Optimal Insurance with Heterogeneous Beliefs and Disagreement about Zero-Probability Events

Abstract

In problems of optimal insurance design, Arrow’s classical result on the optimality of the deductible indemnity schedule holds in a situation where the insurer is a risk-neutral Expected-Utility (EU) maximizer, the insured is a risk-averse EU-maximizer, and the two parties share the same probabilistic beliefs about the realizations of the underlying insurable loss. Recently, Ghossoub re-examined Arrow’s problem in a setting where the two parties have different subjective beliefs about the realizations of the insurable random loss, and he showed that if these beliefs satisfy a certain compatibility condition that is weaker than the Monotone Likelihood Ratio (MLR) condition, then optimal indemnity schedules exist and are nondecreasing in the loss. However, Ghossoub only gave a characterization of these optimal indemnity schedules in the special case of an MLR. In this paper, we consider the general case, allowing for disagreement about zero-probability events. We fully characterize the class of all optimal indemnity schedules that are nondecreasing in the loss, in terms of their distribution under the insured’s probability measure, and we obtain Arrow’s classical result, as well as one of the results of Ghossoub as corollaries. Finally, we formalize Marshall’s argument that, in a setting of belief heterogeneity, an optimal indemnity schedule may take “any”shape.

1. Introduction

The problem of optimal insurance design under uncertainty dates back to the seminal work of Arrow [1] who showed that when the insured, or Decision Maker (DM), is a risk-averse Expected-Utility (EU)-maximizer, the insurer is a risk-neutral EU-maximizer, the two parties assign the same distribution to the insurable random loss and the premium principle depends on the actuarial value (expected value) of the indemnity; then, full insurance above a deductible is optimal for the DM. In particular, the optimal indemnity schedule is a nondecreasing function of the insurable loss1, and its distribution can be fully characterized. This is a fundamental and foundational result that has been extended in several directions, all the while maintaining the assumption of belief homogeneity. We refer to Gollier [3] and Schlesinger [4] for surveys.

Belief heterogeneity is pervasive in insurance markets. On a theoretical level, disagreements about (subjective) beliefs arise naturally in the De Finetti–Savage [5,6] framework from divergent preferences over alternatives. Moreover, disagreement about (posterior) beliefs can be a direct consequence of relaxing the controversial and heavily-criticized common priors assumption in game theory [7,8]. On a practical level, primary and secondary insurance markets do display belief heterogeneity, more often than not: pooling and diversification effects give the insurer a different view of risks than the insured; advances in insurance analytics often lead to disagreements about beliefs. We refer to Ghossoub [9] for a more detailed description discussion.

Although belief heterogeneity is an important consideration in insurance markets, little work has been devoted to the systematic study of optimal insurance design when the insurer and the DM assign different distributions to the random loss, or more generally, entertain different subjective beliefs over the relevant state space. The first formal examination of this problem was done by Marshall [10] (hereafter, Marshall). In the setting of Marshall, the DM assigns a probability density function (pdf) $f\left(t\right)$ to the insurable loss, whereas the insurer attributes the pdf $g\left(t\right)$ to the loss. Marshall assumes that the DM is more optimistic than the insurer, in the sense of second-order stochastic dominance, and argues that at that level of generality, nothing can be said about the optimal contract: the optimal indemnity could take any shape (within the usual constraints on an indemnity schedule). Marshall then considers a special case in which the DM assigns a higher probability to the no-loss event than the insurer; but conditioning on the loss being non-zero, the two parties assign the same conditional distribution to the random loss. In this case, the probability of a zero loss can be seen as a proxy for the DM’s optimism, and Marshall shows that if the insurer is risk-neutral, the optimal insurance indemnity is a deductible contract; and the deductible level increases with the DM’s optimism. However, this is a rather restrictive approach to belief heterogeneity, since this heterogeneity is reduced only to the likelihood that each party attaches to the event of a zero loss. Huang et al. [11] examine a similar problem, but in their framework, risk-attitude and belief heterogeneity are intertwined; and it is not clear how to separate the effect of belief heterogeneity from the effect of risk-aversion on the shape of the optimal indemnity schedule. Jeleva [12], Jeleva and Villeneuve [13] and Anwar and Zheng [14] examine related problems, but their settings are too restrictive to yield either a distributional or an analytical characterization of the optimal indemnity. Recently, Ghossoub [9] re-examined Arrow’s problem in a setting where the two parties have different subjective beliefs about the realizations of the insurable random loss, and he showed that if these beliefs satisfy a certain compatibility condition that is weaker than the Monotone Likelihood Ratio (MLR) condition2, then optimal indemnity schedules exist and are nondecreasing in the loss. However, Ghossoub [9] only gave a characterization of these optimal indemnity schedules in the special case of an MLR: he showed that, in that case, the optimal indemnity schedule is a variable deductible schedule, with a state-contingent deductible that depends on the state of the world only through the likelihood ratio. Arrow’s classical result was then obtained as a special case.

