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IAI Actuarial Core Principles · Risk Modelling and Survival Analysis · Markov chains

A motor insurer uses a No Claims Discount system modelled as a Markov chain with states 0%, 25% and 50% discount. A policyholder with no claim in a year moves up one level (or stays at 50%); a policyholder with a claim moves down one level (or stays at 0%). Which statement about this model is correct?

The chain is irreducible and aperiodic. Every discount level can be reached from every other through runs of claims or claim-free years, and the end states can stay put, which gives period one. Being finite, it therefore has a unique stationary distribution.

  1. AThe chain is reducible because the 0% state cannot be reached from the 50% state
  2. BThe chain is irreducible and aperiodicCorrect
  3. CThe chain is irreducible but periodic with period 2
  4. DThe 50% state is absorbing
  5. The chain has no stationary distribution

Explanation

Any state can be reached from any other through repeated claims or claim-free years, so the chain is irreducible. The 0% and 50% states have self-loops, so the period is 1 and the chain is aperiodic. A finite irreducible chain has a unique stationary distribution, so the other options fail.

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