Expected utility operators and coinsurance problem
aa r X i v : . [ q -f i n . M F ] A ug Expected utility operators and coinsuranceproblem
Irina Georgescu
Academy of Economic StudiesDepartment of Economic CyberneticsPiat¸a Romana No 6 R 70167, Oficiul Postal 22, Bucharest, RomaniaEmail: [email protected]
Abstract
The expected utility operators introduced in a previous paper, offer aframework for a general risk aversion theory, in which risk is modelled bya fuzzy number A .In this paper we formulate a coinsurance problem in the possibilisticsetting defined by an expected utility operator T . Some properties of theoptimal saving T -coinsurance rate are proved and an approximate calcu-lation formula of this is established with respect to the Arrow-Pratt indexof the utility function of the policyholder, as well as the expected valueand the variance of a fuzzy number A . Various formulas of the optimal T -coinsurance rate are deduced for a few expected utility operators in caseof a triangular fuzzy number and of some HARA and CRRA-type utilityfunctions. Keywords : expected utility operators, coinsurance
In most cases, economic and financial activities are accompanied by risk, whichgenerates pecuniary losses for the agents. A risk-averse agent will try to diminishlosses caused by risk by closing an insurance contract. By [12], in the componentof an insurance contract enter a premium P paid by the agent (policyholder)to an insurer and a real function I ( . ) which specifies the part of the loss thatis recovered: if the loss has the size x , then the insurer will pay the agent theamount I ( x ).Usually, the function I ( . ) is defined by setting a coinsurance rate β , if x isthe loss then the policyholder will receive the amount I ( x ) = βx . The agentwill choose that β maximizing the expected utility of her final wealth. So anoptimal problem occurs, called the coinsurance problem .A probabilistic model of risk assumes that this is mathematically representedby a random variable. The coinsurance problem from [12], [26] is such a proba-1ilistic model, in which the loss is a random variable, and the agent is describedby a utility function.In the paper [19] there is studied a probabilistic type coinsurance problem:the agent is represented by a utility function, but the loss caused by risk is afuzzy number.The coinsurance problem from [19] is formulated by maximizing a possibilis-tic expected utility theory, defined in [16], [17] in the framework of a possibilistictreatment of risk aversion. On the other hand, in [16] there is a second notionof possibilistic expected utility, and the expected utility operators from [18]allow the definition of a general notion of possibilistic expected utility whichgeneralizes the two mentioned above. By this, to each expected utility opera-tor T one associates a possibilistic expected utility theory (called T -possibilistic EU -theory), in which various topics on risk theory can be discussed.The aim of this paper is to study the coinsurance problem in the frameworkoffered by expected utility operators. The formulation of an abstract coinsur-ance problem can be done inside an arbitrary T -possibilistic EU -theory, butthe proofs of the optimal coinsurance rate and its approximate computationassume T to fulfill a supplementary property. For this purpose, the D -operatorshave been chosen, a class of expected utility operators introduced in [20] by apreservation condition of derivability of the utility function with respect to aparameter.Section 2 presents two notions of possibilistic expected utility from [15], [16],as well as some operators associated with fuzzy numbers (possibilistic expectedutility, possibilistic variance).In Section 3 the definition of expected utility operators is recalled from [17],[18] and the D -operators are introduced.To formulate the coinsurance problem in the context of a T -possibilistic EU -theory, in Section 4 a few basic notions are defined: the coinsurance contract,the T -premium for insurance indemnity, the T -coinsurance rate, etc. Assumingthat T is a D -operator, the T -coinsurance rate can be computed as a solutionof a first order condition.The results on the optimal T -coinsurance β ∗ are contained in Section 5. Afirst result is a possibilistic version of a Mossin theorem ([26] or [12], p. 51).The main result is an approximate calculation formula of β ∗ , with respect tothe expected value E f ( A ) of the fuzzy number A representing the risk, the T -variance V ar T ( A ) of A and the Arrow-Pratt index of the agent’s utility function.It is also demonstrated a formula that approximates the maximal total expectedutility (obtained by the choice of the T -coinsurance rate β ∗ ).Another result of the section asserts that if an agent u ia more risk aversethan an agent u , then the optimal T -coinsurance rate is higher for u than for u . In Section 6 forms of the approximation formula of the optimal T -coinsurancerate in the particular case of the expected utility operators T , T from [15],[16] and a risk represented by a triangular fuzzy number are obtained. Theseformulas are applied for HARA and CRRA-type utility functions.In the concluding remarks section a few open issues are commented and the2esult from Appendix presents a necessary condition for the positivity of theoptimal T -coinsurance rate. Fuzzy numbers are generalizations of real numbers, able to express impreciseinformation. Using Zadeh’s extension principle [32], the operations with realnumbers can be extended to fuzzy numbers, such most of algebraic propertiesare preserved [9], [10]. At the same time fuzzy numbers can be thought of aspossibilistic distributions [10], [6]. By parallelism with probabilistic distribu-tions, but in a completely different way, with each fuzzy number possibilisticindicators can be associated: expected value, variance, covariance, moments,etc. [5], [7], [11], [13], [15], [33]. All these make the fuzzy numbers