2. Model

In this section, we follow Cao et al. (Reference Cao, Li, Young and Zou2023b) to define a discrete-time insurance market to study the loss reporting problem of an insured (she/her). The key difference from Cao et al. (Reference Cao, Li, Young and Zou2023b) is that we now allow the insured to buy deductible insurance, instead of restricting her coverage to full insurance.

2.1. Insurance market

We model the insured’s risk by a series of independent and identically distributed (i.i.d.) nonnegative random variables $\{Z_t\}_{t =1, 2, \dots}$ , in which $Z_t \stackrel{d}{=} Z$ denotes her loss over the $t^{th}$ period $[t-1, t)$ . As suggested by claim data from short-term insurance, we assume that the distribution of Z is a mixture of a point mass at 0, with probability $\mathbb{P}(Z = 0) = p_0 \in (0,1)$ , and a continuous, positive random variable, with full support over $(0, \infty)$ and a probability density function f. Let F (resp. $S = 1 - F$ ) denote the cumulative distribution function (resp. survival function) of Z, that is, $F(x) = p_0 + \int_0^x f(t) \textrm{d} t$ for $x \ge 0$ . See, for example, Zacks and Levikson (Reference Zacks and Levikson2004) for the same setup.

To mitigate her risk exposure, the insured buys deductible insurance from a representative seller/insurer (it), who adopts a two-class BMS in ratemaking. Under our BMS, if the insured files a claim in the previous period, she moves to or stays in class 2 for the next period and pays premium $\pi_2$ ; otherwise, she moves to or stays in class 1 and pays premium $\pi_1$ .Footnote 10

As supported by common practice, by following related research (i.e., Zacks and Levikson, Reference Zacks and Levikson2004; Charpentier et al., Reference Charpentier, David and Elie2017, and Cao et al., Reference Cao, Li, Young and Zou2023b), we assume that the insured applies a barrier strategy to decide whether she should report a loss. Such a reporting strategy is fully captured by a pair $(b_1, b_2) \in \mathbb{R}_+^2$ ; when the insured is in rate class $i = 1, 2$ , she reports a loss if and only if it is greater than the barrier $b_i$ . Because the payoff of deductible insurance is zero when the loss size is less than the deductible, we assume, without loss of generality, that $b_i \ge d_i$ for $i=1,2$ . Let $\{X_t\}_{t = 0,1, \dots}$ denote the insured’s wealth process under a chosen barrier strategy (with dependence on $(b_1, b_2)$ suppressed for notational simplicity); here, $X_t$ is the insured’s wealth at time t before receiving income and paying premium to the insurer. Then, the dynamics of her wealth when she is in rate class i over $[t, t+1)$ follows:

(2.1) \begin{equation} X_{t+1} = X_t + c - \pi_i - Z_{t+1} + (Z_{t+1} - d_i)_+ \cdot \unicode{x1D7D9}_{\{Z_{t+1} > b_i\}}, \quad i=1,2,\end{equation}

in which $c>0$ is the insured’s per-period income, $\pi_i$ is the premium payable when she is in rate class i, $Z_{t+1}$ is the loss over $[t, t+1)$ , and $\unicode{x1D7D9}_{\{Z_{t+1} > b_i\} }$ equals 1 if and only if $Z_{t+1} > b_i$ and equals 0 otherwise. We impose the following assumption on the income c:

(2.2) \begin{align} c > \max\{\pi_1 + d_1, \, \pi_2 + d_2 \},\end{align}

which implies that the insured is able to pay the premium and deductible regardless of her rate class. This assumption is, though technical, easily satisfied in real life.Footnote 11

2.2. Insured’s preferences

We assume the insured is risk-averse, and her preferences are characterized by the standard expected utility theory with an exponential utility $U(x) = - \textrm{e}^{- \gamma x}$ , in which $\gamma > 0$ is the insured’s (constant) absolute risk aversion (see, e.g., Ghossoub et al., Reference Ghossoub, Li and Shi2023 and Meng et al., Reference Meng, Wei, Zhang and Zhuang2022 for the same preference assumption). The goal of the insured is to choose equilibrium barrier and deductible strategies to maximize the expected utility of her “terminal” wealth (see, e.g., Merton, Reference Merton1969 for a standard reference on this criterion).

However, instead of a finite or an infinite planning horizon, we assume that the insured’s planning horizon $\tau$ is random and exogenously given. Several recent papers (see Cao et al., Reference Cao, Li, Young and Zou2022 and follow-ups) on optimal (re)insurance problems also consider a random horizon; in particular, footnote 18 in Cao et al. (Reference Cao, Li, Young and Zou2023a) discusses in detail the mathematical and economic justifications for the choice of a random horizon.Footnote 12 Because we assumed that the insured adopts a homogeneous barrier strategy in reporting, it is natural to assume that the random horizon $\tau$ follows a geometric distribution, which is the only memoryless discrete distribution. Under this assumption, we have

\begin{align*} \mathbb{P}(\tau = t) = (1 - p) \, p^{t-1}, \quad t = 1,2, \dots,\end{align*}

in which $p = \mathbb{P}(\tau > 1) \in (0,1)$ .

Based on the above setup, the insured’s objective function when she is currently in rate class $i=1,2$ is given by:

(2.3) \begin{equation} J_i(x) = \mathbb{E} \big({-} \textrm{e}^{-\gamma X_\tau} \big| X_0 = x, I_0 = i \big),\end{equation}

in which $X_0 = x \in \mathbb{R}$ is the insured’s initial wealth, and $I_0 = i \in \{1, 2\}$ is her initial rate class. For notational simplicity, we often suppress the dependence of $J_i$ upon $(b_1, b_2, d_1, d_2)$ . Several remarks are due regarding the objective function $J_i$ in (2.3). First, $J_i$ is time-independent because we only consider time-homogeneous deductible and barrier strategies, losses are i.i.d., and $\tau$ follows a geometric distribution. Therefore, the condition in (2.3), though written at time 0, can be equivalently understood as at any time. Second, $J_i$ is a function of both $(b_1, d_1)$ and $(b_2, d_2)$ for $i=1,2$ , and this reflects the game nature embedded in the BMS. To be more concrete, the joint effect of $(b_1, d_1)$ and $(b_2, d_2)$ on $J_i$ is due to the fact that there is a strictly positive probability that the insured will move from one rate class to another. Consequently, the insured plays against herself in a noncooperative Nash game; please see Remark 3.1 in Cao et al. (Reference Cao, Li, Young and Zou2023a) for a detailed discussion on this game feature and Björk et al. (Reference Björk, Murgoci and Zhou2014) and (Reference Björk, Khapko and Murgoci2017) for standard references on such a game-theoretical approach.

We end this section with a useful lemma that reveals the relation between $J_1$ and $J_2$ .

Lemma 2.1. The objectives $J_1$ and $J_2$ defined in (2.3) satisfy

(2.4) \begin{align} J_1(x) &= - (1 - p) \textrm{e}^{-\gamma(x + c - \pi_1)} \, \mathbb{E} \Big( \textrm{e}^{\gamma(Z - (Z - d_1)_+ \cdot \unicode{x1D7D9}_{\{Z > b_1\}})} \Big) + p F(b_1) \mathbb{E} \big(J_1(x + c - \pi_1 - Z) \big| Z \le b_1 \big) \notag \\[4pt] &\quad+ p S(b_1) \mathbb{E} \big( J_2(x + c - \pi_1 - (Z \wedge d_1)) \big| Z > b_1 \big), \end{align}

(2.5) \begin{align} J_2(x) &= - (1 - p) \textrm{e}^{-\gamma(x + c - \pi_2)} \, \mathbb{E} \Big( \textrm{e}^{\gamma(Z - (Z - d_2)_+ \cdot \unicode{x1D7D9}_{\{Z > b_2\}})} \Big) + p F(b_2) \mathbb{E} \big(J_1(x + c - \pi_2 - Z) \big| Z \le b_2 \big) \notag \\[4pt] &\quad+ p S(b_2) \mathbb{E} \big( J_2(x + c - \pi_2 - (Z \wedge d_2)) \big| Z > b_2 \big), \end{align}

in which

(2.6) \begin{align} \mathbb{E} \Big( \textrm{e}^{\gamma \left(Z - (Z - d_i)_+ \cdot \unicode{x1D7D9}_{\{Z > b_i\}}\right) } \Big) = \int_0^{b_i} \textrm{e}^{\gamma z} \textrm{d} F(z) + S(b_i) \textrm{e}^{\gamma d_i}, \quad i = 1, 2. \end{align}

Proof. By a standard argument in renewal theory, the objective function $J_1$ satisfies the following recursion:

\begin{align*} J_1(x) &= - \mathbb{E}_x \big(\textrm{e}^{-\gamma X_\tau} \big| \tau = 1 \big)\mathbb{P}(\tau = 1) \notag \\[4pt] &\quad + \big\{ \mathbb{E}_x(J_1(X_1)|\tau > 1, Z \le b_1)\mathbb{P}(Z \le b_1) + \mathbb{E}_x(J_2(X_1)|\tau > 1, Z > b_1)\mathbb{P}(Z > b_1) \big\} \mathbb{P}(\tau > 1), \end{align*}

in which $\mathbb{E}_x$ denotes conditioning on $X_0 = x$ . The above recursion, along with (2.1), implies (2.4); by a similar argument, we obtain (2.5). The result in (2.6) is straightforward by splitting the integration region into $[0, b_i]$ and $(b_i, \infty)$ . Note that the Riemann integral in (2.6) includes the jump at 0, that is, $\int_0^{b_i} \textrm{e}^{\gamma z} \textrm{d} F(z) = p_0 + \int_{(0, b_i]} \, \textrm{e}^{\gamma z} f(z) \textrm{d} z$ ; we follow this convention throughout this paper when writing integrals.

