Variance Contracts

Insurance is an efficient mechanism to facilitate risk reallocation between two parties. Borch (1960) was the first to study the insurance contract design problem and to prove that given a fixed premium, a stop-loss (or deductible) insurance policy (i.e., full coverage above a deductible) achieves the smallest variance of the insured’s share of payment.¹¹1Borch (1960) presents the problem in a reinsurance setting, in which the ceding insurer corresponds to the insured in an insurance setting. Arrow (1963) assumes that the premium is calculated by the expected value principle (i.e., the insurance cost is proportional to the expected indemnity) and imposes the principle of indemnity (i.e., the insurer’s reimbursement is non-negative and smaller than the loss). Under these specifics, Arrow (1963) shows that the stop-loss insurance is Pareto optimal between a risk-neutral insurer and a risk-averse insured. This is a foundational result that has earned the name of Arrow’s theorem of the deductible in the literature. Mossin (1968) further proves that the deductible is strictly positive if and only if the insurance price is actuarially unfair (i.e., the safety loading is strictly positive).

In Arrow (1963), the insurer is assumed to be risk-neutral. This is based on the assumption that the insurer has a sufficiently large number of independent and homogenous insureds, such that his risk, by the law of large numbers, is sufficiently diversified to be nearly zero. This kind of theoretically ideal situation hardly occurs in practice, even if the insurer does indeed have a huge number of clients. Moreover, it does not apply to tailor-made contracts for insuring one-off events (e.g., the shipment of a highly valuable painting). Theorem 2 in Arrow (1971) stipulates that, when the insurer is risk-averse and insurance cost is absent, an optimal contract must involve coinsurance. Raviv (1979) extends this result to include nonlinear insurance costs and shows that an optimal policy involves both a deductible and coinsurance. Much of the recent research in this area has focused on the insurer’s tail risk exposure. Cummins and Mahul (2004) and Zhou et al. (2010) extend Arrow’s model by introducing an exogenous upper bound on indemnity and thereby limiting the insurer’s liability with respect to catastrophic losses. From a regulatory perspective, Zhou and Wu (2008) propose a model in which the insurer’s expected loss above a prescribed level is controlled, and they conclude that an optimal policy is generally piecewise linear. Doherty et al. (2015) investigate the case in which losses are nonverifiable and deduce that a contract with a deductible and an endogenous upper limit is optimal.

As important as tail risks are for both parties in insurance contracts, in practice insurers are also concerned with other parts of the loss distribution. Kaye (2005), in a Casualty Actuarial Society report, writes

“Different stakeholders have different levels of interest in different parts of the distribution - the perspective of the decision-maker is important. Regulators and rating agencies will be focused on the extreme downside where the very existence of the company is in doubt. On the other hand, management and investors will have a greater interest in more near-term scenarios towards the middle of the distribution and will focus on the likelihood of making a profit as well as a loss” (p. 4).

Assuming the insurer to be risk-averse with a concave utility function indeed takes the whole risk distribution into consideration, as studied in Arrow (1971) and Raviv (1979). However, there are notable drawbacks to the utility function approach. The notion of utility is opaque for many non-specialists, and the benefit–risk tradeoff is only implied, implicitly, through a utility function. Moreover, one can rarely obtain, analytically, optimal policies with a general utility, hindering post-optimality analyses such as comparative statics. For instance, Raviv (1979) derives a differential equation satisfied by the optimal indemnity, which takes a rather complex form depending on the utility function used.

By contrast, variance, as a measure of risk originally put forth in Markowitz’s pioneering work (Markowitz, 1952), is also related to the whole distribution, yet it is more intuitive and transparent. Borch (1960) designs a contract that aims to minimize the variance of the insured’s liability. Kaluszka (2001) extends Borch (1960)’s work by incorporating a variance-related premium principle and shows that the optimal contract to minimize the variance of the insured’s payment can be stop-loss, quota share (i.e., the insurer covers a constant proportion of the loss) or a combination of the two. Vajda (1962) studies the problem from the insurer’s perspective, and shows that a quota share policy minimizes the variance of indemnity in an actuarially fair contract. However, his result depends critically on limiting the admissible contracts to be such that the ratio between indemnity and loss increases as the loss increases, a feature that enables the derivation of a solution through rather simple calculus.²²2Vajda (1962) claims that this feature “agrees with the spirit of (re)insurance, at least in most cases” (p. 259). However, for a larger loss, it is indeed in the spirit of insurance that the insurer should pay more, but it is not clear why he should be responsible for a higher proportion. Interestingly, our results will show that the optimal policies of our model possess this property if the insured is prudent; see Corollary 3.9.

