2.1 Background

We refer to the individual and the life insurance company as the “buyer” and “seller,” respectively. Let $\tau$ denote the random time of death of the buyer, which is a stopping time with respect to a filtered probability space $\big(\Omega, \mathcal{F}, \mathbb{F} = \{\mathcal{F}(t) \}_{t \ge 0}, \mathbb{P} \big)$ . We assume $\tau$ follows an exponential distribution with constant hazard rate $\lambda \gt 0$ , that is,

\begin{align*} \mathbb{P}(\tau \gt t) = \mathrm{e}^{-\lambda t}, \quad t \ge 0.\end{align*}

Moreover, we assume that the individual can purchase term life insurance via a premium payable continuously at the rate of $h \ge 0$ per dollar of insurance.Footnote ² Let $D(x, t) \ge 0$ represent the death benefit that the buyer purchases at time $t^-$ (when her wealth at that time equals x). This means that if the individual spends h D(x, t) at time $t^-$ for this term life insurance, her beneficiaries will receive D(x,t) if she dies at time t. If she survives, there is no payout and the term ends. We also assume that the individual has constant net income at the rate of c, that is, c is her income net of consumption.

Furthermore, we assume that the individual can invest her wealth in a risky asset whose price process $S = \{ S_t \}_{t \ge 0}$ follows geometric Brownian motion:

\begin{equation*}\mathrm{d} S_t = \mu S_t \mathrm{d} t + \sigma S_t \mathrm{d} B_t,\end{equation*}

in which $\mu \gt 0$ , $\sigma \gt 0$ , and $B = \{ B_t \}_{t \ge 0}$ is a standard Brownian motion defined on the probability space $\big(\Omega, \mathcal{F}, \mathbb{F} = \{\mathcal{F}(t) \}_{t \ge 0}, \mathbb{P} \big)$ . Let $\pi(x, t) \in \mathbb{R}$ denote the amount of wealth invested in the risky asset at time t when her wealth at that time equals x. The buyer’s controlled wealth process $X = \{X_t\}_{t \in [0, \tau)}$ , thus, follows:

(2.1) \begin{equation}\begin{cases}\mathrm{d} X_t = \big(c - h D(X_{t}, t) + \mu \pi(X_{t}, t) \big) \mathrm{d} t + \sigma \pi(X_{t}, t) \mathrm{d} B_t, \qquad 0 \le t \lt \tau, \\[3pt] X_\tau = X_{\tau^-} + D(X_{\tau^-}, \tau),\end{cases}\end{equation}

and the seller’s controlled wealth process $Y = \{Y_t\}_{t \in [0, \tau)}$ in relation to the risk and reward arising from the single buyer follows:

(2.2) \begin{equation}\begin{cases}\mathrm{d} Y_t = h D(X_{t}, t) \, \mathrm{d} t, \qquad 0 \le t \lt \tau, \\[3pt] Y_\tau = Y_{\tau^-} - D(X_{\tau^-}, \tau).\end{cases}\end{equation}

Next, we define admissible life insurance and investment strategies for the buyer.

The individual has mean-variance preferences and optimizes over her lifetime $[0, \tau)$ . In consequence, given a life insurance and investment strategy $(\mathcal{D}, \Pi)$ , the buyer’s objective function equals

(2.3) \begin{equation}J(x, t;\, \mathcal{D}, \Pi, h) = \mathbb{E}_{x, t}(X_\tau) - \dfrac{\gamma}{2} \, \mathbb{V}_{x, t}(X_\tau),\end{equation}

in which $\mathbb{E}_{x, t}$ and $\mathbb{V}_{x, t}$ denote expectation and variance, respectively, conditional on $X_{t} = x \in \mathbb{R}$ and $\tau \gt t$ . In (2.3), the parameter $\gamma \gt 0$ represents her risk aversion toward variance. The individual seeks to maximize J in (2.3); however, the mean-variance objective can lead to a time-inconsistent problem because variance does not satisfy the property of iterated expectations. To handle this difficulty, we assume that the buyer essentially plays against future versions of herself and seeks to maximize her criterion, given her future choices and given the premium rate. In the following, we define the time-consistent Stackelberg follower action $\hat{\mathcal{D}}$ and $\hat \Pi$ of the buyer’s game.

Definition 2.2 (Buyer’s time-consistent Stackelberg follower action). Suppose we are given a non-negative premium rate for life insurance h. For a given admissible strategy $(\hat{\mathcal{D}}(h), \hat \Pi(h))$ , fix an arbitrary initial time $t \in [0, \tau)$ , a positive number $\varepsilon$ , a non-negative number D, and a real number $\pi$ . Then, define the strategy ${(\mathcal{D}^\varepsilon(h), \Pi^\varepsilon(h))}$ by:

\begin{equation*}D^\varepsilon(X_{s}, s; h) =\begin{cases}D, &\quad t \le s \lt \varepsilon \wedge \tau, \\[3pt] \hat D(X_{s}, s;\, h), &\quad \varepsilon \wedge \tau \le s \lt \tau,\end{cases}\end{equation*}

and

\begin{equation*}\pi^\varepsilon(X_{s}, s; h) =\begin{cases}\pi, &\quad t \le s \lt \varepsilon \wedge \tau, \\[3pt] \hat \pi(X_{s}, s;\, h), &\quad \varepsilon \wedge \tau \le s \lt \tau.\end{cases}\end{equation*}

The strategy $(\hat{\mathcal{D}}(h), \hat \Pi(h))$ is said to be a time-consistent Stackelberg follower action if, for all $(x, t) \in \mathbb{R} \times [0, \tau)$ , $D \ge 0$ and $\pi \in \mathbb{R}$ ,

\begin{equation*}\liminf_{\varepsilon \to 0^+} \dfrac{J(x, t;\,{\hat{\mathcal{D}}(h), \hat \Pi(h), h}) - J(x, t;\, {\mathcal{D}^\varepsilon(h), \Pi^\varepsilon(h), h})}{\varepsilon} \ge 0,\end{equation*}

and the equilibrium value function V equals

(2.4) \begin{equation}V(x, t;\,h) = J(x, t;\, \hat{\mathcal{D}}(h), \hat \Pi(h), h).\end{equation}

Next, we define the insurance company’s (the leader’s) optimal action.

