Define and interpret state-dependent insurance and annuity present value random variables and identify and calculate their expected values.

Multi-state pricing collapses to one move: write the cash flow at time t, condition on which state the life occupies (or which transition just happened), and integrate against the matching transition probability and discount factor.

Let $\{Y_t{\}}_{t\ge 0}$ be a continuous-time Markov chain on states $0,1,\dots ,n$. State 0 is usually "active" or "healthy", and at least one state is absorbing (typically "dead"). Two probabilities drive every pricing formula:

• Occupancy probability: ${}_t{p}_x^{ij}=\mathrm{Pr}(Y_{x+t}=j\mid Y_x=i)$. Where is the life t years from now?
• Transition intensity: ${\mu }_{x+t}^{jk}={\lim }_{h\to 0}\mathrm{Pr}(Y_{x+t+h}=k\mid Y_{x+t}=j)/h$. Instantaneous rate of jumping from j to k.

KEY: Annuity language is about WHERE the life IS. Insurance language is about a JUMP just made. Read the policy wording, identify which, and pick the matching formula.

A continuous annuity paying at rate $1 per year while the life occupies state j has present value random variable

$$Y={\int }_0^{\infty }{e}^{-\delta t}1\{Y_{x+t}=j\}dt$$

Common mistakes

• Putting an intensity in the annuity formula. Annuity EPVs integrate ${}_t{p}_x^{ij}$ alone. A stray ${\mu }^{kj}$ factor turns it into a transition insurance and undercounts by orders of magnitude.
• Using ${}_t{p}_x^{ij}$ instead of ${}_t{p}_x^{ik}$ before a $k\to j$ jump. The pre-jump state is k, not j. Picking the wrong probability is the single most common sign error in state-dependent EPV.
• Forgetting to discount intensity-driven payments. ${\mu }^{kj}$ is a rate per unit time, not a probability. The integrand $e^{-\delta t}{}_t{p}_x^{ik}{\mu }_{x+t}^{kj}$ needs all three factors.

Bottom line

• Annuity benefits depend on the state the life OCCUPIES; insurance benefits depend on the TRANSITION just made.
• Continuous annuity EPV in state j given start in state i: ${\bar{a}}_x^{ij}={\int }_0^{\infty }{e}^{-\delta t}{}_t{p}_x^{ij}dt$.
• Transition insurance EPV paid on entry to state j from anywhere: ${\bar{A}}_x^{ij}={\int }_0^{\infty }{e}^{-\delta t}{\sum }_{k\ne j}{}_t{p}_x^{ik}{\mu }_{x+t}^{kj}dt$; for non-absorbing j it counts EVERY entry, not one jump.
• Specific-transition benefits restrict both prior state and destination, using the integrand ${}_t{p}_x^{i,i^{\prime }}{\mu }^{i^{\prime }{j}^{\prime }}$.

Exam shortcut

When intensities are constant, every EPV reduces to a sum of integrals of the form ${\int }_0^{\infty }{e}^{-\alpha t}dt=1/\alpha$. Read off $\alpha$ as the sum of all exponential rates in the integrand and finish in seconds. DECISION: Question says "while" → annuity formula, integrate ${}_t{p}_x^{ij}$. Question says "on death" or "on becoming disabled" → insurance formula, multiply in the matching ${\mu }^{kj}$.

The full lesson (about 5,600 words, 37 min read) adds 5 worked examples, all 11 common mistakes, a self-check, free in the app.

Learning objectives

• 2a

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