In this paper, we extend the analysis done in Ghossoub [9] by considering the general case of belief heterogeneity, hence allowing for disagreement about zero-probability events. We fully characterize the class of all optimal indemnity schedules that are nondecreasing in the loss, in terms of their distribution under the insured’s probability measure, and we obtain Arrow’s classical result, as well as one of the results of Ghossoub [9] as corollaries. Specifically, our contribution is three-fold:

(1)

First, under a condition of compatibility between the subjective beliefs of the insurer and the DM introduced by Ghossoub [9], we characterize the optimal indemnity schedule ${\mathcal{Y}}^{*}$ in terms of its distribution under the DM’s subjective probability measure. The importance of characterizing the distribution of an optimal indemnity schedule rather than its actual shape has been stressed by Gollier and Schlesinger [16]. Specifically, we show that ${\mathcal{Y}}^{*}$ has the same distribution (under the DM’s subjective probability measure) as a function of the form:

$$Z:=min\left[X,max\left(0,X-\left[W_0-\Pi -{\left(u^{\prime }\right)}^{-1}\left({\lambda }^{*}h\right)\right]\right)\right],$$

(1)

for some ${\lambda }^{*}\ge 0$ and a nonnegative measurable function h, which is entirely characterized from the subjective probabilities of both parties (Theorem 8). The function u is the DM’s utility function; $W_0$ is the DM’s initial level of wealth; and Π is the premium paid by the DM, as in Arrow’s setting. The function Z can be written in the form $Z=min\left(X,D\right)$, where $D=max\left(X-d\left(h\right),0\right)$ is a deductible indemnity schedule with a state-contingent deductible $d\left(h\right)=W_0-\Pi -{\left(u^{\prime }\right)}^{-1}\left({\lambda }^{*}h\right)$ that depends on the state of the world only through the function h.

Moreover, when the beliefs of both parties coincide, the function h appearing in Equation (1) is the constant function equal to one, and hence, the function Z is simply a deductible contract, which is a nondecreasing function of the loss X. In this case, Corollary 9 states that the main result of this paper (Theorem 8) boils down to the classical result of Arrow (Theorem 5).

(2)

Second, we formalize Marshall’s argument that the optimal insurance indemnity schedule may take “any” form in general (within the usual constraints imposed on an indemnity schedule). Specifically, although Theorem 8 asserts the existence and monotonicity of an optimal indemnity schedule ${\mathcal{Y}}^{*}$ that has a given distribution for the DM’s belief and that satisfies the constraint $0\le {\mathcal{Y}}^{*}\le X$, Corollary 11 states that ${\mathcal{Y}}^{*}$ can take different shapes. Depending on the function h (seen as a proxy for belief heterogeneity) and/or the DM’ s utility function u (seen as a proxy for the DM’s risk aversion), ${\mathcal{Y}}^{*}$ may include a non-zero deductible or a disappearing deductible, whereby losses of high magnitude are fully insured. However, for losses of moderate magnitude, ${\mathcal{Y}}^{*}$ is of the form $f\left(X\right)$, where f is a nondecreasing, Borel-measurable and left-continuous function, such that $0\le f\left(t\right)\le t$, for all t in the range of X. Nothing else can be said about the function f, in terms of concavity, convexity and inflection points, for instance.