an effectivetool in the possibilistic treatment of some topics on risk theory [6], [8], [17], [29],[33].In this section we will present after [15], [16], [17] two notions of expectedutility associated with a triple consisting of a utility function (representing anagent), a fuzzy number (representing the risk) and a weighting function. Also,we will recall the definition of expected value and two variances associated witha fuzzy number [5], [6], [17], [33].We fix a mathematical framework consisting of three entities: • a weighting function f : [0 , → R ( f is a non-negative and increasingfunction that satisfies R f ( γ ) dγ = 1); • a utility function u : R → R of class C ; • a fuzzy number A whose level sets have the form [ A ] γ = [ a ( γ ) , a ( γ )] forall γ ∈ [0 , A will be supp ( A ) = { x ∈ R | A ( x ) > } = ( a (0) , a (0)).Following [17] Section 4 . E ( f, u ( A )) = R [ u ( a ( γ )) + u ( a ( γ ))] f ( γ ) dγ (2.1) E ( f, u ( A )) = R [ a ( γ ) − a ( γ ) R a ( γ ) a ( γ ) u ( x ) dx ] f ( γ ) dγ (2.2)Setting in (2.1) or (2.2) u = 1 R (the identity function of R ) one obtains thepossibilistic expected value ([11], [5], [13], [25]): E f ( A ) = E ( f, R ( A )) = E ( f, R ( A )) = R [ a ( γ ) + a ( γ )] f ( γ ) dγ (2.3)If supp ( A ) ⊆ R + , then from (2.3) it follows that E f ( A ) ≥ u ( x ) = ( x − E f ( A )) two different notions of possibilistic variance follow[5], [13], [15], [33]: V ar ( f, A ) = R [( a ( γ ) − E f ( A )) + ( a ( γ ) − E f ( A )) ] f ( γ ) dγ (2.4) V ar ( f, A ) = R [ a ( γ ) − a ( γ ) R a ( γ ) a ( γ ) ( x − E f ( A )) ] f ( γ ) dγ (2.5)3 Expected utility operators and D -operators In this section we will recall the definitions of the expected utility operatorsand the D -operators, introduced in [16], respectively [20] as an abstraction ofthe two possibilistic expected utilities E ( f, u ( A )) and E ( f, u ( A )) from theprevious section.Let F be the set of fuzzy numbers, C ( R ) the set of real continuous functions(mapped from R to R ) and U a subset of C ( R ) satisfying the following properties:( U ) U contains constant functions and first and second degree polynomialfunctions;( U ) U is closed under linear combinations: if a, b ∈ R and g, h ∈ U then ag + bh ∈ U .For each a ∈ R we denote ¯ a : R → R the constant function ¯ a ( x ) = a , for x ∈ R . R will be the identity function of R . Then ¯ a, R belong to U . Inparticular, we can consider U = C ( R ).We fix a weighting function f : [0 , → R and a family U with the properties( U ) and ( U ). Definition 3.1 [17], [18] An (f-weighted) expected utility operator is a func-tion T : F × U → R such that for any a, b ∈ R , g, h ∈ U and A ∈ F the followingconditions are fulfilled:(a) T ( A, R ) = E f ( A ) ;(b) T ( A, ¯ a ) = a ;(c) T ( A, ag + bh ) = aT ( A, g ) + bT ( A, h ) ;(d) g ≤ h implies T ( A, g ) ≤ T ( A, h ) . Example 3.2 [15], [17] We consider the function T : F × C ( R ) → R definedas follows: for any fuzzy number A and for any g ∈ C ( R ) : T ( A, g ) = E ( f, g ( A )) (3.1)Then T is an expected utility operator. Example 3.3 [15], [17] We consider the function T : F × C ( R ) → R definedas follows: for any fuzzy number A and for any g ∈ C ( R ) : T ( A, g ) = E ( f, g ( A )) (3.2)Then T is an expected utility operator. An expected utility operator T is strictly increasing if for any A ∈ F and g, h ∈ U , g < h implies T ( A, g ) < T ( A, h ). One can prove that the expectedutility operators T , T are strictly increasing.If T is a strictly increasing operator, then for any A ∈ F and h ∈ U , h > T ( A, h ) > T and T . Therefore, the real number T ( A, g ) will becalled generalized possibilistic expected utility (shortly, T -expected utility) andit will represent the starting point of a possibilistic EU -theory associated with T . Sometimes, instead of T ( A, g ) we will use the notation T ( A, g ( x )).4articularizing g , from T ( A, g ) various possibilistic indicators are obtained.By axiom (a), for g = R , the possibilistic expected value E f ( A ) follows. For g ( x ) = ( x − E f ( A )) we have the notion of T -covariance: V ar T ( A ) = T ( A, ( x − E f ( A )) ) (3.3)Using the axiom (d) of Definition 3.1, it follows immediately that V ar T ( A ) ≥ Remark 3.4
As we have seen previously, the two operators T and T introducethe possibilistic variances V ar T ( A ) = V ar ( f, A ) , respectively V ar T ( A ) = V ar ( f, A ) . The two possibilistic variances
V ar T ( A ), V ar T ( A ) have been used in theapplication of some models in possibilistic risk theory [1], [6], [8], [19], [25], [29].In case of probabilistic risk, the risk aversion of an agent is described bythe Arrow-Pratt index [2], [3], [28]: for a utility function u of class C , theArrow-Pratt index is defined by r u ( w ) = − u ′′ ( w ) u ′ ( w ) for w ∈ R (3.4)If u , u are the utility functions of the two agents, then the Arrow-Pratttheorem [2], [3], [28] asserts that ”the agent u is more risk-averse than theagent u ” iff r u ( w ) ≥ r u ( w ) for any w ∈ R .Papers [15], [16] contain two distinct possibilistic treatments of risk aversionwhen the risk is a fuzzy number. These possibilistic theories of risk aversion arebased on the possibilistic utilities T ( A, u ), T ( A, u ). In particular, in both casesa Pratt type theorem is proved. A surprising result is obtained: the possibilisticrisk aversion is characterized in terms of the Arrow-Pratt index. In a certainsense, we could say that ”the possibilistic risk aversion (in the sense of papers[15], [16]) is equivalent to the probabilistic risk aversion” [2], [3], [28].The expected utility operators allow a generalization of possibilistic riskaversion theories from [15], [16].In this general framework it is defined what it means that ”an agent is morerisk-averse than another agent” and it is proved a Pratt-type theorem whichcharacterizes this property. The main tool used in proving this result is theapproximation formula from the following proposition.
