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Moral-hazard-free insurance contract design

under rank-dependent utility theory

Zuo Quan Xu∗

August 4, 2020

Abstract

Bernard et al. (2015) studied an insurance contract design problem under rank-

dependent utility (RDU) theory. Their results, however, suffer from a moral hazard

problem, namely, providing incentives for the insured to falsely report the actual

loss. Xu et al. (2019) investigated the same problem, but took the incentive com-

patibility constraint into consideration to avoid that moral hazard. Mathematically

speaking, the model reduces to a quantile optimisation problem with a compatibil-

ity constraint. They solved the problem by imposing assumptions on the loss and

the probability weighting function. This paper solves the problem completely by

a new quantile optimisation approach under general setting. The optimal solution

is expressed by the solution of an obstacle problem for a semilinear second-order

elliptic operator with mixed boundary conditions. Surprisingly, it is shown that

every reasonable moral-hazard-free contract is optimal for infinitely many RDU

maximisers with different utility functions and probability weighting functions.

Keywords: Rank-dependent utility theory, quantile optimisation, probability

weighting/distortion, relaxation method, insurance contract, obstacle problem, cal-

culus of variations

1 Introduction

Probability weighting (also called distortion) function (see [26, 20]) plays a key role in

many modern behavioural theories of choice under uncertainty, such as Kahneman and

Tversky’s [16, 27] cumulative prospect theory, Yaari’s [33] dual model, Lopes’ [17] SP/A

∗Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong.Email: maxu@polyu.edu.hk. The author acknowledges financial support from NSFC (No.11971409),Hong Kong GRF (No.15204216 and No.15202817), and the Hong Kong Polytechnic University.

1

model and Quiggin’s [21] rank-dependent utility (RDU) theory. These behavioural fi-

nance theories provide satisfactory explanations of the many paradoxes for which classical

expected utility (EU) theory fails to account (see, e.g. [8, 1, 7, 18]).

In recent years, much attention has been paid to the theoretical study of behavioural

finance investment models (including portfolio choice and optimal stopping models) in-

volving probability weighting functions; see, e.g., [15, 11, 14, 31, 29, 28, 24]. A typical

approach to such problems is described as follows. Rather than looking for the opti-

mal strategy directly by analysing the original problem, one reduces the analysis to a

corresponding quantile optimisation problem, in which the decision variable becomes

a quantile function (or simply called a quantile). With this change, solving a dynamic

stochastic control problem reduces to solving a static deterministic optimisation problem.

In the second step, the deduced quantile optimisation problem is solved using optimisa-

tion techniques, such as convex analysis and calculus-of-variations method. The last step

is to recover the optimal strategy for the original problem by appealing to some proper

hedging theories, such as the backward stochastic differential equation theory (for port-

folio choice problems, see [15]) or the Skorokhod embedding theory (for optimal stopping

problems, see [31]). The main difficulty of this approach is solving the quantile opti-

misation problem in the second step. An obvious hurdle is the simple fact that every

quantile function (which is the inverse of probability distribution function) is increasing.

As such, any quantile optimisation problem must take this monotonicity (as a minimum)

constraint into consideration.

Lacking a systematic approach, researchers generally tackle quantile optimisation

problems in isolation and under fairly strong assumptions (see, e.g., [15, 11]). Xia and

Zhou [28] provided an initial systematic approach by calculus-of-variations method. They

demonstrated the utility of their approach by solving a portfolio choice problem under

RDU theory. Shortly after, the author [30] offered an alternative method, the change-of-

variable and relaxation method, to solve the same type of problem. For these problems,

the constraints on the decision variables (namely quantiles) are almost minimum: beyond

the monotonicity constraint (which is necessary for quantile optimisation problems, as

previously noted), the only other constraint arising from the models is the so-called bud-

get constraint, which, mathematically speaking, is a linear constraint that can be easily

dealt by Lagrangian method.

Probability weighting function is also widely used in the risk-sharing literature. In

the context of insurance, the primary risk-sharing problem is to design an insurance

contract between the insurer (an insurance company) and the insured that achieves Pareto

optimality for both parties. Although there has also been work to design an insurance

contract within the RDU theory framework ([5, 6, 4, 3, 32, 9]), these studies assume that

the probability weighting function has a special shape, such as convex, concave or inverse-

S-shaped. As with the aforementioned investment problems, contract design problems

2

can also be solved by translating them into quantile optimisation problems, determining

the optimal solution, and translating back to discern the optimal contracts.

As with investment problems, difficulty is typically encountered in the second step.

However, there is a key difference in how optimisation problems are formulated for in-

vestment models (called “first-type”) and for insurance contract design models (called

“second-type”). When designing an insurance contract, to achieve Pareto optimality,

one has to take both the insurer and the insured into account simultaneously; as a con-

sequence, both the indemnity and retention functions are necessarily increasing for the

optimal contracts. Both Huberman, Mayers and Smith Jr [12] and Picard [19] called the

increasing condition of indemnity and retention the incentive compatibility constraint for

optimal insurance contracts. Mathematically speaking, this leads to a new type of quan-

tile optimisation problems, in which the derivatives of decision quantiles are bounded.

Bernard et al. [3] studied an insurance contract design model within the RDU theory

framework, but ignored the incentive compatibility constraint. Their results suffer from a

severe moral hazard problem: providing incentives for the insured to falsely report actual

losses. Xu et al. [32] examined the same model but took the incentive compatibility

constraint into consideration to avoid this moral hazard; and they partially solved the

problem by imposing assumptions on the loss and the probability weighting function.

Ghossoub [9] revisited the same problem by imposing a state-verification cost that the

insurer can incur in order to verify the loss severity, hence automatically ruling out any

ex post moral hazard that could otherwise arise from possible misreporting of the loss by

the insured.

To the best of our knowledge, no systematic approach has been developed to solve the

second-type quantile optimisation problem. Although the calculus-of-variations method

has been applied in the insurance literature, these applications have failed to take the con-

straint of compatibility into account. For example, Spence and Zeckhauser [25] used this

method to solve an insurance contract design problem within the EU theory framework

without considering the constraint. However, because the optimal contract turns out

to be the classical deductible, it coincidentally satisfies the compatibility constraint. As

mentioned above, when the problem is considered within the behavioural finance (such as

RDU) theory framework, the optimal contract can lead to moral hazard issues. Xu et al.