(3)

On a technical level, a third contribution of this paper is to present a methodology for dealing with disagreement about zero-probability events (i.e., singularity of measures) within an optimal insurance design problem. This methodology, however, can be easily extended to general contracting problems with heterogeneous beliefs that exhibit some singularity or disagreement about zero-probability events.

The rest of this paper is organized as follows. Section 2 introduces the formal model of insurance demand in the presence of belief heterogeneity, Section 3 presents the main results of this paper, and Section 4 concludes. Some proofs and related analyses are collected in Appendix A, Appendix B, Appendix C, Appendix D, Appendix E and Appendix F.

2. The Model

Let S denote the non-empty collection of states of the world. The DM faces a loss X, taken to be a mapping of S onto a closed interval $\left[0,M\right]$, against which she or he seeks insurance coverage. The information generated by observing the loss random variable is the σ-algebra $\sigma \{X\}$ of subsets of S generated by X.

Denote the σ-algebra $\sigma \{X\}$ by Σ; denote by $B\left(\Sigma \right)$ the Banach space (sup norm3.) of all bounded, $\mathbb{R}$-valued and Σ-measurable functions on $\left(S,\Sigma \right)$; and denote by $B^{+}\left(\Sigma \right)$ the collection of all ${\mathbb{R}}^{+}$-valued elements of $B\left(\Sigma \right)$. Then, for any $Y\in B\left(\Sigma \right)$, there exists a Borel-measurable map $\zeta :\mathbb{R}\to \mathbb{R}$, such that $Y=\zeta \circ X$ (e.g., Theorem 4.41 of [17]). For $C\subseteq S$, denote by ${\mathbf{1}}_C$ the indicator function of C. For any $A\subseteq S$ and for any $B\subseteq A$, denote by $A\setminus B$ the complement of B in A.

An insurance market gives the DM the possibility of purchasing insurance coverage for a premium $\Pi >0$. The premium is paid ex ante by the DM in return for receiving the ex post indemnity $I\left(X\left(s\right)\right)$ in the state of the world $s\in S$. The indemnity schedule is a Borel-measurable map $I:\left[0,M\right]\to \left[0,M\right]$. Then, $Y:=I\circ X\in B^{+}\left(\Sigma \right)$, and hence, the collection of all indemnity schedules can be identified with the set $B^{+}\left(\Sigma \right)$.

Both the DM and the insurer have preferences over the elements of $B^{+}\left(\Sigma \right)$ (i.e., over indemnity schedules) that have a subjective expected-utility representation yielding:

(i)

A utility utility function $u:\mathbb{R}\to \mathbb{R}$ and a probability measure P on the measurable space $\left(S,\Sigma \right)$ for the DM;

(ii)

A utility function $v:\mathbb{R}\to \mathbb{R}$ and a probability measure $Q\ne P$ on the measurable space $\left(S,\Sigma \right)$ for the insurer.

Both u and v are unique up to a positive linear transformation (e.g., Theorem 14.1 of [18]). Moreover, as in Arrow’s framework, we suppose that the DM is risk averse and that her or his utility function u satisfies the following.

Assumption 1. 

The DM’s utility function u satisfies Inada’s [19] conditions:

• $u\left(0\right)=0$;
• u is strictly increasing and strictly concave;
• u is continuously differentiable; and
• $u^{\prime }\left(0\right)=+\infty$ and ${\displaystyle \underset{x\to +\infty }{lim}}{u}^{\prime }\left(x\right)=0$.

The DM has initial wealth $W_0>\Pi$, and her or his total state-contingent wealth in each sate of the world $s\in S$ is given by:

$$W\left(s\right):=W_0-\Pi -X\left(s\right)+Y\left(s\right).$$

We will also make the assumption that the random loss X has a nonatomic4 law induced by the probability measure P, that the subjective probability measures P and Q are not mutually singular5 and that the DM is almost certain that the random loss she or he will incur is not larger than her or his remaining wealth after the premium has been paid. Specifically:

Assumption 2. 

