esire. Remarks 3.. i The necessary conition in Proposition 3. is implicitely containe in the argument following formula 4.5 in Hürlimann [9], while the sufficient conition has been formulate in a special case in Hürlimann [0], Section 4.3. ii As shown in Hürlimann [0], Section 4., the fair premium conition 3.4 approximately hols in a "istribution-free" sense in the stop-loss case Z=X- +. Let X be an arbitrary risk, efine on the whole real line, with finite mean an stanar eviation, an let H E Var be the stanar eviation pricing principle. Consier the istribution-free stanar eviation premiums efine by H [Y]=H[Y ], H [Z]=H[Z ], where the risk X has been replace by a risk X with Bowers' istribution function F x x x, x,, 3.6 first consiere in Hürlimann [8]. Furthermore set H [X]=+=H[X] since the variance of X is infinite H[X ] is actually not efine, but this oes not isturb the results. Since E[Z ] coincies with the best stop-loss upper boun by Bowers [6], this choice efines safe stop-loss premiums uniformly for all euctibles. Moreover, from the property Var[Y ]=Var[Z ]= 4 for all, one sees that H [Y]=H[Y ]=E[Y ]+ an H [Z]=H[Z ]=E[Z ]+, which efines a "istribution-free" weak fair premium conition of the form 3.4. In particular, the strong conition 3.4 hols true in case X equals X in istribution. The above situation will be further iscusse in Example 4.5.

11 Some alternative actuarial pricing methos iii The conition 3.4 has the following alternative interpretations. Given P=H[X] is a fair premium, then the reistribution of the risk profit loaing H[X]-E[X] is fair in the long run, or mean fair, for all three contractual members in the insurance agreement consult [5] for a precise mathematical concept. Since the insurer's risk has been eliminate perfect hege conition, no risk profit loaing goes to the insurer. The expecte risk profit goes half an half to the reinsurer neee risk profit loaing to absorb reinsurance payment fluctuations an to the insure in form of the bonus of expect amount equal to half the risk profit loaing. In the situation, where the insurer an the reinsurer operations are exercise by the same insurance group, one can say that half of the risk profit loaing belongs to the insure an half to the shareholers of the group, which provie the economic capital. Example 3.: a fair stop-loss premium moel Let X be a positive risk an let Z=X- + efine a perfectly hege experience rating stop-loss contract with bonus D=-X +. Suppose insurance premiums are set accoring to the stanar eviation pricing principle P=H[X]=+ an that the fair premium conition 3.4 hols. Denote by =E[X- + ] the net stoploss premium calculate accoring to a realistic claims moel an let =E[- X + ] the "conjugate" stop-loss premium representing the expecte bonus of the experience rating contract. By 3.4 one has H Y H Z, If the immunization conition H[Y]= of Proposition 3. is impose, that is the risk loaing satisfies the equality, 3.9 then the stanar eviation premium P=+ is necessarily equal to the fair premium P H Z. 3.0 Observe that if the market price P is known, that is is known, then is uniquely etermine by 3.9. Alternatively, if stop-loss market prices H[Z] are known, then is uniquely etermine by the equivalent implicit equation H X. 3.

12 Werner Hürlimann Further, in an uncertain insurance economy, market prices are not known with certainty. In this situation, we use the istribution-free an parameter-free premium P=+k, where k is the coefficient of variation, as justifie later in Section 4.. Then, one has necessarily =k an the "optimal fair" euctible is unique solution of the equation k 4. Distribution-free CAPM fair premiums. 3. In this Section, the CAPM splitting premium formulas.5,.7 an Proposition. are use to construct relatively safe istribution-free reinsurance premiums an fair premiums for the class S of perfectly hege experience rating contracts. The metho follows closely Hürlimann [9], Section 5. Notations are the same as in the preceing Sections. Consier the biatomic risk X such that the reinsurance payment Z =gx has a maximum expecte value over all biatomic risks: max XD a, b E Z E g X. 4. Stanar real analysis leas to the following result proof in Hürlimann [9], Lemma 5.. Lemma 4.. Let Z=gX be the transform of XD [a,b]. Then, the maximizing biatomic risk X D [a,b] solving the optimization problem 4. with support {x,x }, where x efines an involution on [a,b], satisfies one of the x following conitions: i x, x a,b is solution of the equation g x g x x x g' x g' x 4. ii x=a, x =a iii x=b, x =b =b Example 4.. Let Z=gX=X- + be a stop-loss claim. Then the maximum 4. equals