3. Equilibrium reporting strategy under deductible insurance

This section solves Question 1 from the Introduction to obtain the insured’s equilibrium reporting strategy for a fixed deductible insurance contract. To this end, we first formally define the insured’s equilibrium reporting strategy as follows.

Because the insured’s objectives in (2.3) involve exponential utility and because the wealth dynamics in (2.1) is linear, we conjecture that the solution of (2.4)–(2.5) is of the form:

(3.1) \begin{align} J_i(x) = - A_i \, \textrm{e}^{-\gamma x}, \quad i = 1, 2,\end{align}

in which $A_i > 0$ is yet to be determined. Note that $A_i$ depends on the insured’s barrier strategy $(b_1, b_2)$ and on the deductibles $(d_1, d_2)$ , but we suppress this dependence frequently to simplify notation. By using the ansatz in (3.1), the recursions in (2.4)–(2.5) become

\begin{equation*} \textrm{e}^{\gamma(c - \pi_1)} A_1 = (1 - p) \left\{ \int_0^{b_1} \textrm{e}^{\gamma z} \textrm{d} F(z) + S(b_1) \textrm{e}^{\gamma d_1} \right\} + p A_1 \int_0^{b_1} \textrm{e}^{\gamma z} \textrm{d} F(z) + p S(b_1) \textrm{e}^{\gamma d_1} A_2, \end{equation*}

and

\begin{equation*}\textrm{e}^{\gamma(c - \pi_2)} A_2 = (1 - p) \left\{ \int_0^{b_2} \textrm{e}^{\gamma z} \textrm{d} F(z) + S(b_2) \textrm{e}^{\gamma d_2} \right\} + p A_1 \int_0^{b_2} \textrm{e}^{\gamma z} \textrm{d} F(z) + p S(b_2) \textrm{e}^{\gamma d_2} A_2.\end{equation*}

Solving the above system yields

(3.2) \begin{equation} A_1(b_1;\, b_2) = (1 - p) \, \dfrac{\textrm{e}^{\gamma(c - \pi_2)} \left\{ \int_0^{b_1} \textrm{e}^{\gamma z}\textrm{d} F(z) + e^{\gamma d_1} S(b_1) \right\} + N\big(b_1, b_2 \big) }{D(b_1, b_2)},\end{equation}

and

(3.3) \begin{equation} A_2(b_2;\, b_1) = (1 - p) \, \dfrac{\textrm{e}^{\gamma(c - \pi_1)} \left\{ \int_0^{b_2} \textrm{e}^{\gamma z} \textrm{d} F(z) + \textrm{e}^{\gamma d_2} S(b_2) \right\} + N(b_1, b_2)}{D(b_1, b_2)},\end{equation}

in which N and D equal, respectively,

\begin{equation*} N(b_1, b_2) = p \left\{ \textrm{e}^{\gamma d_1}S(b_1) \int_0^{b_2} \textrm{e}^{\gamma z} \textrm{d} F(z) - \textrm{e}^{\gamma d_2} S(b_2) \int_0^{b_1} \textrm{e}^{\gamma z} \textrm{d} F(z) \right\},\end{equation*}

and

(3.4) \begin{align} D(b_1, b_2) &= \left\{ \textrm{e}^{\gamma(c - \pi_1)} - p \int_0^{b_1} \textrm{e}^{\gamma z} \textrm{d} F(z) \right\} \left( \textrm{e}^{\gamma(c - \pi_2)} - p \textrm{e}^{\gamma d_2} S(b_2) \right) - p^2 \textrm{e}^{\gamma d_1} S(b_1) \int_0^{b_2} \textrm{e}^{\gamma z} \textrm{d} F(z). \end{align}

Because the insured is a utility maximizer, her objective $J_i(x)$ in (2.3) must be strictly increasing with respect to the initial wealth x, which is why we require $A_i$ in (3.1) to be strictly positive. However, the expressions in (3.2) and (3.3) cannot guarantee positiveness of $A_1$ and $A_2$ ; thus, we derive conditions under which both are positive.

Lemma 3.1. $A_1$ in (3.2) is positive if and only if

(3.5) \begin{equation} \left\{ \textrm{e}^{\gamma(c - \pi_1)} - p \int_0^{b_1} \textrm{e}^{\gamma z} \textrm{d} F(z) \right\} \left( \textrm{e}^{\gamma(c - \pi_2)} - p \textrm{e}^{\gamma d_2} S(b_2) \right) > p^2 \textrm{e}^{\gamma d_1} S(b_1) \int_0^{b_2} \textrm{e}^{\gamma z} \textrm{d} F(z).\end{equation}

If inequality (3.5) holds, then $A_2$ in (3.3) is positive.

Proof. For ease of notation in this proof, let

\begin{equation*}\alpha_i = \textrm{e}^{\gamma(c - \pi_i)}, \qquad \beta_i = \textrm{e}^{\gamma d_i} S(b_i), \qquad \delta_i = \int_0^{b_i} \textrm{e}^{\gamma z} \textrm{d} F(z), \quad i = 1, 2,\end{equation*}

and

\begin{equation*}\Delta_1 = \alpha_1 - p\delta_1, \qquad \Delta_2 = \alpha_2 - p \beta_2.\end{equation*}

Note that $\Delta_2 > 0$ because $\textrm{e}^{\gamma(c - (\pi_2 + d_2))} > 1 > p S(b_2)$ by the assumption in (2.2).

First, rewrite $A_1$ and $A_2$ as follows (by factoring out the positive constant $1-p$ ):

\begin{equation*}A_1 \propto \frac{N_1}{D}, \quad A_2 \propto \frac{N_2}{D},\end{equation*}

in which

\begin{equation*}N_1 = \Delta_2\delta_1 + (\alpha_2 + p\delta_2)\beta_1 = \Delta_2 \delta_1 + (\Delta_2 + p\beta_2 + p\delta_2)\beta_1 = \Delta_2(\delta_1 + \beta_1) + (\delta_2 + \beta_2)p\beta_1,\end{equation*}

\begin{equation*}N_2 =\Delta_1\beta_2 + (\alpha_1 + p\beta_1)\delta_2 = \Delta_1\beta_2 + (\Delta_1 + p\delta_1 + p\beta_1)\delta_2 = \Delta_1(\delta_2 + \beta_2) + (\delta_1 + \beta_1)p\delta_2,\end{equation*}

and

\begin{equation*}D = \Delta_1 \Delta_2 - p^2\beta_1 \delta_2.\end{equation*}

Because $\Delta_2 > 0$ , we see that $N_1 > 0$ . $A_1$ is positive if and only if D is positive, which is equivalent to inequality (3.5). Finally, if $D > 0$ , then

\begin{equation*} \Delta_1 = \textrm{e}^{\gamma(c - \pi_1)} - p \int_0^{b_1} \textrm{e}^{\gamma z} \textrm{d} F(z) > 0,\end{equation*}

which implies $N_2 > 0$ and eventually $A_2 > 0$ .

Because the condition in (3.5) is sufficient and necessary in order for both $A_1$ and $A_2$ to be positive, as required by the ansatz in (3.1), we assume in the subsequent analysis that this condition holds, which we formalize next.

For a deductible pair $(d_1, d_2)$ satisfying Assumption 3.1, define the set of admissible barrier strategies by:

\begin{align*} \mathcal{D} \,:\!=\, \big\{ (b_1, b_2) \in [d_1, \infty) \times [d_2, \infty) \, \big|\, D(b_1, b_2) > 0 \big\},\end{align*}

in which D is defined in (3.4). As an aside, because the function D is continuous, Assumption 3.1 implies $\mathcal{D}$ contains a neighborhood of $(d_1, d_2)$ . In what follows, when we consider optimization problems over $b_1$ or $b_2$ , we always implicitly assume the optimization region is a projection of $\mathcal{D}$ onto the corresponding argument space. After we obtain a candidate for the equilibrium barrier strategy $\left(b_1^*, b_2^*\right)$ , we will verify that it is, indeed, in the set $\mathcal{D}$ .

We first analyze the case when the insured is in rate class 1 at time 0, and we solve the problem:

\begin{equation*}\bar{b}_1(b_2) \,:\!=\, \underset{b_1 \ge d_1}{\operatorname{arg \, sup}} \, J_1(x) = \underset{b_1 \ge d_1}{\operatorname{arg \, inf}} \, A_1(b_1;\, b_2),\end{equation*}

for a fixed $b_2 \ge d_2$ . The solution is presented in the next proposition.

