Multistate models in health insurance

Abstract

We illustrate how multistate Markov and semi-Markov models can be used for the actuarial modeling of health insurance policies, focusing on health insurances that are pursued on a similar technical basis to that of life insurance. In the first part, we give an overview of the basic modeling frameworks that are commonly used and explain the calculation of prospective reserves and net premiums. In the second part, we discuss the biometric insurance risk, focusing on the calculation of implicit safety margins. We present new results on implicit margins in the semi-Markov model and on biometric estimation risk in the Markov model, and we explain why there is a need for future research concerning the systematic biometric risk.

Safety Margins for Systematic Biometric and Financial Risk in a Semi-Markov Life Insurance Framework

“…The proof of this theorem is analogous to the proof of Proposition 4.2 in [3], but extended to interest rate risk, see Appendix A. Note that the decomposition in Equation (2) depends on the order of V i (t, u, Φ * , q * ) and V i (t, u, Φ, q).…”

“…Therefore, we give the following theorem, which can be found in [11] for the Markov case and in [3] for the semi-Markov case but without considering interest rate risk.…”

“…In what follows, we give a short introduction to the assumed continuous time semi-Markov framework, which mainly follows [3], and from where we briefly repeat the most important definitions. This framework goes back to [6] and is also discussed in [7].…”

“…As in [3] we assume that b and B are of bounded variation on compacts and B i (s, ·) should be right-continuous. In the next step we define the prospective reserve.…”

“…Pannenberg [2] considers an active-dead model and uses this approach to calculate safety margins for the unsystematic biometric risk based on a yearly time grid. A calculation of the unsystematic biometric risk and the biometric estimation risk within a general multi-state model can be found in [3,4], respectively. Christiansen and Steffensen [5] derive a first order valuation basis by means of worst case scenarios.…”