Natural hedging in continuous time life insurance (Q6201514)

The paper deals with the optimization problem, which asks to find the weights \(w_1(t),\,w_2(t),\,\ldots,\,w_P(t)\) such that \[ \sum_{p=1}^{P}w_{(p)}(t)=1,\\ \sum_{p=1}^{P}w_{(p)}(t)\frac{\partial}{\partial\eta_{jk}}V^i_{(p)}(t,\,\mu+\eta g)\Big|_{\eta=0}=0, \] where \[ V^i(t,\,u)=\sum_{p=1}^{P}w_{(p)}(t)V^i_{(p)}(t,\,\mu) \] denotes the total liability of the portfolio. According to the author ``We suggest an alternative to hedge insurance risks. In a multi-state setup in continuous time life insurance, we describe the concept of natural hedging, which enables us to compose a portfolio of different insurance products where the liabilities are unaffected by shifts in the transition intensities. We describe how to find and how to calculate the natural hedging strategy using directional derivatives (Gateaux derivatives) to measure the sensitivity of the life insurance liabilities with respect to shifts in the transition intensities of a Markov chain governing the state of the insured. Furthermore, we implement the natural hedging strategy in two numerical examples based on the survival model and the disability model, respectively''.