Understand the implications of independence or dependence of future lifetimes in both versions of the joint life model. Identify sources of dependence and understand how they are accommodated in the models.

A husband and wife buy a joint annuity. Price it assuming their deaths are independent and you may be wrong by 5 to 10 percent on the reserve. The error has a sign, and exam questions exploit it.

Let $(x)$ and $(y)$ be two lives with future lifetimes $T_x$ and $T_y$. Two derived statuses dominate exam questions.

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

A joint life annuity pays while both are alive. A last survivor annuity pays while at least one is alive (the classic widow/widower benefit). A joint life insurance pays on the first death; a last survivor insurance pays on the second.

KEY: The min/max identity $\min +\max =T_x+T_y$ gives $a_{xy}+a_{\overline{xy}}=a_x+a_y$ and $A_{xy}+A_{\overline{xy}}=A_x+A_y$. This holds with OR without independence. It is the most-tested algebraic anchor in the topic.

Common mistakes

• Flipping the joint-life sign. Candidates often write that positive dependence raises joint-life annuity AND raises last-survivor annuity. The identity forbids it. One rises, the other falls by the same dollar amount.
• Adding the common-shock rate twice. The joint exit rate from state 0 is ${\mu }_x^{\ast }+{\mu }_y^{\ast }+\lambda$, not $({\mu }_x^{\ast }+\lambda )+({\mu }_y^{\ast }+\lambda )={\mu }_x^{\ast }+{\mu }_y^{\ast }+2\lambda$. The common shock event is shared, not counted on each life.
• Forgetting the broken-heart channel. Dependence is not only $\lambda$. Setting ${\mu }^{13}>{\mu }^{02}$ injects post-bereavement mortality even when $\lambda =0$.

Bottom line

• Two statuses: joint life $(xy)$ fails on the FIRST death; last survivor $\overline{xy}$ fails on the SECOND.
• Identity: $T_{xy}+T_{\overline{xy}}=T_x+T_y$, so $a_{xy}+a_{\overline{xy}}=a_x+a_y$ regardless of dependence.
• Sources of dependence: common shock, common lifestyle, broken-heart effect, shared environment, assortative mating; real couples empirically show positive dependence.
• Sign rules (positive dependence): independence UNDERSTATES ${}_t{p}_{xy}$, $a_{xy}$, $A_{\overline{xy}}$; OVERSTATES ${}_t{p}_{\overline{xy}}$, $a_{\overline{xy}}$, $A_{xy}$.

Exam shortcut

When the question gives one EPV under dependence and asks for the other, use the identity. You almost never need to recompute from scratch. DECISION: "Common shock" in the stem → use ${\mu }^{ij}+\lambda$ bookkeeping. "Increased mortality after spouse dies" → use state-dependent ${\mu }^{13}$. Both phrases → both adjustments. DECISION: SULT-with-shock + need ${}_t{p}^{00}$? Multiply two SULT survivals and tack on $e^{+\lambda t}$. SULT-without-shock + need ${}_t{p}^{00}$?

The full lesson (about 6,492 words, 43 min read) adds 7 worked examples, all 11 common mistakes, a self-check, free in the app.

Learning objectives

• 3b

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