In this paper, we revisit the work of Arrow (1963) by imposing a variance constraint on the insurer’s risk exposure. Unlike Vajda (1962), we consider the general actuarially unfair case and remove the restriction that the proportion of the insurer’s payment increases with the size of the loss. The presence of the variance constraint causes substantial technical challenges in solving the problem. In the literature, there are generally two approaches used to study variants of Arrow’s model: those involving sample-wise optimization and stochastic orders. However, the former fails to work for our problem due to the nonlinearity of the variance constraint, and the latter is not readily applicable either because the presence of the variance constraint invalidates the claim that any admissible contract is dominated by a stop-loss one. The first contribution of this paper is methodological: we develop a new approach by combining the techniques of stochastic orders, calculus of variations and the Lagrangian duality to derive optimal insurance policies. The solutions are semi-analytical in the sense that they can be computed by solving some algebraic equations (as opposed to differential equations in Raviv 1979).

Because the expected value premium principle ensures the expected profit of the insurer, our model is essentially a mean–variance model à la Markowitz for the insurer. Our second contribution is actuarial: we show that the optimal contract is coinsurance above a deductible when the variance constraint is binding. Moreover, the deductible disappears if and only if the insurance price is actuarially fair, consistent with Mossin’s Theorem (Mossin, 1968). These results are qualitatively similar to those of Raviv (1979), who uses a concave utility function for the insurer. A natural question is why one would bother to study the mean–variance version of a problem that would generate contracts with similar characteristics to its expected utility counterpart. This question can be answered in the same way as in the field of financial portfolio selection, where there is an enormously large body of study on the Markowitz mean–variance model along with its popularity in practice, despite the existence of the equally well-studied expected utility maximization models. In other words, expected utility and mean–variance are two different frameworks, and, as argued earlier, the latter underlines a more transparent and explicit return–risk tradeoff, which usually leads to explicit solutions.

Our optimal policies involve coinsurance, which is widely utilized in the insurance industry. As pointed out by Raviv (1979), risk aversion on the part of the insurer could be a cause for coinsurance, but other attributes such as the nonlinearity of the insurance cost function could also lead to coinsurance. Another explanation for coinsurance is to mitigate the moral hazard risk; see Hölmstrom (1979) and Dionne and St-Michel (1991). From the insured’s perspective, Doherty and Schlesinger (1990) argue that default risk of the insurer can motivate the insured to choose coinsurance. Picard (2000) also shows that coinsurance is optimal in order to reduce the risk premium paid to the auditor. In this paper, we prove that optimal policies can turn from full insurance to coinsurance as the variance bound tightens, thereby providing a novel yet simple reason for the prevalent feature of coinsurance in insurance theory and practice: a variance bound on the insurer’s risk exposure.