Proposition 3.5 [16], [17] Let T be an expected utility operator, A a fuzzynumber and u a utility function of class C . Then T ( A, u ) ≈ u ( E f ( A )) + u ′′ ( E f ( A )) V ar T ( A ) (3.5) Proposition 3.5 will be used in this paper to prove the approximation formulafrom Theorem 5.9.Still the treatment of other topics from risk theory in the framework offeredby expected utility operators is not possible without imposing some supplemen-tary conditions on those. In paper [20] the D -operators have been introduced tostudy a possibilistic portfolio choice problem. We will present next the definitionof the D -operators.For a utility function g ( x, λ ), in which λ is a parameter, we consider thefollowing properties: 5i) g ( x, λ ) is continuous with respect to the argument x and derivable withrespect to the argument λ ;(ii) for any λ ∈ R , the function ∂g ( .,λ ) ∂λ : R → R is derivable. Definition 3.6 [20] An expected utility operator T : F × C ( R ) → R is a D - operator if for any fuzzy number A and for any function g ( x, λ ) with the prop-erties (i) and (ii), the following axioms are fulfilled:( D ) The function λ T ( A, g ( ., λ )) is derivable (with respect to λ );( D ) T ( A, ∂g ( .,λ ) ∂λ ) = ddλ T ( A, g ( ., λ )) . By Proposition 1 from [20], T and T are D -operators. Remark 3.7
Conditions ( D ) and ( D ) make it possible the use of first orderconditions in solving some optimization problems in which the objective functionis a T -expected utility. In paper [20], the D -operators offer the framework tofind some approximate solutions of a possibilistic portfolio problem. In the nextsections, the axioms ( D ) and ( D ) will be intensely used to determine theoptimal coinsurance rate in a coinsurance problem formulated in the context ofexpected utility operators. Proposition 3.8
Let
T, S be two expected utility operators and c ∈ R .(a) U = cT + (1 − c ) S is an expected utility operator;(b) If S, T are D -operators then U is a D -operator. Proof. (a) By [17], Proposition 5.2.5.(b) The axiom ( D ) is immediate. We will verify ( D ). Since S, T fulfillcondition ( D ), we will have the following equalities: ddλ U ( A, g ( ., λ )) = c ddλ T ( A, g ( ., λ )) + (1 − c ) ddλ S ( A, g ( ., λ ))= cT ( A, ∂g ( .,λ ) ∂λ ) + (1 − c ) S ( A, ∂g ( .,λ ) ∂λ )= U ( A, ∂g ( .,λ ) ∂λ )for any fuzzy number A and for any utility function g ( x, λ ) which fulfilshypotheses (i) and (ii). In this section we will deal with the coinsurance problem in the context ofexpected utility operators. First we will introduce a few entities by which we willdefine this coinsurance problem, then we will restrict the universe of discussionto D -operators in order to start the study of optimal coinsurance rate.Consider an agent with a utility function u of class C such that u ′ > u ′′ <
0. Assume that the agent possesses an initial wealth subject to risk. Toretrieve a part of the loss caused by this risk, the agent will close an insurancecontract. By [12], p. 46, an insurance contract has two components: • a premium P to be paid by the policyholder;6 an indemnity schedule I ( x ) indicates the amount to be paid by the insurerfor a loss x .We will think of I ( x ) as a utility function, and the premium P will be definedwith respect to the mathematical modeling of risk. In case of a probabilisticmodel, the loss will be a random variable X ≥
0, and P will be defined bymeans of (probabilistic) expected utility EI ( X ) (see [12], p. 49).The possibilistic form of the coinsurance problem from [19] has as hypothesisthe fact that the risk is a fuzzy number A with the property that supp ( A ) ⊆ R + and supp ( A ) does not reduce to a single point. In particular, this hypothesisassures that E f ( A ) > EU -theory associated with an expected utility operator T , we will fix T and a weighting function f : [0 , → R .The T -premium for insurance indemnity is defined by P = (1 + λ ) T ( A, u ) (4.1)where λ ≥ Remark 4.1 ( i) The expression (4.1) of P is inspired from the form of thepremium for insurance indemnity from the probabilistic model ([12], p. 42).