[32] is only study we could identify that tackled this type of problem with the constraint

in mind, and they only obtained partial results because of technical difficulties. It seems

that the calculus-of-variations method simply cannot provide a satisfactory solution for

this type of problem.

In this paper we introduce a new relaxation method, adapted from the relaxation

method introduced in [30], and provide a systematic approach to solving this second-

type problem, subject to the compatibility constraint. Our most significant contribution

is to link the solution of the second-type problem to an obstacle problem for a semilinear

3

second-order elliptic operator with mixed boundary conditions. The obstacle problem

is similar to the Black-Scholes variational inequality for American options, the theory

of which has been extensively examined in the literature. Numerous numerical schemes

for solving such problems are available when closed-form solutions are not available.

Our approach also allows us to express the optimal quantile obtained in [28] and [30]

by an obstacle problem for a linear second-order elliptic operator. We also give an

equivalent condition under which the optimal contract is deductible and show that every

moral-hazard-free contract (that satisfies very mild technical conditions) is optimal for

infinitely many RDU maximisers with different utility functions and probability weighting

functions.

The rest of this paper is organised as follows. In Section 2, we introduce the back-

ground insurance problem and reformulate the related non-concave optimisation as a

concave problem using the change-of-variable method. In Section 3, we propose our

quantile optimisation problem with the compatibility constraint and study its feasibil-

ity. Section 4 is devoted to solving the new problem using a three-step procedure. We

give our main findings on the optimal contract in Section 5, where we derive the closed-

form solution for the exponential utility case, reveal the main result of [32], give an

equivalent condition under which the optimal contract is deductible and show that every

moral-hazard-free contract (that satisfies very mild technical conditions) is optimal. In

Section 6, we give some concluding remarks, offer an alternative approach using dynamic

programming principle and discuss its inability to tackle the new problem.

Notation

Throughout the paper, we fix an atomless probability space. For any random variable

Y > 0, we denote its probability distribution function by FY ; and define its quantile

function (or the left-continuous inverse function of FY ) by

F −1Y (p) := inf

{z > 0

∣∣∣ FY (z) > p}, p ∈ [0, 1],

with the convention that inf ∅ = +∞. By this definition, any quantile function is non-

negative, increasing, left-continuous and F −1Y (0) = 0. Note F −1

Y (1) < ∞ for non-negative

bounded random variables Y , so the collection of quantiles of non-negative bounded ran-

dom variables, denoted by O+, is

O+ :={

G : [0, 1] → [0, ∞)∣∣∣ G is increasing, left-continuous and G(0) = 0

}.

Quantiles are not continuous in general, however, when taking the incentive compati-

bility constraint for the optimal insurance contract into account, we only face absolutely

continuous quantiles with locally bounded derivatives.

In our argument, “almost surely” (a.s.) will be suppressed for notation simplicity in

4

many circumstances, when no confusion occurs.

2 Insurance background

In an insurance contract design problem, one seeks the best way for the insurer (an

insurance company) and the insured to share a potential loss to achieve Pareto optimality

for both parties, which is evidently a multi-objective optimisation problem. In this paper,

we fix the insurance premium for the contract, and aim at finding out the best contract for

the insured. Therefore, the problem reduces to a single objective optimisation problem.

2.1 Non-concave optimisation problem

We use random variable X > 0 to denote potential loss that is covered by the insurance

contract. Let I(x) and R(x) be the losses borne by the insurer and by the insured,

respectively, when a loss x occurs. These are called compensation (or indemnity) and

retention functions, respectively. In the insurance literature, a contract is called full

coverage if I(x) ≡ x; called deductible (with deductible d) if I(x) ≡ max{x − d, 0}.

Economically speaking, one always has

I(0) = R(0) = 0, I(x) + R(x) = x, x > 0. (2.1)

Furthermore, both the insurer and the insured should bear a greater financial responsi-

bility when bigger losses occur. For, if one party borne less responsibility, it could result

in the moral hazard issue. Therefore, mathematically speaking, one also requires that

I(x) > I(y), R(x) > R(y), x > y > 0. (2.2)

This is called the compatibility constraint for insurance contract. In this paper, we call

a contract moral-hazard-free if it satisfies the above constraints. For instance, every

deductible and proportional contract (that is I(x) ≡ cx with some 0 < c < 1) is moral-

hazard-free.

Cleary we can combine the constraints (2.1) and (2.2) into the following one:

I(0) = 0, 0 6 I(x) − I(y) 6 x − y, x > y > 0,

which can be also stated as I ∈ M where

M ={

I : [0, ∞) → [0, ∞)∣∣∣ I is absolutely continuous

with I(0) = 0 and 0 6 I ′ 6 1 almost everywhere (a.e.).}

.

Note that I ∈ M if and only if R ∈ M . Because the derivative of the compensation

5

function I(·) (as well as the retention function R(·)) is bounded, it avoids the potential

severe problem of moral hard. Meanwhile, it also makes the related (moral-hazard-free)

insurance contract design problems extremely hard to solve.

Variations in risk preferences can lead to different insurance contract design models.

In this paper, we follow [32] to consider the problem within the RDU theory framework.

In this framework the insured is a RDU maximiser, whose risk preference for a (random)

final wealth Y > 0 is given by

V (Y ) :=∫

[0,∞)u(z)w′(1 − FY (z)) dFY (z), (2.3)

where u : [0, ∞) → [0, ∞) is her utility function which is second-order continuously

differentiable with u(0) = 0, u′ > 0 and u′′ < 0, and w is her probability weighting

function in the set

W ={

w : [0, 1] → [0, 1]∣∣∣ w is absolutely

continuous with w(0) = 0, w(1) = 1 and w′ > 0 a.e.}

.

Clearly w is strictly increasing. One can easily show, by change-of-variable, that

V (Y ) =∫ 1

0u(F −1

Y (p))

w′(1 − p) dp, (2.4)

where F −1Y (·) denotes the quantile function of Y .

The moral-hazard-free insurance contract design problem for the insured within the

RDU theory framework is formulated as

maxI∈M

V(β − X + I(X)

)

s.t. P

(I(X)

)6 ,

where β > 0 is a constant representing the insured’s final wealth when no loss occurs,

P(I(X)) denotes the premium (also called calculation principle) of the contract I, and

> 0 represents the maximum amount that the insured wants to pay for the contract.

