Moral-hazard-free insurance contract design under rank-dependent utility theory
aa r X i v : . [ q -f i n . M F ] M a r Quantile optimization under derivative constraint
Zuo Quan Xu ∗ March 8, 2018
Abstract
This paper studies a new type of quantile optimization problems arising frominsurance contract design models. This type of optimization problems is charac-terized by a constraint of infinity -dimension, that is, the derivatives of the decisionquantile functions are bounded. Such a constraint essentially comes from the “ in-centive compatibility ” constraint for any optimal insurance contract to avoid thepotential severe problem of moral hazard in insurance contract design models. Bya further development of the author’s relaxation method, this paper provides asystemic approach to solving this new type of quantile optimization problems. Theoptimal quantile is expressed via the solution of a free boundary problem for asecond-order nonlinear ordinary differential equation (ODE), which is similar tothe Black-Scholes ODE for perpetual American options and has been well studiedin literature theoretically and numerically.
Keywords:
Quantile optimization, probability weighting/distortion, relaxationmethod, insurance contract design, free boundary problem, calculus of variations
Probability weighting (also called distortion) function (see [23, 18]) plays a key rolein a lot of theories of choice under uncertainty such as Kahneman and Tversky’s [15, 24]cumulative prospect theory (CPT), Yaari’s [30] dual model, the Lopes’ SP/A model,Quiggin’s [19] (1982) rank-dependent utility theory (RDUT). These theories (often calledbehavioral finance theories) provide satisfactory explanations of many paradoxes that theclassical expected utility theory (EUT) fails to explain (see, e.g. [8, 1, 7, 16]).In recent years, a lot of attentions has been paid to the theoretical study of behavioralfinance investment (including portfolio choice and optimal stopping) models involvingprobability weighting functions, see, e.g., [14, 10, 13, 28, 25, 21]. A typical approach to ∗ The author acknowledges financial supports from NSFC (No.11471276), Hong Kong GRF(No.15204216 and No.15202817), and the Hong Kong Polytechnic University. S -shaped. Similar to tackling theaforementioned investment problems, one can deal with these contract design problemsby the same approach: one first turns them into quantile optimization problems, andthen solves the latter, and lastly recovers the optimal contracts.The main difficulty still lies in the second step, but there is a key difference betweenthe formulations of the quantile optimization problems for investment models (called firsttype) and that for insurance contract design models (called second type). This comes from2he fact that, when designing an insurance contract, one has to take both the insured andinsurer into account simultaneously so as to achieve Pareto optimality for them, whichmathematically results in that both the indemnity and retention functions are necessaryto be increasing. Both Huberman, Mayers, and Smith Jr [11] and Picard [17] call theincreasing condition of indemnity and retention the “incentive compatibility” constraintfor optimal insurance contract. Mathematically speaking, it leads to the new secondtype of quantile optimization problems, in which the derivatives of decision quantiles arebounded. Because this constraint is of infinity -dimension, it makes the second type ofproblems harder than the first type of ones. If one would simply ignore the constraint,then the quantile optimization problems would reduce to the first type, and the deducedoptimal contract may cause the potential severe problem of moral hazard (see [3, 29]).To the best of our knowledge, there is no a general systematic approach to solving thesecond type of quantile optimization problems. Although calculus of variations methodhas been applied in the insurance literature but without taking the constraint of boundedderivatives into account. For example, Spence and Zeckhauser [22] used this method tosolve an insurance contract design problem in the setting of EUT without considering theconstraint, but the optimal contract turns out to be the classical deductible one whichaccidentally satisfies the constraint of bounded derivatives. As earlier mentioned, if theproblem were considered within the behavioral finance theory framework, the optimalcontract may cause the potential severe problem of moral hazard issue. In fact, Xu etal. [29] is the only existing work we know that tackles this type of problems with theconstraint kept in mind and only partial results are obtained. It seems that calculus