(ii) If T is the operator T from Example 3.2, then we obtain the notion ofpossibilistic premium for insurance indemnity from [19]. We will assume that I ( x ) = βx for all x . Following the terminology from[12], β will be called coinsurance rate, and 1 − β will be called retention rate .The coinsurance rate β represents the fraction from the size of the loss theinsured gets following an insurance contract.Similar with [12], p. 49 or [19], we will make the hypothesis that the policy-holder chooses apriori a coinsurance rate β . Then the corresponding T -premiumfor insurance indemnity P ( β ) will have the form: P ( β ) = (1 + λ ) T ( A, βx ) = β (1 + λ ) T ( A, x ).By the axiom (a) from Definition 3.1, P ( β ) will be written: P ( β ) = β (1 + λ ) E f ( A ) (4.2)By denoting P = (1 + λ ) E f ( A ) we will have P ( β ) = βP (4.3)If β is the coinsurance rate and x is the size of the loss then the agent remainsultimately with the following amount: g ( x, β ) = w − P ( β ) − x + βx = w − βP − (1 − β ) x (4.4)Consider the function which gives the utility of amount g ( x, β ): h ( x, β ) = u ( g ( x, β )) = u ( w − βP − (1 − β ) x ) (4.5)Then H ( β ) = T ( A, h ( x, β )) (4.6)is the total T -utility associated with a possibilistic risk A , an initial wealth w and an insurance contract with a coinsurance rate β .Since the agent wants to maximize this total utility, he will choose β as thesolution of an optimization problem:max β H ( β ) (the coinsurance problem) (4.7)7o be able to study the existence and the computation of the solution ofproblem (4.7), we will assume from now on that T is a D -operator.Taking into account the axioms ( D ) and ( D ) of Definition 3.5 we will have H ′ ( β ) = ddβ T ( A, h ( ., β )) = T ( A, ∂∂β h ( x, β )))Taking into account that ∂∂β h ( x, β ) = ∂∂β u ( g ( x, β )) = u ′ ( g ( x, β )) ∂g ( x,β ) ∂β = u ′ ( g ( x, β ))( x − P )the following form of the derivative of H ( β ) follows: H ′ ( β ) = T ( A, u ′ ( g ( x, β ))( x − P )) (4.8)Analogously, we obtain the second derivative: H ′′ ( β ) = T ( A, u ′′ ( g ( x, β ))( x − P ) ) (4.9)By hypothesis, u ′′ ( g ( x, β )) <
0. Applying the axiom (d) of Definition 3.1,from (4.9) it follows H ′′ ( β ) ≤
0, thus H is a concave function. Moreover, if theexpected utility operator T is strictly increasing then H is strictly concave. Wecan consider then the solution β ∗ T of the optimization problem 4.7: H ∗ ( β ∗ T ) =max β H ( β ). The determination of the optimal coinsurance rate β ∗ and the totalutility function H ( β ∗ ) is one of the agent’s important problems. When it exists,the optimal coinsurance rate β ∗ verifies the first order condition H ′ ( β ) = 0.Taking into account (4.8), the first order condition H ′ ( β ) = 0 will be written: T ( A, ( x − P ) u ′ ( g ( x, β ∗ T ))) = 0 (4.10)Let us consider the case of D -operators T and T . By (3.1) and (3.2), thefirst order condition (4.10) gets the following form: • for the D -operator T : E ( f, ( A − P ) u ′ ( g ( A, β ∗ T ))) = 0 (4.11) • for the D -operator T : E ( f, ( A − P ) u ′ ( g ( A, β ∗ T ))) = 0 (4.12)The coinsurance problem formulated in EU -theory associated with the op-erator T has been studied in [19]. In particular, using the first-order condition(4.11), in [19] an approximate calculation formula of the optimal coinsurancehas been proved. Let f : [ a, b ] → R be a weighting function, T a D -operator, u : R → R a utilityfunction and A a fuzzy number. As in the previous section, we will make thefollowing assumptions on u and A : • u is of class C , u ′ > u ′′ < • supp ( A ) ⊆ R + and supp ( A ) is not a point set.According to the second hypothesis, one gets E f ( A ) > β ∗ = β ∗ T be the solution of the insurance problem (4.7). We will keepall notations from Section 4. Proposition 5.1 (i) If λ = 0 then β ∗ = 1 ;(ii) If λ > then β ∗ < . roof. (i) Setting β = 1 in (4.4), we have g ( x,
1) = w − P . Then, by Definition3.1 it follows T ( A, ( x − P ) u ′ ( g ( x, T ( A, ( x − P ) u ′ ( w − P ))= u ′ ( w − P ) T ( A, x − P )= u ′ ( w − P )( T ( A, x ) − P )= u ′ ( w − P )( E f ( A ) − P )= − λE f ( A ) u ′ ( w − P )If λ = 0 then β ∗ = 1 verifies the first order condition (4.10).(ii) Since supp ( A ) ⊆ R + and supp ( A ) is not a point set, it follows E f ( A ) = R [ a ( γ ) + a ( γ )] f ( γ ) dγ > λ > u ′ > H ′ (1) = T ( A, ( x − P ) u ′ ( g ( x, − λE f ( A ) u ′ ( w − P ) < β ∗ ≥
1. Since H is concave, its derivative H ′ isdecreasing, thus H ′ ( β ) ≤ H ′ (1) <
0. This contradicts the first order condition H ′ ( β ∗ ) = 0, thus β ∗ < β ∗ ≤ β (as in [12], Section3.2, for the probabilistic coinsurance rate). Therefore, the solution β ∗ of (4.7)may not satisfy the inequality 0 < β ∗ ≤
1. In the Appendix, we will establish anecessary condition for 0 < β ∗ . Remark 5.2
Proposition 5.1 is a result analogous to Mosin theorem ([26] or[12], Proposition 3.1). Then when T is the operator T one obtains Proposition4 from [19]. An exact solution for the maximization problem (4.7) is difficult to find.Therefore, it is more convenient to find approximate solutions of equation (4.10).Before proving a formula for the approximate calculation of β ∗ , let us denote w = w − P . Then formula (4.4) becomes: g ( x, β ) = w − (1 − β )( x − P ) (5.1) Theorem 5.3
An approximate value of the optimal T -coinsurance rate β ∗ is: β ∗ ≈ u ′ ( w ) u ′′ ( w ) λE f ( A ) V ar T ( A )+ λ E f ( A ) Proof.