Proposition 3.1. For a fixed $b_2 \ge d_2$ , the function $b_1 \mapsto A_1(b_1;\, b_2)$ has a unique minimizer $\bar b_1(b_2)$ on $[d_1, \infty)$ . If

\begin{equation*}\textrm{e}^{\gamma \pi_1} \left\{ \int_0^{d_1} \textrm{e}^{\gamma z}\textrm{d} F(z) + \textrm{e}^{\gamma d_1} S(d_1) \right\} \ge \textrm{e}^{\gamma \pi_2 } \left\{ \int_0^{b_2} \textrm{e}^{\gamma z}\textrm{d} F(z) + \textrm{e}^{\gamma d_2} S(b_2) \right\},\end{equation*}

then $\bar b_1(b_2) = d_1$ . Otherwise, $\bar b_1(b_2)$ is the unique zero of $\unicode{x1D4F0}_1(\cdot \, ;\, b_2)$ in $(d_1, \infty)$ , in which

(3.6) \begin{align}\unicode{x1D4F0}_1(b_1;\, b_2) &= \big( \textrm{e}^{\gamma(b_1 - d_1)} - 1 \big) + p \textrm{e}^{-\gamma(c - \pi_1)} \left\{ \int_0^{b_1} \textrm{e}^{\gamma z}\textrm{d} F(z) + e^{\gamma b_1} S(b_1) \right\} \notag \\&\quad - p \textrm{e}^{-\gamma(c - \pi_2)} \left\{ \int_0^{b_2} \textrm{e}^{\gamma z}\textrm{d} F(z) + \textrm{e}^{\gamma(b_1 - d_1)} \textrm{e}^{\gamma d_2} S(b_2) \right\}.\end{align}

Proof. After factoring $\frac{1 - p}{D^2} \, \textrm{e}^{\gamma(c - \pi_1)} \textrm{e}^{\gamma(c - \pi_2)} \, \textrm{e}^{\gamma d_1} f(b_1) \big(\textrm{e}^{\gamma(c - \pi_2)} - p \textrm{e}^{\gamma d_2} S(b_2)\big) > 0$ from the derivative of $A_1$ with respect to $b_1$ , we obtain that this derivative is positively proportional to $\unicode{x1D4F0}_1$ in (3.6). The derivative of $\unicode{x1D4F0}_1$ with respect to $b_1$ equals

\begin{equation*}\dfrac{\partial \unicode{x1D4F0}_1}{\partial b_1} = \gamma \textrm{e}^{\gamma(b_1 - d_1)} \big( 1 + p \textrm{e}^{-\gamma(c - (\pi_1 + d_1))} S(b_1) - p \textrm{e}^{-\gamma(c - (\pi_2 + d_2))} S(b_2) \big) >0,\end{equation*}

which is positive because $\textrm{e}^{-\gamma(c - (\pi_2 + d_2))} < 1$ . Next,

\begin{align*}&\lim_{b_1 \to \infty} \unicode{x1D4F0}_1(b_1;\, b_2) \\&= \lim_{b_1 \to \infty} \textrm{e}^{\gamma (b_1 - d_1)} \left( 1 + p \textrm{e}^{-\gamma(c - (\pi_1 + d_1))} \left\{ \int_0^{b_1} \textrm{e}^{-\gamma(b_1 - z)} \textrm{d} F(z) + S(b_1) \right\} - p \textrm{e}^{-\gamma(c - (\pi_2 + d_2))} S(b_2) \right) \\&= \infty,\end{align*}

in which the last line follows because $0 < \int_0^{b_1} \textrm{e}^{-\gamma(b_1 - z)} \textrm{d} F(z) + S(b_1) < 1$ for all $b_1$ and because $\textrm{e}^{-\gamma(c - (\pi_2 + d_2))} < 1$ .

Finally,

\begin{equation*}\unicode{x1D4F0}_1(d_1;\, b_2) \propto \textrm{e}^{\gamma \pi_1} \left\{ \int_0^{d_1} \textrm{e}^{\gamma z}\textrm{d} F(z) + \textrm{e}^{\gamma d_1} S(d_1) \right\} - \textrm{e}^{\gamma \pi_2} \left\{ \int_0^{b_2} \textrm{e}^{\gamma z}\textrm{d} F(z) + \textrm{e}^{\gamma d_2} S(b_2) \right\}.\end{equation*}

If this expression is nonnegative, then the minimizer $\bar b_1(b_2) = d_1$ because $\unicode{x1D4F0}_1$ increases to $\infty$ ; otherwise, if this expression is negative, then $\bar b_1(b_2)$ equals the unique zero of $\unicode{x1D4F0}_1(\cdot \, ; b_2)$ in $(d_1, \infty)$ .

In the next proposition, we solve the problem:

\begin{equation*}\underset{b_2 \ge d_2}{\operatorname{arg \, sup}} \, J_2(x) = \underset{b_2 \ge d_2}{\operatorname{arg \, inf}} \, A_2(b_2;\, b_1),\end{equation*}

which corresponds to the case when the insured is in rate class 2.

Proposition 3.2. For a fixed $b_1 \ge d_1$ , the function $b_2 \mapsto A_2(b_2;\, b_1)$ has a unique minimizer $\bar b_2(b_1)$ on $[d_2, \infty)$ . If

\begin{equation*}\textrm{e}^{\gamma \pi_1} \left\{ \int_0^{b_1} \textrm{e}^{\gamma z}\textrm{d} F(z) + \textrm{e}^{\gamma d_1} S(b_1) \right\} \ge \textrm{e}^{\gamma \pi_2} \left\{ \int_0^{d_2} \textrm{e}^{\gamma z}\textrm{d} F(z) + \textrm{e}^{\gamma d_2} S(d_2) \right\},\end{equation*}

then $\bar b_2(b_1) = d_2$ . Otherwise, $\bar b_2(b_1)$ is the unique zero of $\unicode{x1D4F0}_2(\cdot \, ;\, b_1)$ in $(d_2, \infty)$ , in which

(3.7) \begin{align}\unicode{x1D4F0}_2(b_2;\, b_1) &= \big( \textrm{e}^{\gamma(b_2 - d_2)} - 1 \big) + p \textrm{e}^{-\gamma(c - \pi_1)} \left\{ \int_0^{b_1} \textrm{e}^{\gamma z}\textrm{d} F(z) + \textrm{e}^{\gamma(b_2 - d_2)} \textrm{e}^{\gamma d_1} S(b_1) \right\} \notag \\&\quad - p \textrm{e}^{-\gamma(c - \pi_2)} \left\{ \int_0^{b_2} \textrm{e}^{\gamma z}\textrm{d} F(z) + \textrm{e}^{\gamma b_2} S(b_2) \right\}.\end{align}

Proof. After factoring $\frac{1 - p}{D^2} \, \textrm{e}^{\gamma(c - \pi_1)} \textrm{e}^{\gamma(c - \pi_2)} \, \textrm{e}^{\gamma d_2} f(b_2) \big(\textrm{e}^{\gamma(c - \pi_1)} - p \int_0^{b_1} \textrm{e}^{\gamma z} \textrm{d} F(z) \big) > 0$ from the derivative of $A_2$ with respect to $b_2$ , we obtain that this derivative is positively proportional to $\unicode{x1D4F0}_2$ in (3.7). The derivative of $\unicode{x1D4F0}_2$ with respect to $b_2$ equals

\begin{equation*}\dfrac{\partial \unicode{x1D4F0}_2}{\partial b_2} = \gamma \textrm{e}^{\gamma(b_2 - d_2)} \big( 1 + p \textrm{e}^{-\gamma(c - (\pi_1 + d_1))} S(b_1) - p \textrm{e}^{-\gamma(c - (\pi_2 + d_2))} S(b_2) \big),\end{equation*}

which is positive because $\textrm{e}^{-\gamma(c - (\pi_2 + d_2))} < 1$ . We obtain the claimed result by following the same argument as in the proof of Proposition 3.1.

Recall from Definition 3.1 that the insured’s equilibrium barrier strategy, if exists, is a fixed point of the mapping $(b_1, b_2) \mapsto (\bar{b}_1 (b_2), \bar{b}_2(b_1))$ , in which $\bar{b}_1 (b_2) \ge d_1$ and $\bar{b}_2(b_1) \ge d_2$ are obtained in the previous two propositions, respectively. We naturally ask whether the fixed point will be achieved at the boundary corner $(d_1, d_2)$ ; in that case, the insured reports all losses above the deductible, and there is no ex post moral hazard in reporting. The next theorem provides an answer to this question.