Definition 2.3 (Seller’s optimal Stackelberg leader action). The price $\hat h$ is the optimal Stackelberg leader action if, for $(y, t) \in \mathbb{R} \times [0, \tau)$ ,

(2.5) \begin{equation}\hat h = {\mathop{\mathrm{arg\,sup}}\limits_{h \ge 0}} \, \mathbb{E}_{y, t} (Y_\tau),\end{equation}

in which Y follows the process in (2.2) with $(\mathcal{D}, \Pi) = (\hat{\mathcal{D}}(h), \hat \Pi(h))$ , and $\mathbb{E}_{y, t}$ denotes expectation conditional on $Y_{t} = y \in \mathbb{R}$ and $\tau \gt t$ .Footnote ³

Finally, we are ready to define the Stackelberg equilibrium of this term life insurance game.

2.2 Buyer’s problem

Before we find the equilibrium follower action for the buyer, we present a verification lemma for solving the buyer’s problem. For the statement of the verification lemma, we define a differential operator corresponding to our problem $\mathcal{L}^{(D, \pi)}$ , for all $\phi \in \mathcal{C}^{2, 1}(\mathbb{R} \times [0, \tau))$ , $D \ge 0$ , and $\pi \in \mathbb{R}$ , by:

(2.6) \begin{align}\mathcal{L}^{(D, \pi)} \, \phi(x, t) &= \phi_t(x, t) + (c - hD + \mu \pi)\phi_x(x, t) + \dfrac{1}{2} \, \sigma^2 \pi^2 \phi_{xx}(x, t).\end{align}

We omit the proof of the following verification lemma for $V(\cdot, \cdot \, ; h)$ because it is similar to the proof of Theorem 3.1 in Landriault et al. (Reference Landriault, Li, Li and Young2018).

Then, $\tilde{V}(\cdot, \cdot \, ;\, h)$ equals the individual’s equilibrium value function $V(\cdot, \cdot \, ; h)$ defined in (2.4), and $(\hat{\mathcal{D}}(h), \hat \Pi(h))$ is an equilibrium strategy. Moreover, $G(x, t;\,h) = \mathbb{E}_{x, t}(X_\tau)$ under this equilibrium strategy.

In the following theorem, we use Lemma 2.1 to obtain the buyer’s equilibrium strategy and corresponding value function.

Theorem 2.2. For a given value of the premium rate $h \ge 0$ , the equilibrium strategy $(\hat{\mathcal{D}}(h), \hat \Pi(h))=\big( \{ \hat D(X_{t}, t;\,h) \}, \{ \hat \pi(X_{t}, t;\,h) \} \big)_{t \in [0, \tau)}$ equals the constant strategy:

(2.9) \begin{align}& \hat D(X_{t}, t;\,h) \equiv \hat D(h) = \frac{1}{\gamma h}\left(\gamma c - \left( (h - \lambda) - \frac{\mu^2}{\sigma^2} \right) \right)_+, \end{align}

(2.10) \begin{align}& \hat \pi(X_{t}, t;\,h) \equiv \hat \pi = \frac{\mu}{\gamma \sigma^2}, \end{align}

and the corresponding value function equals

\begin{align*}V(x, t;\, h) = x + B(h),\end{align*}

in which

(2.11) \begin{equation}B(h) = \dfrac{1}{\gamma \lambda} \left\{ \gamma c - \gamma (h - \lambda) \hat D(h) - \dfrac{(h - \lambda)^2}{2\lambda} + \dfrac{\mu^2}{2 \sigma^2} \right\}.\end{equation}

Moreover,

\begin{align*}G(x, t;\,h) = \mathbb{E}_{x, t}(X_\tau) = x + b(h),\end{align*}

in which

\begin{equation*}b(h) = \hat D(h) + \dfrac{h - \lambda}{\gamma \lambda}.\end{equation*}

Proof. Because the problem is time-homogeneous, the value function is time-independent. Moreover, from the mean-variance criterion and from the dynamics of X in (2.1), we deduce that the value function equals $x + B$ for some value of B that depends on h. Through this reasoning, we obtain

\begin{align*}V(t,x;\, h)= x + B(h),\quad \text{and}\quad G(t,x;\, h)= x + b(h),\end{align*}

in which b is another value that depends on h. By substituting the above expressions and their derivatives into (2.7), the HJB equation becomes

(2.12) \begin{align}\sup_{D \ge 0, \pi} \left\{ (\mu \pi + c -h D) - \dfrac{\gamma}{2} \sigma^2 \pi^2 - \lambda (B - D + \frac{\gamma}{2} \, (b-D)^2)\right\} = 0.\end{align}

Note that the expression in curly brackets is concave with respect to $\pi$ and D; graphically, it is a downward opening paraboloid. Thus, the critical points yield the maximizers. In particular, the first-order condition for $\pi$ gives us the optimal investment amount in (2.10). Similarly, from (2.12), the first-order condition for $\hat D$ gives us

(2.13) \begin{align}\lambda - h + \gamma \lambda (b - \hat D) = 0.\end{align}

Moreover, by substituting $G(t, x;\,h)= x + b(h)$ into (2.8), we derive another equation for $\hat D$ :

(2.14) \begin{align}\mu \hat \pi + c -h \hat D = \lambda (b - \hat D).\end{align}

By eliminating b(h) from (2.13) and (2.14) and by solving for $\hat D$ (truncated to lie in $\mathbb{R}_+$ ), we obtain $\hat D$ as in (2.9). Furthermore, by substituting $\hat D$ and $\hat \pi$ into the HJB equation, we derive the expression for B(h) in (2.11). The expression for b(h) follows from (2.13).

To check that the proposed value function is, indeed, the value function as defined in (2.4) and that the corresponding strategy is the equilibrium strategy, we verify the conditions given in Lemma 2.1. Conditions (i) and (ii) hold by construction. For the transversality condition, note that, under the candidate constant investment and life insurance strategy, the buyer’s wealth process $X = \{X_t\}_{t \in [0, \tau)}$ in (2.1) reduces to a diffusion process with constant coefficients. Thus, Condition (iii) holds. Hence, we complete the proof.

Theorem 2.2 shows that the time-consistent equilibrium life insurance and investment strategy is a constant strategy, as expected from the form of the problem.

