Perform joint life calculations.

A couple buys a survivor annuity that pays as long as at least one spouse is alive. Pricing it requires moving from a single future-lifetime random variable to a pair, and then to the statuses built on top of that pair.

Two statuses on the same pair. Given lives aged $x$ and $y$ with future-lifetime variables $T_x$ and $T_y$, define the joint-life status $(xy)$ that survives only while both are alive and the last-survivor status $\overline{xy}$ that survives while at least one is alive. The lifetime of the status is the time until it fails.

$T_{xy}=\min (T_x,T_y)$ and $T_{\overline{xy}}=\max (T_x,T_y)$.

KEY: The status is the bookkeeping unit. Once you know which status a contract is written on, every formula you already know for a single life (insurances, annuities, expected lifetime) applies with the subscript changed.

Survival and failure probabilities. Probability the joint-life status survives $t$ years equals the probability both lives survive:

${}_t{p}_{xy}=P(T_x>t,T_y>t)$.

Probability the last-survivor status survives $t$ years equals the probability at least one is alive:

Common mistakes

• Multiplying marginal survivals under common shock. ${}_t{p}_{xy}\ne {}_t{p}_x\cdot {}_t{p}_y$ when a shared hazard is present. Using independence under-prices first-to-die coverage and over-prices reversions.
• Swapping which status fails on which death. Joint-life fails on the first death; last-survivor fails on the second. Candidates who flip this compute the wrong annuity and miss by a factor of two.
• Forgetting the inclusion-exclusion sign. ${}_t{p}_{\overline{xy}}={}_t{p}_x+{}_t{p}_y-{}_t{p}_{xy}$. Adding without subtracting double-counts the "both alive" event.

Bottom line

• Joint-life status $(xy)$ fails on the first death: $T_{xy}=\min (T_x,T_y)$; survival ${}_t{p}_{xy}=P(T_x>t,T_y>t)$.
• Last-survivor status $\overline{xy}$ fails on the second death: $T_{\overline{xy}}=\max (T_x,T_y)$; ${}_t{p}_{\overline{xy}}={}_t{p}_x+{}_t{p}_y-{}_t{p}_{xy}$.
• Identity (always holds): $T_{xy}+T_{\overline{xy}}=T_x+T_y$, so ${\mathring{e}}_{xy}+{\mathring{e}}_{\overline{xy}}={\mathring{e}}_x+{\mathring{e}}_y$.
• Inclusion-exclusion transfers to insurances and annuities: ${\bar{A}}_{xy}+{\bar{A}}_{\overline{xy}}={\bar{A}}_x+{\bar{A}}_y$ and ${\bar{a}}_{xy}+{\bar{a}}_{\overline{xy}}={\bar{a}}_x+{\bar{a}}_y$, but never mix the two.

Exam shortcut

If a problem gives constant forces and constant interest, immediately use ${\bar{a}}_{xy}=1/({\mu }_x+{\mu }_y+\delta )$ and ${\bar{A}}_{xy}=({\mu }_x+{\mu }_y)/({\mu }_x+{\mu }_y+\delta )$; under common shock, replace the sum with ${\mu }_x^{\ast }+{\mu }_y^{\ast }+\lambda$. Whenever three of the four quantities in $\{A_x,A_y,A_{xy},A_{\overline{xy}}\}$ are given, jump straight to the identity instead of recomputing from scratch. For reversionary annuities, write the answer as a difference of two ordinary life-annuity EPVs before reaching for any integral.

The full lesson (about 1,909 words, 13 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.

Learning objectives

• A5

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