Manage Pension Deficit with Heterogeneous Insurance

Abstract

This paper considers a positive and increasing pension deficit of a certain pay-as-you-go (PAYG) pension system, and tries to make up for this deficit by using heterogeneous insurance. The positive pension deficit is formulated as a mathematical function in continuous time. The surplus of an appropriate heterogeneous insurance is described by diffusion approximation of a Cramér-Lundberg process. The system of extended Hamilton-Jacobi-Bellman equations under mean-variance criterion is established. The closed-form solution and optimal surplus-multiplier of heterogenous insurance are obtained. Some interpretations further explain the theoretical values of the results.

Appendices

Appendix A. Proof of Theorem 3.2

Proof

The dynamics of an appropriate heterogenous insurance’s surplus-scale divided by the positive pension deficit is given by

$$\begin{aligned} \begin{aligned} \mathrm{d} X^{k}(t)&= \frac{ \big [~ k_{t} \tilde{G}(t) \theta \lambda \mu - R(t) \big ( p(t) \psi _{2}(t) - c(t) \psi _{1}(t) \big ) ~\big ] }{ \tilde{G}(t)^{2} } ~ \mathrm{d} t + \frac{ k_{t} \sqrt{ \lambda } \sigma }{ \tilde{G}(t) } ~ \mathrm{d}W(t) \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \mathrm{d}R(t) = \theta \lambda \mu \mathrm{d}t + \sqrt{ \lambda } \sigma \mathrm{d}W(t). \end{aligned}$$

Use the definition of infinitesimal generator, we have

$$\begin{aligned} \begin{aligned} \mathscr {A}\kern 0.14em^{k}V(t, x, R)&= V_{t} + V_{x} \frac{ \big [~ k_{t} \tilde{G}(t) \theta \lambda \mu - R \big ( p(t) \psi _{2}(t) - c(t) \psi _{1}(t) \big ) ~\big ] }{ \tilde{G}(t)^{2} } + V_{R} ~ \theta \lambda \mu \\&~~~ ~~~ + \frac{1}{2} V_{xx} \frac{ k_{t}^{2} \lambda \sigma ^{2} }{ \tilde{G}(t)^{2} } + \frac{1}{2} V_{RR} ~ \lambda \sigma ^{2} + V_{xR} \frac{ k_{t} \lambda \sigma ^{2} }{ \tilde{G}(t) } ~ , \end{aligned} \end{aligned}$$

$$\begin{aligned} \begin{aligned}&(\mathscr {A}\kern 0.14em^{k}(G\diamond g))(t, x, R) \\&~~ = G_{y} \big \{~ g_{t} + g_{x} \frac{ \big [~ k_{t} \tilde{G}(t) \theta \lambda \mu - R \big ( p(t) \psi _{2}(t) - c(t) \psi _{1}(t) \big ) ~\big ] }{ \tilde{G}(t)^{2} } + g_{R} ~ \theta \lambda \mu ~\big \} \\&~~~ ~~~ + \frac{1}{2} ( G_{yy} g_{x}^{2} + G_{y} g_{xx} ) \frac{ k_{t}^{2} \lambda \sigma ^{2} }{ \tilde{G}(t)^{2} } + \frac{1}{2} ( G_{yy} g_{R}^{2} + G_{y} g_{RR} ) ~ \lambda \sigma ^{2} + ( G_{yy} g_{x} g_{R} + G_{y} g_{xR} ) \frac{ k_{t} \lambda \sigma ^{2} }{ \tilde{G}(t) } ~ , \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \begin{aligned}&(\mathscr {H}\kern 0.18em^{k}g)(t, x, R) \\&~~ = G_{y} \big \{ ~ g_{t} + g_{x} \frac{ \big [~ k_{t} \tilde{G}(t) \theta \lambda \mu - R \big ( p(t) \psi _{2}(t) - c(t) \psi _{1}(t) \big ) ~\big ] }{ \tilde{G}(t)^{2} } + g_{R} ~ \theta \lambda \mu \\&~~~ ~~~ + \frac{1}{2} g_{xx} \frac{ k_{t}^{2} \lambda \sigma ^{2} }{ \tilde{G}(t)^{2} } + \frac{1}{2} g_{RR} ~ \lambda \sigma ^{2} + g_{xR} \frac{ k_{t} \lambda \sigma ^{2} }{ \tilde{G}(t) } ~ \big \} \end{aligned} \end{aligned}$$