Note β − X + I(X) is the insured’s net wealth after receiving the payment from the

insurer. The constraint P

(I(X)

)6 is called the participation constraint.

In this paper, we assume the insurer uses the pure risk premium with safety loading

(see, e.g. [2, 22, 10]), that is,P(ξ) = (1 + θ)E[ξ] ,

where θ > 0, called the relative safety loading, is a constant. In this case, the problem

6

becomes

maxI∈M

V(β − X + I(X)

)(2.5)

s.t. E[I(X)] 6 π,

whereπ :=

1 + θ> 0.

Remark 2.1. There are many other widely used calculation principles in practice; for

instance,

• The variance principle: P(ξ) = E[ξ] + α Var(ξ) .

• The standard deviation principle: P(ξ) = E[ξ] + α√

Var(ξ).

We hope to address them in our future works.

In insurance practice, very big losses usually are covered by a reinsurance company,

so the insurer cares about the losses up to certain level only. Because of this, following

[32], we put the following standard technical assumptions on X. Let M := F −1X (1) here

and hereafter.

Assumption 2.1. We have 0 < M 6 β. Furthermore, the distribution function FX is

strictly increasing, and the quantile function F −1X is absolutely continuous on [0, 1].

By this assumption X is essentially bounded so that our subsequent arguments only deal

with bounded random variables. Assumption 2.1 also allows X having a mass at 0, which

is the most common case in insurance practice. More discussions on this assumption can

be found in [32].

Remark 2.2. If the assumption β > M failed, then the net wealth of the insured after

claim, β − X + I(X), may be negative. Consequently, the risk preference (2.3) would be

no more suitable to measure it, so a general RDU risk preference, which covers negative

positions, should be used. The related problem becomes complicated. Similarly, one may

also consider random final wealth β, in which case the distribution of (β, X) makes the

related contract design problem extremely challenging. These problems, however, are out

of the scope of this paper.

Remark 2.3. Xu, Zhou and Zhuang [32] solved Problem (2.5) under the following addi-

tional assumptions. First, the probability weighting function w is inverse-S-shaped, that

is, it is strictly concave on [0, a] and strictly convex on [a, 1] for some 0 < a < 1. Second,

the function u′′

u′is increasing. Third, w′′

w′(p) < u′′

u′

(β +− (1+θ)E[X]−F −1

X (p))(

F −1X

)′

(p)

for p ∈ [0, a]. Clearly these assumptions restrict the applications of their results. The

present paper will solve Problem (2.5) without these assumptions.

7

2.2 Equivalent concave optimisation problem

The probability weighting function w within the definition (2.3) of the risk preference V

makes the preference a nonlinear expectation (in fact it is a Choquet expectation), so

that the optimisation problem (2.5) is not concave. We first do a change-of-variable to

find an equivalent concave optimisation problem. This clearly will reduce the difficulty

of solving it.

Let us forget the compatibility constraint I ∈ M temporally. Because the probability

space is atomless, there exists a random variable U , which is uniformly distributed on

(0, 1), such that X = F −1X (U) almost surely. Let

g(p) = R(F −1X (p)), p ∈ [0, 1].

Theng(U) = R(F −1

X (U)) = R(X) = X − I(X).

By definition we have F −1β−g(U)(p) = β −g(1 −p) for a.e. p ∈ [0, 1]. Taking these into (2.4)

yields

V(β − X + I(X)

)= V

(β − g(U)

)=∫ 1

0u(F −1

β−g(U)(p))

w′(1 − p) dp

=∫ 1

0u (β − g(1 − p)) w′(1 − p) dp =

∫ 1

0u (G(p)) w′(1 − p) dp,

whereG(p) := β − g(1 − p), p ∈ [0, 1].

Furthermore,

E[I(X)] = E[X − g(U)] = E[X] −∫ 1

0g(p) dp = E[X] − β +

∫ 1

0G(p) dp.

Hence, Problem (2.5) without the compatibility constraint I ∈ M can be expressed as

maxG(·)

∫ 1

0u (G(p)) w′(1 − p) dp

s.t.∫ 1

0G(p) dp 6 ,

with := β + π − E[X] > 0.

In this problem, the objective functional is concave and the constraint is linear with

respect to G(·), so it is a concave optimisation problem.

We now rewrite the compatibility constraint I ∈ M in terms of the new decision

variable G(·). Notice

G(p) = β − g(1 − p) = β − R(F −1X (1 − p)), p ∈ [0, 1], (2.6)

8

so the constraint I ∈ M , which is equivalent to R ∈ M , can be stated as G ∈ G 1, where

G ={

G : [0, 1] → (−∞, β]∣∣∣ G is absolutely

continuous with G(1) = β and 0 6 G′ 6 h a.e.}

,

and

h(p) :=(F −1

X

)′

(1 − p) > 0, a.e. p ∈ [0, 1]. (2.7)

Then Problem (2.5) under compatibility constraint becomes

maxG∈G

∫ 1

0u (G(p)) w′(1 − p) dp (2.8)

s.t.∫ 1

0G(p) dp 6 ,

where = β +

1 + θ− E[X] > 0.

Note that the aforementioned argument is invertible. In fact, if G ∈ G solves Problem

(2.8), then by I(x) = x − R(x) and (2.6),

I(x) = x − β + G(1 − FX(x)), (2.9)

solves Problem (2.5). As such, we have translated the non-concave optimisation problem

(2.5) with compatibility constraint into an equivalent concave optimisation problem (2.8).

By Assumption 2.1, we have

∫ 1

ph(t) dt =

∫ 1

p

(F −1

X

)′

(1 − t) dt = F −1X (1 − p) 6 M, p ∈ [0, 1]. (2.10)

Remark 2.4. Note that I(·) is a deductible contract with deductible d if and only if

R(x) ≡ min{x, d}. The latter, by (2.6), is equivalent to

G(p) = β − min{F −1

X (1 − p), d}

=

β − F −1X (1 − p0), p 6 p0;

β − F −1X (1 − p), p > p0,

where p0 = sup{p ∈ [0, 1] | F −1X (1 − p) > d}. For simplicity, we also call quantiles G(·)

of the above form deductible contracts. In particular, a quantile is of full coverage if and

only if d = 0, or equivalently G ≡ β.

