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Asset liability management for an ordinary insurance system with proportional reinsurance in a CIR stochastic interest rate and Heston stochastic volatility framework

  • * Corresponding author: Yan Zhang, Yonghong Wu

    * Corresponding author: Yan Zhang, Yonghong Wu 
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  • This paper investigates the asset liability management problem for an ordinary insurance system incorporating the standard concept of proportional reinsurance coverage in a stochastic interest rate and stochastic volatility framework. The goal of the insurer is to maximize the expectation of the constant relative risk aversion (CRRA) of the terminal value of the wealth, while the goal of the reinsurer is to maximize the expected exponential utility (CARA) of the terminal wealth held by the reinsurer. We assume that the financial market consists of risk-free assets and risky assets, and both the insurer and the reinsurer invest on one risk-free asset and one risky asset. By using the stochastic optimal control method, analytical expressions are derived for the optimal reinsurance control strategy and the optimal investment strategies for both the insurer and the reinsurer in terms of the solutions to the underlying Hamilton-Jacobi-Bellman equations and stochastic differential equations for the wealths. Subsequently, a semi-analytical method has been developed to solve the Hamilton-Jacobi-Bellman equation. Finally, we present numerical examples to illustrate the theoretical results obtained in this paper, followed by sensitivity tests to investigate the impact of reinsurance, risk aversion, and the key parameters on the optimal strategies.

    Mathematics Subject Classification: Primary: 91G80, 93E20; Secondary: 39A50.

    Citation:

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  • Figure 1.  Evolutions of the CIR stochastic interest rate $r(t)$ and Heston stochastic volatility $\sigma(t)$ within the investment horizon $[0,\ T]$

    Figure 2.  Evolutions of the risky assets' prices for the insurer and the reinsurer

    Figure 3.  Evolutions of the wealth processes for the insurer and the reinsurer

    Figure 4.  The dynamic behaviour of (a) the optimal reinsurance control strategy $\psi^{*}(t)$, (b) the optimal investment strategy for the insurer $\pi^{*}(t)$ and (c) the optimal investment strategy for the reinsurer $u^{*}(t)$

    Figure 5.  Sensitivities of $\psi^{*}(t)$ with respect to $\zeta(t)$

    Figure 6.  Sensitivities of $\psi^{*}(t)$ with respect to $\gamma$

    Figure 7.  Sensitivities of $\pi^{*}(t)$ with respect to $\gamma$

    Figure 8.  Sensitivities of $\pi^{*}(t)$ with respect to the parameter $\nu$

    Figure 9.  Sensitivities of the optimal investment strategy $u^{*}(t)$ for the reinsurer with respect to the interest rate $\mu$

    Figure 10.  Sensitivities of the optimal investment strategy $u^{*}(t)$ for the reinsurer with respect to the risk aversion coefficient $q$

    Figure 11.  Sensitivities of the optimal investment strategy $u^{*}(t)$ for the reinsurer with respect to the positive correlation coefficient

    Figure 12.  Sensitivities of the optimal investment strategy $u^{*}(t)$ for the reinsurer with respect to the negative correlation coefficient.

    Figure 13.  Sensitivities of the optimal investment strategy $u^{*}(t)$ for the reinsurer with respect to the mean reversion speed $\kappa$ when $q = 4$ and $\rho>0$

    Figure 14.  Sensitivities of the optimal investment strategy $u^{*}(t)$ for the reinsurer with respect to the mean reversion speed $\kappa$ when $q = 2$ and $\rho <0$

    Figure 15.  Sensitivities of the optimal investment strategy $u^{*}(t)$ for the reinsurer with respect to "volatility of volatility" $\xi$ when $q = 2$ and $\rho <0$

    Figure 16.  Sensitivities of the optimal investment strategy $u^{*}(t)$ for the reinsurer with respect to "volatility of volatility" $\xi$ when $q = 2$ and $\rho>0$

    Figure 17.  Sensitivities of the optimal investment strategy $u^{*}(t)$ for the reinsurer with respect to $a$

    Figure 18.  Sensitivities of the optimal investment strategy $u^{*}(t)$ for the reinsurer with respect to $b$

    Table 1.  Parameter values for the original model

    Symbol Value Symbol Value Symbol Value
    $T$ 5 $\nu$ 1 $r_{0}$ 0.05
    $\phi(t)$ 1.2 $\theta$ 0.06 $b_{0}$ 1
    $m(t)$ 0.6 $\kappa$ 2 $b^{re}_{0}$ 1
    $\zeta(t)$ 0.8 $\xi$ 0.1 $\sigma_{0}$ 0.04
    $\alpha$ 0.1 $a$ 1 $s_{0}$ 1
    $\beta$ 0.1 $b$ 1 $s^{re}_{0}$ 1
    $K$ 0.15 $\mu$ 0.1 $l_{0}$ 2
    $\tau$ 1.5 $\rho$ 0.5 $x_{0}$ 5
    $\gamma$ 4 $q$ 0.5 $y_{0}$ 5
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