2.3 Seller’s problem

In this section, we solve the seller’s problem, namely, to choose a value of $h \ge 0$ to maximize its expected wealth at time $\tau$ . To that end, we compute from (2.2) and (2.9)

\begin{align*}U(y, t) & \,:\!=\, \sup_{h \ge 0} \mathbb{E}_{y, t}(Y_{\tau})= \sup_{h \ge 0} \mathbb{E}_{y,t}(Y_{\tau-} - \hat D(h)) \\[3pt]& = \sup_{h \ge 0} \int^{\infty}_0 \lambda e^{-\lambda t}(Y_{t} - \hat D(h)) dt = \sup_{h \ge 0} \int^{\infty}_0 \lambda e^{-\lambda t}(y + h \hat D(h) \cdot t - \hat D(h)) dt\\[3pt]&= y + \sup_{h \ge 0}\left(\frac{h}{\lambda} -1\right) \hat D(h). \end{align*}

U is independent of time t, so we write U(y) in place of U(y, t).

Theorem 2.3. The optimal premium rate equals

(2.15) \begin{align} \hat h = \lambda \, \sqrt{1 + \dfrac{\gamma c + \frac{\mu^2}{\sigma^2}}{\lambda}}.\end{align}

Proof. If $\hat D(h) = 0$ , then $U(y) = y$ , and the seller can achieve average wealth strictly greater than y by choosing $h \in (\lambda, \gamma c + \lambda + \mu^2/\sigma^2)$ to guarantee $\hat D(h) \gt 0$ . Thus, without loss of generality, we assume that h is such that $\hat D(h) \gt 0$ . (In Theorem 2.4, we confirm that $\hat D(\hat h) \gt 0$ .)

Because $\hat D(h)$ in (2.9) is strictly positive, we have

\begin{align*}\mathbb{E}_{y}(Y_{\tau}) = y + \frac{h - \lambda}{\gamma \lambda h}\left( \gamma c - (h - \lambda) + \frac{\mu^2}{\sigma^2} \right).\end{align*}

Let $\unicode{x1D4EF}$ denote the second term in the above expression modulo a factor of $1/(\gamma\lambda)$ , that is,

\begin{align*}\unicode{x1D4EF}\,(h) = \frac{h - \lambda}{h} \left(\gamma c - (h - \lambda) + \frac{\mu^2}{\sigma^2} \right),\end{align*}

so $\mathrm{arg\,sup}_{h \ge 0} \mathbb{E}_{y}(Y_{\tau}) = \mathrm{arg\,sup}_{h \ge 0} \unicode{x1D4EF}\,(h)$ . Simple calculus yields

\begin{align*}\unicode{x1D4EF}\,^{\prime}(h) = \dfrac{\lambda^2 - h^2}{h^2} + \dfrac{\lambda}{h^2} \left( \gamma c + \dfrac{\mu^2}{\sigma^2} \right) \propto (\lambda^2 - h^2) + \lambda \left( \gamma c + \dfrac{\mu^2}{\sigma^2} \right) \,=\!:\, \unicode{x1D4F0}\,(h),\end{align*}

and

\begin{align*} \unicode{x1D4F0}\,^{\prime}(h) = - 2h,\end{align*}

which implies that the critical point equals the maximizer. By solving $\unicode{x1D4EF}\,^{\prime}(h) = 0$ , we obtain $\hat h$ as given in (2.15).

2.4 Stackelberg equilibrium

From Theorems 2.2 and 2.3, we obtain the following theorem, which presents the Stackelberg equilibrium for the term life insurance game (with investment).

In the following corollary, we analyze properties of the Stackelberg equilibrium strategy.

Proof. These properties are easy to see from the forms of $\hat h$ in (2.16), $\hat D(\hat h)$ in (2.18), and $\hat \pi(\hat h)$ in (2.16), except for the relationship between the death benefit and the risk aversion parameter. To complete the proof of this corollary, we show that $\hat D(\hat h)$ decreases with respect to $\gamma$ :

\begin{align*}\dfrac{\partial \hat D(\hat h)}{\partial \gamma} &= - \dfrac{1}{\gamma^2} \left( \sqrt{1 + \dfrac{\gamma c + \frac{\mu^2}{\sigma^2}}{\lambda}} - 1 \right) + \dfrac{1}{\gamma} \cdot \dfrac{\frac{c}{\lambda}}{2\sqrt{1 + \frac{\gamma c + \frac{\mu^2}{\sigma^2}}{\lambda}}} \\[3pt] &\propto - 2 \left( 1 + \frac{\gamma c + \frac{\mu^2}{\sigma^2}}{\lambda} - \sqrt{1 + \frac{\gamma c + \frac{\mu^2}{\sigma^2}}{\lambda}} \right) + \dfrac{\gamma c}{\lambda} \\[3pt] &= 2 \sqrt{1 + \frac{\gamma c + \frac{\mu^2}{\sigma^2}}{\lambda}} - 2 - \frac{\gamma c + \frac{2\mu^2}{\sigma^2}}{\lambda}.\end{align*}

Thus, $\frac{\partial \hat D(\hat h)}{\partial \gamma} \lt 0$ if and only if

\begin{align*}&2 \sqrt{1 + \frac{\gamma c + \frac{\mu^2}{\sigma^2}}{\lambda}} \lt 2 + \frac{\gamma c + \frac{2\mu^2}{\sigma^2}}{\lambda} \\[3pt] \iff & 4 \left( 1 + \frac{\gamma c + \frac{\mu^2}{\sigma^2}}{\lambda} \right) \lt 4 + 4 \, \frac{\gamma c + \frac{2\mu^2}{\sigma^2}}{\lambda} + \left( \frac{\gamma c + \frac{2\mu^2}{\sigma^2}}{\lambda} \right)^2 \\[3pt] \iff &0 \lt \frac{4 \mu ^2}{\lambda \sigma^2} + \left( \frac{\gamma c + \frac{2\mu^2}{\sigma^2}}{\lambda} \right)^2,\end{align*}

which is true.

Note that $\hat h \gt \lambda$ and $\hat D(\hat h) \gt 0$ ; therefore, the seller’s expected wealth at time $\tau$ is strictly greater than y, as expected. It makes sense that $\hat h$ increases with increasing $\lambda$ because as the individual’s mortality hazard rate increases, she is more likely to die, which means life insurance should be more expensive. It is intuitively pleasing that $\hat h$ increases with increasing c because as the individual’s income increases, she is able to spend more for life insurance. The death benefit $\hat D(\hat h)$ also increases with increasing c; thus, we see that the death benefit acts as a type of income replacement, which matches actuarial wisdom. Moreover, as the financial market becomes more favorable, that is, as $\mu$ increases or $\sigma$ decreases, the individual invests more in the risky asset and uses some of those expected earnings to purchase more life insurance.

