# 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 $\upsilon \left(A\cup B\right)+\upsilon \left(A\cap B\right)\ge \upsilon \left(A\right)+\upsilon \left(B\right)$, for all $A,B\in \Sigma $; and,
- submodular (or concave) if $\upsilon \left(A\cup B\right)+\upsilon \left(A\cap B\right)\le \upsilon \left(A\right)+\upsilon \left(B\right)$, for all $A,B\in \Sigma $.

**Definition 2.**

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

**Definition 3.**

**Proposition 1**

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

**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 $\left(S,\Sigma \right)$, such that $Q\circ {X}^{-1}$ is nonatomic;
- (2)
- $T:\left[0,1\right]\to \left[0,1\right]$ 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 $\left(S,\Sigma ,P\right)$ with a uniform distribution on $\left(0,1\right)$.

**Corollary 2.**

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A. More on Capacities and Choquet Integration

**Definition A1.**A capacity υ on $\left(S,\Sigma \right)$ 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({\varphi}_{1}+{\varphi}_{2}\right)\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({\varphi}_{1}+{\varphi}_{2}\right)\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({\varphi}_{1}+{\varphi}_{2}\right)\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, $\tilde{I}\left(\left[0,M\right]\right)\subseteq \left[0,M\right]$;
- (3)
- If ${I}_{1},{I}_{2}:\left[0,M\right]\to \left[0,M\right]$ 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,$$\zeta \left(\left\{t\in \left[0,M\right]:I\left(t\right)\in B\right\}\right)=\zeta \left(\left\{t\in \left[0,M\right]:\tilde{I}\left(t\right)\in B\right\}\right)$$
- (5)
- If $\overline{I}:\left[0,M\right]\to {\mathbb{R}}^{+}$ 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 $\left[0,M\right]$ into itself;
- (2)
- $\tilde{Y}$ is a nondecreasing function of X:$\left[X\left(s\right)\le X\left({s}^{\prime}\right)\right]\Rightarrow \left[\tilde{Y}\left(s\right)\le \tilde{Y}\left({s}^{\prime}\right)\right]$, 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):$P\left(\left\{s\in S:Y\left(s\right)\le \alpha \right\}\right)=P\left(\left\{s\in S:\tilde{Y}\left(s\right)\le \alpha \right\}\right)$, for any $\alpha \in \left[0,M\right]$.