13 Some alternative actuarial pricing methos 3 an is attaine as follows : E x Z x, 4.3 x x Case : a a a x a Case : a a b b, x Case 3:, b b b x b In this special case 4.3 leas to a rigorous safe net stop-loss premium. Inee, it is actually the best upper boun over all risks efine on [a,b] with given mean an variance e.g. DeVyler an Goovaerts [7], Goovaerts et al. [8], p. 36, Jansen et al. [7], Goovaerts et al. [9]. The limiting Case when a -, b is ue to Bowers [6]. Typical often encountere situations inclue positive risks limiting Case a=0, b solve by Case an Case an arbitrary risks limiting Case a -, b solve by Case. Non-life insurance concerns mainly positive risks. Applications, which require the stuy of arbitrary risks, inclue financial risks rate of return, asset an liability management, portfolio theory, etc. an some of the life-insurance risks e.g. mixe portfolios of whole life insurances an life annuities. A istribution-free pricing system for the class of contracts S is now obtaine as follows. Let X be a risk efine on [a,b] with finite mean an variance. In a istribution-free worl any such risk will be associate the same insurance premium P=H[X], where H[] is a pricing principle epening stochastically only on the mean an variance e.g. variance principle, stanar eviation principle, etc.. Consier the biatomic risk X ={x,x } that is solution of the optimization problem 4. escribe in Lemma 4., an let Y =fx, Z =gx be the biatomic splitting risk components such that Y +Z =X. Since X has the same mean an variance as X, its associate insurance premium equals H[X ]=H[X]=P. On the other han, consier Borch's aitive premium principle efine by.5,.7 an enote by B[]. Assume that the compatibility conition B[X ]=B[X]= =P hols an that, 0. Applying Proposition., one gets the following CAPM splitting risk premiums Cov X, Z E Z VarX B Z g x g x g x P x x x P E X 4.4

14 4 Werner Hürlimann B f x f x Y P BZ f x P x x x 4.5 The CAPM base istribution-free pricing system H [] obtaine by setting H [X]:=B[X ], H [Y]:=B[Y ], H [Z]:=B[Z ], satisfies the esire splitting property P=H [X]=H [Y]+H [Z]. It may be viewe as a biatomic approximation to Borch's aitive CAPM premium principle, which satisfies the splitting property P=B[X]=B[Y]+B[Z]. Furthermore the maximum 4., enote by [Z]:=E[Z ] equals g x g x Z g x x x x. 4.6 Using that P=,, 0, x, it follows that H g x g x x x. 4.7 Z BZ g x x Z A generalize version of Theorem 5. in Hürlimann [9] has been shown. Theorem 4.. Given is a risk X efine on [a, b] with finite mean an variance. Let Y,Z=fX,gXS be a perfectly hege experience rating contract with maximum euctible an bonus D=-Y. Then the CAPM base istributionfree pricing system H [X]=B[X ], H [Y]=B[Y ], H [Z]=B[Z ], efine by 4.4, 4.5, satisfies the splitting property P=H [X]=H [Y]+H [Z] as well as the criterion of relative safeness H Z Z max E g X. 4.8 XD a, b To illustrate let us evelop some explicit results in case Z=X- + is a perfectly hege experience rating stop-loss contract with bonus D=-X + for the two typical situations of positive risks an arbitrary risks. Results for a general risk efine on [a,b] an other reinsurance structures can be obtaine similarly. 4.. Positive risks. Consier the limiting case a=0, b of Example 4.. The coefficient of variation is enote by k. Two cases must be istinguishe: Case : 0 k, x 0, x k

15 Some alternative actuarial pricing methos 5 One has the parameter-free pricing formulas: H Z k Z, k 4.9 P, k 4.0 H Y P. 4. k Case : k, x, x Z, 4. H Z P x, 4.3 H Y P x. 4.4 Let us apply the immunization argument of Section 3. In Case an for >0, the immunization conition H [Y]= implies that P=+k, which is thus shown to be a istribution-free an parameter-free fair premium by Proposition 3.. By continuity the same hols true in the limiting case as the euctible goes to zero see also Hürlimann [9], Proposition 4.. This argument provies a new insurance economics interpretation of the "Karlsruhe" pricing principle introuce by Heilmann [] see also Hürlimann [6]. In Case one has x>0 an the conition H [Y]= is equivalent to the relation P / x, 4.5 which restricts the possible euctible choices. Since P= 4.5 is also equivalent to the relation is assume, x. 4.6 In view of Proposition., interesting interpretations in terms of utility theory are possible. By. the following relations hol: x x u' x u' x x x, 4.7 x x