By (5.1), u ′ ( g ( x, β )) = u ′ ( w − (1 − β )( x − P )). We consider the first-order Taylor approximation of u ′ ( w − (1 − β )( x − P )) around w : u ′ ( g ( x, β )) ≈ u ′ ( w ) − (1 − β )( x − P ) u ′′ ( w )from where it follows( x − P ) u ′ ( g ( x, β )) ≈ u ′ ( w )( x − P ) − (1 − β ) u ′′ ( w )( x − P ) Taking into account (4.8) and Definition 3.1, we have H ′ ( β ) = T ( A, ( x − P ) u ′ ( g ( x, β )) ≈ T ( A, u ′ ( w )( x − P ) − (1 − β ) u ′′ ( w )( x − P ) )= u ′ ( w ) T ( A, x − P ) − (1 − β ) u ′′ ( w ) T ( A, ( x − P ) )We notice that 9 ( A, x − P ) = T ( A, x ) − P = E f ( A ) − P . T ( A, ( x − P ) ) = T ( A, x − P x + P )= T ( A, x ) − P T ( A, x ) + P = T ( A, x ) − P E f ( A ) + P = ( T ( A, x ) − E f ( A )) + ( E f ( A ) − P ) = V ar T ( A ) + ( E f ( A ) − P ) Replacing T ( A, x − P ) and T ( A, ( x − P ) ) in the approximate expressionof H ′ ( β ) one obtains: H ′ ( β ) ≈ u ′ ( w )( E f ( A ) − P ) − (1 − β ) u ′′ ( w )[ V ar T ( A ) + ( E f ( A ) − P ) ]Then the first-order condition H ′ ( β ∗ ) = 0 can be written u ′ ( w )( E f ( A ) − P ) − (1 − β ∗ ) u ′′ ( w )[ V ar T ( A ) + ( E f ( A ) − P ) ] ≈ β ∗ ≈ − u ′ ( w ) u ′′ ( w ) E f ( A ) − P V ar T ( A )+( E f ( A ) − P ) Since P = (1 + λ ) E f ( A ), we have E f ( A ) − P = E f ( A ) − (1 + λ ) E f ( A ) = − λE f ( A ). With this, the approximate value of β ∗ gets the form β ∗ ≈ u ′ ( w ) u ′′ ( w ) λE f ( A ) V ar T ( A )+ λ E f ( A ) Taking into account the definition of the Arrow-Pratt index from (3.4) oneobtains
Corollary 5.4 β ∗ ≈ − λr u ( w ) E f ( A ) V ar T ( A )+ λ E f ( A ) (5.3) By particularizing the operator T , different approximation formulas of theoptimal coinsurance rate β are obtained from (5.3). If T is the D -operator T from Example 3.2, then the approximation formula (22) from [19] is obtained. Remark 5.5
The approximate value of β ∗ given by (5.3) gives us the way theoptimal T -insurance depends on the risk aversion of the agent who closes theinsurance contract, as well as it depends on the expected value and the varianceof the fuzzy number representing the risk. The following result will give a moreprecise form of the relation between the risk aversion and the T -coinsurancerate: an increase in risk aversion will generate an increase in coinsurance rate. We consider two agents whose utility functions u , u are of class C andverify the conditions u ′ > u ′ > u ′′ < u ′′ <
0. Let β ∗ , β ∗ be theoptimal T -coinsurance rates associated with the utility functions u , u , theweighting function f , the D -operator T and the fuzzy number A . Proposition 5.6
If the agent u is more risk-averse than u , then β ∗ > β ∗ . Proof.
We consider the Arrow-Pratt indices of the utility functions u , u : r u ( w ) = − u ′′ ( w ) u ′ ( w ) , r u ( w ) = − u ′′ ( w ) u ′ ( w ) By u ′ > u ′ > u ′′ < u ′′ < r u ( w ) > r u ( w ) > w . We apply Corollary 5.4 in case of the coinsurance problem correspondingto the agents u and u : 10 ∗ ≈ − λr u ( w ) E f ( A ) V ar T ( A )+ λ E f ( A ) (5.4) β ∗ ≈ − λr u ( w ) E f ( A ) V ar T ( A )+ λ E f ( A ) (5.5)By hypothesis, r u ( w ) ≥ r u ( w ) >
0, thus 0 < λr u ( w ) < λr u ( w ) . Since E f ( A ) > V ar T ( A ) ≥
0, from (5.4) and (5.5) it follows immediately β ∗ ≥ β ∗ .Let T, S be two T -operators and c ∈ R . By Proposition 3.8, U = cT +(1 − c ) S is a D -operator. We consider the coinsurance problems associated with the D -operators T, S, U and in rest, keeping the same data which define the coinsuranceproblem (4.7).
Proposition 5.7
Let β ∗ T , β ∗ S and β ∗ U the optimal coinsurance rates correspond-ing to the D -operators T, S, U . Then β ∗ U ≈ − c − β ∗ T + − c − β ∗ S (5.6) Proof.
By (5.3) we have the following approximate values of β ∗ T , β ∗ S and β ∗ U : β ∗ T ≈ − λr u ( w ) E f ( A ) V ar T ( A )+ λ E f ( A ) β ∗ S ≈ − λr u ( w ) E f ( A ) V ar S ( A )+ λ E f ( A ) β ∗ U ≈ − λr u ( w ) E f ( A ) V ar U ( A )+ λ E f ( A ) from where it follows: − β ∗ T ≈ r u ( w ) λ V ar T ( A )+ λ E f ( A ) E f ( A ) (5.7) − β ∗ S ≈ r u ( w ) λ V ar S ( A )+ λ E f ( A ) E f ( A ) (5.8) − β ∗ U ≈ r u ( w ) λ V ar U ( A )+ λ E f ( A ) E f ( A ) (5.9)By [17], Proposition 5.1.5, we have V ar U ( A ) = cV ar T ( A ) + (1 − c ) V ar S ( A ).Then, by taking into account (5.7)-(5.9) the following equalities hold: c − β ∗ T + − c − β ∗ S ≈ r u ( w ) λ cV ar T ( A )+(1 − c ) V ar S ( A )+ λ E f ( A ) E f ( A ) ≈ r u ( w ) λ V ar U ( A )+ λ E f ( A ) E f ( A ) ≈ − β ∗ U From here it follows β ∗ U ≈ − c − β ∗ T + − c − β ∗ S . Remark 5.8
The previous proposition allows to obtain the optimal coinsurancerates for all convex combinations of two D -operators. In particular, if we take c = then U = T + S and by (5.6), the optimal coinsurance rate of U willbe: β ∗ U ≈ − − β ∗ T + − β ∗ S (5.10) β ∗ from Corollary 5.4will be used in the following to approximate the total expected utility H ( β ∗ ) = T ( A, h ( x, β ∗ )). Theorem 5.9
The total expected utility H ( β ∗ ) corresponding to the optimalcoinsurance rate β ∗ can be approximated by H ( β ∗ ) ≈ u ( w + r u ( w ) λ E f ( A ) V ar T ( A )+ λ E f ( A ) )+ λ r u ( w ) E f ( A ) V ar T ( A ( V ar T ( A )+ λ E f ( A )) u ′′ ( w + r u ( w ) λ E f ( A ) V ar T ( A )+ λ E f ( A ) ) Proof.
