Apply Markov multiple state models to state-contingent life and long-term health insurance benefits, and Continuing Care Retirement Communities (CCRCs).

A disability rider pays only while you are sick. A CCRC contract follows a resident across independent living, assisted living, and skilled nursing. Valuing either requires a Markov framework that prices each state and each transition explicitly.

The model partitions life into states. Each state $i$ has forces $\mu_x^{ij}$ into every reachable $j$. The Markov property says the future depends only on the current state at age $x+t$, not the path. A semi-Markov extension lets intensities depend on duration in the current state (useful for disability where recovery slows with sickness length).

KEY: Drawing the state diagram first is the highest-leverage move. Label every arrow with its intensity before writing a single integral.

Morbidity products like disability income and long-term care pay only while a policyholder occupies a sickness state, so pricing requires modeling entry, exit, and (where relevant) recovery, which is exactly what a Markov framework delivers and a single-decrement life model cannot.

• Term and whole life: alive, dead.
• Joint life: both alive, only x alive, only y alive, both dead.

Common mistakes

• Confusing ${}_tp_x^{ii}$ with ${}_tp_x^{\overline{ii}}$. When recovery is allowed, the bar version is smaller. Using $e^{-\int\mu^{ij}dt}$ for occupancy is wrong whenever re-entry exists.
• Skipping recovery in DI. Setting $\mu^{10}=0$ overprices the disability annuity. Realistic DI tables show recovery rates 10 to 20 times mortality at early durations.
• Treating CCRC fees as flat under Type C. Type C charges market rate in AL/SNF. Only Type A levels the monthly fee across all live states.

Bottom line

• Markov model: states $0,1,\dots,n$, transition intensities $\mu_x^{ij}$; the future depends only on current state and age.
• Occupancy probability ${}_tp_x^{ij}$ allows any path; sojourn probability ${}_tp_x^{\overline{ii}}$ requires no exit.
• Kolmogorov forward equations evolve ${}_tp_x^{ij}$; Thiele extends reserve recursion to multi-state policies.
• Annuity-in-state EPV uses $\int e^{-\delta t}{}_tp_x^{ij}\,dt$; lump-sum-on-transition EPV multiplies by $\mu^{jk}$.

Exam shortcut

When intensities are constant and the model is acyclic, ${}_tp^{0j}$ has a closed form as a difference of exponentials. Memorize ${}_tp^{01} = \frac{\mu^{01}}{\mu^{1\cdot}-\mu^{0\cdot}}\big(e^{-\mu^{0\cdot}t}-e^{-\mu^{1\cdot}t}\big)$. To check a state diagram, sum outgoing intensities from each state and verify every benefit-bearing state is reachable from the starting state via at least one path. For CCRC questions, identify the contract type (A/B/C) first.

The full lesson (about 4,483 words, 30 min read) adds 3 worked examples, all 10 common mistakes, a self-check, free in the app.

Learning objectives

• 1b

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