Assume that:

• $P\circ X^{-1}$ is nonatomic (i.e., X is a continuous random variable for P);
• $X\le W_0-\Pi ,P$-a.s.In other words, $P\left(\{s\in S:X\left(s\right)>W_0-\Pi \}\right)=0$.
• P and Q are not mutually singular.

Assumption 2 (1) is common (e.g., when it is assumed that a probability density function for X exists). Assumption 2 (2) simply states that the DM is well-diversified so that the particular loss exposure X against which she or he is seeking an insurance coverage is sufficiently small. Assumption 2 (3) means that the insurer and the DM do not have beliefs that are totally incompatible. However, this does not prevent the agents from assigning different probabilities to events, and they typically do not assign same likelihoods to the realizations of the uncertainty X. For instance, they might disagree on zero-probability events.

As in Arrow’s model, we assume that the insurer is risk-neutral. Without loss of generality, the insurer’s utility function v can then be taken to be the identity function. The insurer has initial wealth $W_0^{ins}$, and his or her total state-contingent wealth in each state of the world $s\in S$ is given by:

$$W^{ins}\left(s\right):=W_0^{ins}+\Pi -\left(1+\rho \right)Y\left(s\right),$$

where $\rho >0$ is a loading factor meant to account for the cost associated with handling the insurance indemnity payment, as in the classical model.

The DM seeks an indemnity that will maximize her or his expected utility of wealth, subject to the insurer’s participation constraint and to some constraints on the indemnity function:

Problem 3. 

For a given loading factor $\rho >0$,

$$\begin{array}{c}\hfill \begin{array}{c}\underset{Y\in B^{+}\left(\Sigma \right)}{sup}\left\{\int u\left(W_0-\Pi -X+Y\right)dP\right\}:\hfill \\ \left\{\begin{array}{c}0\le Y\le X\hfill \\ \int YdQ\le R:=\frac{\Pi }{1+\rho }\hfill \end{array}\right.\hfill \end{array}\end{array}$$

The first constraint is standard and says that an indemnity is nonnegative and cannot exceed the loss itself. The latter requirement rules out situations where the DM has an incentive to create damage [2], which would result in ex post moral hazard. The second constraint is the risk-neutral insurer’s participation constraint, restated as a premium constraint. Here, we do not impose an additional monotonicity constraint, and we show that an optimal indemnity will have this property (see Theorem 8).

Now, for any $Y\in B\left(\Sigma \right)$, which is feasible for Problem (3), one has $0\le Y\le X$, and hence, $0\le \int YdQ\le \int XdQ$. It is easily seen that when $R\ge \int XdQ$, the solution to Problem (3) is $Y^{*}=X$, i.e., full insurance. In particular, the solution is comonotonic6 with X and its distribution is the same as that of X, for both the DM and the insurer. Therefore, we will consider the remaining case; that is, we will make the following assumption all throughout:

Assumption 4. 

$0<R<\int XdQ$.

3. The Results

The difference between Problem (3) and the classical problem of Arrow is the fact that the probability measures Q and P differ. At this point, no assumption of absolute continuity is made, and the two parties can disagree about zero-probability events. Clearly, when $P=Q$, we recover the classical framework of Arrow:

Theorem 5 (Arrow). 

If $P=Q$, then there exists some $d>0$, such that $I_d\circ X$ is optimal for Problem (3), where $I_d$ is a deductible indemnity schedule defined by:

$$\begin{array}{c}\hfill I_d\left(t\right)=\left\{\begin{array}{c}0ift<d\hfill \\ t-dift\ge d\hfill \end{array}\right.\end{array}$$

That is, an optimal solution for Problem (3) takes the form $Y^{*}=max\left(0,X-d\right),$ for some $d>0$.