We consider the unidimensional function v ( x ) = h ( x, β ∗ ) = u ( w − (1 − β ∗ )( x − P )) (5.11)One will notice that H ( β ∗ ) = T ( A, v ). v is a utility function of class C thuswe can apply the approximation formula (3.5): H ( β ∗ ) ≈ v ( E f ( A )) + v ′′ ( E f ( A ))2 V ar T ( A ) (5.12)Since E f ( A ) − P = − λE f ( A ) it follows the approximation w − (1 − β ∗ )( E f ( A ) − P ) = w + λ (1 − β ∗ ) E f ( A ) (5.13)By Corollary 5.4 one has 1 − β ∗ ≈ λr u ( w ) E f ( A ) V ar T ( A )+ λ E f ( A ) (5.14)Deriving twice in (5.11) it follows v ′′ ( x ) = (1 − β ∗ ) u ′′ ( w − (1 − β ∗ )( x − P )) (5.15)From (5.11), (5.15) and (5.14) we can deduce v ( E f ( A )) ≈ u ( w + r u ( w ) λ E f ( A ) V ar T ( A )+ λ E f ( A ) ) (5.16) v ′′ ( E f ( A )) ≈ λ r u ( w ) E f ( A )( V ar T ( A )+ λ E f ( A )) u ′′ ( w + r u ( w ) λ E f ( A ) V ar T ( A )+ λ E f ( A ) ) (5.17)Replacing v ( E f ( A )) and v ′′ ( E f ( A )) with their approximate values from(5.14) and (5.17) the approximation formula from the statement of the theoremfollows. In this section we will study the optimal T -coinsurance rate for some particular D -operators, making the following assumptions on the weighting function f andthe fuzzy number A : • f ( t ) = 2 t , for any t ∈ [0 , • A is the triangular fuzzy number ( a, α, β ): A ( t ) = − a − tα a − α ≤ t ≤ a − t − aβ a ≤ t ≤ a + β otherwise As to the D -operator T , we will consider the following particular cases:(a) T is the D -operator T . By [17], Examples 3.3.10 and 3.4.10 we have E f ( A ) = a + β − α (6.1) V ar T ( A ) = α + β + αβ (6.2) 12eplacing E f ( A ) and V ar T ( A ) in (5.3), the optimal T -coinsurance rate β ∗ = β ∗ T gets the form: β ∗ ≈ − λr u ( w ) a + β − α α β αβ + λ ( a + β − α ) (6.3)(b) T is the D -operator T . By [17], Example 3.4.10 we have
V ar T ( A ) = α + β (6.4)Replacing E f ( A ) and V ar T ( A ) in (5.3), the optimal T -coinsurance rate β ∗ T becomes: β ∗ ≈ − λr u ( w ) a + β − α α β + λ ( a + β − α ) (6.5)(c) We consider the D -operator U = T + T (by Proposition 3.8). Forthe computation of the optimal U -coinsurance rate β ∗ u we will recall the formulafrom Remark 5.8. Using (6.3) and (6.5) one obtains: − β ∗ + − β ∗ ≈ r u ( w ) λ α β αβ + α β +2 λ ( a + β − α ) a + β − α = r u ( w ) λ ( α + β )2+2( α β +2 λ ( a + β − α ) a + β − α By Remark 5.8, one gets β ∗ U = 1 − λr u ( w ) a + β − α α + β )2+2( α β +2 λ ( a + β − α ) . (6.6)We ask the problem of comparing the two coinsurance rates β ∗ , β ∗ from(6.3) and (6.5). First we notice that if λ = 0, then by Proposition 5.1 (i), wehave β ∗ = β ∗ = 1. Proposition 6.1 If λ > then there is the following dependence relation be-tween β ∗ and β ∗ : − β ∗ − − β ∗ ≈ ( α + β ) λE f ( A ) r u ( w ) (6.7) Proof.
Formulas (6.3) and (6.5) can be written α + β + αβ + λ E f ( A ) ≈ λE f ( A ) r u ( w )(1 − β ∗ ) α + β + λ E f ( A ) ≈ λE f ( A ) r u ( w )(1 − β ∗ ) (By Proposition 5.1 (ii), 1 − β ∗ > − β ∗ > ( α + β ) ≈ λE f ( A ) r u ( w ) [ − β ∗ − − β ∗ ]which implies (6.7). Corollary 6.2 If λ > then β ∗ > β ∗ . Proof.
