Optimal insurance design of ambiguous risks

Abstract

We examine the characteristics of the optimal insurance contract under linear transaction costs and an ambiguous distribution of losses. Under the standard expected utility model, we know from Arrow (1965) that it contains a straight deductible. In this paper, we assume that the policyholder is ambiguity averse in the sense of Klibanoff et al. (Econometrica 73(6):1849–1892, 2005). The optimal contract depends upon the structure of the ambiguity. For example, if the set of possible priors can be ranked according to the monotone likelihood ratio order, the optimal contract contains a disappearing deductible. We also show that the policyholder’s ambiguity aversion may have the counterintuitive effect to reduce the optimal insurance coverage of an ambiguous risk.

Appendix

1.1 Proof of Proposition 5

Suppose that

$$\begin{aligned} \phi _t (u)=u+t\psi (u), \end{aligned}$$

(19)

for all $u$ in the domain of $\phi$, with $t\in {\mathbb {R}}$ and $\psi$ is a smooth increasing and concave function. We examine the properties of the optimal contract around $t=0$, i.e., when the degree of ambiguity aversion is small. When $t=0$, we know from Proposition 1 that the optimal contract has a straight deductible $D^{*}$. It yields

$$\begin{aligned} u^{\prime }(w_0 -x+P)<\mu (1+\tau ) \end{aligned}$$

(20)

for all $x<D^{*}$. Under this contract, the presence of ambiguity for losses below $D^{*}$ implies that the conditional expected utilities $U_\theta$ vary with $\theta$. Now, consider a positive $t$. The optimal contract solves the set of first-order conditions (10) and (11). By the implicit function theorem, we know that the characteristics of the optimal contract are continuous in $t$ for $t$ close to zero. So are the conditional EU $U_\theta (t)$. Define function $K$ such that

$$\begin{aligned} K(t,x)=\frac{dG(x)}{dH(x,t)}=\frac{\left( {\sum _{\theta =1}^n {q_\theta \phi _t^{\prime } (U_\theta (t))} } \right) \left( {\sum _{\theta =1}^n {q_\theta dF(x,\theta } )} \right) }{\sum _{\theta =1}^n {q_\theta \phi _t^{\prime } (U_\theta (t))dF(x,\theta } )} \end{aligned}$$

(21)

for all $t$ and $x$, where $H(x,t)$ is the distorted probability distribution expressed as a function of parameter $t$. Observe that $K(0,x)=1$ for all $x$. By continuity, condition (20) implies that if $t$ is small enough, then it is still true that

$$\begin{aligned} u^{\prime }(w_0 -x+P)<\mu (1+\tau )K(t,x) \end{aligned}$$

(22)

This means that the first-order condition (10) is still satisfied as a strict inequality for all $x<D^{*}$, or that $I(x)=0$ for this subset of losses. The existence of a unique deductible for unambiguous states is a consequence of Proposition 4. $\square$

1.2 Proof of Proposition 6

Property $i$ is a direct consequence of Proposition 4, since the degree of ambiguity is constant in all unambiguous states $x\notin \left\{ {x_1,x_2}\right\}$. Let $D_0$ be defined by the following condition:

$$\begin{aligned} u^{\prime }(w_0 -D_0 -P)=\mu (1+\tau ). \end{aligned}$$

(23)

We first show that final wealth must be larger in state $x_1$ than in state $x_2$. Suppose by contradiction that $w(x_1)<w(x_2)$. This is possible only if $I(x_2)$ be positive. Suppose without the loss of generality that $p_{2\theta }$ is increasing in $\theta$. This implies that

$$\begin{aligned} U_\theta&= u(w(x_1 ))+p_{2\theta } \left[ {u(w(x_2 ))-u(w(x_1 ))} \right] \nonumber \\&+\int \limits _{x\notin \left\{ {x_1 ,x_2 } \right\} } {\left[ {u(w(w))-u(w(x_1 ))} \right] dG(x)} \end{aligned}$$

(24)

is increasing in $\theta$. Because $\phi ^{\prime }$ is decreasing, the covariance rule implies that

$$\begin{aligned} \sum _{\theta =1}^n {q_\theta \phi ^{\prime }(U_\theta )p_{2\theta } } \le \left( {\sum _{\theta =1}^n {q_\theta \phi ^{\prime }(U_\theta )} } \right) \left( {\sum _{\theta =1}^n {q_\theta p_{2\theta } } } \right) . \end{aligned}$$