Note that since $d>0$, the optimal indemnity schedule can also be written as:

$$Y^{*}=min\left[X,max\left(0,X-d\right)\right].$$

Unlike the classical and the vast majority of the subsequent insurance literature, the insurance model presented here allows for heterogeneity of beliefs. Ghossoub [9] shows that if the analysis is restricted to a class of beliefs Q that are compatible with the DM’s belief P as per the definition below, then optimal indemnity schedules exist and are monotonic. This notion of belief compatibility is introduced in Ghossoub [9] and then extended to risk measures in Ghossoub [20] and to a setting with ambiguous beliefs in Amarante, Ghossoub and Phelps [21,22].

Definition 6 (Ghossoub [9]). 

The probability measure Q is said to be compatible with the probability measure P, or the insurer is said to be compatible, if for any two indemnity schedules $Y_1,Y_2\in B^{+}\left(\Sigma \right)$, such that:

(i)

$Y_1$ and $Y_2$ have the same distribution under P (i.e., $P\circ Y_1^{-1}=P\circ Y_2^{-1}$); and,

(ii)

$Y_2$ and X are comonotonic,

we have $\int Y_{2}dQ\le \int Y_{1}dQ$.

Clearly, the probability measure P is compatible with itself7. Therefore, the classical insurance setup of Arrow can be seen as a special case. Ghossoub [9] showed that when a likelihood ratio can be defined (as a ratio of probability density functions), then belief compatibility is a strictly weaker requirement than an MLR condition. Hence, the restriction on belief heterogeneity imposed by a condition of belief compatibility is general enough to encompass, for instance, cases where these heterogeneous beliefs induce a likelihood ratio that is monotone.

Theorem 7 (Ghossoub [9]). 

If assumptions 1, 2 and 4 hold and if the insurer’s subjective probability measure Q is compatible with the DM’s subjective probability measure P, then there exists an optimal indemnity schedule $Y^{*}$ which is a nondecreasing function of the loss X. Moreover, any other $Z^{*}$ which is nondecreasing in X and which has the same distribution as $Y^{*}$ under P is such that $Z^{*}=Y^{*},P$-a.s. Finally, if the utility function u is strictly concave, then any solution $Z^{*}$ to Problem (3) is such that $Z^{*}=Y^{*}$, P-a.s. In particular, any solution is nondecreasing in X, P-a.s.

The above result shows the existence and monotonicity of optimal indemnity schedules, but it does not provide an analytical or distributional characterization of optima. The main complication in the setting where $Q\ne P$ is, precisely, dealing with the fact that the two parties can disagree about zero-probability events. One insight comes from Lebesgue’s decomposition theorem (e.g., Theorem 4.3.1. of [23]).

By Lebesgue’s decomposition theorem, there exists a unique pair $\left(Q_{ac},Q_s\right)$ of (nonnegative) finite measures on $\left(S,\Sigma \right)$, such that $Q=Q_{ac}+Q_s$, $Q_{ac}\ll P$ and $Q_s\perp P$. That is, for all $B\in \Sigma$ with $P\left(B\right)=0$, one has $Q_{ac}\left(B\right)=0$, and there is some $A\in \Sigma$, such that $P\left(S\setminus A\right)=Q_s\left(A\right)=0$. It then also follows that $Q_{ac}\left(S\setminus A\right)=0$ and $P\left(A\right)=1$. Note also that for all $Z\in B^{+}\left(\Sigma \right)$, $\int ZdQ={\int }_{A}Zd{Q}_{ac}+{\int }_{S\setminus A}Zd{Q}_s$. Furthermore, by the Radon–Nikodým theorem (e.g., Theorem 4.2.2 of [23]), there exists a P-a.s. unique Σ-measurable and P-integrable function:

$$h:S\to \left[0,+\infty \right)$$

such that $Q_{ac}\left(C\right)={\int }_{C}hdP$, for all $C\in \Sigma$.