Since $G(y)= \frac{\gamma }{2} y^{2}$, $G_{y} = \gamma y$ and $G_{yy} = \gamma$ , so that

$$\begin{aligned} \begin{aligned}&\sup \limits _{k\in \Lambda } \big \{~ (\mathscr {A}\kern 0.14em^{k}V)(t, x, R) - (\mathscr {A}\kern 0.14em^{k}(G\diamond g))(t, x, R) + (\mathscr {H}\kern 0.18em^{k}g)(t, x, R) ~ \big \} \\&= \sup \limits _{k\in \Lambda } \big \{ V_{t} + V_{x} \frac{ \big [~ k_{t} \tilde{G}(t) \theta \lambda \mu - R \big ( p(t) \psi _{2}(t) - c(t) \psi _{1}(t) \big ) ~\big ] }{ \tilde{G}(t)^{2} } + V_{R} ~ \theta \lambda \mu \\&~~~ ~~~ + \frac{1}{2} V_{xx} \frac{ k_{t}^{2} \lambda \sigma ^{2} }{ \tilde{G}(t)^{2} } + \frac{1}{2} V_{RR} ~ \lambda \sigma ^{2} + V_{xR} \frac{ k_{t} \lambda \sigma ^{2} }{ \tilde{G}(t) } \\&~~~ ~~~ - \frac{\gamma }{2} g_{x}^{2} \frac{ k_{t}^{2} \lambda \sigma ^{2} }{ \tilde{G}(t)^{2} } - \frac{\gamma }{2} g_{R}^{2} ~ \lambda \sigma ^{2} - \gamma g_{x} g_{R} \frac{ k_{t} \lambda \sigma ^{2} }{ \tilde{G}(t) } \big \} = 0 . \end{aligned} \end{aligned}$$

According to the first order necessary condition of extremes, we get the following optimal surplus-multiplier $k^{*}$ from the above equation,

$$\begin{aligned} ~ k^{*} = \frac{ ( \theta \mu V_{x} + \sigma ^{2} V_{xR} - \gamma \sigma ^{2} g_{x} g_{R} ) \tilde{G}(t) }{ ( g_{x}^{2} - V_{xx} ) \sigma ^{2} } \end{aligned}$$

(8)

and substitute $k^{*}$ into the HJB equation of g(t, x, R), which gives

$$\begin{aligned} \begin{aligned}&g_{t} + g_{x} \frac{ \big [~ k_{t}^{*} \tilde{G}(t) \theta \lambda \mu - R \big ( p(t) \psi _{2}(t) - c(t) \psi _{1}(t) \big ) ~\big ] }{ \tilde{G}(t)^{2} } + \theta \lambda \mu g_{R} ~ \\&~~~ ~~~ + \frac{1}{2} g_{xx} \frac{ ( k_{t}^{*} )^{2} \lambda \sigma ^{2} }{ \tilde{G}(t)^{2} } + \frac{1}{2}~ \lambda \sigma ^{2} g_{RR} + g_{xR} \frac{ k_{t}^{*} \lambda \sigma ^{2} }{ \tilde{G}(t) } = 0. \end{aligned} \end{aligned}$$

Therefore, the extended HJB system of equations as required are obtained.

Appendix B. Proof of Theorem 3.3

Proof

The derivatives of (6) are as follows

$$\begin{aligned} ~ \begin{aligned}&V_{t} = H^{'}(t) x + L^{'}(t) R + Q^{'}(t) , ~~ V_{x} = H(t), ~~ V_{R} = L(t), ~~ V_{xx} = V_{xR} = V_{RR} = 0 , \\&g_{t} = h^{'}(t) x + l^{'}(t) R + q^{'}(t) , ~~~ g_{x} = h(t) , ~~~ g_{R} = l(t), ~~~ g_{xx} = g_{xR} = g_{RR} = 0 . \end{aligned} \end{aligned}$$

(9)

Plugging (9) into (4), it gives that

$$\begin{aligned} ~ \begin{aligned}&\sup \limits _{k\in \Lambda } \big \{ H'(t) x + L'(t) R + Q'(t) + H(t) \frac{ \big [~ k_{t} \tilde{G}(t) \theta \lambda \mu - k_{t} R \big ( p(t) \psi _{2}(t) - c(t) \psi _{1}(t) \big ) ~\big ] }{ \tilde{G}(t)^{2} } + L(t) ~ \theta \lambda \mu \\&~~~ ~~~ - \frac{\gamma }{2} h(t)^{2} \frac{ k_{t}^{2} \lambda \sigma ^{2} }{ \tilde{G}(t)^{2} } - \frac{\gamma }{2} l(t)^{2} ~ \lambda \sigma ^{2} - \gamma h(t) l(t) \frac{ k_{t} \lambda \sigma ^{2} }{ \tilde{G}(t) } \big \} = 0 . \end{aligned} \end{aligned}$$

(10)

Plugging (9) into (8), it follows

$$\begin{aligned} ~ k_{t}^{*} = \frac{ \theta \mu H(t) - \gamma h(t) l(t) \sigma ^{2} }{ h(t)^{2} } \cdot \frac{ \tilde{G}(t) }{ \gamma \sigma ^{2} } . \end{aligned}$$