ofvariations method simply cannot provide a satisfactory solution for this type of problems.In this paper we further develop the author’s [27] relaxation method and provide asystemic approach to solving this new type of problems that are subject to the constraintof bounded derivatives. The most novel part of this paper is that we link the problem to afree boundary problem for a second-order nonlinear ordinary differential equation (ODE),which is similar to the Black-Scholes ODE for perpetual American options and has beenwell studied in literature theoretically and numerically. The optimal quantile is expressedin terms of the solution of the ODE. To the best of our knowledge, we have never seenanyone else has done this. This also allows us to give a similar ODE interpretation forthe optimal quantiles obtained in [25] and [27].The rest of this paper is organized as follows. In Section 2, we introduce the insurancebackground of the problem. In Section 3 we propose our new type of quantile optimizationproblems. Section 4 is devoted to making change-of-variable to simplify the formulation ofthe problem and study its feasibility issue. In Section 5, we further develop the relaxationmethod so as to solve the new type of problems completely. Some concluding remarksare given in Section 6, where we point out another possible way (that is using dynamicprogramming principle) and its limitations in tackling the new type of problems.3 otation Generally speaking, quantiles are always increasing and may not be continuous. Depend-ing on the definition, they may be left-or right-continuous.Let F be the probability distribution function of a random variable. In this paper,we define the quantile function (or simply called quantile) of the random variable (or theleft-continuous inverse function of F ) as F − ( p ) := inf { z ∈ R | F ( z ) > p } , ∀ p ∈ (0 , F − (0) := F − (0+) so that it is continuous at 0. By thisdefinition, a quantile is always increasing and left-continuous. We denote by Q the setof quantiles for random variables.On the other hand, if the incentive compatibility constraint for optimal insurancecontract is taken into account, it turns out that only absolutely continuous quantiles willbe interested in. In an insurance contract design problem, one seeks for the best way to share a potentialloss by an insured and an insurer so as to achieve Pareto optimality for them.Let I ( X ) and R ( X ) be the losses borne by the insurer and by the insured, respectively,when a potential loss X > I ( X ) + R ( X ) = X, I (0) = R (0) = 0 . (2.1)Furthermore, both of the insurer and the insured shall bear more if a bigger loss happens.If one of them would bear less, it may potentially cause the severe problem of moral hazardas pointed out earlier. Therefore, mathematically speaking, it is a must to require bothof I ( X ) and R ( X ) are increasing with respect to X , that is, I ( x ) > I ( y ) , R ( x ) > R ( y ) , ∀ x > y > . (2.2)This is the incentive compatibility constraint for optimal insurance contract.It is easily seen that we can express the joint constraints of (2.1) and (2.2) via thefollowing single one R (0) = 0 , R ( x ) − R ( y ) x − y, ∀ x > y > . (2.3) The quantile of the random variable is right-continuous if defined as F − ( p ) = inf { z ∈ R | F ( z ) > p } for p ∈ [0 , R is increasing, absolutely continuous, R (0) = 0 and 0 R ′ Y > V ( Y ) := Z ∞ u ( z ) w ′ (1 − F Y ( z )) d F Y ( z ) . (2.5)Here u is a second-order differentiable utility function mapping R + onto itself with u ′ > u ′′ < w is a probability weighting function in the set of probability weighting functions D = { w : [0 , [0 , | bijection, continuously differentiable } ;and F Y is the probability distribution function of Y . One can easily show that V ( Y ) = Z u (cid:16) F − Y ( p ) (cid:17) w ′ (1 − p ) d p, (2.6)We can now formulate the insurance contract design problem for the insured asmax I V ( β − X + I ( X )) (2.7)s . t . E [ I ( X )] π, where β represents the insured’s (constant) final wealth if the loss X does not occur; π denotes an upper bound for the value of the contract. We see that β − X + I ( X ) is thenet wealth that the insured will have after the loss X happened and the claim amountof I ( X ) was received from the insurer. In this model, we also assume that the insurer isrisk neutral (see, e.g. [2, 20, 9]), so the value of the contract is simply given by E [ I ( X )].The constraint E [ I ( X )] π is called the budget constraint in this paper.For simplicity of the presentation we put the following technical assumption on X .One may release this assumption by employing the ideas of [26, 29]. Assumption 2.1.
