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. ${}_tp_{xy} = {}_tp_x \cdot {}_tp_y$ requires independence. With a common shock, this overstates ${}_tp_{xy}$ 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: ${}_tp_{xy} = {}_tp_x \cdot {}_tp_y$ and $\mu_{xy} = \mu_x + \mu_y$; then ${}_tp_{\overline{xy}} = {}_tp_x + {}_tp_y - {}_tp_{xy}$.
• 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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