Theorem 3.1. If $d_1$ and $d_2$ satisfy

(3.8) \begin{equation}\textrm{e}^{\gamma \pi_1} \left\{ \int_0^{d_1} \textrm{e}^{\gamma z}\textrm{d} F(z) + \textrm{e}^{\gamma d_1} S(d_1) \right\} \ge \textrm{e}^{\gamma \pi_2} \left\{ \int_0^{d_2} \textrm{e}^{\gamma z}\textrm{d} F(z) + \textrm{e}^{\gamma d_2} S(d_2) \right\},\end{equation}

then the equilibrium strategy $\left(b_1^*, b_2^*\right)$ is $(d_1, d_2)$ .

Proof. Note that

(3.9) \begin{equation}\dfrac{\partial}{\partial b}\left[ \textrm{e}^{\gamma \pi_1} \left\{ \int_0^{b} \textrm{e}^{\gamma z}\textrm{d} F(z) + \textrm{e}^{\gamma d} S(b) \right\} \right] = \big(\textrm{e}^{\gamma b} - \textrm{e}^{\gamma d}\big) f(b) \ge 0,\end{equation}

for all $b \ge d$ . Thus, if (3.8) is satisfied, then, by using (3.9) and $b_1 \ge d_1$ , we get

\begin{align*}\textrm{e}^{\gamma \pi_1} \left\{ \int_0^{b_1} \textrm{e}^{\gamma z}\textrm{d} F(z) + \textrm{e}^{\gamma d_1} S(b_1) \right\}&\ge \textrm{e}^{\gamma \pi_2} \left\{ \int_0^{d_2} \textrm{e}^{\gamma z}\textrm{d} F(z) + \textrm{e}^{\gamma d_2} S(d_2) \right\}.\end{align*}

Under the above condition, Propositions 3.1 and 3.2, along with Assumption 3.1, imply that $(d_1, d_2)$ is the unique equilibrium barrier strategy.

One can interpret inequality (3.8) as follows: Consider the expected utility of the per-period outgo of the insured when she is in rate class i and when she files claims for all losses in excess of the deductible (or equivalently, $b_i = d_i$ ), that is,

\begin{equation*} \mathbb{E} \left({-} \textrm{e}^{-\gamma({-} \pi_i - (Z \wedge d_i))} \right) = - \textrm{e}^{\gamma \pi_i} \left\{ \int_0^{d_i} \textrm{e}^{\gamma z}\textrm{d} F(z) + \textrm{e}^{\gamma d_i} S(d_i) \right\}.\end{equation*}

Inequality (3.8) means that the expected utility of this outgo is (weakly) less when the insured is in rate class 1 than when she is in rate class 2. Therefore, the insured (weakly) prefers to be in rate class 2 and has no incentive to hide her losses in excess of the deductible. This possible preference for rate class 2 defeats its purpose as a less desirable rate class, or “punishment” for filing claims, so for the remainder of this section, we assume that $d_1$ and $d_2$ satisfy

(3.10) \begin{equation} \textrm{e}^{\gamma \pi_1} \left\{ \int_0^{d_1} \textrm{e}^{\gamma z}\textrm{d} F(z) + \textrm{e}^{\gamma d_1} S(d_1) \right\} < \textrm{e}^{\gamma \pi_2} \left\{ \int_0^{d_2} \textrm{e}^{\gamma z}\textrm{d} F(z) + \textrm{e}^{\gamma d_2} S(d_2) \right\}.\end{equation}

Given (3.10), we show in the next theorem that the insured employs a nontrivial barrier strategy in equilibrium and has incentives to hide certain losses above the deductible, which is in stark contrast to the finding of Theorem 3.1.

Theorem 3.2. Assume $d_1$ and $d_2$ satisfy (3.10). The insured’s equilibrium barrier strategy is uniquely given by $\left(b_1^*, b_2^*\right) = (b_1^*, b_1^* - d_1 + d_2) \in \mathcal{D}$ , in which $b_1^*$ equals the unique zero of $\unicode{x1D4F1}$ on $(d_1, \infty)$ , with $\unicode{x1D4F1}$ defined by:

(3.11) \begin{align}\unicode{x1D4F1}(b) &= \big( \textrm{e}^{\gamma(b - d_1)} - 1 \big) + p \textrm{e}^{-\gamma(c - \pi_1)} \left\{ \int_0^{b} \textrm{e}^{\gamma z}\textrm{d} F(z) + e^{\gamma b} S(b) \right\} \notag \\&\quad - p \textrm{e}^{-\gamma(c - \pi_2)} \left\{ \int_0^{b - d_1 + d_2} \textrm{e}^{\gamma z}\textrm{d} F(z) + \textrm{e}^{\gamma(b - d_1 + d_2)} S(b - d_1 + d_2) \right\}.\end{align}

Proof. We start by showing that $b_i^* \neq d_i$ for $i=1,2$ , if the inequality in (3.10) holds. First, (3.9) and (3.10) imply, for $b_2 \ge d_2$ ,

\begin{equation*}\textrm{e}^{\gamma \pi_1} \left\{ \int_0^{d_1} \textrm{e}^{\gamma z}\textrm{d} F(z) + \textrm{e}^{\gamma d_1} S(d_1) \right\} < \textrm{e}^{\gamma \pi_2} \left\{ \int_0^{b_2} \textrm{e}^{\gamma z}\textrm{d} F(z) + \textrm{e}^{\gamma d_2} S(b_2) \right\}.\end{equation*}

It follows from Proposition 3.1 that $\bar b_1(b_2) > d_1$ for all $b_2 \ge d_2$ .

Next, we show that $(\bar b_1(d_2), d_2)$ is not an equilibrium. Assuming the contrary holds, we have $\unicode{x1D4F0}_1(\bar b_1(d_2);\, d_2) = 0$ , which, by (3.6), is equivalent to

\begin{align*}0 &= \big( \textrm{e}^{\gamma(\bar b_1(d_2) - d_1)} - 1 \big) + p \textrm{e}^{-\gamma(c - \pi_1)} \left\{ \int_0^{\bar b_1(d_2)} \textrm{e}^{\gamma z}\textrm{d} F(z) + e^{\gamma \bar b_1(d_2)} S(\bar b_1(d_2)) \right\} \notag \\&\quad - p \textrm{e}^{-\gamma(c - \pi_2)} \left\{ \int_0^{d_2} \textrm{e}^{\gamma z}\textrm{d} F(z) + \textrm{e}^{\gamma(\bar b_1(d_2) - d_1)} \textrm{e}^{\gamma d_2} S(d_2) \right\}.\end{align*}

Using $\bar b_1(b_2) > d_1$ , (3.9), and the above equality, we obtain

\begin{align*}&\textrm{e}^{\gamma \pi_1} \left\{ \int_0^{\bar b_1(b_2)} \textrm{e}^{\gamma z}\textrm{d} F(z) + \textrm{e}^{\gamma d_1} S(\bar b_1(b_2)) \right\} \\&\le \textrm{e}^{\gamma \pi_1} \left\{ \int_0^{\bar b_1(b_2)} \textrm{e}^{\gamma z}\textrm{d} F(z) + \textrm{e}^{\gamma \bar b_1(b_2)} S(\bar b_1(b_2)) \right\} \\&= - \, \dfrac{\textrm{e}^{\gamma c}}{p} \, \big( \textrm{e}^{\gamma(\bar b_1(b_2) - d_1)} - 1 \big) + \textrm{e}^{\gamma \pi_2} \left\{ \int_0^{d_2} \textrm{e}^{\gamma z}\textrm{d} F(z) + \textrm{e}^{\gamma(\bar b_1(b_2) - d_1)} \textrm{e}^{\gamma d_2} S(d_2) \right\} \\&= \textrm{e}^{\gamma \pi_2} \left\{ \int_0^{d_2} \textrm{e}^{\gamma z}\textrm{d} F(z) + \textrm{e}^{\gamma d_2} S(d_2) \right\} - \big( \textrm{e}^{\gamma(\bar b_1(b_2) - d_1)} - 1 \big) \left( \dfrac{\textrm{e}^{\gamma c}}{p} - \textrm{e}^{\gamma(\pi_2 + d_2)} S(d_2) \right) \\&< \textrm{e}^{\gamma \pi_2} \left\{ \int_0^{d_2} \textrm{e}^{\gamma z}\textrm{d} F(z) + \textrm{e}^{\gamma d_2} S(d_2) \right\},\end{align*}

in which the last inequality follows from $c > \pi_2 + d_2$ in (2.2). Thus, Proposition 3.2 implies $\bar b_2\big(\bar b_1(d_2) \big) > d_2$ , a contradiction! Therefore, $(\bar b_1(d_2), d_2)$ is not an equilibrium.

It follows that an equilibrium $\left(b_1^*, b_2^*\right)$ , if it exists, simultaneously solves the two equations $\unicode{x1D4F0}_1(b_1;\, b_2) = 0$ and $\unicode{x1D4F0}_2(b_2;\, b_1) = 0$ , in which $\unicode{x1D4F0}_1$ and $\unicode{x1D4F0}_2$ are given in (3.6) and (3.7), respectively. By subtracting those two equations, we obtain

\begin{equation*}0 = \big(\textrm{e}^{\gamma (b_1 - d_1)} - \textrm{e}^{\gamma(b_2 - d_2)} \big) \left\{1 + p \left( \textrm{e}^{-\gamma(c - (\pi_1 + d_1))} S(b_1) - \textrm{e}^{-\gamma(c - (\pi_2 + d_2))} S(b_2) \right) \right\},\end{equation*}

which implies $b_1^* - d_1 = b_2^* - d_2$ because the factor in curly brackets is positive.