(25)

Similarly, we have that

$$\begin{aligned} \sum _{\theta =1}^n {q_\theta \phi ^{\prime }(U_\theta )p_{1\theta } } \ge \left( {\sum _{\theta =1}^n {q_\theta \phi ^{\prime }(U_\theta )} } \right) \left( {\sum _{\theta =1}^n {q_\theta p_{1\theta } } } \right) . \end{aligned}$$

(26)

Combining these two equalities with first-order condition (10)—which must be an equality for $x=x_2$—yields

$$\begin{aligned} u^{\prime }(w(x_2 ))&= \mu (1+\tau )\frac{\left( {\sum \nolimits _{\theta =1}^n {q_\theta \phi ^{\prime }(U_\theta )} } \right) \left( {\sum \nolimits _{\theta =1}^n {q_\theta p_{2\theta } } } \right) }{\left( {\sum \nolimits _{\theta =1}^n {q_\theta \phi ^{\prime }(U_\theta )p_{2\theta } } } \right) }\nonumber \\&\ge \mu (1+\tau ) \nonumber \\&\ge \mu (1+\tau )\frac{\left( {\sum \nolimits _{\theta =1}^n {q_\theta \phi ^{\prime }(U_\theta )} } \right) \left( {\sum \nolimits _{\theta =1}^n {q_\theta p_{1\theta } } } \right) }{\left( {\sum \nolimits _{\theta =1}^n {q_\theta \phi ^{\prime }(U_\theta )p_{1\theta } } } \right) }\ge u^{\prime }(w(x_1 )). \end{aligned}$$

(27)

This implies in turn that $w(x_2)\le w(x_1)$, a contradiction. Thus, we must have that final wealth in the smaller ambiguous state is larger than in the larger ambiguous state. From (24), this is possible only if $U_\theta$ and $p_{2\theta }$ are anti-comonotone. This implies that

$$\begin{aligned} u^{\prime }(w(x_2 ))&\le \mu (1+\tau )\frac{\left( {\sum \nolimits _{\theta =1}^n {q_\theta \phi ^{\prime }(U_\theta )} } \right) \left( {\sum \nolimits _{\theta =1}^n {q_\theta p_{2\theta } } } \right) }{\left( {\sum \nolimits _{\theta =1}^n {q_\theta \phi ^{\prime }(U_\theta )p_{2\theta } } } \right) }\nonumber \\&\quad \le \mu (1+\tau )=u^{\prime }(w_0 -D_0 -P). \end{aligned}$$

(28)

This means that final wealth in the larger ambiguous state cannot be smaller than when applying deductible $D_0$ to the corresponding loss, or equivalently, that $D_2 \le D_0$. At this stage, we know that

$$\begin{aligned} w_0 -D_0 -P\le w(x_2 )\le w(x_1 ). \end{aligned}$$

(29)

We now show that $I(x_1)=0$. Suppose by contradiction that $I(x_1)>0$. Because condition (10) must hold as an equality for $x=x_1$, and because $U_\theta$ and $p_{1\theta }$ are comonotone, we have that

$$\begin{aligned} u^{\prime }(w(x_1 ))&= \mu (1+\tau )\frac{\left( {\sum \nolimits _{\theta =1}^n {q_\theta \phi ^{\prime }(U_\theta )} } \right) \left( {\sum \nolimits _{\theta =1}^n {q_\theta p_{1\theta } } } \right) }{\left( {\sum \nolimits _{\theta =1}^n {q_\theta \phi ^{\prime }(U_\theta )p_{1\theta } } } \right) }\nonumber \\&\ge \mu (1+\tau )=u^{\prime }(w_0 -D_0 -P). \end{aligned}$$

(30)

This would mean that $w_0-D_0-P\ge w(x_1)$. This inequality is compatible with inequalities in (29) only if $w_0-D_0-P=w(x_2)=w(x_1)$. The second equality requires that $U_\theta$ be independent of $\theta$, in which case a straight deductible is optimal. But the optimal deductible is $D^{*}$, which is by assumption larger than $x_1$. This is a contradiction. Thus, $I(x_1 )=0$.