1For more details we refer to [32].

9

3 New type of quantile optimisation problems

Both Xia and Zhou [28] and the author [30] studied the same type of quantile optimisation

problem (in the context of financial investment models under RDU theory), as follows.

maxG∈O+

∫ 1

0u(G(p))w′(1 − p) dp (3.1)

s.t.∫ 1

0G(p)φ(p) dp 6 , (3.2)

where φ > 0 is a given integrable function. In this problem, other than the constraint

(3.2) (called budget constraint in the financial economic literature), only the minimum

monotonicity requirement is placed on the decision quantiles G(·).

Although the insurance problem (2.8) is very similar to the investment problem (3.1),

there are two notable differences. First, φ ≡ 1 in the former.2 Second, the former re-

quires the compatibility constraint, which is much stronger than the simple monotonicity

requirement in the latter.

In this paper, we introduce the following new type of quantile optimisation problem:

maxG∈G

∫ 1

0u(G(p))w′(1 − p) dp (3.3)

s.t.∫ 1

0G(p)φ(p) dp 6 .

Clearly, Problem (3.1) can be regarded as its special case: h ≡ +∞, and Problem (2.8)

as its special case: φ ≡ 1. Because u is strictly concave, the solution of Problem (3.3), if

exists, must be unique.

In the subsequent analysis, we first introduce a new relaxation method based on

the relaxation method introduced by the author in [30]. Then link Problem (3.3) to a

well studied obstacle problem in differential equation literature. Finally, we express the

optimal quantile via the solution of the obstacle problem. To the best of our knowledge,

this approach has never appeared in the finance and insurance literature.

3.1 Feasibility issue

Before tackling Problem (3.3), we first study its feasibility issue, namely, whether it has

at least one feasible solution.3 By a feasible solution, we mean a quantile that satisfies

all constraints.

2If the calculation principle in (2.5) was not the pure risk premium, then the constraint of (2.8) mightbe replaced by (3.2) with some φ 6≡ 1.

3For general discussions of feasibility issue, we refer to [13] for the EU theory framework, and [30] forthe RDU theory framework.

10

By (2.10), we have, for any G ∈ G ,

G(p) = G(1) −∫ 1

pG′(t) dt > β −

∫ 1

ph(t) dt = β − F −1

X (1 − p), p ∈ [0, 1].

Since φ > 0, it follows

∫ 1

0G(p)φ(p) dp >

∫ 1

0

(β − F −1

X (1 − p))

φ(p) dp.

From this, we immediately see that Problem (3.3) has

no feasible solution, if <∫ 1

0

(β − F −1

X (1 − p))

φ(p) dp;

a unique feasible (thus optimal) solution, if =∫ 1

0

(β − F −1

X (1 − p))

φ(p) dp;

infinity many feasible solutions, if >∫ 1

0

(β − F −1

X (1 − p))

φ(p) dp.

Furthermore, if and only if

> β∫ 1

0φ(p) dp,

the full contract G∗ ≡ β is a feasible solution of Problem (3.3) and thus an optimal

solution by the monotonicity of its objective functional. So it is only left to study the

case:

∫ 1

0

(β − F −1

X (1 − p))

φ(p) dp < < β∫ 1

0φ(p) dp, (3.4)

which is assumed from now on. In this case, there are infinite many feasible solutions

(but the full contract is excluded).

Remark 3.1. For the insurance contract design problem (2.8), the condition (3.4) reads

0 < < (1 + θ)E[X] .

It means the insured can not pay an amount enough to buy the full coverage contract.

This, of course, is the most interesting and important case in practice.

4 Optimal solution

We now tackle Problem (3.3). Our first result confirms

Lemma 1. Problem (3.3) admits a unique optimal solution. When (3.4) holds, the con-

straint in Problem (3.3) becomes effective, that is, it is an equation for the optimal solu-

tion.

Proof. Let Gn ∈ G be a maximising sequence. Then, for any n and p ∈ [0, 1],

Gn(p) = Gn(1) −∫ 1

pG′

n(t) dt > β −∫ 1

0h(t) dt = β − M,

11

and Gn(p) 6 Gn(1) = β, so the sequence is uniformly bounded on [0, 1]. For any n and

0 6 p1 < p2 6 1,

|Gn(p2) − Gn(p1)| =∫ p2

p1

G′

n(t) dt 6∫ p2

p1

h(t) dt = F −1X (1 − p1) − F −1

X (1 − p2),

hence the sequence is uniformly equicontinuous. By Arzelà-Ascoli theorem Gn has a

subsequence that converges uniformly to some G∗. Moreover, it is not hard to verify that

G∗ ∈ G , so G∗ is the optimal solution.

Suppose (3.4) holds. Then G∗ 6≡ β. Define

Hε(p) = (1 − ε)G∗(p) + εβ, p ∈ [0, 1].

Then Hε ∈ G and Hε > G∗ for every ε ∈ [0, 1], and Hε is increasing in ε. If∫ 1

0 G∗(p)φ(p) dp <

, then by (3.4), there exists ε∗ ∈ (0, 1) such that∫ 1

0 Hε∗(p)φ(p) dp = . This further

implies

∫ 1

0u(Hε∗(p))w′(1 − p) dp >

∫ 1

0u(H0(p))w′(1 − p) dp =

∫ 1

0u(G∗(p))w′(1 − p) dp,

which contradicts the optimality of G∗. The proof is complete.

With the help of this result, we can prove the following duality principle.

Lemma 2. Suppose (3.4) holds. Then G∗ is the optimal solution of Problem (3.3) if and

only if, there exists a constant λ∗ > 0 such that G∗ ∈ G satisfies

∫ 1

0G∗(p)φ(p) dp =

and it is the optimal solution of

maxG∈G

( ∫ 1

0u(G(p))w′(1 − p) dp − λ∗

∫ 1

0G(p)φ(p) dp + λ∗

). (4.1)

Proof. This is a well known result. We prove the sufficiency only, and the proof of

necessity is left for the interested readers. By assumption, G∗ is a feasible solution of

Problem (3.3). For any feasible solution G of Problem (3.3), we have

∫ 1

0u(G(p))w′(1 − p) dp 6

∫ 1

0u(G(p))w′(1 − p) dp − λ∗

∫ 1

0G(p)φ(p) dp + λ∗

6

∫ 1

0u(G∗(p))w′(1 − p) dp − λ∗

∫ 1

0G∗(p)φ(p) dp + λ∗

=∫ 1

0u(G∗(p))w′(1 − p) dp.