We are somewhat surprised that the equilibrium death benefit decreases with respect to $\gamma$ because we expected the death benefit to increase with increasing risk aversion toward variance. However, there is variability associated with the effect of the death benefit upon $X_\tau$ through the time of death $\tau$ , which is the length of time the individual will have to pay premium. Thus, because increasing $\gamma$ leads to an increase in premium, this increased premium decreases the value of $X_\tau$ and might add to the variance. That said, the amount invested in the risky asset behaves as one expects with respect to $\gamma$ ; that is, $\hat \pi$ decreases with increasing $\gamma$ .

Note that $\hat D(\hat h)$ and $\hat \Pi$ are independent of both $X_{t}$ and t; that is, even though the buyer continually decides how much life insurance to purchase and how much money to invest in the risky asset (as she plays the mean-variance “game” against future versions of herself), she purchases a constant amount of life insurance and invests a constant amount. In other words, she effectively buys whole life insurance via a continuous premium $\hat h$ , payable for life, and continually rebalances her stock portfolio to keep its balance constant. Thus, in the following section, we consider a Stackelberg game similar to the one in this section, except that we assume the individual purchases whole life insurance and invests a constant amount in the risky asset, choosing both at time 0 to maximize her mean-variance criterion at her time of death. In other words, we assume the buyer “pre-commits” to her constant strategy at time 0.

3.1 Buyer’s problem

By contrast with Section 2, in which we solve the buyer’s problem from a time-consistent perspective, in this section, we maximize the mean-variance objective J in (2.3) assuming the individual commits to buying a constant amount of whole life insurance and to investing a constant amount in the risky asset. Under a slight abuse of notation, we write $J(x;\, D, \pi, h)$ in place of $J(x, t;\, \mathcal{D}, \Pi, h)$ because $D_t = D$ and $\pi_t = \pi$ for all $t \in [0, \tau)$ , for some constants $D \ge 0$ and $\pi \in \mathbb{R}$ .Footnote ⁴

We begin with a verification lemma that we use to obtain an explicit expression for the objective function J. Afterward computing J, we maximize it directly over all-non-negative D and $\pi$ to obtain the optimal life insurance and investment strategy and the corresponding value function.

Then, $\tilde{J}({\cdot} \, ;\, D, \pi, h)$ equals the individual’s objective function $J({\cdot} \, ;\, D, \pi, h)$ defined in (2.3) for a constant strategy. Moreover, $g(x;\, D, \pi, h) = \mathbb{E}_{x}(X_\tau)$ .

Proof. Suppose $\tilde J$ and g satisfy equations (3.1) and (3.2).

Step 1. We first show that $g(x;\, D, \pi, h) = \mathbb{E}_{x}(X_\tau) = \mathbb{E}_{x} \big( \int_0^\infty \lambda \, \mathrm{e}^{- \lambda t} \, \big(X_{t} + D) \, \mathrm{d} t \big)$ . Let $k \gt 0$ be a fixed number. By applying Itô’s formula to $\mathrm{e}^{- \lambda k} \, g \big( X_{k};\, D, \pi, h \big)$ , we obtain

\begin{align*}\mathrm{e}^{- \lambda k} \, g( X_{k};\, D, \pi, h) & = g(x;\, D, \pi, h) + \int_0^k \mathrm{e}^{- \lambda t} \left( \mathcal{L}^{(D, \pi)} g(X_{t};\, D, \pi, h) - \lambda g(X_{t};\, D, \pi, h) \right) \mathrm{d} t\\[3pt]& \quad + \int^{k}_0 e^{-\lambda t}\sigma \pi g_x(X_{t};\, D, \pi, h) dB_t. \end{align*}

By taking the expectation of the above expression, conditional on $X_{0} = x$ , we obtain

\begin{align*}{\mathbb{E}}_{x} \Big[ \mathrm{e}^{- \lambda k} \, g(X_{k};\, D, \pi, h) \Big] &= g(x;\, D, \pi, h) + {\mathbb{E}}_{x} \bigg[ \int_0^k \mathrm{e}^{- \lambda t} \left( \mathcal{L}^{(D, \pi)} g(X_{t};\, D, \pi, h) - \lambda g(X_{t};\, D, \pi, h) \right) \mathrm{d} t \bigg] \notag \\[3pt] & = g(x;\, D, \pi, h) - \mathbb{E}_{x} \bigg[ \int_0^k \lambda \, \mathrm{e}^{- \lambda t} \, (X_{t} + D) \, \mathrm{d} t \bigg],\end{align*}

in which the second equality follows from (3.2), and in which the arguments of the integrals are interpreted via their left limits.

By letting $k \to \infty$ and using (3.3), we obtain

\begin{align*} g(x;\, D, \pi, h) = \mathbb{E}_{x} \bigg[ \int_0^\infty \lambda \, \mathrm{e}^{- \lambda t} \, \big(X_{t} + D \big) \, \mathrm{d} t \bigg] = \mathbb{E}_{x}(X_\tau). \end{align*}

Here, we applied Lebesgue’s dominated convergence theorem:

\begin{align*} \mathbb{E}_{x} \bigg[ \int_0^k \lambda \, \mathrm{e}^{- \lambda t} \, |X_{t} + D| \, \mathrm{d} t \bigg]= \mathbb{E}_{x} \bigg[ \int_0^{\infty} \lambda \, \mathrm{e}^{- \lambda t} \, |X_{t} + D| 1_{\{t\lt k\}}\, \mathrm{d} t \bigg]\lt \mathbb{E}_{x} \bigg[ \int_0^{\infty} \lambda \, \mathrm{e}^{- \lambda t} \, |X_{t} + D|\, \mathrm{d} t \bigg] \lt \infty.\end{align*}

Step 2. In this step, we show that, if $\tilde J$ solves (2.7), then $J(x;\, D, \pi, h) = \tilde J(x;\, D, \pi, h)$ . From equation (3.2), we rewrite (2.7) as follows:

\begin{align*}&\mathcal{L}^{(D, \pi)} \Big( \tilde J(x;\, D, \pi, h) - \frac{\gamma}{2} \, g^2(x;\, D, \pi, h) \Big) + \gamma g(x;\, D, \pi, h) \lambda \big( g(x;\, D, \pi, h) - (x + D) \big) \end{align*}

\begin{align*}& = \lambda \left( \tilde J(x;\, D, \pi, h) - (x + D) + \frac{\gamma}{2} \, \big(g(x;\, D, \pi, h) - (x + D) \big)^2 \right),\end{align*}

or equivalently,

(3.4) \begin{align}&\mathcal{L}^{(D, \pi)} \Big( \tilde J(x;\, D, \pi, h) - \frac{\gamma}{2} \, g^2(x;\, D, \pi, h) \Big) \\[3pt]&= \lambda \left( \tilde J(x;\, D, \pi, h) - (x + D) - \frac{\gamma}{2} \, \big(g^2(x;\, D, \pi, h) - (x + D)^2 \big) \right). \nonumber \end{align}