We have β > X almost surely (a.s.), the distribution F X of X isstrictly increasing up to and the quantile F − X of X is absolutely continuous on [0 , . The above assumption is satisfied if, for example, X is uniformly distribution on (0 , β/ X has a mass at 0, which is the most common case in insurance practice.More discussions on this assumption can be found in [29].5ow rewrite the problem (2.7) in terms of R asmax R V ( β − R ( X )) (2.8)s . t . E [ R ( X )] > E [ X ] − π. We notice that there exists a random variable U which is uniformly distributed on (0 , X = F − X ( U ) a.s.. Let g ( p ) = R ( F − X ( p )) for p ∈ [0 , g V ( β − g ( U )) (2.9)s . t . E [ g ( U )] > E [ X ] − π, Using (2.6) we have V ( β − g ( U )) = Z u (cid:16) F − β − g ( U ) ( p ) (cid:17) w ′ (1 − p ) d p = Z u ( β − g (1 − p )) w ′ (1 − p ) d p = Z u ( G ( p )) w ′ (1 − p ) d p, where G ( p ) := β − g (1 − p ) for p ∈ [0 , E [ g ( U )] = Z g ( p ) d p = Z g (1 − p ) d p = β − Z G ( p ) d p. Hence the problem (2.9) can be expressed asmax G Z u ( G ( p )) w ′ (1 − p ) d p (2.10)s . t . Z G ( p ) d p ̟, with ̟ = β + π − E [ X ] being a given constant.In the above argument, we have not considered the constraint (2.4) yet. Notice G ( p ) = β − g (1 − p ) = β − R ( F − X (1 − p )) , so the constraint (2.4) in terms of G can be stated as G is absolutely continuous, G (1) = β and 0 G ′ h a.e. on [0 , h = (cid:16) F − X (cid:17) ′ (1 − p ). We denote by G the subset of quantiles that satisfies thisconstraint. For more details we refer to [29].
6e remark that the above argument is invertible, so solving the insurance contractdesign problem (2.7) reduces to solving the quantile optimization problem (2.10) subjectto the constraint (2.11).
Xia and Zhou [25] and the author [27] respectively studied the same type of quantileoptimization problems as follows.max G Z u ( G ( p )) w ′ (1 − p ) d p (3.1)s . t . Z G ( p ) φ ( p ) d p ̟, where φ is a given nonnegative and integrable function. In this problem, other than thebudget constraint, only the minimum monotonicity requirement on the decision quantiles G has been put.The problem (2.10) is very similar to the above problem (3.1), but there are twonotably differences. Firstly, there is no φ involved in the insurance problem. If theinsurer in the insurance problem (2.7) was not risk-neutral, then there might have φ involved in the problem (2.10). Secondly, the insurance problem requires the constraint(2.11) which is much stronger than the simple monotonicity requirement.In this paper, we investigate following new type of quantile optimization problemsmax G ∈G Z u ( G ( p )) w ′ (1 − p ) d p (3.2)s . t . Z G ( p ) φ ( p ) d p ̟. The problem (3.1) can be regarded as its special case where h → + ∞ ; while the problem(2.10) can also be regarded as its special case where φ ≡ We first simplify the problem (3.2). The following change-of-variable argument is similarto [27]. We put it here for the completeness of the paper.We first make a change of variable to remove w from the objective function. Let ν : [0 , [0 ,
1] be the inverse map of p − w (1 − p ), given by ν ( p ) := 1 − w − (1 − p ) , ∀ p ∈ [0 , . ν ∈ D is also a probability weighting function. It follows that Z u ( G ( p )) w ′ (1 − p ) d p = Z u ( G ( p )) d (1 − w (1 − p ))= Z u ( G ( p )) d ( ν − ( p )) = Z u ( G ( ν ( p ))) d p = Z u ( Q ( p )) d p, where Q ( p ) := G ( ν ( p )) , ∀ p ∈ [0 , . Note that G is a quantile if and only if so is Q . Moreover, G ′ h a.e. on [0 ,