Hence G∗ is the optimal solution of Problem (3.3).

12

In the subsequent, we solve the following relaxed problem for each λ > 0,

maxG∈G

(∫ 1

0u(G(p))w′(1 − p) dp − λ

∫ 1

0G(p)φ(p) dp + λ

), (4.2)

by a three-step procedure. Similar to the proof of Lemma 1, one can show the above

problem admits a unique optimal solution.

4.1 Step 1: Change-of-variable

We first simplify Problem (4.2) to remove w from the objective functional. We borrow

the following change-of-variable argument from [30].

Let ν : [0, 1] → [0, 1] be the inverse map of p 7→ 1 − w(1 − p), namely,

ν(p) := 1 − w−1(1 − p), p ∈ [0, 1].

Note ν is also a probability weighting function in W . By virtue of 1 − w(1 − ν(p)) = p,

differentiating it yields

w′(1 − ν(p))ν ′(p) = 1, a.e. p ∈ [0, 1]. (4.3)

From now on we denote, for p ∈ [0, 1],

Q(p) := G(ν(p)), (4.4)

ϕ(p) :=∫ p

0φ(ν(t))ν ′(t) dt =

∫ ν(p)

0φ(t) dt. (4.5)

Then

∫ 1

0u(G(p))w′(1 − p) dp − λ

∫ 1

0G(p)φ(p) dp

=∫ 1

0u(G(ν(p)))w′(1 − ν(p)) dν(p) − λ

∫ 1

0G(ν(p))φ(ν(p)) dν(p)

=∫ 1

0u(Q(p))w′(1 − ν(p))ν ′(p) dp − λ

∫ 1

0Q(p)φ(ν(p))ν ′(p) dp

=∫ 1

0u(Q(p)) dp − λ

∫ 1

0Q(p)ϕ′(p) dp.

Note that G is increasing and left-continuous if and only if so is Q. Moreover, G′(p) 6 h(p)

for a.e. p ∈ [0, 1] if and only if Q′(p) = G′(ν(p))ν ′(p) 6 h(ν(p))ν ′(p) for a.e. p ∈ [0, 1].

Therefore, G ∈ G if and only if Q ∈ Q, where

Q :={

Q : [0, 1] → (−∞, β]∣∣∣ Q is absolutely

continuous with Q(1) = β and 0 6 Q′ 6 ~ a.e.}

,

13

and

~(p) := h(ν(p))ν ′(p) > 0, a.e. p ∈ [0, 1]. (4.6)

By changing variables, Problem (4.2) has now been reduced to the following simple form

maxQ∈Q

( ∫ 1

0u(Q(p)) dp − λ

∫ 1

0Q(p)ϕ′(p) dp + λ

), (4.7)

in which the probability weighting function is not a part of the objective functional.

By (2.10),

∫ 1

p~(t) dt =

∫ 1

ν(p)h(t) dt = F −1

X (w−1(1 − p)) 6 M, p ∈ [0, 1]. (4.8)

4.2 Step 2: Relaxation method

This section outlines our main technical contribution. We introduce an obstacle prob-

lem for a semilinear second-order elliptic operator with mixed boundary conditions and

express the optimal quantile for Problem (4.7) by the solution of the obstacle problem.

Because the relaxation method used in [30] cannot be directly applied to incorporate

the compatibility constraint, we now introduce a new way to overcome this setback.

For any Q ∈ Q, we use integration by parts to obtain

∫ 1

0u(Q(p)) − λQ(p)ϕ′(p) dp

=∫ 1

0u(Q(p)) + λQ′(p)ϕ(p) dp + λQ(0)ϕ(0) − λQ(1)ϕ(1)

=∫ 1

0u(Q(p)) + λ(Q′(p) − ~(p))ϕ(p) dp +

∫ 1

0λ~(p)ϕ(p) dp − βλϕ(1). (4.9)

Let δ : [0, 1] → R be an absolutely continuous function (to be determined) such that

δ(0) = 0 and δ 6 λϕ. (4.10)

By virtue of λ > 0 and Q′ 6 ~ a.e., the right-hand side of (4.9) is

6

∫ 1

0u(Q(p)) + (Q′(p) − ~(p))δ(p) dp +

∫ 1

0λ~(p)ϕ(p) dp − βλϕ(1) (4.11)

=∫ 1

0u(Q(p)) + Q′(p)δ(p) dp +

∫ 1

0~(p)(λϕ(p) − δ(p)) dp − βλϕ(1),

which, by applying integration by parts again, is

=∫ 1

0u(Q(p)) − Q(p)δ′(p) dp +

∫ 1

0~(p)(λϕ(p) − δ(p)) dp + β(δ(1) − λϕ(1)). (4.12)

BecauseQ(p) = (u′)−1(δ′(p)),

14

maximises the integrand in the first integral in (4.12) at p ∈ [0, 1], (4.12) is

6

∫ 1

0u(Q(p)) − Q(p)δ′(p) dp +

∫ 1

0~(p)(λϕ(p) − δ(p)) dp + β(δ(1) − λϕ(1)). (4.13)

This plus λ gives an upper bound for Problem (4.7). We hope to chose δ in a wise

way such that Q becomes the optimal solution. In other words, the inequities (4.11) and

(4.13) become equations when Q is replaced by Q.

If Q is an optimal solution of Problem (4.7), it should belong to Q at least. This

requires 0 6 Q′6 ~ a.e., that is,

0 6δ′′

u′′

((u′)−1(δ′)

) 6 ~,

which can also be expressed as

δ′′6 0 and δ′′ − ~u′′

((u′)−1(δ′)

)> 0. (4.14)

It also requires that Q(1) = β, that is

δ′(1) = u′(β). (4.15)

We conclude that δ should satisfy the requirements (4.10), (4.14), and (4.15).

4.3 Step 3: Optimal solution

Summarising the above intuitive argument leads to the following verification result.