Intriguingly, our optimal insurance polices automatically satisfy the so-called incentive-compatible condition that both the insured and the insurer pay more for a larger loss (or, equivalently, the marginal indemnity is between 0 and 1).³³3The incentive-compatible condition is termed the no-sabotage condition in Carlier and Dana (2003). In Arrow (1963)’s setting, the optimal contract – the stop-loss one – turns out to be incentive-compatible; however, this is generally untrue. Gollier (1996) considers an insured facing an additional background risk that is not insurable. Under the expected value principle, he discovers that the optimal insurance, which relies heavily on the dependence between the background risk and the loss, may render the marginal indemnity strictly larger than 1. Bernard et al. (2015) generalize the insured’s risk preference from expected utility to rank-dependent utility involving probability distortion (weighting), and also find that the optimal indemnity may decrease when the loss increases. In both of these papers, the derived optimal contracts would incentivize the insured to misreport the actual losses, leading to ex post moral hazard. Equally absurd would be the case in which the insurer pays less for a larger loss. To address this issue, Huberman et al. (1983) propose the incentive-compatible condition as a hard constraint on admissible insurance policies, in addition to the principle of indemnity. Xu et al. (2019) add this constraint to the model of Bernard et al. (2015), painstakingly developing a completely different approach in order to overcome the difficulty arising out of this additional constraint and deriving qualitatively very different contracts. On the other hand, Raviv (1979) discovers that his optimal solution is incentive-compatible, assuming that the loss has a strictly positive probability density function. Carlier and Dana (2005) use a Hardy–Littlewood rearrangement argument to prove that any optimal contract is dominated by an incentive-compatible contract, establishing the optimality of the latter. However, their approach relies heavily on the assumption that the loss is non-atomic. Both of these studies rule out the important and practical case in which the loss is atomic at $0$. By contrast, in the presence of the variance constraint, we show that the optimal policy is naturally incentive-compatible even without the corresponding hard constraint or the assumption of an atom-less loss.

Our final contribution is a comparative statics analysis examining the respective effects of the insured’s wealth and the variance constraint on insurance demand under actuarially fair pricing. Our results indicate that the presence of a variance bound fundamentally changes the insurance strategy – it makes the insured’s wealth relevant and it changes the way in which the two parties share the risk. In particular, the expected coverage is always larger for a wealthier insured who has strictly decreasing absolute prudence (DAP), rendering the insurance product a normal good. This finding provides some theoretical foundation for the empirical observations of Millo (2016) and Armantier et al. (2018). Moreover, we show that the insurer has less downside risk when contracting with a wealthier insured with strictly DAP.⁴⁴4Menezes et al. (1980) introduce the notion of downside risk to compare two risks with the same mean and variance. The formal definition is given in Section 2.2. This result reconciles with the well-documented phenomenon that more economically advanced regions or countries have higher insurance densities and penetrations. On the other hand, we establish that the variance bound significantly changes a prudent insured’s risk transfer decision – she would consistently transfer more losses as the variance bound loosens. A corollary of this result is, rather surprisingly, that the insurer can reduce the tail risk by simply tightening the variance constraint. This suggests that our variance contracts do, after all, address the issue of tail exposure.

The rest of the paper proceeds as follows. In Section 2 we formulate the problem and present some preliminaries about risk preferences. In Section 3 we develop the solution approach and present the optimal insurance contracts. In Section 4 we conduct a comparative analysis by examining the effects of the insured’s initial wealth and the variance constraint on insurance demand. Section 5 concludes the paper. Some auxiliary results and all proofs are relegated to the appendices.

In this section, we present our approach to solving Problem (2.1).

First, consider Problem (2.1) without the variance constraint:

$$\max_{{I\in\mathfrak{C}}}\ \ \ {\mathbb{E}}[U(W_{I}(X))].$$

(3.1)

This is the classical Arrow (1963)’s model, for which the optimal contract is a deductible one of the form $(x-d^{*})_{+}$ for some non-negative deductible $d^{*}$, where $(x)_{+}:=\max\{x,0\}$. This contract automatically satisfies the incentive-compatible condition. Moreover, Chi (2019) (see Theorem 4.2 therein) was the first to derive an analytical form of the optimal deductible level $d^{*}$. More precisely, define

$$VaR_{\frac{1}{1+\rho}}(X):=\inf\left\{x\in[0,\mathcal{M}]:F_{X}(x)\geqslant\frac{\rho}{1+\rho}\right\}$$

and

$$\varphi(d):=\frac{{\mathbb{E}}[U^{\prime}(W_{(x-d)_{+}}(X))]}{U^{\prime}(w_{0}-d-\pi((X-d)_{+}))},\quad 0\leqslant d<\mathcal{M},$$

where $\inf\emptyset:=\mathcal{M}$ by convention. Then the optimal $d^{*}$ is

$$d^{*}=\sup\left\{VaR_{\frac{1}{1+\rho}}(X)\leqslant d<\mathcal{M}:\,\varphi(d)\geqslant\frac{1}{1+\rho}\right\}\vee VaR_{\frac{1}{1+\rho}}(X),$$

(3.2)

where $\sup\emptyset:=0$ and $x\vee y:=\max\{x,y\}$.⁸⁸8The number $d^{*}$ can be numerically computed easily, because $\varphi(d)$ is decreasing over $[VaR_{\frac{1}{1+\rho}}(X),\mathcal{M})$; see Chi (2019). This leads immediately to the following proposition.

