Understand how joint-life mortality can be modelled using (i) a time-to-status-failure random variable, and (ii) a multiple state model.

A couple buys a survivor annuity that pays until both are gone. Pricing it requires modeling two lives together, not adding two single-life prices. Two frameworks dominate: a status-failure clock that stops on a defined event, and a four-state Markov model that tracks who is alive.

You can model two lives by defining a single random variable that marks when a chosen status ends, or by tracking the joint state of the couple over time. The status approach is compact and produces clean formulas under independence. The multiple-state approach is more general and handles dependent mortality.

Let $T_x$ and $T_y$ denote the future lifetimes of lives currently aged $x$ and $y$. Two statuses appear constantly on the exam.

The joint-life status $(xy)$ is alive while BOTH lives are alive. It fails at the first death.

$$T_{xy}=\min (T_x,T_y)$$

The last-survivor status $\overline{xy}$ is alive while AT LEAST ONE life is alive. It fails at the second death.

Common mistakes

• Multiplying probabilities when lives are dependent. ${}_t{p}_{xy}={}_t{p}_x\cdot {}_t{p}_y$ requires independence. With a common shock, this overstates ${}_t{p}_{xy}$ because it misses simultaneous deaths.
• Forgetting the common-shock term in ${\mu }_{xy}$. Under common shock, ${\mu }_{xy}={\mu }^{01}+{\mu }^{02}+{\mu }^{03}$, not ${\mu }_x+{\mu }_y$. Skipping ${\mu }^{03}$ underprices joint-life insurance.
• Confusing $xy$ with $\overline{xy}$. Joint-life pays while BOTH live and fails at the first death. Last-survivor pays while at least one lives and fails at the second death.

Bottom line

• Joint-life status $(xy)$ fails at the first death: $T_{xy}=\min (T_x,T_y)$. Last-survivor status $\overline{xy}$ fails at the second death: $T_{\overline{xy}}=\max (T_x,T_y)$.
• Independence shortcuts: ${}_t{p}_{xy}={}_t{p}_x\cdot {}_t{p}_y$ and ${\mu }_{xy}={\mu }_x+{\mu }_y$; then ${}_t{p}_{\overline{xy}}={}_t{p}_x+{}_t{p}_y-{}_t{p}_{xy}$.
• Identity: $T_x+T_y=T_{xy}+T_{\overline{xy}}$, so $a_x+a_y=a_{xy}+a_{\overline{xy}}$ and the same for $A$ values.
• Multiple-state model: 4 states (both alive, only $x$, only $y$, both dead). Transition intensities ${\mu }^{ij}$ drive Kolmogorov ODEs.

Exam shortcut

When given two constant forces and constant $\delta$, every continuous EPV is $1/(\text{total exit rate from the paying state}+\delta )$. ${\bar{a}}_{xy}=1/({\mu }_x+{\mu }_y+{\mu }^{03}+\delta )$ handles common shock instantly. DECISION: Problem says "independent" → status formulas. Problem shows a state diagram or mentions common shock or broken heart → multiple-state model and Kolmogorov. Use the identity ${\bar{a}}_{\overline{xy}}={\bar{a}}_x+{\bar{a}}_y-{\bar{a}}_{xy}$ whenever ${\bar{a}}_{xy}$ is easy and ${\bar{a}}_{\overline{xy}}$ looks ugly. Same trick works for $\bar{A}$ values.

The full lesson (about 6,449 words, 43 min read) adds 4 worked examples, all 13 common mistakes, a self-check, free in the app.

Learning objectives

• 3a

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