By following the proof of Step 1, we obtain both

(3.5) \begin{equation}\tilde J(x;\, D, \pi, h) = \mathbb{E}_{x} \bigg[ \int_0^\infty \mathrm{e}^{- \lambda t} \left( \lambda \tilde J \big(X_{t};\, D, \pi, h \big) - \mathcal{L}^{(D, \pi)} \tilde J \big(X_{t};\, D, \pi, h \big) \right) \mathrm{d} t \bigg],\end{equation}

and

(3.6) \begin{equation}g^2(x;\, D, \pi, h) = \mathbb{E}_{x} \bigg[ \int_0^\infty \mathrm{e}^{- \lambda t} \left( \lambda g^2 \big(X_{t};\, D, \pi, h \big) - \mathcal{L}^{(D, \pi)} g^2 \big(X_{t};\, D, \pi, h \big) \right) \mathrm{d} t \bigg] .\end{equation}

By linearly combining (3.5) and (3.6) to form $\tilde J - \gamma g^2/2$ , and by using (3.4), we obtain the following expression:

\begin{align*}&\tilde J(x;\, D, \pi, h) - \frac{\gamma}{2} \, g^2(x;\, D, \pi, h) \\[3pt] &= \mathbb{E}_{x} \bigg[ \int_0^\infty \mathrm{e}^{- \lambda t} \left( \lambda \left(\tilde J \big(X_{t};\, D, \pi, h \big) - \frac{\gamma}{2} \, g^2(X_{t};\, D, \pi, h) \right) \right. \\[3pt] & \qquad \qquad \qquad \qquad \left.- \mathcal{L}^{(D, \pi)} \left(\tilde J \big(X_{t};\, D, \pi, h \big) - \frac{\gamma}{2} \, g^2(X_{t};\, D, \pi, h) \right) \right) \mathrm{d} t \bigg] \\[3pt] &= \mathbb{E}_{x} \bigg[ \int_0^\infty \lambda \mathrm{e}^{- \lambda t} \left( \big(X_{t} + D\big) - \dfrac{\gamma}{2} \, \big( X_{t} + D \big)^2 \right) \mathrm{d} t \bigg] \\[3pt] &= \mathbb{E}_x \big( X_\tau \big) - \dfrac{\gamma}{2} \, \mathbb{E}_x \big( \big(X_\tau)^2 \big),\end{align*}

which implies

\begin{align*}\tilde J(x;\, D, \pi, h) = \mathbb{E}_x(X_\tau) - \dfrac{\gamma}{2} \, \mathbb{V}_x(X_\tau) = J(x;\, D, \pi, h),\end{align*}

and completes the proof of this lemma.

By using Lemma 3.1, we obtain the objective function in the following theorem.

Theorem 3.2 Under a constant pre-commitment strategy, the objective function J in (2.3) equals

\begin{equation*}J(x;\, D, \pi, h) = x + A(D, \pi;\, h),\end{equation*}

for all $(x, D, \pi) \in \mathbb{R} \times \mathbb{R}_+ \times \mathbb{R}$ , in which

(3.7) \begin{align}A(D, \pi;\, h) = a(D, \pi;\, h) - \dfrac{\gamma}{2} \, \nu(D,\pi;\, h),\end{align}

with

(3.8) \begin{align}a(D, \pi;\, h) = D + \dfrac{c - hD + \mu\pi}{\lambda}, \end{align}

and

(3.9) \begin{align}\nu(D, \pi;\, h) = \dfrac{\sigma^2 \pi^2}{\lambda} + \left( \dfrac{c - hD + \mu \pi}{\lambda} \right)^2.\end{align}

Moreover,

\begin{equation*}g(x;\, D, \pi, h) = \mathbb{E}_{x}(X_\tau) = x + a(D, \pi;\, h),\end{equation*}

for all $(x, D, \pi) \in \mathbb{R} \times \mathbb{R}_+ \times \mathbb{R}$ , in which a is given in (3.8).

Proof. Because the dynamics for X in (2.1) are independent of initial wealth, and because we are considering the mean-variance criterion, we hypothesize that $J(x;\, D, \pi, h) = x + A(D, \pi;\, h)$ and $g(x;\, D, \pi, h) = x + a(D, \pi;\, h)$ for some functions A and a on $\mathbb{R}_+ \times \mathbb{R}$ . By substituting $g(x;\, D, \pi, h) = x + a(D, \pi;\, h)$ into (3.2) (and by omitting the dependence of a on D and $\pi$ for simplicity), we obtain

(3.10) \begin{equation} c - hD + \mu \pi = \lambda(a - D),\end{equation}

which leads to the expression for $a(D,\pi;\, h)$ in (3.8).

Next, consider the equation in (3.1). Under our ansatz $J(x;\, D, \pi, h) = x + A(D, \pi;\, h)$ and $g(x;\, D, \pi, h) = x + a(D, \pi;\, h)$ , we obtain

(3.11) \begin{equation}\mathcal{L}^{(D, \pi)} \, J(x;\, D, \pi, h) = \mathcal{L}^{(D, \pi)} \, g(x;\, D, \pi, h) = c - hD + \mu \pi,\end{equation}

and

(3.12) \begin{equation}\mathcal{L}^{(D, \pi)} \, g^2(x;\, D, \pi, h) = 2 (c - hD + \mu \pi)(x + a) + \sigma^2 \pi^2.\end{equation}

By substituting (3.11) and (3.12) into (3.1) and by using (3.10), we get

\begin{align*}A = a - \dfrac{\gamma}{2 }\left(\frac{\sigma^2 \pi^2}{\lambda} + (a - D)^2 \right),\end{align*}

which implies the expressions for $A(D,\pi;\, h)$ in (3.7) and $\nu(D,\pi;\, h)$ in (3.9).