Intuitively, if the variance bound $\nu$ is set sufficiently high, then the variance constraint in Problem (2.1) is redundant and the problem reduces to the classical Arrow (1963)’s problem. Proposition 3.1 tells exactly and explicitly what the bound should be for the variance constraint to be binding.

Therefore, it suffices to solve Problem (2.1) for the case in which $\nu<var[(X-d^{*})_{+}]$, which we now set as an assumption.

The main thrust for finding the solution is to first restrict the analysis with a fixed level of expected indemnity and then find the optimal level of expected indemnity. To this end, we need to first identify the range in which the optimal expected indemnity possibly lies. Noting that $var[(X-d)_{+}]$ is strictly decreasing and continuous in $d$ over $[{\rm ess\ inf}\;X,\mathcal{M})$, we define

$$d_{L}:=\inf\{d\geqslant d^{*}:var[(X-d)_{+}]\leqslant\nu\}\quad\text{and}\quad m_{L}:={\mathbb{E}}[(X-d_{L})_{+}].$$

Intuitively, the insurer would demand a deductible higher than Arrow’s level $d^{*}$ due to the additional risk control reflected by the variance constraint, and $d_{L}$ is the smallest deductible that makes this constraint binding.

Therefore, we can rule out any contract whose expected indemnity is strictly smaller than $m_{L}$; in other words, $m_{L}$ is a lower bound of the optimal expected indemnity. In particular, no-insurance (i.e., $I^{*}(x)\equiv 0$) is never optimal under Assumption 3.1.

Next, we are to derive an upper bound of the optimal expected indemnity. Consider a loss-capped contract $X\wedge k$, where $x\wedge y:=\min\{x,y\}$ and $k\geqslant 0$, which pays the actual loss up to the cap $k$.⁹⁹9A loss-capped contract is also called “full insurance up to a (policy) limit” or “full insurance with a cap.” Define

$$K_{U}:=\inf\{k\geqslant 0:var[X\wedge k]\geqslant\nu\}\quad\text{and}\quad m_{U}:={\mathbb{E}}[X\wedge K_{U}].$$

In the above, $K_{U}$ is well-defined because $X\wedge k-{\mathbb{E}}[X\wedge k]$ is increasing in $k$ in the sense of convex order, according to Lemma A.2 in Chi (2012).¹⁰¹⁰10A random variable $Y$ is said to be greater than a random variable $Z$ in the sense of convex order, denoted as $Z\leqslant_{cx}Y$, if $${\mathbb{E}}[Y]={\mathbb{E}}[Z]\ \text{and}\ \ {\mathbb{E}}[(Z-d)_{+}]\leqslant{\mathbb{E}}[(Y-d)_{+}]\;\;\forall d\in\mathbb{R},$$ provided that the expectations exist. Obviously, $Z\leqslant_{cx}Y$ implies $var[Z]\leqslant var[Y]$. Clearly, both $K_{U}$ and $m_{U}$ depend on the variance bound $\nu$. Since $var[X]\geqslant var[(X-d^{*})_{+}]>\nu$, we have

$$K_{U}<\mathcal{M},\quad m_{U}<{\mathbb{E}}[X]\quad\text{and}\quad var\left[X\wedge K_{U}\right]=\nu.$$

This lemma stipulates that $m_{U}$ is an upper bound of the optimal expected indemnity. Moreover, any admissible contract achieving this upper bound is equivalent to the loss-capped contract $X\wedge K_{U}$. An immediate corollary of the lemma is $m_{L}\leqslant m_{U}$, noting that $var[(X-d_{L})_{+}]=\nu$.

The following result identifies the case $m_{L}=m_{U}$ as a trivial one.