1] if andonly if Q ′ ( p ) = G ′ ( ν ( p )) ν ′ ( p ) h ( ν ( p )) ν ′ ( p ) a.e. on [0 , G ∈ G if and onlyif Q belongs to Q := { Q | Q is absolutely continuous, Q (1) = β and 0 Q ′ ~ a.e. on [0 , } , where ~ ( p ) := h ( ν ( p )) ν ′ ( p ) > , ∀ p ∈ [0 , . Notice Z G ( p ) φ ( p ) d p = Z G ( ν ( p )) φ ( ν ( p )) ν ′ ( p ) d p = Z Q ( p ) ϕ ′ ( p ) d p, where ϕ ( p ) := Z p φ ( ν ( t )) ν ′ ( t ) d t = Z p φ ( ν ( t )) d ν ( t )= Z ν − ( p ) ν − (0) φ ( t ) d t = Z − w (1 − p )0 φ ( t ) d t, ∀ p ∈ [0 , , is increasing as φ and w ′ are both nonnegative. Moreover, ϕ is uniformly bounded0 = ϕ (0) ϕ ϕ (1) = Z φ ( p ) d p. By making the above change-of-variable, solving the problem (3.2) has now reducedto solving the problem max Q ∈Q Z u ( Q ( p )) d p (4.1)s . t . Z Q ( p ) ϕ ′ ( p ) d p ̟, in which the probability weighting function does not appear in the objective.Before solving the problem (4.1), we would like to study its feasibility issue, thatis, whether it has a feasible solution. By a feasible solution, we mean a quantile that For general discussions of feasibility and other issues, we refer to [12] for EUT framework, and [27]for RDUT framework. Q ∈ Q , Q ( p ) = Q (1) − Z p Q ′ ( t ) d t > β − Z p ~ ( t ) d t, so Z Q ( p ) ϕ ′ ( p ) d p > Z (cid:18) β − Z p ~ ( t ) d t (cid:19) ϕ ′ ( p ) d p = β Z φ ( p ) d p − Z ϕ ( p ) ~ ( p ) d p, where we used the fact that ϕ is increasing and integration by parts. Therefore, theproblem (4.1) has no solution , if ̟ < β R φ ( p ) d p − R ϕ ( p ) ~ ( p ) d p ;a unique feasible (thus optimal) solution , if ̟ = β R φ ( p ) d p − R ϕ ( p ) ~ ( p ) d p ;infinity many feasible solutions , if ̟ > β R φ ( p ) d p − R ϕ ( p ) ~ ( p ) d p .The first two cases are trivial, so from now on we focus on the last one ̟ > β Z φ ( p ) d p − Z ϕ ( p ) ~ ( p ) d p. (4.2) The most novel part of this paper is this section. We will introduce an ODE throughwhich we can express the optimal quantile for the problem (4.1), provided (4.2) holds.Recall that we have assumed (4.2) so that the problem (4.1) has infinity many fea-sible solutions. Under this condition, because u is strictly concave, the problem (4.1) isequivalent to max Q ∈Q Z u ( Q ( p )) − λQ ( p ) ϕ ′ ( p ) d p (5.1)for some Lagrange multiplier λ >
0. Recall that Q = { Q | Q is absolutely continuous, Q (1) = β and 0 Q ′ ~ a.e. on [0 , } . We now modify the relaxation method [27] so as to incorporate the constraint ofbounded derivatives.For any Q ∈ Q , an application of integration by parts leads to Z u ( Q ( p )) − λQ ( p ) ϕ ′ ( p ) d p = Z u ( Q ( p )) + λQ ′ ( p ) ϕ ( p ) d p − βλϕ (1)= Z u ( Q ( p )) + λ ( Q ′ ( p ) − ~ ( p )) ϕ ( p ) d p + Z λ ~ ( p ) ϕ ( p ) d p − βλϕ (1) . (5.2)9et δ be an absolutely continuous function (to be determined) such that δ (0) = 0 and δ λϕ on [0 , Q ′ ~ , Z u ( Q ( p )) + ( Q ′ ( p ) − ~ ( p )) δ ( p ) d p + Z λ ~ ( p ) ϕ ( p ) d p − βλϕ (1) (5.3)= Z u ( Q ( p )) + Q ′ ( p ) δ ( p ) d p + Z ~ ( p )( λϕ ( p ) − δ ( p )) d p − βλϕ (1) , which is, by applying integration by parts again,= Z u ( Q ( p )) − Q ( p ) δ ′ ( p ) d p + Z ~ ( p )( λϕ ( p ) − δ ( p )) d p + β ( δ (1) − λϕ (1)) Z u ( Q ( p )) − Q ( p ) δ ′ ( p ) d p + Z ~ ( p )( λϕ ( p ) − δ ( p )) d p + β ( δ (1) − λϕ (1)) , (5.4)where Q ( p ) = ( u ′ ) − ( δ ′ ( p )) , ∀ p ∈ [0 , , maximizes the integrand point wisely.We hope Q is the optimal solution of the problem (5.1). It shall be a feasible solution,which requires 0 Q ′ ~ , that is,0 δ ′′ u ′′ (( u ′ ) − ( δ ′ )) ~ , which can also be expressed as δ ′′ δ ′′ − ~ u ′′ (( u ′ ) − ( δ ′ )) > Q (1) = ( u ′ ) − ( δ ′ (1)) = β , that is δ ′ (1) = u ′ ( β ) . Summarizing the results obtained thus far, we conclude that
