Calculate premiums for insurance and annuities on joint lives using the equivalence principle.

A husband age 65 and wife age 62 want a policy that pays $500,000 at the second death and a reversionary annuity to whichever spouse survives. Pricing it requires two different "lives," each built from the same two underlying mortalities.

A status is a life-like entity that is "alive" while a condition holds and "dies" when it fails. Two statuses dominate joint pricing:

• Joint life status $(xy)$: alive while both $x$ and $y$ are alive. Time to failure $T_{xy}=\min (T_x,T_y)$.
• Last survivor status $(\overline{xy})$: alive while at least one is alive. Time to failure $T_{\overline{xy}}=\max (T_x,T_y)$.

Because the min plus the max equals the sum, you get the most useful identity in the topic:

$$A_{xy}+A_{\overline{xy}}=A_x+A_y,{\ddot{a}}_{xy}+{\ddot{a}}_{\overline{xy}}={\ddot{a}}_x+{\ddot{a}}_y$$

KEY: Compute the easier of each pair (usually joint-life under independence), then back out the other via the identity.

When $T_x$ and $T_y$ are independent:

Common mistakes

• Using ${}_t{p}_x\cdot {}_t{p}_y$ under dependence. The product rule needs independence. With common shock rate $\lambda$, multiply by $e^{-\lambda t}$ or your ${}_t{p}_{xy}$ overstates joint survival and the premium is too low. Common lifestyle and broken-heart effects also violate independence and require explicit modeling.
• Wrong premium-paying status. Plugging ${\ddot{a}}_{\overline{xy}}$ into the denominator when premiums actually stop at the first death undercharges by 20 to 40 percent. Read the contract twice.
• Sign slip in the additive identity. $A_{\overline{xy}}=A_x+A_y-A_{xy}$, not plus. Forgetting the minus inflates the last-survivor insurance by the full $A_{xy}$. The same sign error breaks the term-version identity.

Bottom line

• Joint life status $(xy)$ fails at the first death, i.e. $\min (T_x,T_y)$; last survivor $(\overline{xy})$ fails at the second.
• Additive identity: $T_{xy}+T_{\overline{xy}}=T_x+T_y$ gives $A_{\overline{xy}}=A_x+A_y-A_{xy}$ (minus, not plus) and ${\ddot{a}}_{xy}+{\ddot{a}}_{\overline{xy}}={\ddot{a}}_x+{\ddot{a}}_y$; holds for term versions too.
• Independence: ${}_t{p}_{xy}={}_t{p}_x\cdot {}_t{p}_y$ and ${\mu }_{xy}={\mu }_x+{\mu }_y$; common shock adds $\lambda$ (multiply by $e^{-\lambda t}$); lifestyle and broken-heart also break independence.
• Recursions: ${}_{n+m}{E}_{xy}={}_n{E}_{xy}\cdot {}_m{E}_{x+n:y+n}$, ${}_n{E}_{xy}=v^n{}_n{p}_x{}_n{p}_y$, and ${\ddot{a}}_{xy:\overline{n+m}|}={\ddot{a}}_{xy:\overline{n}|}+{}_n{E}_{xy}{\ddot{a}}_{x+n:y+n:\overline{m}|}$; use ${}_n{E}_{xy}$, not ${}_n{p}_{xy}$.

Exam shortcut

Build a 2×2 table of single-life and joint-life EPVs first. Last-survivor values fall out by subtraction; you almost never integrate twice. DECISION: The premium denominator follows the premium-paying status, not the benefit status. Read "premiums payable until..." and match the annuity to it. When a problem gives ${\ddot{a}}_x$, ${\ddot{a}}_y$, and ${\ddot{a}}_{xy}$, you already know ${\ddot{a}}_{\overline{xy}}$, the reversionary annuity, and (via $A=1-d\ddot{a}$) all three insurances. Convert once, then assemble.

The full lesson (about 11,634 words, 78 min read) adds 10 worked examples, all 17 common mistakes, a self-check, free in the app.

Learning objectives

• 3c

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