(11)

Replacing $k_{t}$ in (10) with (11), the Eq. (10) becomes

$$\begin{aligned} ~ \begin{aligned}&H'(t) x + L'(t) R + Q'(t) + \frac{ \big [ \theta \mu H(t) - \gamma h(t) l(t) \sigma ^{2} \big ]^{2} }{ h(t)^{2} } \cdot \frac{ \lambda }{ 2 \gamma \sigma ^{2} } \\&~~~ ~~~- H(t) \frac{ R \big ( p(t) \psi _{2}(t) - c(t) \psi _{1}(t) \big ) }{ \tilde{G}(t)^{2} } + L(t) ~ \theta \lambda \mu - \frac{\gamma }{2} l(t)^{2} ~ \lambda \sigma ^{2} = 0 \end{aligned} \end{aligned}$$

(12)

Plugging (9) into (5) and Replacing $k_{t}$ with $k^{*}$, it gives that

$$\begin{aligned} ~ h'(t) x + l'(t) R + q'(t) + h(t) \frac{ \big [~ k_{t}^{*} \tilde{G}(t) \theta \lambda \mu - k_{t}^{*} R \big ( p(t) \psi _{2}(t) - c(t) \psi _{1}(t) \big ) ~\big ] }{ \tilde{G}(t)^{2} } + l(t)~ \theta \lambda \mu = 0. \end{aligned}$$

(13)

Observing the Eq. (12) and the Eq. (13), it is easy to obtain that

$$\begin{aligned} H'(t) x = 0 , ~~ h'(t) x = 0 \end{aligned}$$

So it gives

$$\begin{aligned} H(t) = h(t) \equiv 1 . \end{aligned}$$

Thus, the two equations, the Eq. (12) and the Eq. (13), are split further to the following equations

$$\begin{aligned} \left\{ \begin{aligned}&L'(t) - \frac{ \big ( p(t) \psi _{2}(t) - c(t) \psi _{1}(t) \big ) }{ \tilde{G}(t)^{2} } = 0 , \\&Q'(t) + \frac{ \lambda \big [ \theta \mu - \gamma l(t) \sigma ^{2} \big ]^{2} }{ 2 \gamma \sigma ^{2} } + L(t) ~ \theta \lambda \mu - \frac{\gamma }{2} l(t)^{2} ~ \lambda \sigma ^{2} = 0 , \\&l'(t) - \frac{ \big ( p(t) \psi _{2}(t) - c(t) \psi _{1}(t) \big ) }{ \tilde{G}(t)^{2} } = 0 , \\&q'(t) + \frac{ \theta \mu - \gamma l(t) \sigma ^{2} }{ h(t)^{2} } \cdot \frac{ \theta \lambda \mu }{ \gamma \sigma ^{2} }+ l(t)~ \theta \lambda \mu = 0. \end{aligned} \right. \end{aligned}$$

Solving the above four ordinary differential equations, we obtained

$$\begin{aligned} \left\{ \begin{aligned}&L(t) = l(t) = \frac{1}{\tilde{G}(T)} - \frac{1}{\tilde{G}(t)} , \\&Q(t) = \frac{ \lambda \theta ^{2} \mu ^{2} }{ 2 \gamma \sigma ^{2} } ( t - T ) , \\&q(t) = \frac{ \lambda \theta ^{2} \mu ^{2} }{ \gamma \sigma ^{2} } ( t - T ) . \end{aligned} \right. \end{aligned}$$

Replacing the expressions of H(t),  h(t),  l(t) into (11), the optimal surplus-multiplier $k_{t}^{*}$ is obtained in explicit form

$$\begin{aligned} \begin{aligned} k_{t}^{*}&= \big ( \theta \mu - \gamma l(t) \sigma ^{2} \big )\cdot \frac{ \tilde{G}(t) }{ \gamma \sigma ^{2} } \\&= \frac{ \theta \mu }{ \gamma \sigma ^{2} } \cdot \tilde{G}(t) - \frac{ \tilde{G}(t) }{\tilde{G}(T)} + 1 \\&= \big ( \frac{ \theta \mu }{ \gamma \sigma ^{2} } - \frac{ 1}{\tilde{G}(T)} \big ) \cdot \tilde{G}(t) + 1 . \end{aligned} \end{aligned}$$

Substituting the expressions of H(t),  L(t),  Q(t) into V(t, x, R), the optimal value function is also obtained explicitly

$$\begin{aligned} V(t, x, R) = x + ( \frac{1}{\tilde{G}(T)} - \frac{1}{\tilde{G}(t)} ) R + \frac{ \lambda \theta ^{2} \mu ^{2} }{ 2 \gamma \sigma ^{2} } ( t - T ) . \end{aligned}$$