Fundamentals of Actuarial Mathematics, 3rd Edition

19 Multi-state models

19.1 Introduction

Multi-state models are an attempt to look at a variety of life insurance and annuity contracts in a unified manner, by making use of Markov processes. As motivation, take an individual now age x and consider a two-state Markov chain, where the person is in state 0 (alive) or state 1 (deceased) at any time. Life insurance contracts provide benefits upon transfer from state 0 to state 1, while life annuity contracts provide benefits as long as the process remains in state 0.

More generally, consider a multiple-decrement model for (x) with m causes of failure. We can consider a chain with m + 1 states. State 0 means that (x) has not succumbed to any cause and is often referred to as the active state. State j refers to having succumbed first to cause j. The insurance benefits discussed in Chapter 11 can be viewed as payments upon transfer from state 0 to other states.

For still another example, consider a joint-life contract issued to (x) and (y). We can now take a chain with four states as illustrated in Figure 19.1. The arrows indicate that there are possible transitions from state 0 into state 1 or state 2, occasioned by the death of (y) or (x) respectively, and then further transitions from state 1 or state 2 into state 3 when the second death occurs. The dotted line, showing a transition directly from state 0 to state 3, would not be present in our original model, but would be there if we wanted to consider a common shock model as discussed ...