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

**Proof.**

**Lemma A14.**

**Proof.**

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

## References

- K. Arrow. Essays in the Theory of Risk-Bearing. Chicago, IL, USA: Markham Publishing Company, 1971. [Google Scholar]
- B. De Finetti. “La Prévision: Ses Lois Logiques, Ses Sources Subjectives.” Ann. l’Inst. Henri Poincaré 7 (1937): 1–68. [Google Scholar]
- L. Savage. The Foundations of Statistics, 2nd ed. New York, NY, USA: Dover Publications, 1972. [Google Scholar]
- J. Von Neumann, and O. Morgenstern. Theory of Games and Economic Behavior. Princeton, NJ, USA: Princeton University Press, 1944. [Google Scholar]
- F. Knight. Risk, Uncertainty, and Profit. Boston, MA, USA; New York, NY, USA: Houghton Mifflin, 1921. [Google Scholar]
- D. Ellsberg. “Risk, Ambiguity, and the Savage Axioms.” Q. J. Econ. 75 (1961): 643–669. [Google Scholar] [CrossRef]
- C. Camerer. “Individual Decision Making.” In Handbook of Experimental Economics. Edited by J.H. Kagel and A.E. Roth. Princeton, NJ, USA: Princeton University Press, 1995. [Google Scholar]
- D. Schmeidler. “Subjective Probability and Expected Utility without Additivity.” Econometrica 57 (1989): 571–587. [Google Scholar] [CrossRef]
- I. Gilboa, and D. Schmeidler. “Maxmin Expected Utility with a Non-Unique Prior.” J. Math. Econ. 18 (1989): 141–153. [Google Scholar] [CrossRef]
- P. Ghirardato, F. Maccheroni, and M. Marinacci. “Differentiating Ambiguity and Ambiguity Attitude.” J. Econ. Theory 118 (2004): 133–173. [Google Scholar] [CrossRef]
- M. Amarante. “Foundations of Neo-Bayesian Statistics.” J. Econ. Theory 144 (2009): 2146–2173. [Google Scholar] [CrossRef]
- I. Gilboa, and M. Marinacci. “Ambiguity and the Bayesian Paradigm.” In Advances in Economics and Econometrics: Theory and Applications, Tenth World Congress of the Econometric Society. Edited by D. Acemoglu, M. Arellano and E. Dekel. New York, NY, USA: Cambridge University Press, 2013. [Google Scholar]
- R. Hogarth, and H. Kunreuther. “Risk, Ambiguity, and Insurance.” J. Risk Uncertain. 2 (1989): 5–35. [Google Scholar] [CrossRef]
- M. Amarante, M. Ghossoub, and E. Phelps. “Ambiguity on the Insurer’s Side: The Demand for Insurance.” J. Math. Econ. 58 (2015): 61–78. [Google Scholar] [CrossRef]
- D. Alary, C. Gollier, and N. Treich. “The Effect of Ambiguity Aversion on Insurance and Self-Protection.” Econ. J. 123 (2013): 1188–1202. [Google Scholar] [CrossRef]
- P. Klibanoff, M. Marinacci, and S. Mukerji. “A Smooth Model of Decision Making under Ambiguity.” Econometrica 73 (2005): 1849–1892. [Google Scholar] [CrossRef]
- C. Gollier. “Optimal Insurance Design of Ambiguous Risks.” Economic Theory 57 (2014): 555–576. [Google Scholar] [CrossRef]
- M. Jeleva. “Background Risk, Demand for Insurance, and Choquet Expected Utility Preferences.” GENEVA Pap. Risk Insu.-Theory 25 (2000): 7–28. [Google Scholar] [CrossRef]
- V. Young. “Optimal Insurance under Wang’s Premium Principle.” Insur. Math. Econ. 25 (1999): 109–122. [Google Scholar] [CrossRef]
- C. Bernard, X. He, J. Yan, and X. Zhou. “Oprimal Insurance Design under Rank-Dependent Expected Utility.” Math. Financ. 25 (2015): 154–186. [Google Scholar] [CrossRef]
- J. Quiggin. “A Theory of Anticipated Utility.” J. Econ. Behav. 3 (1982): 323–343. [Google Scholar] [CrossRef]
- M. Yaari. “The Dual Theory of Choice under Risk.” Econometrica 55 (1987): 95–115. [Google Scholar] [CrossRef]
- N. Doherty, and L. Eeckhoudt. “Optimal Insurance without Expected Utility: The Dual Theory and the Linearity of Insurance Contracts.” J. Risk Uncertain. 10 (1995): 157–179. [Google Scholar] [CrossRef]
- E. Karni. “Optimal Insurance: A Nonexpected Utility Analysis.” In Contributions to Insurance Economics. Edited by G. Dionne. Boston, MA, USA: Kluwer Academic Publishers, 1992. [Google Scholar]
- M. Machina. “Non-Expected Utility and the Robustness of the Classical Insurance Paradigm.” GENEVA Pap. Risk Insur.-Theory 20 (1995): 9–50. [Google Scholar] [CrossRef]
- H. Schlesinger. “Insurance Demand without the Expected-Utility Paradigm.” J. Risk Insur. 64 (1997): 19–39. [Google Scholar] [CrossRef]
- G. Carlier, R. Dana, and N. Shahidi. “Efficient Insurance Contracts under Epsilon-Contaminated Utilities.” GENEVA Pap. Risk Insur.-Theory 28 (2003): 59–71. [Google Scholar] [CrossRef]