By construction $J(x;\, D,\pi;\, h) = x + A(D,\pi;\, h)$ and $g(x;\, D,\pi;\, h) = x + a(D,\pi;\, h)$ satisfy (2.7) and (3.2). Thus, it remains to prove the transversality condition in (3.3) for $\phi = J$ , g, and $g^2$ . First, note that

\begin{align*} X_t = x + (\mu \pi + c - hD) t + \sigma \pi B_t. \end{align*}

Thus,

\begin{align*} \lim \limits_{t \to \infty} \mathrm{e}^{-\lambda t} \left( \mathbb{E}_{x}(X_t) + A(D, \pi;\, h) \right) = 0 = \lim \limits_{t \to \infty} \mathrm{e}^{-\lambda t} \left( \mathbb{E}_{x} (X_t) + a(D, \pi;\, h) \right), \end{align*}

thereby proving (3.3) for $\phi = J$ and g. Similarly, $\mathbb{E}_x \big(g^2(X_t; D, \pi, h) \big)$ behaves like $t^2$ as t approaches infinity, which is dominated by $\mathrm{e}^{-\lambda t}$ ; thus, (3.3) also holds for $\phi = g^2$ . It follows from Lemma 3.1 that J and g are as stated.

In the following theorem, we present the optimal amount of investment and life insurance that the individual will buy.

Theorem 3.3. For a given value of the premium rate $h \ge 0$ , the optimal constant pre-commitment amount of life insurance equals

(3.13) \begin{equation}D^*(h) = \dfrac{1}{\gamma h} \left(\gamma c - \dfrac{\lambda}{h} \left( (h - \lambda) - \dfrac{\mu^2}{\sigma^2} \right) \right)_+,\end{equation}

and the optimal constant pre-commitment amount to invest in the risky asset equals

(3.14) \begin{equation}\pi^*(h) = \begin{cases}\dfrac{\lambda}{h} \cdot \dfrac{\mu}{ \gamma \sigma^2}, &\quad \text{if} \quad \gamma c \gt \dfrac{\lambda}{h} \left( (h - \lambda) - \dfrac{\mu^2}{\sigma^2} \right) \iff D^*(h) \gt 0, \\[3pt] \dfrac{\mu}{\mu^2 + \lambda \sigma^2} \left(\dfrac{\lambda}{\gamma} - c\right), &\quad \text{if} \quad \gamma c \le \dfrac{\lambda}{h} \left( (h - \lambda) - \dfrac{\mu^2}{\sigma^2} \right) \iff D^*(h) = 0.\end{cases}\end{equation}

Proof. First, we maximize $J(x;\, D, \pi, h)$ (or equivalently, $A(D, \pi;\, h)$ ) with respect to $\pi \in \mathbb{R}$ to obtain the maximizer $\pi^*(h;\, D)$ ; then, we maximize $A(D, \pi^*(h;\, D);\, h)$ with respect to $D \in \mathbb{R}_+$ . To that end, define $\unicode{x1D4F3}\,$ by:

(3.15) \begin{equation}\unicode{x1D4F3}\,(D, \pi) = \lambda A(D, \pi;\, h) = \lambda D + (c - h D + \mu \pi) - \dfrac{\gamma}{2} \left( \sigma^2 \pi^2 + \dfrac{(c - hD + \mu \pi)^2}{\lambda} \right),\end{equation}

in which we omit the dependence on h for simplicity.

By differentiating $\unicode{x1D4F3}\,(D, \pi)$ in (3.15) with respect to $\pi$ , we obtain

\begin{align*}\dfrac{\partial \unicode{x1D4F3}\,(D, \pi)}{\partial \pi} & = - {\gamma}\left(\sigma^2 + \frac{\mu^2}{\lambda}\right)\pi + {\mu} \left( 1 - \dfrac{\gamma (c - hD)}{\lambda} \right),\end{align*}

with the second derivative strictly negative. Thus, the maximizer $\pi^*(h;\, D)$ equals the critical point, that is,

\begin{equation*}\pi^*(h;\, D) = \dfrac{\mu}{\mu^2 + \lambda \sigma^2} \cdot \dfrac{\lambda - \gamma c + \gamma h D}{\gamma}.\end{equation*}

Next, define $\unicode{x1D4F4}\,$ by

\begin{equation*}\unicode{x1D4F4}\,(D) = \unicode{x1D4F3}\,(D, \pi^*(h;\, D)).\end{equation*}

By differentiating $\unicode{x1D4F4}\,$ , we obtain

\begin{align*}\unicode{x1D4F4}\,^{\prime}(D) &= \dfrac{\partial \unicode{x1D4F3}\,(D, \pi^*(h;\, D))}{\partial D} + \dfrac{\partial \unicode{x1D4F3}\,(D, \pi^*(h;\, D))}{\partial \pi} \cdot \dfrac{\partial \pi^*(h;\, D)}{\partial D} \\[3pt]&= \dfrac{\partial \unicode{x1D4F3}\,(D, \pi^*(h;\, D))}{\partial D} \\[3pt]&= (\lambda - h) + \dfrac{\gamma h}{\lambda} \, \big(c - hD + \mu \pi^*(h;\, D) \big), \end{align*}

and the second derivative is strictly negative. Indeed, the second derivative equals

\begin{align*}\unicode{x1D4F4}\,^{\prime\prime}(D) = -\dfrac{\gamma h^2}{\lambda} + \dfrac{\gamma h \mu}{\lambda}\cdot \dfrac{\mu h}{\mu^2 + \lambda \sigma^2} = - \dfrac{\gamma h^2 \sigma^2}{\mu^2 + \lambda \sigma^2} \lt 0.\end{align*}

Thus, the maximizer $D^*(h)$ equals the critical point truncated to lie in $\mathbb{R}_+$ . By substituting for $\pi^*(h;\, D)$ in $\unicode{x1D4F4}\,^{\prime}(D) = 0$ and solving for D, we get (3.13). Finally, we compute $\pi^*(h) = \pi^*(h;\, D^*(h))$ and obtain (3.14).