The Lebesgue decomposition of Q with respect to P suggests a re-writing of the premium constraint appearing in Problem (3) as:

$$\Pi \left(1+\rho \right)\ge \int YdQ={\int }_{A}YhdP+{\int }_{S\setminus A}YdQ;$$

and one can then re-write Problem (3) as follows: for a given loading factor $\rho >0$,

$$\begin{array}{c}\underset{Y\in B^{+}\left(\Sigma \right)}{sup}\left\{{\int }_{A}u\left(W_0-\Pi -X+Y\right)dP+{\int }_{S\setminus A}u\left(W_0-\Pi -X+Y\right)dP\right\}:\hfill \\ \left\{\begin{array}{c}0\le Y{\mathbf{1}}_A+Y{\mathbf{1}}_{S\setminus A}\le X{\mathbf{1}}_A+X{\mathbf{1}}_{S\setminus A}\hfill \\ \Pi \left(1+\rho \right)\ge {\int }_{A}YhdP+{\int }_{S\setminus A}YdQ\hfill \end{array}\right.\hfill \end{array}$$

This then suggests a splitting of Problem (3) into two problems. Each one of these problems is then solved separately, and the individuals solutions hence obtained are then combined appropriately so as to obtain a solution for Problem (3). All details are provided in Appendix C, but for now, consider heuristically the problems:

$$\underset{Y\in B^{+}\left(\Sigma \right)}{sup}\left\{{\int }_{A}u\left(W_0-\Pi -X+Y\right)dP:Y{\mathbf{1}}_A\le X{\mathbf{1}}_A,{\int }_{A}YhdP=\beta \right\},$$

(2)

for an appropriately chosen β; and,

$$\underset{Y\in B^{+}\left(\Sigma \right)}{sup}\left\{{\int }_{S\setminus A}u\left(W_0-\Pi -X+Y\right)dP:Y{\mathbf{1}}_{S\setminus A}\le X{\mathbf{1}}_{S\setminus A},{\int }_{S\setminus A}YdQ\le \alpha \right\},$$

(3)

for an appropriately chosen α. Since $P\left(S\setminus A\right)=0$, any feasible Y for the problem given in Equation (3) is also optimal for that problem. Since $P\left(A\right)=1$, the problem given in Equation (2) can be written as:

$$\underset{Y\in B^{+}\left(\Sigma \right)}{sup}\left\{\int u\left(W_0-\Pi -X+Y\right)dP:Y{\mathbf{1}}_A\le X{\mathbf{1}}_A,\int YhdP=\beta \right\}.$$

(4)

Solving Problem (3) then boils down to solving the problem given in Equation (4). This is a considerably simpler problem than Problem (3), since dealing with the heterogeneity of beliefs appearing in Problem (3) has been reduced to dealing simply with the function h. The following theorem characterizes an optimal solution of Problem (3). Its proof is given in Appendix C.

Theorem 8. 

Suppose that the previous assumptions hold, and for each $\lambda \ge 0$, define the function $Y_{\lambda }^{*}\in B^{+}\left(\Sigma \right)$ by:

$$Y_{\lambda }^{*}:=min\left[X,max\left(0,X-\left[W_0-\Pi -{\left(u^{\prime }\right)}^{-1}\left(\lambda h\right)\right]\right)\right].$$

(5)

If the insurer’s subjective probability measure Q is compatible with the DM’s subjective probability measure P, then there exist:

• some ${\lambda }^{*}\ge 0$; and
• an optimal solution ${\mathcal{Y}}^{*}$ to Problem (3), such that:

  (1) 

  ${\mathcal{Y}}^{*}$ is a nondecreasing function of the loss X and

  (2) 

  ${\mathcal{Y}}^{*}$ has the same distribution as $Y_{{\lambda }^{*}}^{*}$ under P.

Moreover, any other ${\mathcal{Z}}^{*}$ which is nondecreasing in X and which has the same distribution as $Y_{{\lambda }^{*}}^{*}$ under P is such that ${\mathcal{Z}}^{*}={\mathcal{Y}}^{*},P$-a.s. Finally, if the utility function u is strictly concave, then any solution ${\mathcal{Z}}^{*}$ to Problem (3) is such that ${\mathcal{Z}}^{*}={\mathcal{Y}}^{*}$, P-a.s. In particular, any solution is nondecreasing in X, P-a.s.