By substituting $b_2^* = b_1^* - d_1 + b_2$ into $\unicode{x1D4F0}_1(b_1;\, b_2) = 0$ , we get $\unicode{x1D4F1}(b_1^*) = 0$ , in which $\unicode{x1D4F1}$ is given by (3.11). Next, we show that $\unicode{x1D4F1}$ has a unique positive zero in $(d_1, \infty)$ . To that end, note

\begin{equation*}\unicode{x1D4F1}(d_1) = p \textrm{e}^{-\gamma(c - \pi_1)} \left\{ \int_0^{d_1} \textrm{e}^{\gamma z}\textrm{d} F(z) + e^{\gamma d_1} S(d_1) \right\} - p \textrm{e}^{-\gamma(c - \pi_2)} \left\{ \int_0^{d_2} \textrm{e}^{\gamma z}\textrm{d} F(z) + \textrm{e}^{\gamma(d_2)} S(d_2) \right\} < 0,\end{equation*}

in which the inequality follows from (3.10). Also,

\begin{align*}\lim_{b \to \infty} \unicode{x1D4F1}(b) & = \lim_{b \to \infty} \textrm{e}^{\gamma(b - d_1)} \left[ 1 + p \textrm{e}^{-\gamma(c - (\pi_1 + d_1))} \left\{ \int_0^b \textrm{e}^{-\gamma(b - z)} \textrm{d} F(z) + S(b) \right\} \right. \\&\;\quad \qquad \qquad \qquad \left. - p \textrm{e}^{-\gamma(c - (\pi_2 + d_2))} \left\{ \int_0^{b - d_1 + d_2} \textrm{e}^{-\gamma(b - d_1 + d_2 - z)} \textrm{d} F(z) + S(b - d_1 + d_2) \right\} \right] \\&= \infty,\end{align*}

in which the limit follows from $c > \max(\pi_1 + d_1, \pi_2 + d_2)$ and $\int_0^b \textrm{e}^{-\gamma(b - z)} \textrm{d} F(z) + S(b) \in (0, 1)$ . Finally,

\begin{equation*}\unicode{x1D4F1}'(b) = \gamma \textrm{e}^{\gamma(b - d_1)}\left\{1 + p \left( \textrm{e}^{-\gamma(c - (\pi_1 + d_1))} S(b) - \textrm{e}^{-\gamma(c - (\pi_2 + d_2))} S(b - d_1 + d_2) \right) \right\} > 0,\end{equation*}

proving the claim.

It remains to show that $\left(b_1^*, b_2^*\right) \in \mathcal{D}$ , that is, $D\left(b_1^*, b_2^*\right) > 0$ , which, by Lemma 3.1, is equivalent to (3.5). Setting

\begin{equation*}\mathcal{L}(b) = \int_0^{b} \textrm{e}^{-\gamma(b- z)} \textrm{d} F(z) + S(b),\end{equation*}

which lies in (0, 1), we obtain the following equivalence relations:

\begin{align*}&D\left(b_1^*, b_2^*\right) > 0 \\&\iff \left\{ \textrm{e}^{\gamma(c - \pi_1)} - p \int_0^{b_1^*} \textrm{e}^{\gamma z} \textrm{d} F(z) \right\} \left( \textrm{e}^{\gamma(c - \pi_2)} - p \textrm{e}^{\gamma d_2} S(b_2^*) \right) > p^2 \textrm{e}^{\gamma d_1} S(b_1^*) \int_0^{b_2^*} \textrm{e}^{\gamma z} \textrm{d} F(z) \\&\iff \textrm{e}^{\gamma(b_1^* + d_2)} \left[ \left( \textrm{e}^{\gamma(c - \pi_1 - b_1^*)} - p \mathcal{L}(b_1^*) \right) + p S(b_1^*) \right] \left\{ \left( \textrm{e}^{\gamma(c - (\pi_2 + d_2))} - p \mathcal{L}(b_2^*) \right) + p \int_0^{b_2^*} \textrm{e}^{-\gamma(b_2^* - z)} \textrm{d} F(z) \right\} \\&\qquad \quad > p^2 \textrm{e}^{\gamma (d_1 + b_2^*)} S(b_1^*) \int_0^{b_2^*} \textrm{e}^{-\gamma(b_2^* - z)} \textrm{d} F(z) \\&\iff \left( \textrm{e}^{\gamma(c - \pi_1 - b_1^*)} - p \mathcal{L}(b_1^*) \right) \left\{ \left( \textrm{e}^{\gamma(c - (\pi_2 + d_2))} - p \mathcal{L}(b_2^*) \right) + p \int_0^{b_2^*} \textrm{e}^{-\gamma(b_2^* - z)} \textrm{d} F(z) \right\} \\&\qquad \quad + p S(b_1^*) \left( \textrm{e}^{\gamma(c - (\pi_2 + d_2))} - p \mathcal{L}(b_2^*) \right) > 0,\end{align*}

which holds if $p\textrm{e}^{\gamma b_1^*} \mathcal{L}(b_1^*) < \textrm{e}^{\gamma(c - \pi_1)}$ . Note that $b \mapsto \textrm{e}^{\gamma b} \mathcal{L}(b)$ strictly increases. If $p \textrm{e}^{\gamma b} \mathcal{L}(b) < \textrm{e}^{\gamma(c - \pi_1)}$ for all $b \ge d_1$ , then we are done. Thus, without loss of generality, let $\tilde{b} \ge d_1$ be the unique value that solves

(3.12) \begin{equation}p \, \textrm{e}^{\gamma \tilde{b}} \mathcal{L}(\tilde{b}) = \textrm{e}^{\gamma(c - \pi_1)}.\end{equation}

As an aside, note that $\tilde{b} > d_1$ ; indeed, (3.12) implies $p \textrm{e}^{\gamma(\tilde{b} - d_1)} \mathcal{L}(\tilde{b}) = \textrm{e}^{\gamma(c - (\pi_1 + d_1))}$ , which is greater than 1 because $c > \pi_1 + d_1$ . Thus, because $p \mathcal{L}(\tilde{b}) \in (0, 1)$ , we must have $\textrm{e}^{\gamma(\tilde{b} - d_1)} > 1$ , which implies $\tilde{b} > d_1$ .

Now, $p\textrm{e}^{\gamma b_1^*} \mathcal{L}(b_1^*) < \textrm{e}^{\gamma(c - \pi_1)}$ if and only if $b_1^* < \tilde{b}$ , which is equivalent to $\unicode{x1D4F1}(\tilde{b}) > 0$ because $\unicode{x1D4F1}$ is strictly increasing. We compute

\begin{align*}\unicode{x1D4F1}(\tilde{b}) &= \big( \textrm{e}^{\gamma(\tilde{b} - d_1)} - 1 \big) + p \textrm{e}^{-\gamma(c - \pi_1)} \textrm{e}^{\gamma \tilde{b}} \mathcal{L}(\tilde{b}) - p \textrm{e}^{-\gamma(c - \pi_2)} \textrm{e}^{\gamma (\tilde{b} - d_1 + d_2)} \mathcal{L}(\tilde{b} - d_1 + d_2) \\&= \textrm{e}^{\gamma(\tilde{b} - d_1)} \left(1 - p \textrm{e}^{-\gamma(c - (\pi_2 + d_2))} \mathcal{L}(\tilde{b} - d_1 + d_2) \right),\end{align*}

which is positive because $c > \pi_2 + d_2$ and $p \mathcal{L}(\tilde{b} - d_1 + d_2) \in (0, 1)$ . It follows that $b_1^* < \tilde{b}$ , which implies $p\textrm{e}^{\gamma b_1^*} \mathcal{L}(b_1^*) < \textrm{e}^{\gamma(c - \pi_1)}$ . Therefore, $D\left(b_1^*, b_2^*\right) > 0$ holds, and $\left(b_1^*, b_2^*\right) \in \mathcal{D}$ is verified.

Remark 3.4. We return to an earlier comment regarding the choice of formulating the insured’s loss reporting problem as a noncooperative Nash game, as in Definition 3.1. The same approach was first adopted by Thomas Björk and his coauthors (see Björk et al., Reference Björk, Murgoci and Zhou2014, Reference Björk, Khapko and Murgoci2017) to handle time-inconsistent stochastic control problems, in which the term “time inconsistency” roughly means that an optimal strategy derived by an agent at time s might not be optimal for the same agent at a later time $t > s$ . One possible, certainly not the only, solution is to “partition” the single agent into many selfs (players) indexed by time and formulate a noncooperative Nash game, in which each player only controls during her own time period. If an equilibrium for such a game can be found, no player has an incentive to deviate from it, and thus such a solution can be adopted by the agent without imposing further commitment. With that in mind, the insured in our context faces the same challenge; that is, the best decision based on her current rate class might not be the best when she moves to a different rate class. In other words, $\arg\max_{(b_1, b_2)} J_1 \neq \arg\max_{(b_1, b_2)} J_2$ is possible in general. As such, following the game-theoretical approach, we interpret the insured as two players: player i represents the insured in rate class i and chooses her own action $b_i$ , $i=1,2$ .