Since u ′ ( w ) > u ′′ ( w ) < r u ( w ) = − u ′′ ( w ) u ′ ( w ) >
0. We have λ > E f ( A ) >
0, therefore the right hand side of (6.7) is positive. Further, using(6.7) one obtains the inequality − β ∗ > − β ∗ , from where it follows β ∗ > β ∗ .Formulas (6.3), (6.5) and (6.6) may get different forms with respect to theutility function u . 13 xample 6.3 Assume that the utility function u is HARA -type ([21], Section3.6) u ( w ) = ζ ( η + wγ ) − γ , for η + wγ > (6.8)By [21], Section 3.6, r u ( w ) = ( η + wγ ) − , thus formulas (6.3), (6.5) and(6.6) will get the form: β ∗ ≈ − λ ( η + wγ ) a + β − α α β αβ + λ ( a + β − α ) (6.9) β ∗ ≈ − λ ( η + wγ ) a + β − α α β + λ ( a + β − α ) (6.10) β ∗ U ≈ − λ ( η + wγ ) a + β − α α + β )2+2( α β +2 λ ( a + β − α ) (6.11)If A is a symmetric triangular fuzzy number ( a, α ) , then, setting β = α in(6.9)-(6.11) we find the following forms of the three optimal coinsurance rates: β ∗ ≈ − λ ( η + wγ ) aα +6 λ a (6.12) β ∗ ≈ − λ ( η + wγ ) aα +18 λ a (6.13) β ∗ U ≈ − λ ( η + wγ ) a α +18 λ a (6.14) Example 6.4
Assume that the utility function u is CRRA-type: u ( w ) = ( w − γ − γ γ > ln ( w ) γ = 1 (6 . Then, by [12], p. 21, r u ( w ) = γw for γ > and r u ( w ) = w for γ = 1 . Then,by (6.3), (6.5) and (6.6) the following formulas for the optimal coinsurance rates β ∗ , β ∗ , β ∗ U follow: • for γ > : β ∗ ≈ − λwγ a + β − α α β αβ + λ ( a + β − α ) (6.16) β ∗ ≈ − λwγ a + β − α α β + λ ( a + β − α ) (6.17) β ∗ U ≈ − λwγ a + β − α α + β )2+2( α β +2 λ ( a + β − α ) (6.18) • for γ = 1 : β ∗ ≈ − λw a + β − α α β αβ + λ ( a + β − α ) (6.19) β ∗ ≈ − λw a + β − α α β + λ ( a + β − α ) (6.20) β ∗ U ≈ − λw a + β − α α + β )2+2( α β +2 λ ( a + β − α ) (6.21) Example 6.5
We consider the T -coinsurance problem with the following initialdata: • the weighting function is f ( t ) = 2 t , t ∈ [0 , ; • A is the triangular fuzzy number A = (6 , , ; • u is the utility function of CRRA-type u ( w ) = ln ( w ) , thus r u ( w ) = 1 ; • the initial wealth is w = 40 and the loading factor is λ = .Formulas (6.1)-(6.3) give the following indicators of the fuzzy number A : E f ( A ) = , V ar T ( A ) = , V ar T ( A ) = y (4.1), P = (1 + λ ) E f ( A ) = , thus w = w − P = .Applying formulas (6.3), (6.5) or (6.7), the two optimal coinsurance rateshave the approximate values: β ∗ ≈ − λw E f ( A ) V ar T ( A )+ λ E f ( A ) = − . β ∗ ≈ − λw E f ( A ) V ar T ( A )+ λ E f ( A ) = − . T -coinsurance problems in which the T -coinsurance rated have been strictly negative. In the Appendix we will find anecessary condition for the T -coinsurance rate β ∗ T to be strictly positive. We donot know a necessary and sufficient condition for β ∗ T >
0. For a particular caseof the utility function, the following property will give us a sufficient conditionfor β ∗ T > Proposition 6.6
Assume that the utility function u is defined by: u ( x ) = − e − x , for x ∈ R . If λ > E f ( A ) then β ∗ T > . Proof.
One notices immediately that E f ( A ) >
0, thus λ >
0. Also, r u ( x ) = − u ′′ ( x ) u ′ ( x ) = 1 for any x ∈ R , thus, according to (5.3), the optimal T -coinsurancerate β ∗ = β ∗ T can be approximated as: β ∗ ≈ − λE f ( A ) V ar T ( A )+ λ E f ( A ) (6.22)By hypothesis, λ > E f ( A ) , we will have E f ( A ) λ − λE f ( A )+ V ar T ( A ) = ( E f ( A ) λ − + λE f ( A ) − V ar T ( A ) > V ar T ( A ) + λ E f ( A ) > β ∗ > Example 6.7
We consider the following hypotheses: • the weighting function is f ( t ) = 2 t , t ∈ [0 , ; • the risk is represented by the triangular fuzzy number A = (2 , , ; • the utility function is u ( x ) = − e − x , for x ∈ R ; • the loading factor is λ > .Using the formulas (6.1)-(6.3) we obtain the following indicators: E f ( A ) = ; V ar T ( A ) = ; V ar T ( A ) = (6.23)Then, by applying the approximation (6.22) of β ∗ T in case of D -operators T , T we find: β ∗ T ≈ − λ λ ; β ∗ T ≈ − λ λ .In this case the condition of Proposition 6.6 is λ > . In particular for λ = 1 we obtain β ∗ T = , β ∗ T = . The basic idea of the paper is the study of the coinsurance problem by theexpected utility operators from [16], [17]. The main contributions of the paperare: 15 to build a coinsurance model in the framework offered by the possibilistic EU -theory associated with an expected utility operator; • the use of D -operators defined in [20] to study the properties of the optimalcoinsurance and its approximate calculation; • the application of the general results to the computation of the coinsurancerates in a few remarkable cases and their comparison.We report next a few open problems:(a) We assume that we have a data set representing values of a probabilisticrisk (random variables) which appears as a parameter in the context of a proba-bilistic model (for example, in the coinsurance problem). Based on the existingdata, one could determine those indicators by which we know the phenomenondescribed by the probabilistic model. Thereby, in case of the coinsurance prob-lem, from data one obtains the statistic mean value and variance, then we cancompute the optimal coinsurance (by an approximate calculation formula simi-lar to (5.3)). In [30], Vercher et al. present a method by which from a datasetone can build a trapezoidal fuzzy number. By applying Vercher et al.’s method,the probabilistic model of the coinsurance turns into a possibilistic model, inwhich risk is described by this trapezoidal fuzzy number. We compute then theexpected value and the variance associated with this trapezoidal fuzzy number,then by formula (5.3) on can obtain the optimal coinsurance associated with the T -possibilistic model. An open problem is to find those formulas describing theway the probabilistic model of coinsurance is turned into a possibilistic model(by Vercher et al’s method), which allows a comparison of the two models.