Remark 3.1. Compare the two expressions for the time-consistent equilibrium and optimal constant pre-commitment life insurance, respectively, $\hat D(h)$ in (2.9) and $D^*(h)$ in (3.13):

\begin{align*} \hat D(h) = \frac{1}{\gamma h}\left(\gamma c - \left( (h - \lambda) - \frac{\mu^2}{\sigma^2} \right) \right)_+, \end{align*}

and

\begin{align*} D^*(h) = \dfrac{1}{\gamma h} \left(\gamma c - \dfrac{\lambda}{h} \left( (h - \lambda) - \dfrac{\mu^2}{\sigma^2} \right) \right)_+.\end{align*}

Both begin with the term $c/h$ , which we discuss in Remark 2.1, and they both adjust $c/h$ by $\frac{1}{\gamma h} \left( (h - \lambda) - \frac{\mu^2}{\sigma^2} \right)$ , except that $D^*$ applies an additional factor of $\lambda/h$ to this adjustment term. We generally expect $h \gt \lambda$ ; if so, in the pre-commitment setting, the buyer adjusts $c/h$ by a smaller absolute amount than in the time-consistent setting. When $0 \lt h - \lambda \lt \frac{\mu^2}{\sigma^2}$ , then $\hat D(h) \gt D^*(h)$ ; in other words, when the premium loading for life insurance is less than the square of the Sharpe ratio, then the buyer purchases more term life insurance than whole life insurance. Also, in this case, the death benefits $\hat D(h)$ and $D^*(h)$ are both greater than $c/h$ because life insurance is a good deal, as measured by a relatively small value of the premium loading $h - \lambda$ . On the other hand, when the premium loading is greater than the square of the Sharpe ratio, the buyer purchases less term life insurance than whole life insurance, and in this case, both death benefits are less than $c/h$ because life insurance is not as good a deal due to the larger premium loading.

Similarly, compare the investment in the risky asset for the time-consistent equilibrium and for the optimal constant pre-commitment setting, respectively, $\hat \pi(h)$ in (2.10) and $\pi^*(h)$ in (3.14):

\begin{align*} \hat \pi(h) = \dfrac{\mu}{\gamma \sigma^2}, \end{align*}

and, when $\gamma c \gt \frac{\lambda}{h} \left( (h - \lambda) - \frac{\mu^2}{\sigma^2} \right)$ ,

\begin{align*} \pi^*(h) = \dfrac{\lambda}{h} \cdot \dfrac{\mu}{\gamma \sigma^2}. \end{align*}

Again, in the pre-commitment setting, we see this additional factor of $\lambda/h$ , and if $h \gt \lambda$ (as we expect in equilibrium), then $\pi^*(h) \lt \hat \pi(h)$ . In other words, when $h \gt \lambda$ , the buyer will invest more in the risky asset when buying term life insurance than when buying whole life insurance. Because investment is risky and a life insurance death benefit will be paid upon death with probability 1, under the pre-commitment setting, the individual is less willing to commit to investing a larger amount in the risky asset than under the time-consistent setting.

3.2 Seller’s problem

For the seller’s problem, recall the computations in Section 2.3, and define K by:

(3.16) \begin{align}K(y) \,:\!=\, \sup_{h \ge 0} \mathbb{E}_{y}(Y_{\tau}) = y + \sup_{h \ge 0} \left(\frac{h}{\lambda} -1\right) D^*(h).\end{align}

Theorem 3.4. When $\gamma c \lt 2 \lambda + \frac{\mu^2}{\sigma^2}$ , the optimal premium rate equals

(3.17) \begin{align} h^* = \dfrac{2\lambda \left( \lambda + \frac{\mu^2}{\sigma^2} \right)}{2\lambda + \frac{\mu^2}{\sigma^2} - \gamma c}.\end{align}

When $\gamma c \ge 2 \lambda + \frac{\mu^2}{\sigma^2}$ , the supremum in (3.16) is attained at $h^* = \infty$ .

Proof. If $D^*(h) = 0$ , then $K(y) = y$ , and the seller can achieve average wealth strictly greater than y by choosing $h = \lambda + \varepsilon$ (for some $\varepsilon \gt 0$ small enough) to guarantee $D^*(h) \gt 0$ . Thus, without loss of generality, we assume that h is such that $D^*(h) \gt 0$ . (In Theorem 3.5 below, we confirm that $D^*(h^*) \gt 0$ .)

Because $D^*(h)$ in (3.13) is strictly positive, we have

(3.18) \begin{align}\mathbb{E}_{y}(Y_{\tau}) = y + \frac{h - \lambda}{\gamma \lambda h}\left(\gamma c - \dfrac{\lambda(h - \lambda)}{h} + \dfrac{\lambda}{h} \cdot \dfrac{\mu^2}{\sigma^2} \right).\end{align}

Let $\varpi$ denote the second term in the above expression modulo a factor of $1/(\gamma \lambda)$ , that is,

\begin{align*}\varpi(h) = \frac{h - \lambda}{h} \left(\gamma c - \dfrac{\lambda(h - \lambda)}{h} + \dfrac{\lambda}{h} \cdot \dfrac{\mu^2}{\sigma^2} \right).\end{align*}

By differentiating $\varpi$ , we obtain

\begin{align*}\varpi^{\prime}(h) &= \dfrac{\lambda}{h^2} \left(\gamma c - \dfrac{\lambda(h - \lambda)}{h} + \dfrac{\lambda}{h} \cdot \dfrac{\mu^2}{\sigma^2} \right) + \frac{h - \lambda}{h} \left( - \dfrac{\lambda^2}{h^2} - \dfrac{\lambda}{h^2} \cdot \dfrac{\mu^2}{\sigma^2} \right)\\[3pt]&\propto \gamma c - 2 \, \dfrac{\lambda(h - \lambda)}{h} - \dfrac{h - 2\lambda}{h} \cdot \dfrac{\mu^2}{\sigma^2} \,=\!:\, \ell(h),\end{align*}

and

\begin{align*} \ell^{\prime}(h) = - 2 \, \dfrac{\lambda^2}{h^2} - \dfrac{2\lambda}{h^2} \cdot \dfrac{\mu^2}{\sigma^2} \lt 0, \end{align*}

which implies that the critical point in $\mathbb{R}_+$ equals the maximizer, if it exists. The critical point in $\mathbb{R}_+$ exists if and only if $\gamma c \lt 2 \lambda + \frac{\mu^2}{\sigma^2}$ . In this case, $\varpi^{\prime}(h) = 0$ has a solution in $\mathbb{R}_+$ , which is given in (3.17).

On the other hand, when $\gamma c \ge 2 \lambda + \frac{\mu^2}{\sigma^2}$ , the function $\varpi(h)$ is strictly increasing for $h \in \mathbb{R}_+$ , so the supremum is achieved at $h^* = \infty$ .

3.3 Stackelberg equilibrium

From Theorems 3.3 and 3.4, we obtain the following theorem, which presents the Stackelberg equilibrium for the whole life insurance game (with investment).