In what follows, we consider the general and interesting case in which $m_{L}<m_{U}$. For $m\in(m_{L},m_{U})$, define

$$\mathfrak{C}_{m}:=\left\{I\in\mathfrak{C}:var[I(X)]\leqslant\nu,\,{\mathbb{E}}[I(X)]=m\right\}.$$

We now focus on the following optimization problem

$$\max_{I\in\mathfrak{C}_{m}}\ \ {\mathbb{E}}[U(W_{I}(X))],$$

(3.3)

which is a “cross section” of the original problem (2.1) where the expected indemnity is fixed as $m$.

For $\lambda\in{\mathbb{R}}$ and $\beta\geqslant 0$, denote

$\displaystyle I_{\lambda,\beta}(x):=\sup\Big{\{}y\in[0,x]:U^{\prime}\left(w_{0}-x+y-(1+\rho)m\right)-\lambda-2\beta y\geqslant 0\Big{\}},\;x\in[0,\mathcal{M}].$

(3.4)

Actually, $I_{\lambda,\beta}$ is a contract that coinsures above a deductible or coinsures following full insurance, depending on the relative values between $\lambda$ and $U^{\prime}(w_{0}-(1+\rho)m)$. To see this, when $\lambda\geqslant U^{\prime}(w_{0}-(1+\rho)m)$, we have

$$I_{\lambda,\beta}(x)=\left\{\begin{array}[]{ll}0,&0\leqslant x\leqslant w_{0}-(1+\rho)m-(U^{\prime})^{-1}(\lambda),\\ f_{\lambda,\beta}(x),&w_{0}-(1+\rho)m-(U^{\prime})^{-1}(\lambda)<x\leqslant\mathcal{M},\end{array}\right.$$

(3.5)

and when $\lambda<U^{\prime}(w_{0}-(1+\rho)m)$, we have

$$I_{\lambda,\beta}(x)=\left\{\begin{array}[]{ll}x,&0\leqslant x\leqslant\frac{U^{\prime}(w_{0}-(1+\rho)m)-\lambda}{2\beta},\\ f_{\lambda,\beta}(x),&\frac{U^{\prime}(w_{0}-(1+\rho)m)-\lambda}{2\beta}<x\leqslant\mathcal{M},\end{array}\right.$$

(3.6)

where $f_{\lambda,\beta}(x)$ satisfies the following equation in $y$:¹¹¹¹11It can be shown easily that this equation has a unique solution.

$$U^{\prime}(w_{0}-x+y-(1+\rho)m)-\lambda-2\beta y=0.$$

(3.7)

Moreover, it is easy to see that $0\leqslant f_{\lambda,\beta}(x)\leqslant x$ either when $\lambda\geqslant U^{\prime}(w_{0}-(1+\rho)m)$ and $w_{0}-(1+\rho)m-(U^{\prime})^{-1}(\lambda)<x\leqslant\mathcal{M}$, or when $\lambda<U^{\prime}(w_{0}-(1+\rho)m)$ and $\frac{U^{\prime}(w_{0}-(1+\rho)m)-\lambda}{2\beta}<x\leqslant\mathcal{M}$. Furthermore,

$\displaystyle f^{\prime}_{\lambda,\beta}(x)=\frac{-U^{\prime\prime}(w_{0}-x+f_{\lambda,\beta}(x)-(1+\rho)m)}{2\beta-U^{\prime\prime}(w_{0}-x+f_{\lambda,\beta}(x)-(1+\rho)m)}\in(0,1].$

(3.8)

The following result indicates that there exists an optimal solution to Problem (3.3) that is in the form of $I_{\lambda,\beta}(x)$ and binds both the mean and variance constraints.

Combining Lemma 3.2, Lemma 3.3 and Proposition 3.5 yields that we can always find an optimal contract in one of the following three types: a deductible one of the form $I_{L}(x)=(x-d_{L})_{+}$, a loss-capped one of the form $I_{U}(x)=x\wedge K_{U}$ and a general one of the form $I_{\lambda^{*}_{m},\beta^{*}_{m}}(x)$. In other words, the optimal solutions of the following maximization problem

$$\max_{I\in\big{\{}I_{L},\ I_{U},\ I_{\lambda^{*}_{m},\beta^{*}_{m}}\,\text{for}\,m\in(m_{L},m_{U})\big{\}}}{\mathbb{E}}[U(W_{I}(X))],$$

(3.10)

where $m_{L}<m_{U}$, also solve Problem (2.1).

