# Optimal Insurance for a Minimal Expected Retention: The Case of an Ambiguity-Seeking Insurer

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## Abstract

**:**

## 1. Introduction

#### 1.1. Ambiguity in Optimal Insurance Design

#### 1.2. Related Literature

#### 1.3. This Paper’s Contribution

- (1)
- We first examine the problem of insurance design when the insurer’s beliefs are a concave distortion of a probability measure that differs from the insurer’s probability measure. We show that there is an event A to which the insurer assigns full probability and on which the optimal indemnity schedule is a state-contingent deductible schedule, with a state-contingent deductible that is a function of the insurer’s distortion function and of a likelihood ratio between the two parties’ beliefs. On the complement of the event A, the optimal indemnity is full insurance. However, the insurer assigns zero probability to the complement of A.
- (2)
- As a special case of the above, we examine the problem of insurance design when the insurer’s beliefs are a concave distortion of the insured’s probability measure. We show that in this case, the result of AGP [14] mentioned above still holds: the optimal indemnity schedule is a state-contingent deductible schedule, in which the deductible depends on the state of the world only through the insurer’s distortion function. Arrow’s result then obtains as a special case when the insurer does not distort probabilities.

#### 1.4. Outline

## 2. Optimal Insurance: The Classical Case

#### 2.1. Setup and Preliminaries

**Assumption 1.**

#### 2.2. The Insurance Design Problem

**Theorem 1 (Arrow).**

#### 2.3. Premium Constraint vs. Minimal Expected Retention Constraint

## 3. The Case of an Ambiguity-Seeking Insurer

#### 3.1. Preliminaries: Capacities, Choquet Integration and the CEU Model

**Definition 1.**

- (1)
- $\upsilon \left(\varnothing \right)=0$;
- (2)
- $\upsilon \left(S\right)=1$; and,
- (3)
- υ is monotone: for any $A,B\in \Sigma ,\phantom{\rule{4pt}{0ex}}A\subseteq B\Rightarrow \upsilon \left(A\right)\le \upsilon \left(B\right)$.

- supermodular (or convex) if $\upsilon \left(\right)open="("\; close=")">A\cup B+\upsilon \left(\right)open="("\; close=")">A\cap B$, for all $A,B\in \Sigma $; and,
- submodular (or concave) if $\upsilon \left(\right)open="("\; close=")">A\cup B+\upsilon \left(\right)open="("\; close=")">A\cap B$, for all $A,B\in \Sigma $.

**Definition 2.**

- The core of ${\upsilon}_{1}$, denoted by $core\left(\right)open="("\; close=")">{\upsilon}_{1}$, is the collection of all probability measures Q on $\left(\right)open="("\; close=")">S,\Sigma $, such that $Q\left(A\right)\ge \upsilon \left(A\right),\forall A\in \Sigma $.
- The anti-core of ${\upsilon}_{2}$, denoted by $acore\left(\right)open="("\; close=")">{\upsilon}_{2}$, is the collection of all probability measures Q on $\left(\right)open="("\; close=")">S,\Sigma $, such that $Q\left(A\right)\le \upsilon \left(A\right),\forall A\in \Sigma $.

**Definition 3.**

**Proposition 1**

- If υ is supermodular, then $\int Y\phantom{\rule{4pt}{0ex}}d\upsilon =min\left(\right)open="\{"\; close="\}">\int YdP:P\in core\left(\right)open="("\; close=")">{\upsilon}_{1}$;
- If υ is submodular, then $\int Y\phantom{\rule{4pt}{0ex}}d\upsilon =max\left(\right)open="\{"\; close="\}">\int YdP:P\in acore\left(\right)open="("\; close=")">{\upsilon}_{2}$.

**Definition 4.**

#### 3.2. The Insurance Design Problem

**Assumption 2.**

- (1)
- $u\left(0\right)=0$;
- (2)
- u is strictly increasing and strictly concave;
- (3)
- u is continuously differentiable.
- (4)
- The first derivative satisfies ${u}^{\prime}\left(0\right)=+\infty $ and $\underset{x\to +\infty}{lim}}{u}^{\prime}\left(x\right)=0$.