Proposition 3 (Verification Proposition). Suppose δ : [0, 1] → R is an absolute con-

tinuous function and satisfies the following variational inequality (VI) with mixed bound-

ary conditions

min{δ′′(p) − ~(p)u′′

((u′)−1(δ′(p))

), λϕ(p) − δ(p)

}= 0, a.e. p ∈ [0, 1],

δ(0) = 0, δ′(1) = u′(β).

(4.16)

If δ is concave, then

G(p) := (u′)−1(δ′(1 − w(1 − p))), a.e. p ∈ [0, 1],

is the unique optimal solution of Problem (4.2).

Proof. First G defined above is an optimal solution of Problem (4.2) if and only if

Q(p) = G(ν(p)) = (u′)−1(δ′(p)), a.e. p ∈ [0, 1],

is an optimal solution of Problem (4.7). Because u′′ < 0 and δ′′ 6 0, we see that Q′> 0

15

a.e.. Moreover, we can rewrite (4.16) as

min

−

δ′′(p)

u′′

((u′)−1(δ′(p))

) + ~(p), λϕ(p) − δ(p)

= 0, a.e. p ∈ [0, 1].

that is

min{−Q

′(p) + ~(p), λϕ(p) − δ(p)

}= 0, a.e. p ∈ [0, 1]. (4.17)

This implies Q′6 ~ a.e.. Together with Q(1) = (u′)−1(δ′(1)) = β, we conclude that

Q ∈ Q is a feasible solution of Problem (4.7).

We now show that Q is indeed an optimal solution. It follows from (4.17) that

(Q′(p) − ~(p))(λϕ(p) − δ(p)) = 0, a.e. p ∈ [0, 1],

so

∫ 1

0λ(Q

′(p) − ~(p))ϕ(p) dp =

∫ 1

0(Q

′(p) − ~(p))δ(p) dp.

From this, we see that the inequalities in (4.11) and (4.13) become equations when Q is

replaced by Q. In another words, the upper bound (4.13) is achieved by Q, that is,

∫ 1

0u(Q(p)) − λQ(p)ϕ′(p) dp 6

∫ 1

0u(Q(p)) − λQ(p)ϕ′(p) dp.

Therefore, Q is an optimal solution of Problem (4.7). This completes the proof.

Corollary 4. If ϕ is concave, then the solution of the VI (4.16) is automatic concave.

Proof. If δ was not concave, then the region {p ∈ [0, 1] | δ′′(p) > 0} was not empty. In this

region, we would have δ′′ > 0 > ~u′′((u′)−1(δ′)

), so λϕ = δ by (4.16); and consequently,

ϕ′′ = δ′′/λ > 0. This would contradict the concavity of ϕ.

Corollary 5. If w is concave in Problem (2.5), then the solution of the VI (4.16) is

automatic concave.

Proof. For the insurance problem (2.5), we have φ ≡ 1; and thus by (4.5),

ϕ(p) =∫ ν(p)

0φ(t) dt = ν(p), p ∈ [0, 1].

If w is concave, then by (4.3), so is ν. Consequently, the claim follows from the previous

lemma.

We now state the main result of this paper as follows.

16

Theorem 6. Suppose (3.4) holds. Suppose δλ is a concave solution of (4.16), and set

Gλ(p) = (u′)−1(δ′

λ(1 − w(1 − p))), a.e. p ∈ [0, 1].

If there exists a constant λ∗ > 0 such that

∫ 1

0Gλ∗(p)φ(p) dp = .

Then Gλ∗ is the unique optimal solution of Problem (3.3). In particular, when φ ≡ 1,

I(x) = x − β + Gλ∗(1 − FX(x)), x ∈ [0, M ], (4.18)

is an optimal solution of Problem (2.5).

Proof. By Proposition 3, Gλ∗ is the unique optimal solution of Problem (4.2) with λ

replaced by λ∗. Therefore, by Lemma 2, Gλ∗ is the unique optimal solution of Problem

(3.3).

When φ ≡ 1, Problem (3.3) reduces to Problem (2.8), so Gλ∗ is an optimal solution

of Problem (2.8). It follows from the relation (2.9) that

I(x) = x − β + Gλ∗(1 − FX(x))

is an optimal solution of Problem (2.5). This completes the proof.

Remark 4.1. In [28] and [30], the optimal solution is expressed via ϕ which is the con-

cave envelope of some known function ϕ. In fact, ϕ can also be interpreted as the solution

of the following obstacle problem for a linear second-order elliptic operator

min{

− ϕ′′(p), ϕ(p) − ϕ(p)}

= 0, a.e. p ∈ [0, 1],

with some proper boundary conditions. Therefore, roughly speaking, the compatibility

constraint involved in the new problem turns the linear operator into a non-linear one.

As a consequence, closed-form solutions are no more available in general. But we will

show below it is possible to have a closed-form solution in some special case.

We may simplify (4.16) for the most widely used three utility functions.

• For the exponential utility u(x) = −α−1e−αx, α > 0, the VI (4.16) reduces to

min {δ′′(p) + α~(p)δ′(p), λϕ(p) − δ(p)} = 0, a.e. p ∈ [0, 1],

δ(0) = 0, δ′(1) = u′(β).

(4.19)

We will provide a closed-form solution in this case in the next section.

17

• For the power utility u(x) = α−1xα, 0 < α < 1, the VI (4.16) reduces to

min{

δ′′(p) + (1 − α)~(p)(δ′(p))2−α

1−α , λϕ(p) − δ(p)}

= 0, a.e. p ∈ [0, 1],

δ(0) = 0, δ′(1) = u′(β).

(4.20)

• For the logarithmic utility u(x) = log x, the VI (4.16) reduces to

min {δ′′(p) + ~(p)(δ′(p))2, λϕ(p) − δ(p)} = 0, a.e. p ∈ [0, 1],

δ(0) = 0, δ′(1) = u′(β).

This can be regarded as the special case α = 0 in (4.20).

5 On the optimal solution

In this section we discuss some interesting properties of the optimal solution.

5.1 Closed-form solution for exponential utility

Generally, there is no closed-form solution to the VI (4.16), which therefore calls for a

numerical scheme to solve it. However, we can provide a closed-form solution when u is

the exponential utility u(x) = −α−1e−αx, α > 0. In this case (4.16) reduces to (4.19).

By (4.8), we can define

f(p) =

∫ p0 e−α

∫ t

0~(s) ds dt

∫ 10 e−α

∫ t

0~(s) ds dt

, p ∈ [0, 1].