Theorem 8 characterizes a class of solutions to Problem (3) in terms of their distribution8 for the DM, that is, for the probability measure P. Of course, when $P=Q$, so that there is perfect homogeneity of beliefs as in the classical model, then $h=1$, and so:

$$Y_{\lambda }^{*}=min\left[X,max\left(0,X-\left[W_0-\Pi -{\left(u^{\prime }\right)}^{-1}\left(\lambda \right)\right]\right)\right]=min\left[X,{\left(X-d_{\lambda }\right)}^{+}\right],$$

where $d_{\lambda }:=W_0-\Pi -{\left(u^{\prime }\right)}^{-1}\left(\lambda \right)$. Since $Y_{\lambda }^{*}$ is then a nondecreasing function of X; Theorem 8 simply says that there is some ${\lambda }^{*}$, such that an optimal indemnity schedule ${\mathcal{Y}}^{*}$ for the DM is such that ${\mathcal{Y}}^{*}=min\left[X,{\left(X-d_{{\lambda }^{*}}\right)}^{+}\right],P$-a.s., which is a result similar to Arrow’s classical theorem (Theorem 5). This is stated below, and the proof is omitted.

Corollary 9. 

Suppose that the previous assumptions hold, and suppose also that $P=Q$. For each $\lambda \ge 0$, let $d_{\lambda }:=W_0-\Pi -{\left(u^{\prime }\right)}^{-1}\left(\lambda \right)$, and define the function $Y_{\lambda }^{*}\in B^{+}\left(\Sigma \right)$ by:

$$Y_{\lambda }^{*}:=min\left[X,max\left(0,X-d_{\lambda }\right)\right].$$

(6)

Then, there exists a ${\lambda }^{*}\ge 0$ and an optimal solution ${\mathcal{Y}}^{*}$ to Problem (3), such that ${\mathcal{Y}}^{*}=Y_{{\lambda }^{*}}^{*},P$-a.s.

As a second consequence of Theorem 8, we obtain one of the results of Ghossoub [9]. Namely, suppose that the probability measure Q is absolutely continuous with respect to the probability measure P, with a Radon–Nikodým derivative $h:S\to \left[0,+\infty \right)$, given by $\tilde{h}=dQdP=\Psi \circ X$, for some Borel-measurable and $P\circ X^{-1}$-integrable map $\Psi :X\left(S\right)\to \left[0,+\infty \right)$. The function Ψ can be interpreted as a likelihood ratio. Ghossoub [9] showed that under an assumption of monotonicity on the likelihood ratio Ψ, the optimal indemnity schedule is a variable deductible schedule:

Corollary 10 (Ghossoub [9]). 

If assumptions 1, 2 and 4 hold and if the function Ψ is nonincreasing, then Problem (3) admits a solution which is nondecreasing on the range of the random loss X, and an optimal such indemnity schedule for the DM takes the form:

$$Y_{{\lambda }^{*}}^{*}=min\left[X,max\left(0,X-d\left(\tilde{h}\right)\right)\right],$$

where $d(\tilde{h})=\left[W_0-\Pi -{\left(u^{\prime }\right)}^{-1}\left({\lambda }^{*}\tilde{h}\right)\right]$ and ${\lambda }^{*}>0$ is chosen, so that the premium constraint binds.

Corollary 10 is an immediate consequence of Theorem 8, since in this case, the optimal indemnity is a nondecreasing function of the loss X (by concavity of the utility function and by the fact that Ψ is nonincreasing). Corollary 10 states that monotonicity of the Radon–Nikodým derivative yields that the optimal indemnity schedule for the DM takes the form of a variable deductible schedule, with a state-contingent deductible $d(\tilde{h})$ that depends on the sate of the world only through the Radon–Nikodým derivative $\tilde{h}$. Therefore, when $P=Q$, $\tilde{h}$ is the constant function that equals one, and hence, one recovers Arrow’s result.

