Optimal Decision on Dynamic Insurance Price and Investment Portfolio of an Insurer with Multi-dimensional Time-Varying Correlation

Abstract

In this article, we propose a model of optimal insurance pricing and investment strategies, in which the insurance price, investment returns and insured losses are assumed to be correlated stochastic processes. The investment portfolio discussed in this paper contains more than one risky assets following multi-Vasicek model with time-varying correlation. To solve the optimal investment problem, we take into account the demand of insurance contracts which impacts the price of the contracts; the utility is a performance process for a specific time; we determine dynamical optimal price of an insurance contract and the optimal investment portfolio of an insurer simultaneously by maximizing the performance of the insurer. We carry out numerical analysis with an example. The results show that the Treasury Bill, generally considered as a risk-free asset, is examined to follow the similar pattern as other risky assets in terms of volatility/mean ratio; multi-Vasicek model is an appropriate model to describe the change pattern of the return of risky assets in the investment. We analyze the sensitivity of the change in important parameters of the optimal solutions. It is worth noticing that the equally weighted investment portfolio is proved to be an optimal investment strategy under some conditions; the proposed model in this paper can be used to obtain optimal solutions easily in the situation of multi-dimensional investment portfolio.

Appendix

We calculate $\sigma_{ij} (t)$, the time-varying covariance of the returns of $i$th and $j$th risky

Assets, in a manner similar to Korn and Koziol (2006), and Mamon (2004):

$$ \begin{aligned} &{\text{Vasicek model: }}dr = a\left( {b - r} \right)dr + \sigma dW_{u} , \hfill \\& Let \, X_{i} \left( u \right) = r_{iu} - b, \hfill \\& dX_{i} \left( t \right) = - a_{i} X_{i} \left( t \right) + \, \sigma_{i} dW_{t} , \hfill \\& X_{i} \left( u \right) = e^{{ - a_{i} u}} \left( {X_{i} \left( 0 \right) + \int_{0}^{u} {\sigma_{i} e^{{a_{i} s}} dw_{s} } } \right), \hfill \\& then,r_{iu} = e^{{ - a_{i} u}} \left( {r_{iu} - b + \int_{0}^{u} {\sigma_{i} e^{{a_{i} s}} dw_{s} } } \right) + b. \hfill \\& \end{aligned} $$

$$ E(r_{iu} ) = E\left( {e^{{ - a_{i} u}} \left( {r_{iu} - b + \int_{0}^{u} {\sigma_{i} e^{{a_{i} s}} dw_{s} } } \right) + b} \right) = e^{{ - a_{i} u}} (r_{iu} - b) + b. $$

where subscript $u$ stands for the time and the subscript $i$ for the $i$th risky asset.

In the following, we calculate the time-varying covariance, $\sigma_{ij} (t)$ for $i$th and $j$th risky assets. By Itô isometry, we have:

$$ \begin{aligned} \sigma_{ij} (t) & = Cov\left( {r_{it} (t),r_{jt} (t)} \right) = E(r_{it} (t)r_{jt} (t)) - E\left( {r_{it} (t)} \right)E\left( {r_{jt} (t)} \right) \\ & = E\left( \left( {b_{i} + (r_{i0} - b_{i} )e^{{ - a_{i} t}} + \sigma_{i} e^{{ - a_{i} t}} \int\limits_{0}^{t} {e^{{a_{i} s}} } dw_{s} } \right)\right. \\ &\quad \left. \times\, \left( {b_{j} + (r_{j0} - b_{j} )e^{{ - a_{j} t}} + \sigma_{j} e^{{ - a_{j} t}} \int\limits_{0}^{t} {e^{{a_{j} s}} } dw_{s} } \right) \right) \\ & \quad - \left( {b_{i} + (r_{i0} - b_{i} )e^{{ - a_{i} t}} } \right)\left( {b_{j} + (r_{j0} - b_{j} )e^{{ - a_{j} t}} } \right) \\ & = (b_{i} + (r_{i0} - b_{i} )e^{{ - a_{i} t}} )(b_{j} + (r_{j0} - b_{j} )e^{{ - a_{j} t}} ) \\ &\quad + E\left( {(b_{i} + (r_{i0} - b_{i} )e^{{ - a_{i} t}} )\sigma_{j} e^{{ - a_{j} t}} \int\limits_{0}^{t} {e^{{a_{j} s}} } dw_{s} } \right) \\ & \quad + E\left( {(b_{j} + (r_{j0} - b_{j} )e^{{ - a_{j} t}} )\sigma_{i} e^{{ - a_{i} t}} \int\limits_{0}^{t} {e^{{a_{i} s}} } dw_{s} } \right) \\ &\quad + \sigma_{i} \sigma_{j} e^{{ - a_{i} t - a_{j} t}} E\left( {\int\limits_{0}^{t} {e^{{a_{i} s}} } dw_{s} \int\limits_{0}^{t} {e^{{a_{j} s}} dw_{s} } } \right) \\ & \quad - \left( {b_{i} + (r_{i0} - b_{i} )e^{{ - a_{i} t}} } \right)\left( {b_{j} + (r_{j0} - b_{j} )e^{{ - a_{j} t}} } \right) \\ & = \sigma_{i} \sigma_{j} e^{{ - a_{i} t - a_{j} t}} E\left( {\int\limits_{0}^{t} {e^{{a_{i} s}} } dw_{s} \int\limits_{0}^{t} {e^{{a_{j} s}} dw_{s} } } \right) \\ & = \sigma_{i} \sigma_{j} e^{{ - a_{i} t - a_{j} t}} \int\limits_{0}^{t} {e^{{(a_{i} + a_{j} )s}} } ds \\ & = \frac{{\sigma_{i} \sigma_{j} }}{{a_{i} + a_{j} }}\left( {1 - e^{{ - (a_{i} + a_{j} )t}} } \right). \\ \end{aligned} $$

Please note that $\sigma_{i}^{2} (t) = \sigma_{ij} (t)$; when $i = j$, $\sigma_{i}^{2} (t) = \frac{{\sigma_{i}^{2} }}{{2a_{i} }}\left( {1 - e^{{ - 2a_{i} t}} } \right).$