- S. Anwar, and M. Zheng. “Competitive Insurance Market in the Presence of Ambiguity.” Insur. Math. Econ. 50 (2012): 79–84. [Google Scholar] [CrossRef]
- G. Carlier, and R. Dana. Insurance Contracts with Deductibles and Upper Limits. Paris, France: Universite Paris Dauphine, 2002, Preprint, Ceremade. [Google Scholar]
- G. Carlier, and R. Dana. “Core of Convex Distortions of a Probability.” J. Econ. Theory 113 (2003): 199–222. [Google Scholar] [CrossRef]
- G. Carlier, and R. Dana. “Two-persons Efficient Risk-sharing and Equilibria for Concave Law-invariant Utilities.” Econ. Theory 36 (2008): 189–223. [Google Scholar] [CrossRef]
- A. Chateauneuf, R. Dana, and J. Tallon. “Optimal Risk-sharing Rules and Equilibria with Choquet-expected-utility.” J. Math. Econ. 34 (2000): 191–214. [Google Scholar] [CrossRef]
- A. Balbás, B. Balbás, R. Balbás, and A. Heras. “Optimal Reinsurance under Risk and Uncertainty.” Insur. Math. Econ. 60 (2015): 61–74. [Google Scholar] [CrossRef]
- E. Ert, and S. Trautmann. “Sampling Experience Reverses Preferences for Ambiguity.” J. Risk Uncertain. 49 (2014): 31–42. [Google Scholar] [CrossRef]
- M. Kocher, A. Lahno, and S. Trautmann. Ambiguity Aversion is the Exception. Munich, Germany: CESifo Group, 2015, Preprint, CESifo Working Paper Series No. 5261. [Google Scholar]
- S. Trautmann, and G. van de Kuilen. “Ambiguity Attitudes.” In The Wiley Blackwell Handbook of Judgment and Decision Making. Edited by G. Keren and G. Wu. Oxford, UK: Wiley-Blackwell, 2016. [Google Scholar]
- C. Aliprantis, and K. Border. Infinite Dimensional Analysis, 3rd ed. Heidelberg, Germany: Springer-Verlag, 2006. [Google Scholar]
- A. Pichler. “Insurance Pricing under Ambiguity.” Eur. Actuar. J. 4 (2014): 335–364. [Google Scholar] [CrossRef]
- M. Ghossoub. “Vigilant Measures of Risk and the Demand for Contingent Claims.” Insur. Math. Econ. 61 (2015): 27–35. [Google Scholar] [CrossRef]
- D. Denneberg. Non-Additive Measure and Integral. Dordrecht, Netherlands: Kluwer Academic Publishers, 1994. [Google Scholar]
- M. Marinacci, and L. Montrucchio. “Introduction to the Mathematics of Ambiguity.” In Uncertainty in Economic Theory: Essays in Honor of David Schmeidlers 65th Birthday. Edited by I. Gilboa. London, UK: Routledge, 2004, pp. 46–107. [Google Scholar]
- D. Schmeidler. “Integral Representation without Additivity.” Proc. Am. Math. Soc. 97 (1986): 255–261. [Google Scholar] [CrossRef]
- D. Cohn. Measure Theory. Boston, MA, USA: Birkhauser, 1980. [Google Scholar]
- H. Föllmer, and A. Schied. Stochastic Finance: An Introduction in Discrete Time, 3rd ed. Berlin, Germany: Walter de Gruyter, 2011. [Google Scholar]
- H. Jin, and X. Zhou. “Behavioral Portfolio Selection in Continous Time.” Math. Financ. 18 (2008): 385–426. [Google Scholar] [CrossRef]
- X. He, and X. Zhou. “Portfolio Choice via Quantiles.” Math. Financ. 21 (2011): 203–231. [Google Scholar] [CrossRef]
- H. Jin, and X. Zhou. “Greed, Leverage, and Potential Losses: A Prospect Theory Perspective.” Math. Financ. 23 (2013): 122–142. [Google Scholar] [CrossRef]
- G. Carlier, and R. Dana. “Optimal Demand for Contingent Claims when Agents Have Law Invariant Utilities.” Math. Financ. 21 (2011): 169–201. [Google Scholar] [CrossRef]
- R. Dudley. Real Analysis and Probability. New York, NY, USA: Cambridge University Press, 2002. [Google Scholar]
- W. Rudin. Principles of Mathematical Analysis, 3rd ed. New York, NY, USA: McGraw-Hill Book Company, 1976. [Google Scholar]
- M. Ghossoub. “Equimeasurable Rearrangements with Capacities.” Math. Oper. Res. 40 (2015): 429–445. [Google Scholar] [CrossRef]
- M. Ghossoub. “Contracting under Heterogeneous Beliefs.” Ph.D. Thesis, Department of Statistics & Actuarial Science, University of Waterloo, Waterloo, ON, Canada, May 2011. [Google Scholar]

^{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 $\left(\Omega ,\mathcal{A}\right)$ 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, $\frac{{T}^{\prime \left(1-t\right)}}{\varphi \left({F}_{X}^{-1}\left(t\right)\right)}=\frac{1}{\varphi \left({F}_{X}^{-1}\left(t\right)\right)}$. Moreover, the function $t\mapsto {F}_{X}^{-1}\left(t\right)$ is nondecreasing, since ${F}_{X}^{-1}$ is a quantile function. Therefore, the function $t\mapsto \frac{{T}^{\prime \left(1-t\right)}}{\varphi \left({F}_{X}^{-1}\left(t\right)\right)}$ 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