Theorem 8 above asserts the existence of an optimal indemnity schedule and shows that optimal indemnity schedules are nondecreasing functions of the loss X. It also characterizes a class of solutions to Problem (3) in terms of their distribution for the DM, that is, for the probability measure P. However, Theorem 8 does not give any indication as to what an optimal indemnity schedule looks like. It turns out that an optimal indemnity schedule might take “any” form, as long it has the same distribution as that specified in Theorem 8 and as long as it satisfies the imposed constraints. This can be seen as a formalisation of the results of Marshall [10].

Corollary 11 (A Formalisation of Marshall’s Argument). 

Under the previous assumptions and provided the insurer’s subjective probability measure Q is compatible with the DM’s subjective probability measure P, there exists an optimal solution ${\mathcal{Y}}^{*}$ to Problem (3), which is nondecreasing in the loss X and such that for P-a.a. $s\in S$,

$$\begin{array}{c}\hfill {\mathcal{Y}}^{*}\left(s\right)=\left\{\begin{array}{c}0iffX\left(s\right)\in \left[0,a^{*}\right),\hfill \\ f\left(X\left(s\right)\right)iffX\left(s\right)\in \left[a^{*},b^{*}\right],\hfill \\ X\left(s\right)iffX\left(s\right)\in \left(b^{*},M\right],\hfill \end{array}\right.\end{array}$$

(7)

for some $a^{*}$ and $b^{*}$, such that $0\le a^{*}\le b^{*}\le M$, and a nondecreasing, left-continuous and Borel-measurable function $f:\left[0,M\right]\to \left[0,M\right]$, such that $0\le f\left(t\right)\le t$ for each $t\in \left[a^{*},b^{*}\right]$.

The proof of Corollary 11 is given in Appendix D. When $a^{*}>0$, the indemnity schedule ${\mathcal{Y}}^{*}$ includes a deductible provision, whereby no indemnification is paid to the DM for a loss of amount less than $a^{*}$. Sufficient conditions for $a^{*}$ appearing in Equation (7) to be strictly positive are given in Appendix E. When $b^{*}<M$, the indemnity schedule ${\mathcal{Y}}^{*}$ fully reimburses9 losses of magnitude larger than $b^{*}$. Sufficient conditions for $b^{*}$ appearing in Equation (7) to be strictly less than M are given in Appendix F. For losses of magnitude in the range $\left[a^{*},b^{*}\right]$, ${\mathcal{Y}}^{*}$ is of the form $f\left(X\right)$, where f is a nondecreasing, Borel-measurable and left-continuous function such that $0\le f\left(t\right)\le t$, for all t in the range of X. Nothing else can be said about the function f, in terms of concavity, convexity and inflection points, for instance. In this sense, the optimal indemnity schedule ${\mathcal{Y}}^{*}$ may take any form. This is reminiscent of the results of Marshall [10].

4. Conclusions

The classical approach to problems of optimal insurance design assumes that the insurer and the insured assign the same distribution to the insurable random loss. Recently, Ghossoub [9] considered a setting in which the two parties have different subjective beliefs about the realisations of the insurable loss. Under a requirement of compatibility between the insurer’s and the insured’s subjective beliefs that is weaker than the Monotone Likelihood Ratio (MLR) condition, Ghossoub [9] showed the existence and monotonicity of optimal indemnity schedules. However, Ghossoub [9] only provided an analytical characterization of the optimal indemnity in the special case of an MLR.

In this paper, we extended the analysis of Ghossoub [9] to the general case of belief heterogeneity, allowing for bona fide disagreement about zero-probability events. We gave a characterization of the class of optimal indemnity schedules in terms of their distribution for the DM’s belief, and we showed how Arrow’s classical result on the optimality of a deductible contract can be obtained as a special case. We also showed that even though we can characterize the distribution of an optimal indemnity, we cannot give an exact characterization of its shape: this is a formalization of Marshall’s [10] argument that in the general case, an optimal insurance schedule may take “any” form.