As already hinted above, there are alternative formulations of the insured’s loss reporting problem. For example, one could directly maximize $J_1$ and $J_2$ separably, over both $b_1$ and $b_2$ . Recall from (3.1) that $J_i = - A_i \, \textrm{e}^{-\gamma x}$ , with $A_1$ and $A_2$ given by (3.2) and (3.3), respectively; thus, we would solve

\begin{align*} \inf_{(b_1, b_2) \in [d_1, \infty) \times [d_2, \infty)} \, A_i(b_1, b_2), \quad i=1,2. \end{align*}

When maximizing $J_1(b_1, b_2)$ and $J_2(b_1, b_2)$ over both $b_1$ and $b_2$ , we obtain the identical pair of first-order necessary conditions for the two maximization problems, which necessarily equal those in Propositions 3.1 and 3.2. Moreover, numerical work shows that the maximizers of $J_1$ and $J_2$ are equal to each other and are equal to the equilibrium $\left(b_1^*, b_2^*\right)$ as stated in Theorems 3.1 and 3.2. This nice coincidence, though not expected in general, indicates that maximizing the bivariate objectives $J_1$ and $J_2$ yields the same solution as our Nash equilibrium.

4. Equilibrium insurance under strategic underreporting

In this section, we study an optimal insurance problem for an insured who follows the equilibrium barrier strategy obtained in Theorem 3.1 or 3.2 to strategically underreport her losses. We pose this problem in Question 2 in the Introduction.

We start with a formal definition of the insured’s equilibrium deductible insurance under strategic underreporting as follows. Recall that $(b_1^*(d_1,d_2), b_2^*(d_1,d_2))$ denotes the insured’s equilibrium barrier strategy obtained in Theorem 3.1 or 3.2, for a fixed deductible pair $(d_1, d_2) \in \mathbb{R}_+^2$ .

The premium will depend on the deductible chosen, and for concreteness, we assume that the insurer applies the expected-value principle to calculate premiums, but the premium loading varies by rate class. To be precise, if the insured is currently in rate class i and chooses a deductible $d_i \ge 0$ , her (per-period) premium is given by:

\begin{equation*} \pi_i = (1 + \theta_i) \, \mathbb{E}(Z - d_i)_+, \quad i = 1, 2,\end{equation*}

in which $\theta_i > 0$ is the premium loading for rate class i, and $x_+ \,:\!=\, \max\{x, 0\}$ . Because class 2 is considered as “punishment” and rate class 1 as “reward,” we assume $\theta_1 < \theta_2$ throughout the paper.

To obtain the equilibrium insurance contract as stated in Definition 4.1, we solve the following problem for each rate class:

(4.1) \begin{align} \sup_{d_i \ge 0} \; \unicode{x1D4D9}_i(x;\,d_1, d_2), \qquad i = 1, 2,\end{align}

in which $\unicode{x1D4D9}_i$ equals

\begin{equation*}\unicode{x1D4D9}_i(x;\, d_1, d_2) = J_i\big(x;\, b_1^*(d_1,d_2), b_2^*(d_1,d_2), d_1, d_2\big).\end{equation*}

Recall $J_i$ is defined in (2.3), and $\big(b_1^*(d_1,d_2), b_2^*(d_1,d_2) \big)$ is obtained as in Theorem 3.1 or 3.2. If an equilibrium $\left(d_1^*, d_2^*\right)$ exists, we will write $b_1^*$ (resp. $b_2^*$ ) to denote $b_1^*\left(d_1^*, d_2^*\right)$ $\big($ resp. $b_2^*\left(d_1^*, d_2^*\right)\big)$ .

Although Theorems 3.1 and 3.2 provide a complete characterization of $\big(b_1^*(d_1,d_2), b_2^*(d_1,d_2) \big)$ , the results are semi-explicit, and the related inequalities in (3.8) and (3.10) strongly depend on the loss distribution F. In consequence, finding an analytical solution to the optimization problems in (4.1) is likely impossible. So, we resort to a numerical approach for finding the equilibrium deductible startegy. To start, we assume in the rest of this section that the insured’s per-period loss Z is a mixture of a point mass at zero and a continuous random variable $\tilde{Z} \sim Gamma(\kappa, \lambda)$ , with weights $p_0 \in (0,1)$ and $1 - p_0$ , respectively. Recall that the $Gamma(\kappa, \lambda)$ distribution is characterized by its probability density function:

\begin{equation*}f(x) = \dfrac{x^{\kappa - 1} \, \textrm{e}^{-x/\lambda}}{\lambda^{\kappa} \Gamma(\kappa)},\quad x > 0,\qquad \text{with} \quad \Gamma(\kappa) = \int_0^\infty t^{\kappa-1} \textrm{e}^{-t} \textrm{d} t.\end{equation*}

The Gamma distribution belongs to the exponential family of distributions and can capture “fat tails” that are often encountered in insurance data; see, for instance, Section 17.3.1 in Frees (Reference Frees2009). In Appendix A, we consider another popular fat-tailed distribution, specifically, the Pareto distribution.

We set the parameter values for the model as listed in Table 2, which constitutes the base case of our numerical study. The key purpose of the subsequent numerical analysis is to obtain qualitative findings on the equilibrium deductible strategy and its implications. In addition, we conduct extensive sensitivity analysis to understand how model parameters affect the insured’s equilibrium decisions on insurance and reporting. When we investigate the impact of a particular parameter on the insured’s equilibrium strategy, we allow this parameter to vary over a reasonable range but keep the remaining parameters unchanged, with values as in Table 2.

Table 2. Parameter values in the base case.

Under the base case with the parameters as in Table 2, we compute

(4.2) \begin{align} \left(d_1^*, d_2^*\right) = (7.9029, 12.2407) \quad \text{and} \quad \left(b_1^*, b_2^*\right) = (7.9515, 12.2893).\end{align}

As a side note, the expected loss size is $\mathbb{E} Z = (1-p_0) \mathbb{E}\tilde{Z} = 9$ . We summarize the key common findings from the base case and the subsequent sensitivity studies as follows.

• First, the equilibrium deductibles are always strictly positive (i.e., $d_1^* > 0$ and $d_2^* > 0$ ). In consequence, when strategic underreporting is considered and allowed, full insurance is not optimal; therefore, the full insurance assumption made in Cao et al. (Reference Cao, Li, Young and Zou2023b) and several other papers are questionable. As underreporting is a form of ex post moral hazard, this result strengthens the conclusion that “full insurance is never optimal in the presence of moral hazard” (Winter, Reference Winter and Dionne2013, p.212).

• Second, the inequality $d_2^* > d_1^*$ always holds. This result implies that the insured’s risk classification negatively affects the insurance demand; that is, the worse the rate class, the less the insurance coverage. Several empirical studies obtain the same conclusion; see, for instance, Li et al. (Reference Li, Liu and Yeh2007) for auto insurance and Dombrowski et al. (Reference Dombrowski, Pace and Ratnadiwakara2020) for flood insurance.

• Third, the numerical solutions confirm

  (4.3) \begin{align} b_1^* - d_1^* = b_2^* - d_2^* \,=\!:\, b^* - d^* > 0. \end{align}
  Since the difference is the same in both rate classes, we denote such a difference by $b^* - d^*$ and name it the amount of hidden losses. Recall that in the presence of a deductible d and a reporting barrier b, losses greater than d but less than b are the ones that are strategically hidden by the insured. Note that we know, from Theorem 3.1, that given an insurance contract $(d_1, d_2)$ , it is possible for the insured to report full losses, that is, $b_i^* - d_i =0$ for $i = 1, 2$ . So, the essential takeaway from (4.3) is that the amount of hidden losses is strictly positive in equilibrium, which provides theoretical explanation for the insured’s underreporting and shows that inequality (3.10) holds in equilibrium. It also generalizes the result in Cao et al. (Reference Cao, Li, Young and Zou2023b), in which the authors show that the hidden losses are positive under full insurance; our work shows that this result still holds when the insured can choose her deductibles freely.

