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:

${}_tp_{xy} = P(T_x>t,\,T_y>t)$.

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

Common mistakes

• Multiplying marginal survivals under common shock. ${}_tp_{xy} \ne {}_tp_x\cdot{}_tp_y$ 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. ${}_tp_{\overline{xy}} = {}_tp_x + {}_tp_y - {}_tp_{xy}$. 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 ${}_tp_{xy} = P(T_x>t,\,T_y>t)$.
• Last-survivor status $\overline{xy}$ fails on the second death: $T_{\overline{xy}} = \max(T_x, T_y)$; ${}_tp_{\overline{xy}} = {}_tp_x + {}_tp_y - {}_tp_{xy}$.
• 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 $\mu_x^*+\mu_y^*+\lambda$. 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 2,598 words, 17 min read) adds 8 worked examples, all 6 common mistakes, a self-check, free in the app.

Learning objectives

• A5

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