We acknowledge that the premium rate of $h^* = \infty$ when $\gamma c \ge 2 \lambda + {\mu^2}/{\sigma^2}$ is not realistic, and in the following corollary, we offer an $\varepsilon$ -optimal strategy.

Corollary 3.6. Suppose $\gamma c \ge 2 \lambda + {\mu^2}/{\sigma^2}$ , and let $\varepsilon \gt 0$ be arbitrary. Then, there exists $M(\varepsilon) \gt 0$ such that, if $h \gt M(\varepsilon)$ , then

\begin{equation*}\left(\frac{h}{\lambda} -1\right) D^*(h) + \varepsilon \gt \dfrac{\gamma c - \lambda}{\gamma \lambda},\end{equation*}

that is, the expected wealth of the seller at time $\tau$ when offering the premium h is within $\varepsilon$ of the optimal expected wealth.

Proof. When $\gamma c \ge 2 \lambda + {\mu^2}/{\sigma^2}$ , the expectation of $Y_\tau$ in (3.18) increases with respect to h on $\mathbb{R}_+$ and approaches $y + {(\gamma c - \lambda)}/{\gamma \lambda}$ as h approaches infinity. Thus, the statement of the corollary holds when $M(\varepsilon)$ solves

\begin{align*} \left(\frac{M(\varepsilon)}{\lambda} -1\right) D^*(M(\varepsilon)) + \varepsilon = \dfrac{\gamma c - \lambda}{\gamma \lambda}, \end{align*}

specifically,

\begin{align*} M(\varepsilon) = \dfrac{1}{2 \gamma \varepsilon} \left[ \left( \gamma c - 2 \lambda - \dfrac{\mu^2}{\sigma^2} \right) + \sqrt{\left( \gamma c - 2 \lambda - \dfrac{\mu^2}{\sigma^2} \right)^2 + 4 \gamma\lambda\varepsilon \left( \lambda + \dfrac{\mu^2}{\sigma^2} \right)} \right]. \end{align*}

Remark 3.2. Alternatively, suppose the premium rate $h$ is constrained to lie in $[0, \bar{h}]$ , in which $\bar{h}$ is an upper bound for $h$ and $\bar{h} \gt \lambda$ . Then, when $\gamma c \ge 2 \lambda + {\mu^2}/{\sigma^2}$ , the equilibrium premium rate $h^* = \bar{h}$ , and the seller’s expected wealth at time $\tau$ equals

\begin{align*}K(y) = y + \frac{\bar{h} - \lambda}{\gamma \lambda \bar{h}}\left(\gamma c - \dfrac{\lambda(\bar{h} - \lambda)}{\bar{h}} + \dfrac{\lambda}{\bar{h}} \cdot \dfrac{\mu^2}{\sigma^2} \right).\end{align*}

In response, the buyer pre-commits to purchasing the following constant amount of whole life insurance:

\begin{equation*}D^*(h^*) = \dfrac{1}{\gamma \bar{h}} \left(\gamma c - \dfrac{\lambda}{\bar{h}} \left( (\bar{h} - \lambda) - \dfrac{\mu^2}{\sigma^2} \right) \right) \gt 0,\end{equation*}

and pre-commits to investing the following constant amount in the risky asset:

\begin{equation*}\pi^*(h^*) = \dfrac{\lambda}{\bar{h}}\cdot\dfrac{ \mu}{ \gamma \sigma^2}.\end{equation*}

In this case, the buyer’s value function equals

\begin{align*}J(x; D^*(h^*), \pi^*(h^*), h^*) = x + \dfrac{c}{\bar{h}} + \dfrac{1}{2 \gamma} \left( \dfrac{\bar{h} - \lambda}{\bar{h}} \right)^2 + \dfrac{\lambda}{2 \gamma (\bar{h})^2} \cdot \dfrac{\mu^2}{\sigma^2}.\end{align*}

In the following corollary, we show how the Stackelberg equilibrium varies with some of the parameters. We omit the proof because the properties are straightforward to demonstrate from the expressions in Theorem 3.5.

1. 1. The equilibrium premium rate for whole life insurance $h^*$ increases with respect to $\gamma$ and c. Moreover, if $\lambda - \gamma c \gt 0$ $( \lt 0)$ , then $h^*$ increases (decreases) with respect to $\mu$ and decreases (increases) with respect to $\sigma$ . Also, if $\lambda - \gamma c \gt 0$ , then $h^*$ increases with respect to $\lambda$ , but if $\gamma c$ is close enough to $2 \lambda + {\mu^2}/{\sigma^2}$ , then $h^*$ decreases with respect to $\lambda$ .

2. 2. The equilibrium death benefit of whole life insurance $D^*(h^*)$ increases with respect to $\mu$ , and it decreases with respect to $\gamma$ and $\sigma$ . Moreover, if $\lambda - \gamma c \gt 0$ $( \lt 0)$ , then $D^*(h^*)$ increases (decreases) with respect to c.

3. 3. The equilibrium amount invested in the risky asset $\pi^*(h^*)$ decreases to 0 as $\gamma c$ increases to $2 \lambda + {\mu^2}/{\sigma^2}$ . Moreover, $\pi^*(h^*)$ increases with respect to $\mu$ , and it decreases with respect to $\gamma$ and $\sigma$ .

As for the Stackelberg equilibrium for term life insurance, the equilibrium premium rate increases with increasing risk aversion, as measured by $\gamma$ , and with increasing income rate c. However, the behavior of the premium rate with respect to $\mu/\sigma$ depends on whether $\gamma c \lt \lambda$ or $\gamma c \gt \lambda$ . If $\gamma c$ is small enough, then $h^*$ increases with more favorable financial markets, as measured by increasing values of $\mu/\sigma$ , and vice versa if $\gamma c$ is large enough. Also, the behavior of the premium rate with respect to $\lambda$ also depends on the magnitude of $\gamma c$ ; if $\gamma c$ is small enough, then $h^*$ increases with increasing $\lambda$ .

Similarly to the equilibrium death benefit for term life insurance, $D^*(h^*)$ increases with more favorable financial markets and decreases with increasing risk aversion. As we discuss following Corollary 2.5, an increase in $\gamma$ allows for an increase in $h^*$ , which leads to a decrease in the death benefit. But, the equilibrium amount invested in the risky asset decreases with increasing risk aversion or decreasing financial markets, which one expects.

In the next section, we perform numerical experiments to compare the time-consistent equilibrium controls (for term life insurance plus investment) with the constant pre-commitment equilibrium controls (for whole life insurance plus investment).