Then it is a strictly increasing continuous concave function and satisfies

f ′′(p) = −α~(p)f ′(p), p ∈ [0, 1].

Let f : [0, 1] → [0, 1] be the inverse function of f . By differentiating f(f(p)) = p twice,

we obtain

f ′(f(p)) = (f ′(p))−1, f ′′(f(p))f ′(p) = −f ′′(p)(f ′(p))−2, p ∈ [0, 1].

Hence,α~(f(p)) = −f ′′(f(p))/f ′(f(p)) = f ′′(p)(f ′(p))−2, p ∈ [0, 1]

orf ′′(p) = α~(f(p))(f ′(p))2, p ∈ [0, 1].

18

Define ∆(p) = δ(f(p)) for p ∈ [0, 1]. Then

∆′(p) = δ′(f(p))f ′(p), p ∈ [0, 1],

and

∆′′(p) = (f ′(p))2(δ′′(f(p)) + α~(f(p))δ′(f(p))

), p ∈ [0, 1].

By the above change-of-variable, we see that Problem (4.19) is equivalent to

min {∆′′(p), λϕ(f(p)) − ∆(p)} = 0, a.e. p ∈ [0, 1],

∆(0) = 0, ∆′(1) = u′(β)f ′(1).(5.1)

The solution of this problem is clearly the biggest convex function that is dominated

by λϕ(f(p)) on [0, 1] and satisfies ∆(0) = 0 and ∆′(1) = u′(β)f ′(1). Furthermore,

δ(p) = ∆(f(p)) solves the VI (4.16).

Remark 5.1. Suppose the conditions of Theorem 5.8 [32] holds true (see Remark 2.3).

Then φ ≡ 1 and ϕ ≡ ν. Moreover, as w is inverse-S-shaped, then by (4.3), so is ν. One

can check that, provided proper parameters,

∆(p) =

c1f(p), p ∈ [0, p′1];

cp, p ∈ [p′1, p′

2];

u′(β)f(p), p ∈ [p′2, 1],

is a convex solution of the VI (5.1) (where the third condition in Remark 2.3 plays a

critical role). Consequently,

δ′(p) =(∆(f(p))

)′

=

u′(β − d2), p ∈ [0, p1];

cf ′(p), p ∈ [p1, p2];

u′(β), p ∈ [p2, 1].

As u′(x) = e−αx, one has

cf ′(p) = ce−α∫ p

0~(s) ds = ce−α(M−F −1

X(w−1(1−p)) = u′(β + d1 − F −1

X (1 − ν(p))).

Then

(u′)−1(δ′(p)) =

β − d2, p ∈ [0, p1];

β + d1 − F −1X (1 − ν(p)), p ∈ [p1, p2];

β, p ∈ [p2, 1],

19

and

G(1 − FX(x)) = (u′)−1(δ′(1 − w(FX(x)))) =

β, x ∈ [0, x1];

β + d1 − x, x ∈ [x1, x2];

β − d2, x ∈ [x2, M ].

Finally, one concludes from Theorem 6 that the optimal solution of Problem (2.5) has the

following form

I(x) = x − β + G(1 − FX(x)) =

x, x ∈ [0, x1];

d1, x ∈ [x1, x2];

x − d2, x ∈ [x2, M ].

This reveals the main result of [32], Theorem 5.8.

5.2 Optimal contract of deductible type

Within the EU theory framework, we know that the optimal contract must be of de-

ductible type. In this part we want to investigate under what condition the optimal

contract for the behavioural model (3.3) is also of deductible type.4 The following result

answers this question completely.

Theorem 7. Suppose (3.4) holds. The optimal solution of Problem (4.2) is of deductible

type if and only if there exists p0 ∈ [0, 1) such that

λϕ(p)

= u′(β − F −1

X (w−1(1 − p0)))

p, p 6 p0;

>∫ p

p0u′(β − F −1

X (w−1(1 − t)))

dt + u′(β − F −1

X (w−1(1 − p0)))

p0, p > p0.

(5.2)

In this case the deductible is F −1X (w−1(1 − p0)).

Proof. By Remark 2.4, we see that an optimal contract G is of deductible type if and

only if

G(p) =

β − F −1X (1 − p′), p 6 p′;

β − F −1X (1 − p), p > p′,

for some p′ ∈ [0, 1). Denote p0 = 1 − w(1 − p′). Then ν(p0) = p′ and the above condition

becomes

G(p) =

β − F −1X (w−1(1 − p0)), p 6 ν(p0);

β − F −1X (1 − p), p > ν(p0).

4 Under the condition (3.4), we know that the full contract is not a feasible solution to Problem (3.3)and thus cannot be optimal.

20

Set δ′(p) = u′(G(ν(p))) with δ(0) = 0. Clearly, δ′ is decreasing, so δ is concave. Moreover,

we have the explicit expression

δ(p) =

u′(β − F −1

X (w−1(1 − p0)))

p, p 6 p0;∫ p

p0u′(β − F −1

X (w−1(1 − t)))

dt + u′(β − F −1

X (w−1(1 − p0)))

p0, p > p0.

This yields δ′(1) = u′(β),

δ′′(p) − ~(p)u′′((u′)−1(δ′(p))

)= −~(p)u′′

((u′)−1(δ′(p))

)> 0, p < p0,

and

δ′′(p) − ~(p)u′′((u′)−1(δ′(p))

)

= u′′(β − F −1

X (w−1(1 − p)))[(

β − F −1X (w−1(1 − p))

)′

− ~(p)]

= u′′(β − F −1

X (w−1(1 − p)))[(

F −1X

)′

(1 − ν(p)) − h(ν(p))]ν ′(p) = 0, p > p0.

Hence δ is a concave solution of (4.16) if and only if (5.2) holds.

Remark 5.2. One can use this idea to examine when the optimal contract is of three-fold

as in [32].

The following result reveals the classical result for the insurance contract design prob-

lem within the EU theory framework.

Corollary 8. If there is no probability distortion in the insurance problem (2.5), then

the optimal contract is deductible.

Proof. By assumption, we have φ ≡ 1, w(p) ≡ p. Hence

ϕ(p) =∫ ν(p)

0φ(t) dt = p, p ∈ [0, 1].

Then by the concavity of u, the condition (5.2) is satisfied by setting

p0 = 1 − w(FX(β − (u′)−1(λ))

).