• Last, we can show that the “double equilibrium” contract under strategic underreporting in (4.2) yields higher welfare (as measured by expected utility) for the insured than the “single equilibrium” contract under full reporting. To arrive at this conclusion, we can force the reporting barriers to be equal to the contract deductibles, $b_i = d_i$ for $i=1,2$ , which effectively prevents underreporting. Under this constraint, we revisit the insured’s optimal deductible insurance problem in (4.1) and obtain the corresponding equilibrium deductibles $\,\tilde{\!d}_1$ and $\,\tilde{\!d}_2$ :

  \begin{align*} \,\tilde{\!d}_1 = 7.9030 > d_1^* \quad \text{and} \quad \,\tilde{\!d}_2 = 12.2409 > d_2^*. \end{align*}
  Further, set $x=0$ and write the insured’s expected utility $J_i$ under $(d_1, d_2, b_1, b_2) = (d_1^*, d_2^*, b_1^*, b_2^*)$ as $J_i^*$ and under $(d_1, d_2, b_1, b_2) = (\,\tilde{\!d}_1, \,\tilde{\!d}_2, \,\tilde{\!d}_1, \,\tilde{\!d}_2)$ as $\tilde{J}_i$ , $i=1,2$ . Then, a direct computation shows that $J_i^* > \tilde{J}_i$ holds for both $i=1,2$ , although the two quantities are close. For instance, under the scaling from $J_i$ to $-100 \ln ({-}J_i)$ , we obtain
  \begin{align*} -100 \ln ({-}J^*_1) &= 403.6277 > 403.6268 =-100 \ln ({-} \tilde{J}_1), \\ -100 \ln ({-}J^*_2) &= 396.5032 > 396.5025 = -100 \ln ({-} \tilde{J}_2) .\end{align*}

Figure 1. Impact of insured’s risk aversion $\gamma$ on the equilibrium deductibles and hidden losses.

In the first sensitivity study, we investigate how the insured’s risk aversion $\gamma$ affects her equilibrium deductibles $\left(d_1^*, d_2^*\right)$ and the amount of hidden losses $b^* - d^*$ , as defined in (4.3). The left panel of Figure 1 shows that both $d_1^*$ and $d_2^*$ are decreasing in $\gamma$ . This finding is consistent with the classic literature without moral hazard, namely, that a more risk-averse agent demands more insurance coverage; see Arrow (Reference Arrow1974).

Surprisingly, the amount of hidden losses $b^* - d^*$ is not monotone in $\gamma$ , as shown in the right panel of Figure 1. When the risk aversion parameter $\gamma$ increases from 0 to 1, $b^* - d^*$ first increases and then decreases. To understand this non-monotonicity of $b^* - d^*$ with respect to $\gamma$ , one can show that for the function $\unicode{x1D4F1}$ defined in (3.11), $\lim \limits_{\gamma \to 0^+} \frac{\unicode{x1D4F1}(b_1)}{\gamma} = b_1 - d_1 - p(\pi_2 - \pi_1)$ . As $\gamma \to 0^+$ , the insured becomes risk-neutral and has no demand for insurance, this implies both $d_1^*$ and $d_2^*$ converge to infinity, which implies $\lim \limits_{\gamma \to 0^+} \pi_2 = \lim \limits_{\gamma \to 0^+} \pi_1 = 0$ and $\lim \limits_{\gamma \to 0^+} (b_1^* - d_1^*) =0$ . On the other hand, as $\gamma$ gets arbitrarily large, the individual becomes very risk-averse, and her equilibrium deductibles and barriers all approach 0. Thus, applying Occam’s razor, we expect $b^* - d^*$ either (1) to increase from 0 to some positive number and, then, decrease to 0 as $\gamma$ increases, or (2) to stay identically 0. Our numerical work shows that the former occurs.

Figure 2. Impact of premium loading $\theta_1$ on equilibrium deductibles and hidden losses.

Figure 3. Impact of premium loading $\theta_2$ on equilibrium deductibles and hidden losses.

We next consider the impact of the premium loadings $\theta_1$ and $\theta_2$ on the equilibrium deductibles and hidden losses. From the two left panels in Figures 2 and 3, we see that the equilibrium deductible in each rate class increases with respect to the premium loading of that rate class and does not change much with respect to the premium loading for the other rate class. Therefore, the insured purchases less insurance as the premium loading for her current rate class increases. Also, as the penalty decreases (i.e., as $\theta_2 - \theta_1$ decreases), then $d_2^* - d_1^*$ decreases.

From the two right panels in Figures 2 and 3, we deduce that the amount of hidden losses decreases in $\theta_1$ and increases in $\theta_2$ . The reason for this effect is that, with a smaller $\theta_1$ or larger $\theta_2$ , the difference in premium loading between the two rate classes is larger, or in other words, the penalty for changing from rate class 1 to rate class 2 (or the reward for moving in the other direction) is larger. As a result, the insured has more incentive to hide her losses. Also, note that $d_1^*=0$ when $\theta_1=0$ , which means the insured purchases full insurance in rate class 1 when the premium loading is 0. However, this result also depends on other model parameters; for example, in work not shown here, if the per-period income rate is low enough, the insured will not necessarily purchase full insurance even if $\theta_1 = 0$ .

Figure 4. Impact of probability mass at zero $p_0$ on equilibrium deductibles and hidden losses.

In Figure 4, we study the impact of $p_0 = \mathbb{P}(Z = 0)$ , the probability of no loss per period, on the insured’s equilibrium strategies. The right panel shows that the amount of hidden losses decreases in $p_0$ . The reason lies behind the fact that the motivation for underreporting comes from the difference in the premiums between the two rate classes. When $p_0$ increases, a positive loss is less likely to occur and the difference in the premiums between the two rate classes shrinks. Thus, the insured will hide less of her loss. The left panel of Figure 4 shows that the two equilibrium deductibles $d_1^*$ and $d_2^*$ both decrease in $p_0$ , suggesting the insured will purchase more insurance for more unlikely losses, which seems counterintuitive. However, note that with a larger value of $p_0$ , the premium also decreases, and for the same amount spent on insurance premium, the insured could purchase more insurance via a lower deductible.

Figure 5. Impact of renewal probability p on equilibrium deductibles and hidden losses.

Figure 5 presents the effect of p on the equilibrium deductibles and hidden losses. Recall that $p = \mathbb{P}(\tau>1)$ , with $\tau$ being the insured’s planning horizon, measures the probability of receiving the reward for underreporting (or penalty for reporting). When $p = 0$ , the contract ends at the end of the current period with probability one; thus, there is no need to consider the possible penalty or reward for the next period. In this case, the insured reports her full loss, explaining why $b^* - d^* = 0$ when $p = 0$ . As p increases, the probability of reward for underreporting increases, which motivates the insured to hide more losses. This argument explains the increase of $b^* - d^*$ with respect to p observed in the right panel of Figure 5. We observe from the left panel of Figure 5 that both $d_1^*$ and $d_2^*$ increase in p, meaning the insured will buy less insurance in both rate classes if the game is more likely to continue. The reason for this phenomenon is, as p increases, the expected time horizon $\mathbb{E} \tau$ also increases. If the insured were to purchase the same amount of insurance, the per-period premium would stay the same but the premium over the entire time horizon would increase, which would reduce the insured’s terminal wealth. Thus, the insured is better-off purchasing less insurance and relying more on self-insuring (because both $b_1^*$ and $b_2^*$ increase) for a longer expected time horizon.

Figure 6. Impact of income rate c on equilibrium deductibles and hidden losses.

Finally, Figure 6 demonstrates the impact of the per-period income rate c on equilibrium strategies. The left panel shows that both equilibrium deductibles $d_1^*$ and $d_2^*$ decrease with respect to c. Thus, an insured with higher income will purchase more insurance, because the insurance is relatively less expensive. Similarly, with a higher income rate, the reward for underreporting is less significant, which induces the insured to hide less losses, as shown in the right panel of Figure 6.

5. Impact of underreporting on insurer’s profit

We analyzed strategic loss reporting for a risk-averse insured in Section 3 and examined how underreporting affects her insurance demand in Section 4. In this section, we shift our attention to study how the insured’s underreporting affects the insurer’s profit.

Suppose the insured has chosen a pair of deductibles $\vec{d}\,:\!=\,(d_1, d_2) \in \mathbb{R}_+^2$ and a pair of reporting barriers $\vec{b}\,:\!=\,(b_1, b_2) \in [d_1, \infty) \times [d_2, \infty)$ , both of which can be arbitrarily chosen from their respective domain. Let $Y_t$ denote the insurer’s cumulative profit over [0, t] from this insurance policy for $t = 1, 2, \dots$ . Given that the insured is in rate class i, $i=1,2$ , during $[t, t+1)$ , the dynamics of the insurer’s profit is governed by:

\begin{equation*} Y_{t+1} = Y_t + \pi_i - (Z _{t+1}- d_i) \,\cdot \unicode{x1D7D9}_{\{Z_{t+1}> b_i \}}, \quad Y_0 = y \in \mathbb{R},\end{equation*}

in which $Z_{t+1}$ is the insured’s loss in this period. We now define the insurer’s total expected profit at the insured’s exit time $\tau$ by:

(5.1) \begin{equation} V_i(y)= \mathbb{E}( Y_\tau | Y_0 = y, I_0 = i), \quad i = 1, 2,\end{equation}

in which $I_0$ denotes the insured’s initial rate class. It is clear the $V_i(y)$ depends on both $\vec{d}$ and $\vec{b}$ , and if this dependence relations need to be emphasized, we will expand the arguments accordingly. For instance, $V_i(y, b_1, b_2)$ emphasizes the dependence on the barrier strategy $(b_1, b_2)$ . Similar to the treatment of the insured’s objectives $J_i$ in Section 3, we obtain an expression for $V_i(y)$ , as summarized in the next lemma.