**Assumption 3.**

**Assumption 4.**

- (1)
- Q is a probability measure on $\left(\right)open="("\; close=")">S,\Sigma $, such that $Q\circ {X}^{-1}$ is nonatomic;
- (2)
- $T:\left(\right)open="["\; close="]">0,1\to \left(\right)open="["\; close="]">0,1$ is increasing, concave and twice differentiable; and
- (3)
- $T\left(0\right)=0$ and $T\left(1\right)=1$.

#### 3.3. A Characterization of the Optimal Indemnity Schedule

**Definition 5.**

- (1)
- $f\left(z\right)\le {F}_{X}^{-1}\left(z\right)$, for each $0<z<1$;
- (2)
- $f\left(z\right)\ge 0$, for each $0<z<1$.

- $P={P}_{ac}+{P}_{s}$;
- ${P}_{ac}<<Q$ (${P}_{ac}$ is absolutely continuous with respect to Q); and,
- ${P}_{s}\perp Q$ (${P}_{s}$ and Q are mutually singular).

**Assumption 5.**

**Theorem 2.**

**Assumption 6.**

**Assumption 7.**

**Corollary 1 (The Shape of an Optimal Indemnity Schedule).**

#### 3.4. A Special Case

- $P={P}_{ac}=Q$;
- $h=\varphi \circ X=d{P}_{ac}dQ$ is the constant function equal to one. Hence, Assumptions 5 and 6 trivially hold;
- $A=S$, and hence, $S\backslash A=\varnothing $;
- $U={F}_{X}\left(X\right)$ is a random variable on the probability space $\left(\right)open="("\; close=")">S,\Sigma ,P$ with a uniform distribution on $\left(\right)open="("\; close=")">0,1$.

**Corollary 2.**

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A. More on Capacities and Choquet Integration

**Definition A1.**A capacity υ on $\left(\right)open="("\; close=")">S,\Sigma $ is continuous from above (respectively below) if for any sequence ${\left\{{A}_{n}\right\}}_{n\ge 1}\subseteq \Sigma $, such that ${A}_{n+1}\subseteq {A}_{n}$ (respectively ${A}_{n+1}\supseteq {A}_{n}$) for each n, it holds that:

**Proposition A1.**

- (1)
- If ${\varphi}_{1},{\varphi}_{2}\in B\left(\Sigma \right)$, then in general, $\int \left(\right)open="("\; close=")">{\varphi}_{1}+{\varphi}_{2}\phantom{\rule{4pt}{0ex}}d\upsilon \ne \int {\varphi}_{1}\phantom{\rule{4pt}{0ex}}d\upsilon +\int {\varphi}_{2}\phantom{\rule{4pt}{0ex}}d\upsilon $.
- (2)
- If ${\varphi}_{1},{\varphi}_{2}\in B\left(\Sigma \right)$ are comonotonic, then $\int \left(\right)open="("\; close=")">{\varphi}_{1}+{\varphi}_{2}\phantom{\rule{4pt}{0ex}}d\upsilon =\int {\varphi}_{1}\phantom{\rule{4pt}{0ex}}d\upsilon +\int {\varphi}_{2}\phantom{\rule{4pt}{0ex}}d\upsilon $.
- (3)
- If ${\varphi}_{1},{\varphi}_{2}\in B\left(\Sigma \right)$ are such that ${\varphi}_{1}\le {\varphi}_{2}$, then $\int {\varphi}_{1}\phantom{\rule{4pt}{0ex}}d\upsilon \le \int {\varphi}_{2}\phantom{\rule{4pt}{0ex}}d\upsilon $.
- (4)
- For all $\varphi \in B\left(\Sigma \right)$ and all $c\ge 0$, then $\int c\varphi \phantom{\rule{4pt}{0ex}}d\upsilon =c\int \varphi \phantom{\rule{4pt}{0ex}}d\upsilon $.
- (5)
- If υ is submodular, then for any ${\varphi}_{1},{\varphi}_{2}\in B\left(\Sigma \right)$, $\int \left(\right)open="("\; close=")">{\varphi}_{1}+{\varphi}_{2}\phantom{\rule{4pt}{0ex}}d\upsilon \le \int {\varphi}_{1}\phantom{\rule{4pt}{0ex}}d\upsilon +\int {\varphi}_{2}\phantom{\rule{4pt}{0ex}}d\upsilon $.