Two cases are still possible concerning state $x_2$. A first possibility is that $I(x_2)=0$, as illustrated in Proposition 5. We can set $D_2 =x_2$ in that case, thereby yielding $x_1 \le D_2 \le D_0$. Suppose alternatively that $I(x_2 )=x_2 -D_2 \ge 0$. In that case, condition (29) can be rewritten as $x_1 \le D_2 \le D_0$. $\square$

1.3 Proof of Proposition 7

Suppose that $I(x_1)$ is positive, so that condition (10) holds as an equality for $x=x_1$. Suppose by contradiction that $w(x_1)$ is larger than the certainty equivalent wealth $w^{m}$ conditional to $x\ne x_1$, which is defined as follows:

$$\begin{aligned} u(w^{m})=\int \limits _{x\ne x_1 } {u(w(x))dM(x)} , \end{aligned}$$

(31)

where $w(x)$ is final wealth under the optimal contract. Observe that

$$\begin{aligned} U_\theta =p_{1\theta } u(w(x_1 ))+(1-p_{1\theta } )u(w^{m}). \end{aligned}$$

(32)

Because $w(x_1 )>w^{m}$, $U_\theta$ is increasing in $p_{1\theta }$. By ambiguity aversion, it implies in turn that $\phi ^{\prime }(U_\theta )$ and $p_{1\theta }$ are anti-comonotone. By the covariance rule, we obtain that

$$\begin{aligned} u^{\prime }(w(x_1 ))&= \mu (1+\tau )\frac{dG(x_1 )}{dH(x_1 )} \nonumber \\&= \mu (1+\tau )\frac{\left( {\sum _{\theta =1}^n {q_\theta p_{1\theta } } } \right) \left( {\sum _{\theta =1}^n {q_\theta \phi ^{\prime }(U_\theta )} } \right) }{\left( {\sum _{\theta =1}^n {q_\theta \phi ^{\prime }(U_\theta )p_{1\theta } } } \right) }\nonumber \\&> \mu (1+\tau ). \end{aligned}$$

(33)

Similarly, we know that the optimal deductible $D^{a}$ applied to all losses $x\ne x_1$ is such that, for all $x\ne x_1$ larger than $D^{a}$

$$\begin{aligned} u^{\prime }(w_0 -D^{a}-P)&= \mu (1+\tau )\frac{\left( {\sum _{\theta =1}^n {q_\theta dF(x,\theta )} } \right) \left( {\sum _{\theta =1}^n {q_\theta \phi ^{\prime }(U_\theta )} } \right) }{\left( {\sum _{\theta =1}^n {q_\theta \phi ^{\prime }(U_\theta )dF(x,\theta )} } \right) }\nonumber \\&= \mu (1+\tau )\frac{\left( {\sum _{\theta =1}^n {q_\theta (1-p_{1\theta } )} } \right) \left( {\sum _{\theta =1}^n {q_\theta \phi ^{\prime }(U_\theta )} } \right) }{\left( {\sum _{\theta =1}^n {q_\theta \phi ^{\prime }(U_\theta )(1-p_{1\theta } )} } \right) }\nonumber \\&< \mu (1+\tau ). \end{aligned}$$

(34)

The last inequality comes from the fact that $\phi ^{\prime }(U_\theta )$ and $1-p_{1\theta }$ are comonotone. Combining inequalities (33) and (34) implies that $u^{\prime }(w(x_1 ))>u^{\prime }(w_0-D^{a}-P)$, or that $w(x_1)<w_0-D^{a}-P$. It implies that $w(x_1)<w_0-\min (x,D^{a})-P=w(x)$ for all $x$. It implies in turn that $w(x_1)$ is smaller than the certainty equivalent $w^{m}$, a contradiction. Thus, $w(x_1)$ must be smaller than $w^{m}$. Under this condition, inequalities (33) and (34) are reversed, which implies that $w(x_1 )\ge w_0 -D^{a}-P$. This means that the deductible applied to $x_1$ is smaller than $D^{a}$. $\square$

1.4 Proof of Proposition 8

In the following Lemma, we take the distorted cdf $H$ as exogenous, and we explore the link that exists between the likelihood ratio $dG(x)/dH(x)$ and the design of the optimal contract.

Lemma 1

Suppose that $K(x)=dG(x)/dH(x)$ is non-increasing. Then, there exists a deductible $D^{a}$ such that no indemnity is paid if the loss is below $D^{a}$, and the indemnity is positive for all losses above $D^{a}$, with the net loss $x-I(x)$ being non-increasing in $x$.