16 6 Werner Hürlimann P u' x u' x. 4.8 x x Example 4.: linear utility If the representative insurer is risk-neutral with utility function ux=ax+b, a>0, then one has =a, =0, which implies that P=H[X]=aE[X] is an expecte value principle. In Case all euctibles are feasible while in Case only those 's satisfying 4.5 with P=a are feasible. After calculation one fins that the euctible must satisfy the conition P a k, 4.9 which is feasible only in case a>+k. If P=+k no euctible is feasible in Case. Example 4.3: quaratic utility with saturation Let the representative insurer has marginal utility x, x b u' x b 4.0 0, x b In Case one sees that 4.7, 4.8 are equivalent to,. 4. b b Feasible euctibles are solutions of the equation 4.5, that is P x b. 4. b In case P=+k one has necessarily k b, 4.3 k which implies the parameter values

17 Some alternative actuarial pricing methos 7 k k, k k After calculation one obtains two solutions to 4.: k k, k 4 k. k 4.5 Only the first solution satisfies the require inequality > k of Case. In particular, a non-trivial pricing principle P=H[X]= with >, >0, fins herewith an actuarial application. Note that this example oes not exist if the normalization = is mae as in Aase [] an Hürlimann [9]. Remark 4.. Similar results can be erive for other utility functions of common use, e.g. exponential utility an power utility functions. 4.. Arbitrary risks. In the limiting case a -, b of Example 4., only Case occurs, for which the formulas 4. to 4.4 hol. Example 4.4: An insurance market istribution-free stop-loss premium formula Invoking the immunization conition H [Y]= in the special case =0, for which the whole premium an claim goes to the reinsurer respectively the insurer acts itself as reinsurer, one sees that the following equations must hol: It follows that P H Z P x. 4.6 P k. 4.7 Solving for an inserting the result in the right-han sie of 4.6, one gets the istribution-free stop-loss premium

18 8 Werner Hürlimann k P Z H, 4.8 where enotes Bowers' best stop-loss upper boun. Since by assumption, one recovers the main result from Hürlimann [6], which fins herewith a rigorous theoretical justification. Furthermore, the arbitrage-free stop-loss premium rate H [Z]/P is a istribution-free premium rate epening only on, an. The problem of the partition of the risk profit loaing between insurer an reinsurer see Amsler [3] is herewith solve in a natural way the stop-loss premium is couple with the market risk premium an the inequality H [Z] guarantees safeness for the reinsurer. In the table below, we compare the istribution-free stop-loss premium with the net stop-loss premium obtaine from the Erlang approximation of the probability ensity function. To get an iea of the probability of occurrence of a stop-loss claim, the values PrX> for the Erlang approximation are isplaye. The require formulas are N x x x x g, exp! ;, probability ensity 4.9 G x g k x k, ;, ; cumulative istribution 4.30 ;, ;, g G X E. 4.3 In the numerical example, the parameters are,.

19 Some alternative actuarial pricing methos 9 Table 4.: istribution-free stop-loss premiums vs. other approximations Parameters a =00 =5 b =00 =0 c =00 =0 =00 =5 case 00H [Z]/P PrX> in % a b c , < 0-6 < < < 0-5 < < Remark 4.. Since the formula 4.8 is now establishe rigorously, all the implications mae in Hürlimann [6], Sections 4 to 7, are vali when working in a istribution-free environment of arbitrary risks. It remains to be checke if an uner which conitions the same or similar conclusions may be true or approximately true in a istribution-free environment of positive risks. An illustration of this phenomenon follows in Example 4.6. Example 4.5: a istribution-free fair stanar eviation pricing principle Given is the situation escribe in ii of Remark 3., which also concerns an environment of arbitrary risks. However, observe that the efinition of H [] iffers from that of Theorem 4.. A calculation shows that the conition H [Y]= hols if, an only if, one has