## Appendix B. Rearrangements and Supermodularity

#### Appendix B.1. The Nondecreasing Rearrangement

**Definition A2.**

**Definition A3.**

**Proposition A2.**

- (1)
- $\tilde{I}$ is left-continuous, nondecreasing and Borel-measurable;
- (2)
- $\tilde{I}\left(0\right)=0$ and $\tilde{I}\left(M\right)\le M$. Therefore, $\tilde{I}\left(\right)open="("\; close=")">\left(\right)open="["\; close="]">0,M$;
- (3)
- If ${I}_{1},{I}_{2}:\left(\right)open="["\; close="]">0,M\to \left(\right)open="["\; close="]">0,M$ are such that ${I}_{1}\le {I}_{2},\phantom{\rule{4pt}{0ex}}\zeta $-a.s., then ${\tilde{I}}_{1}\le {\tilde{I}}_{2}$;
- (4)
- $\tilde{I}$ is ζ-equimeasurable with I, in the sense that for any Borel set B,$$\zeta \left(\left(\right),t,\in ,\left(\right),0,,,M\right):I\left(t\right)\in B$$
- (5)
- If $\overline{I}:\left(\right)open="["\; close="]">0,M\to {\mathbb{R}}^{+}$ is another nondecreasing, Borel-measurable map, which is ζ-equimeasurable with I, then $\overline{I}=\tilde{I},\phantom{\rule{4pt}{0ex}}\zeta $-a.s.

- (1)
- $Y,\tilde{Y}\in {B}^{+}\left(\Sigma \right)$, since I and $\tilde{I}$ are Borel-measurable mappings of $\left(\right)open="["\; close="]">0,M$ into itself;
- (2)
- $\tilde{Y}$ is a nondecreasing function of X:$\left(\right)open="["\; close="]">X\left(s\right)\le X\left(\right)open="("\; close=")">{s}^{\prime}$, for all $s,{s}^{\prime}\in S$; and
- (3)
- Y and $\tilde{Y}$ have the same distribution under P (i.e., they are P-equimeasurable):$P\left(\right)open="("\; close=")">\left(\right)open="\{"\; close="\}">s\in S:Y\left(s\right)\le \alpha $, for any $\alpha \in \left(\right)open="["\; close="]">0,M$.

**Lemma A1.**

#### Appendix B.2. Supermodularity

**Definition A4.**

**Lemma A2.**

**Example A1.**If $g:\mathbb{R}\to \mathbb{R}$ is concave and $a\in \mathbb{R}$, then the function ${L}_{1}:{\mathbb{R}}^{2}\to \mathbb{R}$ defined by ${L}_{1}\left(\right)open="("\; close=")">x,y=g\left(\right)open="("\; close=")">a-x+y$ is supermodular. If, moreover, g is strictly concave, then ${L}_{1}$ is strictly supermodular.

**Lemma A3 (Hardy–Littlewood).**

## Appendix C. Proof of Theorem 2

**Assumption A1.**

#### Appendix C.1. “Splitting”

**Remark A1.**

**Definition A5.**

**Definition A6.**

**Lemma A4.**

**Proof.**

**Lemma A5.**

**Proof.**

**Remark A2.**

**Lemma A6.**

**Proof.**

**Lemma A7.**

**Proof.**

**Lemma A8.**

#### Appendix C.2. Solving Problem (A11)

**Lemma A10.**

**Proof.**

**Definition A7.**

- (1)
- $\int u\left(\right)open="("\; close=")">{W}_{0}-\Pi -{Z}_{2}\phantom{\rule{4pt}{0ex}}h\phantom{\rule{4pt}{0ex}}dQ\ge \int u\left(\right)open="("\; close=")">{W}_{0}-\Pi -{Z}_{1}$; and,
- (2)
- $\int {Z}_{2}\phantom{\rule{4pt}{0ex}}dT\circ Q=\int {Z}_{1}\phantom{\rule{4pt}{0ex}}dT\circ Q$.