Note that $I_{L}$, $I_{U}$ and $I_{\lambda^{*}_{m},\beta^{*}_{m}}$ all satisfy the incentive-compatible condition (see (3.8)); hence, so does at least one of the optimal contracts $I^{*}$ of (2.1). That is, $I^{*}(0)=0$ and $0\leqslant{I^{*}}^{\prime}(x)\leqslant 1$ almost everywhere.¹²¹²12As will be evident in the sequel, the values of $I^{\prime}$ on a set with zero Lebesgue measure have no impact on $I$. Therefore, we will often omit the phrase “almost everywhere” in statements regarding the marginal indemnity function ${I}^{\prime}$ throughout this paper. Therefore, it suffices to solve the following maximization problem

$$\begin{array}[]{ll}\underset{I\in\mathcal{IC}}{\text{max}}&\ \ \ {\mathbb{E}}[U(W_{I}(X))]\\ \mbox{subject to}&\ \ \ var\left[e_{I}(X)\right]\leqslant\nu,\end{array}$$

(3.11)

where

$$\mathcal{IC}:=\left\{I:I(0)=0,\,0\leqslant I^{\prime}(x)\leqslant 1,\,\forall x\in[0,\mathcal{M}]\right\}\subsetneq\mathfrak{C},$$

(3.12)

to obtain an optimal contract for Problem (2.1).

Notice that $\mathcal{IC}$ is convex on which ${\mathbb{E}}[U(W_{I}(X))]$ is strictly concave. Using the convex property of variance and applying arguments similar to those in the proof of Proposition 3.1 in Chi and Wei (2020), we obtain the following proposition:

Note that the assumptions in Proposition 3.6-(ii) are satisfied in most situations of practical interest because either an insurer naturally sets a positive safety loading, or a loss actually never occurs with a positive probability, or both happen. On the other hand, since any optimal solution to Problem (3.11) also solves Problem (2.1), Proposition 3.6-(i) establishes the existence of optimal solutions to the latter.¹³¹³13It is difficult to prove the existence of solutions to Problem (2.1) directly because its feasible set is not compact only under the principle of indemnity. Moreover, the argument proving Proposition 3.1 in Chi and Wei (2020) can be used to show that Proposition 3.6-(ii) holds true for Problem (2.1) as well. Finally, now that we have the existence and uniqueness of the optimal solutions for both Problems (3.11) and (2.1), we conclude that these two problems are indeed equivalent under the assumptions of Proposition 3.6-(ii).

While the analysis of Problem (2.1) is simplified to Problem (3.10), it remains challenging to solve this problem because $\lambda^{*}_{m}$ and $\beta^{*}_{m}$ are implicit functions of $m$. Before attacking this problem, we introduce a useful result that provides a general qualitative structure for the optimal indemnity function in Problem (3.11) or, equivalently, Problem (2.1).

Note that (3.13) does not entail an explicit expression of ${I^{*}}^{\prime}$ because its right hand side also depends on $I^{*}$ as well as on an unknown parameter $\beta^{*}$. While deriving the optimal solution $I^{*}$ directly from (3.13) seems challenging, the equation reveals the important property that ${I^{*}}^{\prime}$ must take a value of either 0 or 1, except at point(s) $x$ where $\Phi_{I^{*}}(x)=0$.¹⁴¹⁴14From the control theory perspective, (3.13) corresponds to an optimal control problem in which ${I}^{\prime}$ is taken as the control variable. Moreover, the optimal control turns out to be of the so-called “bang-bang” type, whose values depend on the sign of the discriminant function $\Phi_{I}$. This type of optimal control problem arises when the Hamiltonian depends linearly on control and the control is constrained between an upper bound and a lower bound. It is usually hard to solve for optimal control when the discriminant function is complex, which is the case here. This property will in turn help us to decide whether the optimal contract is of the form $(x-d_{L})_{+},\ x\wedge K_{U}$, or $I_{\lambda^{*}_{m},\beta^{*}_{m}}$.