20 0 Werner Hürlimann. 4.3 In the special case =0, the stanar eviation loaing equals k k, 4.33 which yiels the istribution-free an parameter-free fair stanar eviation pricing principle P H X k, 4.34 first observe in Hürlimann [0], Section 4.. In a istribution-free environment of arbitrary risks, 4.34 is always less than the "Karlsruhe" price P=+k obtaine in a istribution-free environment of positive risks. Note that 4.34 justifies the most competitive an unique choice =, which can be mae in practical applications of Example 4.4. Example 4.6: the "Karlsruhe" principle in a istribution-free environment Consier the moelling situation of Hürlimann [0], Section 4.. Suppose the ceant operates accoring to the istribution-free stanar eviation principle H [Y] as in Example 4.5. On the other han, suppose the reinsurer sets premiums following the principle 4.8: P Z H R k The conition P k is equivalent to the inequality k k As in Example 4.5, one has H Y Since P=H [Y]+H R [Z] the following equation must hol:

21 Some alternative actuarial pricing methos R H Z If the ceant wants to guarantee exactly the surplus D=-X + complete immunization argument, then 4.3 hols. Solving simultaneously the pair of equations 4.3, 4.37 in the unknowns,, yiels after algebraic calculation two feasible solutions, namely k Solution : =0, k Solution : k, k Although the insurer an the reinsurer operate accoring to ifferent pricing principles, the above moel specification contains as special case the pricing principle 4.34 of Example 4.5. Again the lower boun in 4.36 is attaine. The secon solution ientifies the premium with the "Karlsruhe" premium, giving to it an arbitrary risk base interpretation. Solution guarantees a bonus D=- X + D k X. As seen in Section 4., a while solution guarantees market premium P=+k suffices to guarantee a bonus D= - X + for all 0 k in a istribution-free environment of positive risks. In D k X particular, for the same premium P=+k, the bonus can be guarantee inepenently of whether X is a positive or an arbitrary risk cf. Remark Optimal long-term perfectly hege experience rating We show how perfectly hege experience rating strategies, which are "optimal" in the long-run an in a competitive environment, can be obtaine for the class S from Section 3. Optimality of a given pair Y,ZS is consiere with respect to the following two relevant properties: P Acceptable for the ceant are only contracts, which are mean self-financing. P The insurance premium of the contract shoul be competitive To motivate the first property, consier the perfect hege relation +Z=X+D. A time epenent mean self-financing ynamic strategy for the ceant can be formulate as follows. At the beginning of the first perio, the ceant puts asie the maximum euctible an pays the reinsurance premium E[Z], which in general is ajuste by some loaing. At the en of the first perio, the reinsurance payment Z together with the maximum euctible permits to pay the claims an there remains the guarantee bonus, which can be use to finance the maximum euctible an the reinsurance premium of the next perio, an so on. To be mean

22 Werner Hürlimann self-financing, the expecte bonus payment must at least be equal to the expecte reinsurance payment, that is see also Hürlimann [5] E[D] - E[Z] = E[ - X] = Therefore, the maximum euctible shoul be greater or equal to the mean amount of claims. Suppose premiums are set accoring to the variance principle. Taking into account the bonus, which belongs to the ceant resp. the shareholers an/or the insures, the neee perioic ranom payment of the ceant equals P Y, Z; E Z Var Z D E Z Var Z Y. 5. The neee variance premium ifferent from the fair premium of Section 3 is where the risk quantity P E P Y, Z; Var P Y, Z; R Y, Z, 5.3 RX Y, Z Var Y Var Z Var X Cov Y, Z 5.4 is calle "total splitting risk" as measure by the variance of the contract. By Chebychev's inequality one has Cov[Y,Z]0 with equality sign if, an only if, Y or Z is a constant e.g. Hary, Littlewoo an Polya [0], no. 43, or alternatively note that the pair X,X is positively quarant epenent, which implies Cov[fX,gX]0 for all non-ecreasing functions fx, gx. It follows that the neee variance premium is strictly less than the variance premium of the original risk X. The part of the risk stemming from the epenence between the retaine amount Y an the reinsurance payment Z has been eliminate by the perfectly hege experience rating contract variance reuction through iversification. One shoul note that the premium formula 5.3 efines a new "reinsurance base" pricing principle. To satisfy also property P, the premium 5.3 must be minimize. Therefore "optimal" experience rating strategies Y,ZS are obtaine as solutions of the constraine optimization problem Equivalently by 5.4 one has R X [Y,Z] = min. uner the constraint. 5.5 Cov[Y,Z] = max. uner the constraint. 5.6 Let us etermine the possible minimum an maximum premiums. Proposition 5.. For all Y,ZS the following bouns hol: X