Theorem 5.1. If δ ∈ C [0 , is concave, and satisfies the free boundary problem min n δ ′′ ( p ) − ~ ( p ) u ′′ (( u ′ ) − ( δ ′ ( p ))) , λϕ ( p ) − δ ( p ) o = 0 , a.e. p ∈ [0 , , (5.5) with boundary values δ (0) = 0 , δ ′ (1) = u ′ ( β ) . Then Q ( p ) := ( u ′ ) − ( δ ′ ( p )) , ∀ p ∈ [0 , , is the optimal solution of the problem (5.1) . roof. Because u ′′ < δ ′′
0, we see that Q ′ >
0. Moreover, we can rewrite (5.5) asmin ( − δ ′′ ( p ) u ′′ (( u ′ ) − ( δ ′ ( p ))) + ~ ( p ) , λϕ ( p ) − δ ( p ) ) = 0 , a.e. p ∈ [0 , . that is min n − Q ′ ( p ) + ~ ( p ) , λϕ ( p ) − δ ( p ) o = 0 , a.e. p ∈ [0 , . (5.6)This implies Q ′ ~ a.e., together with Q (1) = ( u ′ ) − ( δ ′ (1)) = β , we have proved that Q ∈ Q , thus a feasible solution of the problem (5.1).We see from (5.6) that( Q ′ ( p ) − ~ ( p ))( λϕ ( p ) − δ ( p )) = 0 , a.e. p ∈ [0 , , giving Z λ ( Q ′ ( p ) − ~ ( p )) ϕ ( p ) d p = Z ( Q ′ ( p ) − ~ ( p )) δ ( p ) d p. From this, we conclude that the inequalities (5.3) and (5.4) are equations when Q isreplaced by Q . In another words, the upper bound (5.4) is reached at Q , that is, Z u ( Q ( p )) − λQ ( p ) ϕ ′ ( p ) d p Z u ( Q ( p )) − λQ ( p ) ϕ ′ ( p ) d p, proving the claim.If δ ′′ > δ ′′ > > ~ u ′′ (( u ′ ) − ( δ ′ )) and hence λϕ = δ by (5.5),consequently ϕ ′′ > Corollary 5.2. If ϕ is concave on [0 , , then the solution of the free boundary problem (5.5) is also concave. Since the insurer is risk-neutral in the insurance contract design problem (2.10), we have ϕ ≡ Remark 5.1.
In order to let the problem (5.6) have a classical solution, we require somegrowth condition on u ′′ (( u ′ ) − ( · )) , but this can be easily satisfied with mild conditions, atleast for widely used power, logarithm and exponential utilities. Remark 5.2.
To solve the original quantile optimization problem (3.2) , it is left to de-termine the Lagrange multiplier λ . This can be done by numerical calculation by notingthe fact that λ is monotonic with respect to ̟ . Remark 5.3.
In [25] and [27], the optimal solution is expressed via δ , the concave en-velope of some known function ϕ . In fact δ can be interpreted as the solution of the ollowing free boundary problem min {− δ ′′ ( p ) , δ ( p ) − ϕ ( p ) } = 0 , a.e. p ∈ [0 , , with some proper boundary conditions. It is unknown to us how to modify Xia and Zhou’s [25] calculus of variations methodto solve the present problem. But one can interpret the problem (5.1) as a deterministcontrol problem, where c t = Q ′ ( t ) is regarded as the control variable in the constraint set C = { c : 0 c t ~ ( t ) , a.e. t ∈ [0 , } , and Q as the state process. Then the dynamic programming principle leads to thefollowing Hamilton-Jacobi-Bellman equation v t + sup c ∈C t H ( t, x, c, v x ) = 0 ,v | t =1 = 0 . Here C t = [0 , ~ ( t )] and H ( t, x, c, p ) = pc + u ( x ) − λxϕ ′ ( t ) . There seems, however, no easy way to determine the value function and solve the problemfrom this equation. In fact, wether it has a classical solution is still an open problem.In this paper, we demonstrate the systemic approach within the RDUT framework.But similar to [25, 27], the method also works for problems within other frameworks. Asan example, one can first apply our method to consider the loss and gain parts separatelyfor CPT model, and then combine the results. We encourages reads to give the details.12 eferences [1]
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