(b) In papers [4], [14], [27] it is studied the effect of absolute risk aversion,prudence and temperance on the optimal solution for the standard portfoliochoice problem ([12], Section 4.1). A similar problem is investigated in [20]in the context of EU -theory associated with a D -operator. It would be in-terested to study refinements of Theorems 5.3 and 5.9 such that the optimal T -coinsurance rate and the standard expected utility to be expressed accord-ing to the indicators of risk aversion, prudence and temperance as well as the T -moments of the possibilistic risk represented by the fuzzy number A .(c) A third problem is the study of a coinsurance problem with two typesof risk: besides the investment risk a background risk might appear. Both theinvestment risk and the background risk can be probabilistic (random variables)or possibilistic (fuzzy numbers). Besides the purely probabilistic coinsurancemodel in which both risks are random variables we have: • the possibilistic model, in which risks are fuzzy; • two mixed models, in which a risk is a fuzzy number, and the other is arandom variable.To define such bidimensional coinsurance models it is necessary for the no-tions of multidimensional possibilistic expected utility ([17], Section 6.1) andthe mixed expected utility ([17], Section 7.1) to be generalized for some ”mul-tidimensional expected utility operators”.16 Appendix
In the following we will prove a necessary condition for the optimal T -coinsurance β ∗ = β ∗ T to be strictly positive. We will keep the notations from Sections 4 and5. Lemma 8.1 If T is an expected utility operator, A a fuzzy number and u, v twocontinuous utility functions then T ( A, [ u ( x ) − T ( A, u ( x ))][ v ( x ) − T ( A, v ( x ))]) = T ( A, u ( x ) v ( x )) − T ( A, u ( x )) T ( A, v ( x )) Proof.
One uses axioms (b) and (c) from Definition 3.1.
Proposition 8.2
Let T be a strictly increasing expected utility operator. As-sume that λ > . Then from β ∗ T > the following inequality follows: λ < T ( A, ( x − E f ( A ))[ u ′ ( w − x ) − T ( A,u ′ ( w − x ))]) E f ( A ) T ( A,u ′ ( w − x )) Proof.
We will denote β ∗ = β ∗ T . From (4.4) we have g ( x,
0) = w − x , thus, by(4.8): H ′ (0) = T ( A, u ′ ( w − x )( x − P )) (a)Since T is strictly increasing, H ( β ) is a strictly concave function, thus H ′ ( β )is a strictly decreasing function. Then the following implication holds: β ∗ > ⇒ H ′ ( β ∗ ) < H ′ (0) (b)Applying (a) and Lemma 8.1 it follows H ′ (0) = T ( A, [ u ′ ( w − x ) − T ( A, u ( w − x ))][ x − P − T ( A, x − P )]) + T ( A, u ′ ( w − x )) T ( A, x − P )Noticing that T ( A, x − P ) = − λE f ( A ), the previous inequality gets theform H ′ (0) = T ( A, ( x − E f ( A ))[ u ′ ( w − x ) − T ( A, u ′ ( w − x ))]) − λE f ( A ) T ( A, u ′ ( w − x )) Then the inequality H ′ (0) > T ( A, ( x − E f ( A ))[ u ′ ( w − x ) − T ( A, u ′ ( w − x ))]) > λE f ( A ) T ( A, u ′ ( w − x )).Since T is strictly increasing and u ′ ( w − x ) >
0, we have T ( A, u ′ ( w − x )) >
0. Also E f ( A ) >
0, thus the last inequality from above is equivalent with λ < T ( A, ( x − E f ( A ))[ u ′ ( w − x ) − T ( A,u ′ ( w − x ))] E f ( A ) T ( A,u ′ ( w − x )) (c)From (b) and (c) the implication which we had to prove follows. References [1] S. S. Appadoo, A. Thavaneswaran, Possibilistic moment generating func-tions of fuzzy numbers with Garch applications, Advances in Fuzzy Setsand Systems, Volume 6, Issue 1, 33 - 62, June 2010172] K. J. Arrow, Aspects of the theory of risk bearing, Helsinki: Yri¨oJahnssonin S¨a¨ati¨o, 1965[3] K.J. Arrow, Essays in the theory of risk bearing, North–Holland, Amster-dam, 1970[4] G. Athayde, R. Flores, Finding a Maximum Skewness Portfolio GeneralSolution to Three-Moments Portfolio Choice, Journal of Economic Dy-namics and Control, 28, 2004, 1335–1352.[5] C. Carlsson, R. Full´ e r, On possibilistic mean value and variance of fuzzynumbers, Fuzzy Sets Syst., 122, 2001, 315–326[6] C. Carlsson, R. Full´ e r, Possibility for decision, Springer, 2011[7] C. Carlsson, R. Full´ e r, P. Majlender, On possibilistic correlations, FuzzySets Syst., 155, 2005, 425–445[8] M. Collan, M. Fedrizzi, P. Luukka, Possibilistic risk aversion in groupdecisions: theory with applications in the insurance of giga-investmentsvalued through the fuzzy pay-off method, Soft Computing, 21(15), 2017,4375-4386[9] D. Dubois, H. Prade, Fuzzy sets and systems: theory and applications,Academic Press, New York, 1980[10] D. Dubois, H. Prade, Possibility theory, Plenum Press, New York, 1988[11] D. Dubois, H. Prade, The mean value of a fuzzy number, Fuzzy Sets Syst.,24, 1987, 279–300[12] L. Eeckhoudt, C. Gollier, H. Schlesinger, Economic and Financial Decisionunder Risk, Princeton University Press, 2005[13] R. Full´ ee