23 Some alternative actuarial pricing methos 3 P P R Y, Z P. 5.7 min X Proof. The upper boun follows from the mentione inequality by Chebychev. The lower boun is a Corollary to Proposition. an Proposition.3 in Hürlimann [9]. In Proposition., it is shown that the minimum of R X [Y,Z] over all transforme ranom variables Y, Z is attaine by a linear transformation application of the Cauchy-Schwarz inequality. In Proposition.3, this lower boun is also attaine for a stop-loss experience rating contract Z=X- + for a biatomic risk, which maximizes the net stop-loss premium. In a istribution-free environment with positive risks, one can assume that P max =+k as justifie in Section 4.. In this case one has necessarily. Corollary 5.. Let X0 be a positive risk an let Y,ZS. If P max =+k, then the following bouns hol: R X Y, Z Pmin k P Pmax k. 5.8 Remark 5.. The above minimum an maximum premium bouns are reminiscent of the experience rating metho introuce by Ammeter [] an which, as our starting point, has been given ifferent equivalent characterizations see Hürlimann [4]. On the other han, if one requires aitionally that the premium 5.3 shoul be a fair premium in the sense of Section 3, then one must have max P E D Var Z. 5.9 Now, both premiums can be equal only if the following relation hols: Var[Y] = E[D]. 5.0 Therefore "optimal" strategies Y,ZS with fair variance premiums of the form R X Y, Z P Var Y E D 5. solve the constraine minimization problem

24 4 Werner Hürlimann R Y Z X, Var Y E D = min. uner the constraint. 5. Two illustrate the many applications of the introuce methoology, let us search for "optimal" perfectly hege experience rating contracts in two common situations. 5.. An optimal long-term stop-loss pricing strategy. Let us reprouce the results sketche in Hürlimann [0], Section 3. For Z=X- + use the "conjugate" notations =E[Z], E D, Fx=PrXx, F x F x, Var X, Var X X. The total stop-loss risk equals the univariate function R. A minimum uner the constraint satisfies the necessary conition R' F F 0,. 5.3 Using that F E X X, F E X X, one obtains the equivalent conitional expecte equation E X X E X X,. 5.4 In orer that a stop-loss euctible is optimal, it is necessary that the conitional expecte amount of stop-loss claims equals the conitional expecte bonus. Rearrangement shows that alternatively the following fixe-point equation must hol: E X X E X X. 5.5 In case a fixe-point has been foun, this will be a guarantee local minimum provie R' ' F F 0 f. 5.6 In moern analysis general conitions uner which a fixe-point equation like 5.5 has a solution are well-known. However, in our context there is an elementary proof, which shows that the above necessary conition is, in many cases of practical importance, fulfille. Proposition 5.. Let Fx an x be continuous real functions. If F, then there exists at least one [, such that R'=0.

25 Some alternative actuarial pricing methos 5 Proof. First of all one has from 5.3 an by assumption R'=-F0. By continuity it remains to check that there exists such that g:=r'0. Elementary calculation shows that F F g. With the inequality of Bowers [6], one has for all : F F F g For a fixe 0 such that F 0, it follows that for all 4, max 0 0 F, one has g F 0 0, as was to be shown. In contrast to the above result, the uniqueness of a solution to the fixe-point equation 5.5 cannot in general be guarantee without further assumptions. An interesting useful example illustrates this fact. Example 5.: the istribution of Bowers In Hürlimann [8] the following istribution function has been consiere:.,, x x x x F 5.7 Integrating the ifferential equation 'x= - -Fx, one sees that the associate net stop-loss premium 5.8 coincies with the best upper boun of Bowers given above. One checks the "uniform invariant conjugate" property