**Lemma A11.**

**Proof.**

#### Appendix C.3. Quantile Reformulation

**Lemma A12.**

**Proof.**

## Appendix D. Proof of Corollary 1

**Corollary A1.**

**Proof.**

**Lemma A13.**

- (1)
- ${\int}_{0}^{1}{T}^{\prime}\left(\right)open="("\; close=")">1-t{f}^{*}\left(t\right)\phantom{\rule{4pt}{0ex}}dt={R}_{0}$;
- (2)
- There exists $\lambda \le 0$, such that for all $t\in \left(\right)open="("\; close=")">0,1\backslash \{t:\varphi \circ {F}_{x}^{-1}\left(t\right)=0\}$,$${f}^{*}\left(t\right)=\underset{0\le y\le {F}_{X}^{-1}\left(t\right)}{arg\; max}\left(\right)open="["\; close="]">u\left(\right)open="("\; close=")">{W}_{0}-\Pi -y\varphi \left(\right)open="("\; close=")">{F}_{X}^{-1}\left(t\right)$$

**Proof.**

**Lemma A14.**

**Proof.**

- (i)
- $0\le Y\le X$; and,
- (ii)
- $\int \left(\right)open="("\; close=")">X-Y\phantom{\rule{4pt}{0ex}}dT\circ Q={R}_{0}$

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^{1.}We thank one of the reviewers for bringing this to our attention.^{2.}This is a standard assumption, and it holds in many instances, such as when it is assumed that a probability density function for X exists.^{3.}A finite nonnegative measure η on a measurable space $\left(\right)open="("\; close=")">\Omega ,\mathcal{A}$ is said to be nonatomic if for any $A\in \mathcal{A}$ with $\eta \left(A\right)>0$, there is some $B\in \mathcal{A}$, such that $B\u228aA$ and $0<\eta \left(B\right)<\eta \left(A\right)$.^{4.}Indeed, suppose that $T\left(t\right)=t$, the identity function. Then, ${T}^{\prime}=1$, and so, $\frac{{T}^{\prime \left(\right)open="("\; close=")">1-t}}{}\varphi \left(\right)open="("\; close=")">{F}_{X}^{-1}\left(t\right)$. Moreover, the function $t\mapsto {F}_{X}^{-1}\left(t\right)$ is nondecreasing, since ${F}_{X}^{-1}$ is a quantile function. Therefore, the function $t\mapsto \frac{{T}^{\prime \left(\right)open="("\; close=")">1-t}}{}\varphi \left(\right)open="("\; close=")">{F}_{X}^{-1}\left(t\right)$ is nondecreasing if and only if the function ϕ is nonincreasing.^{5.}Any monotone function is Borel-measurable and, hence, “almost continuous”, in view of Lusin’s Theorem ([49] Theorem 7.5.2). Furthermore, any monotone function is almost surely continuous for the Lebesgue measure.^{6.}Note that this variable deductible is anti-comonotonic with the loss X, since T is concave, u is increasing and concave (and hence, ${\left({u}^{\prime}\right)}^{-1}$ is decreasing by the inverse function theorem ([50] pp. 221–223)), $X={F}_{X}^{-1}\left(V\right)$, P − a.s., and ${F}_{X}$ is a nondecreasing function. However, the indemnity schedule is commonotonic with X

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**MDPI and ACS Style**

Amarante, M.; Ghossoub, M.
Optimal Insurance for a Minimal Expected Retention: The Case of an Ambiguity-Seeking Insurer. *Risks* **2016**, *4*, 8.
https://doi.org/10.3390/risks4010008

**AMA Style**

Amarante M, Ghossoub M.
Optimal Insurance for a Minimal Expected Retention: The Case of an Ambiguity-Seeking Insurer. *Risks*. 2016; 4(1):8.
https://doi.org/10.3390/risks4010008

**Chicago/Turabian Style**

Amarante, Massimiliano, and Mario Ghossoub.
2016. "Optimal Insurance for a Minimal Expected Retention: The Case of an Ambiguity-Seeking Insurer" *Risks* 4, no. 1: 8.
https://doi.org/10.3390/risks4010008