Proof

Define function $h$ such that

$$\begin{aligned} h(x)=u^{\prime -1}\left( {\mu (1+\tau )K(x)} \right) \end{aligned}$$

(35)

for all $x$. Observe that this function is non-decreasing in $x$. Define $D^{a}$ as the single root of equation $h(D^{a})=w_0-D^{a}-P$. For all $x\ge D^{a}$, condition (10) is satisfied $w(x)=w_0 -x+I(x)-P=h(x)$. Because $h$ is non-decreasing, the net loss $x-I(x)$ is non-decreasing. For all losses $x<D^{a}$, we have $I(x)=0$ and condition (10) is satisfied as an inequality. $\square$

In Lemma 2, we link the MLR condition to the property that $dG/dH$ is decreasing when $U_\theta$ is decreasing.

Lemma 2

Suppose that $U_\theta$ is decreasing in $\theta$ and that $dF(x,\theta +1)/dF(x,\theta )$ is non-decreasing (non-increasing) in $x$ for all $\theta =1,\ldots ,n-1$. Then, $dG(x)/dH(x)$ is non-increasing (non-decreasing). This result is reversed if $U_\theta$ is increasing in $\theta$.

Proof

We have that

$$\begin{aligned} k(x)=\left( {\sum \limits _{\theta =1}^n {q_\theta \phi ^{\prime }(u_\theta )} } \right) ^{-1}\frac{dG(x)}{dH(x)}=\frac{\sum _{\theta =1}^n {q_\theta dF(x,\theta )} }{\sum _{\theta =1}^n {q_\theta \phi ^{\prime }(U_\theta )dF(x,\theta )} } \end{aligned}$$

(36)

is non-increasing if and only if for all $x_L$ and $x_H >x_L$, we have that

$$\begin{aligned}&\sum \limits _{\theta =1}^n {q_\theta \left( {\phi ^{\prime }(U_\theta )-k^{-1}(x_L )} \right) } dF(x_L ,\theta )=0\nonumber \\&\quad \Rightarrow \quad \sum \limits _{\theta =1}^n {q_\theta \left( {\phi ^{\prime }(U_\theta )-k^{-1}(x_L )} \right) } dF(x_H ,\theta )\ge 0. \end{aligned}$$

(37)

We know that function $g(\theta )=\phi ^{\prime }(U_\theta )-k^{-1}(x_L )$ is non-decreasing in $\theta$. Therefore, it satisfies the single-crossing property. Moreover, by assumption, $dF(x,\theta )$ is log-supermodular in $(x,\theta )$. As is well-known (see Karlin 1968; Jewitt 1987; Athey 2002; Gollier 2001, Proposition 16), these are necessary and sufficient condition for (37). $\square$

Proof of Proposition 8

Suppose that $n=2$ and that $dF(x,2)/dF(x,1)$ is non-decreasing (non-increasing) in $x$. We first show that this must imply that $U_2$ is smaller than $U_1$. Suppose by contradiction that $U_2>U_1$. By Lemma 2, it implies that dG/dH is non-decreasing in $x$. It implies that $I^{\prime }(x)\le 1$ for all $x$ such that $I(x)>0$ since we must have that

$$\begin{aligned} u^{\prime }(w_0 -x+I(x)-P)=\mu (1+\tau )\frac{dG(x)}{dH(x)} \end{aligned}$$

(38)

for these $x$. By risk aversion, we must have that $-x+I(x)$ must be non-increasing. Hence, $I^{\prime }(x)\le 1$. This implies in turn that final wealth $w(x)$ must be non-increasing in the entire loss domain. But because $F(.,1)$ dominates $F(.,2)$ in the sense of MLR, this implies that $U_1$ is larger than $U_2$. This is a contradiction. So, it must be that $U_2$ is smaller than $U_1$. By Lemma 2, this implies that dG/dH is non-increasing. By Lemma 1, there must exist a deductible $D^{a}$ such that no indemnity is paid if the loss is below $D^{a}$, and the indemnity is positive for all losses above $D^{a}$, with the net loss $x-I(x)$ being non-increasing in $x$. $\square$

1.5 Proof of Proposition 9

By Proposition 5, we know that $I(x)=\max (0,x-D^{a})$. We need to prove that $D^{a}\ge D^{*}$. The optimal deductible maximizes the following objective function:

$$\begin{aligned} W(d)=\sum _{\theta =1}^n {q_\theta \phi \left( {\int {u(w_0 -\min (x,d)-P(d))dF(x,\theta )} } \right) } , \end{aligned}$$