26 6 Werner Hürlimann F F for all, 5.9 which in particular shows that R'=0 for all. From 5.7 an A., A. in the Appenix, one borrows the inequality F F R max,. 5.0 F F But 4 uniformly for all, which shows in particular that the minimum is uniformly attaine for all. We have shown that the istribution of Bowers satisfies the following two extremal properties. Inepenently of the euctible it maximizes the net stop-loss premium an minimizes the total stoploss risk. The simultaneous use of this istribution as "risk valuation function" by the ceant an the reinsurer can thus be mathematically justifie through optimal properties. Our first application in this irection is given in Hürlimann [8]. Similarly, the "optimal" choice =, which playe up to now only a guiing role in an extene version of the classical mean-variance approach to portfolio theory by Markowitz see Hürlimann [5], [4], is justifie below using the notion of "total stop-loss risk". Example 5.: the normal istribution Let Nx be the stanar normal istribution, x=n'x the normal ensity, an assume that F=Nz with z=-/. Then, one has R'=0 an R '' N N. 5. If < / the retention level = is a local minimum of the total stop-loss risk function. It is important to observe that the require technical conition about the stanar eviation or similar volatility is almost always fulfille in applications. Example 5.3: the istribution of a whole life insurance portfolio Computations with the exact istribution have shown that an optimal euctible lies above but often close to the mean. 5.. On the linear combination of proportional an stop-loss reinsurance. The perfectly hege experience rating contract efine by Z=-rX+rX-s +, r,s0,]x[0,, with =rs the maximum euctible an D=rs-X + the bonus, has been stuie in Hürlimann []. In particular, its fair premium, when the

27 Some alternative actuarial pricing methos 7 reinsurer uses the variance principle, has been etermine an shown to be boune by the variance premium in case r0,] an s. Properties of the minimum fair premium with respect to the proportional retention level r have also been iscusse. We show that the optimization problem 5.6 has no extremal solution except perhaps for the egenerate biatomic istribution of Subcase below in the inner of the omain { r,s0,]x[0, : =rs }. Therefore the optimal contract lies necessarily on the bounary of this omain an is thus the stop-loss contract r=, =s, stuie in Section 5.. For the consiere contract one has, Z r r r s s r s C r, s : Cov Y. 5. Using the erivatives ' s F s, ' s F s, s' F s s, s' F s s, one gets the partial erivatives Cr s s r s s s, 5.3 C s r F s s r F s s. 5.4 To show that a stationary point r,s satisfying C r =C s =0 oes not exist for r 0, let us istinguish between two cases : F s Case : s s s s F s In this situation, the inequality of Kremer [8] simple probabilistic proof in the Appenix is strict. In particular the inequality s s s of Hürlimann [7] is strict an F s s 0 F s s 0 correspons to Case an Case in the proof of the inequality of Schmitter [30] in the Appenix, that is one has also s s 0 an F s s 0. The system of equations C r =C s =0 is equivalent to the system F s s r F s s, 5.5 F s s F s s s s. 5.6 s s s Solving for s in 5.6 an using that s s s, one gets

28 8 Werner Hürlimann F s s s s s, 5.7 F s which contraicts the assumption mae. F s Case : s s s s F s This means that the upper boun of the inequality of Kremer [8] is attaine. By the inequality of Schmitter [30], prove in the Appenix, the upper boun is attaine in the space of all risks with given mean, variance an net stop-loss premium s, by a biatomic istribution an equals max s s s 4 s s. 5.8 Subcase : s s 0 Case in Appenix One has s 0, s, Cr 0, Cs rf s s 0. The solution s= implies, using the restriction =rs, that r=. Subcase : s s 0 Case in Appenix One has s 0, s, Cr 0, Cs r r F s s 0 a egenerate limiting iatomic istribution for which F s 0. for Subcase 3 : s s 0 As observe in Case this implies F s s 0. Set z F s s F s s, s s With 5.8 one sees that if the upper boun of Kremer [8] is attaine, one must have the relation multiply with 4 s s : z Solving for z one gets z, 5.3 where by the inequality of Bowers [6]. Now by 5.4 one shoul have r z, hence z. By 5.3 this is only possible if =, hence z= an r=.