Lemma 5.1. The insurer’s expected profit $V_i$ , defined in (5.1), equals

(5.2) \begin{align} V_i(y) = y + \dfrac{M_i}{G}, \quad i = 1, 2, \end{align}

in which

(5.3) \begin{align} M_1 &= p S(b_1) \left\{ \pi_2 - \int_{b_2}^\infty (z - d_2) \textrm{d} F(z) \right\} + (1 - p S(b_2)) \left\{ \pi_1 - \int_{b_1}^\infty (z - d_1) \textrm{d} F(z) \right\}, \end{align}

(5.4) \begin{align} M_2 &= (1 - p F(b_1)) \left\{ \pi_2 - \int_{b_2}^\infty (z - d_2) \textrm{d} F(z) \right\} + p F(b_2) \left\{ \pi_1 - \int_{b_1}^\infty (z - d_1) \textrm{d} F(z) \right\}, \end{align}

(5.5) \begin{align} G &= (1 - p) \big( 1 + p(S(b_1) - S(b_2)) \big). \end{align}

Proof. By following a similar proof to Lemma 2.1, one can show that $V_1$ and $V_2$ satisfy the following system of equations:

\begin{align*} V_1(y) & = (1 - p) \big( y + \pi_1 - S(b_1) \, \mathbb{E} (Z - d_1 | Z > b_1) \big) \notag \\ &\quad + p \big( F(b_1) V_1(y + \pi_1) + S(b_1) \mathbb{E} \big(V_2(y + \pi_1 - Z + d_1) \big| Z > b_1 \big) \big),\\ V_2(y)&= (1 - p) \big( y + \pi_2 - S(b_2) \, \mathbb{E} (Z - d_2|Z > b_2) \big) \notag \\ &\quad + p \big( F(b_2) V_1(y + \pi_2) + S(b_2) \mathbb{E} \big(V_2(y + \pi_2 - Z + d_2) \big| Z > b_2 \big) \big). \end{align*}

To solve the above system, we consider an ansatz of $V_i$ of the form $V_i(y) = y + B_i$ , with $B_i \in \mathbb{R}$ yet to be determined. After a lengthy, but straightforward, computation, we verify that $B_i$ is uniquely given by $B_i = \frac{M_i}{G}$ , in which $M_1$ , $M_2$ , and G are defined in (5.3), (5.4), and (5.5), respectively. In addition, we easily verify that $M_1, M_2, G > 0$ and thus $V_i(y) - y >0$ .

Although Lemma 5.1 finds a fully explicit expression for the insurer’s expected profit $V_i$ from an insured in rate class i, it is still challenging to obtain analytical results about the sensitivity of $V_i$ with respect to the insured’s decisions $\vec{d}$ and $\vec{b}$ . As such, we proceed with a numerical study to achieve this goal. Without the loss of generality, set $x = 0$ and $y=0$ in the subsequent analysis and suppress the y-dependence in $V_i$ . Indeed, recall from (3.1) and (5.2) that the objectives are separable in the initial wealth x and initial profit y, and, in particular, the insured’s equilibrium reporting strategy is independent of x and y. In the numerical study of this section, we set the model parameters as in Table 2.

First, fix the insurance contract (i.e., fix the deductible pair $\vec{d} = (d_1, d_2)$ ) and study how the reporting barriers $b_1$ and $b_2$ affect the insurer’s expected profit. For comparison, we consider two contracts: full insurance with $(d_1, d_2) = (0, 0)$ , and a nontrivial deductible contract with $(d_1, d_2) = (3, 5)$ . In Figure 7, we plot the insurer’s expected profit $V_1 \,:\!=\, V_1(b_1, b_2)$ as a function of the reporting barriers $b_1$ and $b_2$ in a neighborhood of the equilibrium point $(b_1^*(d_1, d_2), b_2^*(d_1, d_2))$ . (The graphs for $V_2$ are similar and all the results discussed below hold for $V_2$ as well.) The most important finding is that, for both contracts, $V_1(d_1, d_2) > V_1 (b_1^*(d_1, d_2), b_2^*(d_1, d_2))$ ; equivalently, the insurer’s expected profit when the insured follows the equilibrium reporting strategy is strictly less than that when the insured fully reports all losses above the deductibles. This result makes intuitive sense because the savings from strategic underreporting to the insured is connected with lost profit to the insurer. In addition, we observe that the maximum value of the insurer’s expected profit over the considered region is achieved when the insured does not hide any losses (above the deductible), that is, $\arg\max V_1 (b_1, b_2) = (d_1, d_2)$ .

Figure 7. Insurer’s expected profit $V_1$ as a function of the insured’s reporting strategy $(b_1, b_2)$ .

Next, we study how the insured’s two decisions, deductibles $\vec{d}$ and reporting barriers $\vec{b}$ , affect the insurer’s expected profit, along with her own expected utility. We consider three different contracts $(d_1, d_2)$ : Contract I is the insured’s equilibrium contract, and Contract II (resp., Contract III) has higher (resp., lower) deductibles than those of Contract I. For each fixed contract $(d_1, d_2)$ , we further consider three different reporting strategies $(b_1, b_2)$ : the first strategy is full reporting with $(b_1, b_2) = (d_1, d_2)$ , the second strategy is the corresponding equilibrium reporting strategy with $(b_1, b_2) = (b_1^*(d_1, d_2), b_2^*(d_1, d_2)) > (d_1, d_2)$ , and the third strategy is the pair of integers obtained by rounding up from the equilibrium strategy, that is, $(b_1, b_2) = (\lceil b_1^*(d_1, d_2) \rceil, \lceil b_2^*(d_1, d_2) \rceil)$ .Footnote 14 Under each strategy, we report the insured’s scaled expected utility $({-}100\ln({-}J_1), -100\ln({-}J_2))$ (we scale $(J_1, J_2)$ only to make the comparison easier) and the insurer’s expected profit $(V_1, V_2)$ in Table 3, with the largest value highlighted in blue. Taking the insurer’s standpoint, we observe a consistent result that $V_1(d_1, d_2) > V_1 (b_1^*(d_1, d_2), b_2^*(d_1, d_2))$ for all three contracts, also confirmed by Figure 7 previously. As indicated by the third strategy under each contract, $V_1(b_1, b_2) > V_1(d_1, d_2)$ is possible when $(b_1, b_2)$ is above the equilibrium choice, implying that, in theory, the insurer could benefit from underreporting when the insured hides large losses. However, the reporting decision is entirely the insured’s, and the best outcome to the insured is always achieved by the equilibrium reporting strategy under each fixed contract. Table 3 further shows that the best among all nine combinations for the insured is the “double equilibrium,” that is, taking the equilibrium contract $\left(d_1^*, d_2^*\right)$ and following the equilibrium reporting strategy $(b_1^*\left(d_1^*, d_2^*\right), b_2^*\left(d_1^*, d_2^*\right))$ , as expected. Therefore, we conclude that the insured’s strategic underreporting negatively impacts the insurer’s profit.

Table 3. Insured’s expected utility and insurer’s expected profit under different strategies.

6. Conclusions

Underreporting losses is well documented in the empirical insurance literature, but very few theoretical insurance models exist to explain this behavior, let alone its impact on insurance demand. In our previous work Cao et al. (Reference Cao, Li, Young and Zou2023b), we propose a BMS insurance model with two rate classes and consider a utility-maximizing insured with full insurance who chooses a barrier strategy for reporting losses. However, as argued in Introduction, the assumption of full insurance is questionable and, in particular, prevents us from studying the impact of strategic underreporting on insurance demand. Therefore, we extend the work of Cao et al. (Reference Cao, Li, Young and Zou2023b) from full insurance to deductible insurance and make two significant contributions. First, for a fixed insurance contract (characterized by a pair of deductibles $(d_1, d_2)$ ), we solve for the equilibrium barrier strategy $(b_1^*(d_1, d_2), \, b_2^*(d_1, d_2))$ that maximizes the expected exponential utility of the insured’s terminal wealth. We obtain the equilibrium barrier strategy in semi-closed form, via the solution of a nonlinear equation. Second, we propose a novel optimal insurance problem in which the insured takes into account the fact that she will follow the equilibrium barrier strategy $(b_1^*(d_1, d_2), \, b_2^*(d_1, d_2))$ for reporting. Extensive numerical analysis confirms that the equilibrium contract deductibles $d_1^*$ and $d_2^*$ are strictly positive, justifying the extension from full insurance to deductible insurance. We also find that $b_1^*\left(d_1^*, d_2^*\right) > d_1^*$ and $b_2^*\left(d_1^*, d_2^*\right) > d_2^*$ ; that is, under the equilibrium insurance contract, the insured strategically underreports losses. We expect this last result to hold under premium principles other than the expected-value principle assumed in this paper.