The following theorem presents a complete solution to Problem (2.1).

Theorem 3.8 provides a complete solution to Problem (2.1). It indicates that the optimal contract can not be a pure deductible of the form $(x-d_{L})_{+}$, nor a pure loss-capped of the form $x\wedge K_{U}$. It can only be in the form $I_{\lambda^{*}_{m},\beta^{*}_{m}}$ of (3.5) (rather than (3.6)). The optimal policies can be computed by solving a system of three algebraic equations; so the result is semi-analytic.

Actuarially, Theorem 3.8 reveals how the variance bound impacts the contract. When the bound $\nu$ is sufficiently low so that it is binding (hence the model does not degenerate into the classical Arrow 1963’s model), the optimal policy is always genuine coinsurance if there is no safety loading. Here, by “genuine” we mean the strict inequalities $0<I^{*}(x)<x$ for all $x\in(0,\mathcal{M}]$, namely both the insurer and the insured pay positive portions of the loss incurred. When the safety loading coefficient is positive, the optimal contract demands genuine coinsurance above a positive deductible. So the variance bound translates into a change from the part of the full insurance in Arrow’s contract to coinsurance. Our contracts are similar qualitatively to those of Raviv (1979), in which a utility function is in the place of a variance bound; however, ours are quantitatively different from Raviv (1979)’s.

On the other hand, the deductible $\tilde{d}$ is positive if and only if the safety loading coefficient is positive. So the existence of the deductible is completely determined by the loading coefficient in the insurance premium. This result is consistent with Mossin’s Theorem (Mossin 1968).

So, with a prudent insured, the insurer pays more not only absolutely but also relatively as loss increases. Vajda (1962) restricts his study on a variance contracting problem to policies that have this feature of the insurer covering proportionally more for larger losses. Corollary 3.9 uncovers ex post this feature in our optimal policies, provided that the insured is prudent.

Thanks to the semi-analytic results derived in the previous section, we are able to analyze the impacts of the insured’s initial wealth and the variance bound on the insurance demand.

We make the following assumptions for our comparative statics analysis:

Assumption 4.1-(i) is standard in the literature that accommodates most of the used distributions by actuaries, such as exponential, lognormal, gamma, and Pareto distributions. Assumption 4.1-(ii) is not necessarily plausible in practice, but it is meaningful in theory, as it describes a state in competitive equilibrium, in which insurers break even and insurance policies are actuarially fair for representative insureds (see e.g., Rothschild and Stiglitz, 1976; Viscusi, 1979). It is important to carry out comparative statics analyses in such a “fair” state in order to rule out any impact emanating from an unfair price. Such an assumption is indeed often imposed when conducting comparative statics in the literature of insurance economics. For example, the comparative statics results of Ehrlich and Becker (1972) and Viscusi (1979) deal exclusively with actuarially fair situations. Many recent studies, such as Eeckhoudt et al. (2003), Huang and Tzeng (2006) and Teh (2017), also impose this assumption for their comparative statics analyses.

Finally, we will assume $\nu<var[X]$ throughout this section, as otherwise the variance constraint is redundant and the optimal solution is trivially full insurance.

In this paper, we have revisited the classical Arrow (1963)’s model by adding a variance limit on the insurer’s risk exposure. This constraint is motivated by the insurer’s desire to manage underwriting risk; at the same time, it poses considerable technical challenges for solving the problem. We have developed an approach to derive optimal contracts semi-analytically, in the form of coinsurance above a deductible when the variance constraint is active. The final policies automatically satisfy the incentive-compatible condition, thereby eliminating potential ex post moral hazard. We have also conducted a comparative analysis to examine the impact of the insured’s wealth and of the variance bound on insurance demand.

This work can be extended in a couple of directions. First, we have restricted the comparative analysis to actuarially fair insurance. Analyzing for the general unfair case calls for a different approach than the one presented here. Second, a model incorporating probability distortion (weighting) is of significant interest, both theoretically and practically. This is because probability distortion, a phenomenon well documented in psychology and behavioral economics, is related to tail events, about which both insurers and insureds have great concerns.