29 Some alternative actuarial pricing methos 9 Appenix: Dual inequalities of Kremer [8] an the inequality of Schmitter [30] First, a simple probabilistic erivation of a slightly generalize version of the inequality by Kremer [8] on the stop-loss variance is presente. Base on this result, it is shown that among all risks with a given mean, variance an net stoploss premium, a biatomic istribution maximizes the stop-loss variance, a result ue to Schmitter [30]. We use the following "conjugate" notations. For a risk X with finite mean an variance, let Fx=PrXx the corresponing istribution function, F x F x the survival function, =E[X- + ] the net stop-loss premium, =E[-X + ]=-+ the "conjugate" net stop-loss premium, =Var[X-+ ] the stop-loss variance an Var X X the variance of the retaine claims. The "conjugate" attribute has two motivations. First in the language of "life contingencies" the quantities F an F E X X are associate to the "mortality" of the risk X at "age", while the quantities F an F E X X are given the corresponing "survival" metaphoric interpretations. Secon, there is a "conjugate" property relating the inequalities for an, which rener them easy to remin of. To pass from one inequality to the other, it suffices to take "conjugate" quantities with the convention that, F F,. Proposition A.. Kremer [8] Given is a risk X with known,,, an F. Then, in the "conjugate" notation, the following inequalities hol: F F, A. F F F F. A. F F Proof. Conitioning on the event {X>} one has F Var E X F E X X X E X X X A.3 F Var X X. F Rearranging, one obtains the relation

30 30 Werner Hürlimann F F Var X X, A.4 F which implies the lower boun in A.. On the other han, combining the relation E X F E X X F E X X with A.3, one shows similarly that A.5 F F Var X X, A.6 F which implies the upper boun in A.. The inequalities A. follow from A. an the relation. Remark A.. In case the probability F is not known, one has the simpler "self-conjugate" upper bouns see Hürlimann [7], Hesselager [3]:. A.7. A.8 The following result is originally ue to Schmitter [30]. We obtain a slightly more general version base on a simpler an more rigorous proof. Proposition A.. The maximal stop-loss variance to the euctible for a risk X with mean, variance an net stop-loss premium, is given by max 4, A.9 an is attaine by a biatomic ranom variable Z with support {a,b} such that p a p, A.0 p b p, where the probability p=prz=a satisfies the relation p 4 p A.

31 Some alternative actuarial pricing methos 3 Proof. It suffices to consier istributions which, for given p=f, maximize, that is which satisfy equality in the upper boun A.. A maximum is obtaine if we etermine p so that the right han sie of A. is maximal. Since the erivative of that right han sie 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 han, the variation of p is restricte by the lower boun constraint in A.: p p, A. where equality hols for biatomic istributions with given an, which are always of the type A.0. Moreover the right han sie of A. is maximum in case p is greatest possible. Provie equality hols in A. an inserting this relation into the upper boun A. with equality sign, it follows that the maximizing probability p is the greatest solution of the quaratic equation p p p p. A.3 But, this equation is equivalent to the quaratic equation for p p : p p 0, A.4 p p whose greatest solution is A.. Inserting this value into A. with equality sign one gets the maximum A.9. It remains to show that the maximum is attaine by a biatomic istribution of the type A.0, with p from A., such that the quantities,, are the given ones. Using the well-known properties of A.0, it remains only to check that E[Z- + ]= is the given net stop-loss premium. We istinguish between three cases. Case : =-0 One has =0 an thus max{ }=. The appropriate biatomic istribution of the type A.0, A. is a b p Pr Z a A.5

32 3 Werner Hürlimann Case : =-0 One has =0 an thus max{ }=. The maximizing biatomic istribution of the type A.0, A. is the limiting biatomic istribution obtaine setting p, a, b. Case 3: - It suffices to consier the case a<b. Otherwise, one has a<b with E[Z + ]=0, leaing to the Case, or <a<b with E[Z- + ]=-, which is Case. One has to satisfy which is seen to be equivalent to E Z p p p, A.6 p p p p. A.7 Taking squares one sees that p must be solution of A.3. Remarks A.. Schmitter [30] oes not provie A. an verify that the maximizing istribution has as net stop-loss premium the given. Case reveals a statement similar to that of Proposition A. for the weaker inequality A.7. Provie, the equality sign in A.7 is attaine by the biatomic istribution A.5 an gives also the maximum of the stop-loss variance, namely max{ }=. This fact fins an interesting application see Hürlimann []. References [ K.K. Aase, Equilibrum in a reinsurance synicate; existence, uniqueness an characterization, ASTIN Bulletin, 3 993, [] H. Ammeter, Risikotheoretische Grunlagen er Erfahrungstarifierung, Bulletin of the Swiss Association of Actuaries, 96, [3] M.-H. Amsler, Sur les marges e sécurité es primes "stop-loss", Bulletin of the Swiss Association of Actuaries, 986, [4] K. Borch, Aitive insurance premiums: a note, The Journal of Finance, 37 98, 95-98, Reprinte in Borch, 990,