(39)

where $P(d)=(1+\tau )\int {I(x)dG(x)}$ is the premium as a function of the deductible $d$. Under ambiguity neutrality, the first-order condition of this problem can be written as follows:

$$\begin{aligned} -u^{\prime }(w_{D^{*}} (D^{*}))+(1+\tau )\int {u^{\prime }(w_{D^{*}} (x))dG(x)=0,} \end{aligned}$$

(40)

with $w_D (x)=w_0 -\min (x,D)-P(D)$. Because the objective function $W$ is concave in $d$, we know that $D^{a}\ge D^{*}$ if and only if

$$\begin{aligned} W^{\prime }(D^{*})&= \sum _{\theta =1}^n {q_\theta \phi ^{\prime }\left( {U_\theta } \right) } (1-F(D^{*},\theta ))\bigg [-u^{\prime }(w_{D^{*}} (D^{*}))\nonumber \\&+\,(1+\tau )\int {u^{\prime }(w_{D^{*}} (x))dF(x,\theta )} \bigg ]\ge 0. \end{aligned}$$

(41)

Because the ambiguity is concentrated on losses below $D^{*}$, we have that $F(D^{*},\theta )=G(D^{*})$ for all $\theta =1,\ldots ,n$. Thus, inequality (41) can be rewritten as follows:

$$\begin{aligned} (1+\tau )\sum _{\theta =1}^n {q_\theta \phi ^{\prime }\left( {U_\theta } \right) } \int {u^{\prime }(w_{D^{*}} (x))dF(x,\theta )} \ge u^{\prime }(w_{D^{*}} (D^{*}))\sum _{\theta =1}^n {q_\theta \phi ^{\prime }\left( {U_\theta } \right) }. \end{aligned}$$

(42)

Using equation (40), this is equivalent to

$$\begin{aligned}&\sum _{\theta =1}^n {q_\theta \phi ^{\prime }\left( {U_\theta } \right) } \int {u^{\prime }(w_{D^{*}} (x))dF(x,\theta )}\nonumber \\&\quad \ge \left( {\sum _{\theta =1}^n {q_\theta \phi ^{\prime }\left( {U_\theta } \right) } } \right) \left( {\sum _{\theta =1}^n {q_\theta } \int {u^{\prime }(w_{D^{*}} (x))dF(x,\theta )} } \right) \end{aligned}$$

(43)

We first show that condition 1 in the proposition is sufficient for (43). Under condition 1, suppose without the loss of generality that $F(.,\theta +1)$ dominates $F(.,\theta )$ in the sense of FSD, for $\theta =1,\ldots ,n-1$. Given that $u(w)$ is non-decreasing in $w$ and $w_{D^{*}} (x)=w_0 -\min (x,D^{*})-P(D^{*})$ is non-increasing in $x$, we have that $U_\theta$ is non-increasing in $\theta$, and $\phi ^{\prime }(U_\theta )$ is non-decreasing in $\theta$. Symmetrically, because $u^{\prime }(w_{D^{*}}(x))$ is non-decreasing in $x$ under risk aversion, we have that $\int {u^{\prime }(w_{D^{*}} (x))dF(x,\theta )}$ is non-decreasing in $\theta$. Thus, $\phi ^{\prime }(U_\theta )$ and $\int {u^{\prime }(w_{D^{*}}(x))dF(x,\theta )}$ are comonotone. By the covariance rule, this implies inequality (43).

We then show that condition 2 is also sufficient for (43). Under condition 2, suppose without the loss of generality that $F(.,\theta +1)$ is riskier than $F(.,\theta )$ in the sense of Rothschild and Stiglitz (1970), for $\theta =1,\ldots ,n-1$. Given that $u(w)$ is concave in $w$ and $w_{D^{*}}(x)$ is linear in $x$ in the relevant domain $x<D^{*}$, we have that $U_\theta$ is non-increasing in $\theta$, and $\phi ^{\prime }(U_\theta )$ is non-decreasing in $\theta$. Symmetrically, because $u^{\prime }(w_{D^{*}}(x))$ is convex in $x$ in the relevant domain $x<D^{*}$, we have that $\int {u^{\prime }(w_{D^{*}} (x))dF(x,\theta )}$ is non-decreasing in $\theta$. Thus, $\phi ^{\prime }(U_\theta )$ and $\int {u^{\prime }(w_{D^{*}}(x))dF(x,\theta )}$ are again comonotone. By the covariance rule, this implies inequality (43). $\square$