We derive a matrix representation for formulas of moments of cash value of future payment streams arising from multistate insurance contract, where the evolution of the insured risk and the interest rate are random. As an application, we derive formulas for net single and period premiums. The general theory is illustrated with a case where the evolution of the insured risk is modeled by a Markov chain. Appl. Stochastic Models Bus. Ind. 2013, 29 224-240 J. DȨ BICKAwhere 1I A is the indicator of event A. Observe that the information X.k/Dj is not enough to determine the future cash flow at time k arising from a multistate insurance contract in discrete-time model (because of P i 2Snfj g c ij .k/1I fX.k 1/Di g ). In the next section, we cope with this problem by a proper extension of the state space. Appl. Stochastic Models Bus. Ind. 2013, 29 224-240Assumption 6 All moments of the random discounting function e Y.t/ are finite.Note that for a portfolio of L policies, the random variables Z 1 ; Z 2 ; : : : Z L are not independent because the discount function is the same for each Z l .
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We model the probabilistic structure and cash flows arising from marriage reverse annuity contracts in the case of the joint-life status and the last surviving status. In contrast to the classical approach, we take into consideration that future lifetimes between spouses are dependent. The structure of dependence of the length of spouses' lives is modelled using copulas. The term structure of interest rate is modelled using a time-dependent function. The numerical results are based on actual Polish data covering both the structure of the probabilistic model and the interest rate.
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