When (5.2) is not satisfied, the optimal contract must not be of deductible type. We

are interested in which situation a moral-hazard-free contract is optimal for at least one

RDU maximiser; in other words, when such a contract is acceptable in market. This will

be addressed in the following section.

5.3 Is every moral-hazard-free contract optimal?

In this section, we consider the following reverse problem. For a given moral-hazard-free

contract, is it possible to find at least one RDU maximiser such that the contract is

21

optimal for her? The problem can be mathematically stated as follows: for any given

reasonable functions φ, FX , and δ, can we find a triple (u, w, λ) such that the VI (4.16)

is satisfied? Here u is a utility function, w is probability weighting function and λ is

a positive constant. Clearly it is necessarily to assume that δ is increasing, concave,

δ(0) = 0 and δ′(0) < ∞5. We also assume FX is continuous.

We first fix the utility function u (such that u′(β) = δ′(1) as a must). Now look for

(w, λ) to satisfy (4.16).

We begin with the case that the potential loss can be relatively large compared to the

final wealth β, namely,

M > β − (u′)−1(δ′(0)). (5.3)

In this case, the answer is affirmative. To show this, let

H(p) = M − F −1X (w−1(1 − p)), p ∈ [0, 1].

Then by (4.5), λϕ > δ if and only if H(p) > κ(p, λ), where

κ(p, λ) = inf

{q ∈ [0, M ] : λ

∫ 1−FX(M−q)

0φ(t) dt > δ(p)

}∧ M, p ∈ [0, 1],

is a bounded continuous function in p and λ. It is not hard to verify

H ′(p) = ~(p),((u′)−1(δ′(p))

)′

=δ′′(p)

u′′

((u′)−1(δ′(p))

) , p ∈ [0, 1].

Moreover, the condition w(0) = 0 and w(1) = 1 is equivalent to that H(0) = 0 and

H(1) = M . Therefore, finding (w, λ) to satisfy (4.16) becomes finding a constant λ > 0

and an increasing absolutely continuous function H such that

min{

H ′(p) −((u′)−1(δ′(p))

)′

, H(p) − κ(p, λ)}

= 0, a.e. p ∈ [0, 1],

H(0) = 0, H(1) = M.(5.4)

LetHλ(p) = sup

t6p

(κ(t, λ) − (u′)−1(δ′(t))

)+ (u′)−1(δ′(p)), p ∈ [0, 1].

Because δ(0) = 0, we have κ(0, λ) = 0; and consequently, Hλ(0) = 0 for any λ > 0.

As δ′(1) = u′(β) > 0 and δ is increasing, we get δ(1) > 0. If 0 < λ < δ(1)∫ 1

0φ(t) dt

, then

Hλ(1) > κ(1, λ) = M . On the other hand, since κ(p, λ) is increasing in p,

Hλ(1) 6 supt61

(κ(1, λ) − (u′)−1(δ′(t))

)+ (u′)−1(δ′(1)) = κ(1, λ) − (u′)−1(δ′(0)) + β.

5Note (4.16) implies that δ′(0+) 6 λϕ′(0+) < ∞.

22

Because limλ→+∞ κ(1, λ) = 0, we obtain

limλ→+∞

Hλ(1) 6 −(u′)−1(δ′(0)) + β < M.

Therefore, there exists λ∗ > 0 such that Hλ∗(1) = M . By Skorokhod lemma (see, e.g.,

Lemma VI.2.1, Revuz and Yor [23]), Hλ∗ is a solution of Problem (5.4). Therefore, we find

one RDU maximiser to accept the moral-hazard-free contract. Economically speaking,

when the potential loss can be relatively large, the probability weighting function can

distort insured’s behaviours such that she will accept the moral-hazard-free contract.

Now we assume the the potential loss is relatively small, namely, M < β−(u′)−1(δ′(0)).

In this case, the answer is negative. In fact,

Hλ(1) > supt61

(−(u′)−1(δ′(t))

)+ (u′)−1(δ′(1)) = −(u′)−1(δ′(0)) + β > M,

so Problem (5.4) has no solution. Economically speaking, when the potential loss is

relatively small, the probability weighting function can not distort insured’s behaviours

too much.

In the above argument, we fixed the utility function u. Now we fix the probability

weighting function w. In this case, the answer is also negative, because one may fail to

find a pair (u, λ) to satisfy (4.16). This indicates that the probability weighting function

rather than the utility function plays the most important role in distorting insured’s

behaviours in the present model.

Now suppose we can freely choose the triple (u, w, λ). Clearly we can first choose u to

satisfy u′(β) = δ′(1) and (5.3), then choose a pair (w, λ) to satisfy (4.16). Consequently,

the moral-hazard-free contract is optimal for the insured (u, w, λ). In this sense, the

utility function is of equivalent importance as the probability weighting function. Since

there are infinity many u that satisfy (5.3), the moral-hazard-free contract is accepted by

infinity many RDU maximisers with different utility functions and probability weighting

functions.

6 Concluding remarks

In this paper, we present a systematic approach to solving the moral-hazard-free insurance

contract design problem within the RDU theory framework. Similar to [28, 30], the

approach also works for problems within other behavioural finance frameworks. For

instance, one could use our method to consider the loss and gain parts of the cumulative

prospect theory model separately, and then combine the results.

Whether Xia and Zhou’s [28] calculus-of-variations method can be adapted to solve

Problem (4.7) remains open. On the other hand, we propose another possible approach

23

as follows. Let ct = Q′(t) be the control variable in the constraint set

C ={c : 0 6 ct 6 ~(t), a.e. t ∈ [0, 1]

},

and Q(t) = β −∫ 1

t cs ds be the corresponding state process. Then one can interpret Prob-

lem (4.7) as a deterministic optimal control problem with control constraint. Applying

the dynamic programming principle, one obtains the following Hamilton-Jacobi-Bellman

equation

vt + supc∈Ct

H(t, x, c, vx) = 0,

v|t=1 = 0,

where Ct = [0, ~(t)] and

H(t, x, c, p) = pc + u(x) − λxϕ′(t).

However, this approach has its own setback. For instance, this problem may not have a

classical solution. Also if there is one, then there is no easy way to determine it from the

equation, which is necessary in order to solve the control problem completely.

24

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