Calculate policy values for insurance and annuities on joint lives.

A husband-and-wife annuity that pays $50,000 while both are alive and $30,000 to the survivor needs a reserve that responds to which spouse is still living. Joint-life policy values track exactly that: one prospective reserve per surviving state.

Two lives $(x)$ and $(y)$ generate four states: both alive (0), only $x$ alive (1), only $y$ alive (2), both dead (3). Most exam problems assume independent lives, so the transition intensities reduce to ${\mu }_{x+t}$ and ${\mu }_{y+t}$ acting separately.

KEY: Carry a separate reserve in each non-absorbing state. ${}_t{V}^{(0)}$ is the reserve while both lives survive; ${}_t{V}^{(1)}$ and ${}_t{V}^{(2)}$ are the single-survivor reserves.

Real spouses share environments, diets, and accidents, so their future lifetimes drift together even without a literal common shock. The cleanest test for dependence reads the transition intensities right off the multi-state diagram. If the rate of $y$ dying while $x$ is alive equals the rate of $y$ dying after $x$ has already died, the lifetimes are independent.

$$\text{Independence}⟺{\mu }_{x+t,y+t}^{01}={\mu }_{y+t}^{23}\text{AND}{\mu }_{x+t,y+t}^{02}={\mu }_{x+t}^{13}$$

Common mistakes

• Setting ${}_t{V}^{(1)}=0$ for last-survivor insurance. The benefit is unpaid; the single-survivor state still owes $S\cdot A_{x+t}$ minus any remaining premium EPV.
• Dropping premiums after the first death. Most joint contracts collect premiums while either life survives. Premium EPV in state 1 uses ${\ddot{a}}_{x+t}$, not zero.
• Using ${\ddot{a}}_x+{\ddot{a}}_y-{\ddot{a}}_{xy}$ for the survivor leg. That formula values the last-survivor annuity, which pays in ALL three living states. A pure survivor benefit (pays only when exactly one is alive) uses ${\ddot{a}}_x+{\ddot{a}}_y-2{\ddot{a}}_{xy}$.

Bottom line

• Prospective reserve: ${}_{t}V=\text{EPV future benefits}-\text{EPV future premiums}$, computed in EACH non-absorbing state (a 4-state contract carries one reserve per non-absorbing state).
• Thiele recursion: $\frac{d}{dt}{}_{t}V=\delta \cdot {}_{t}V+P_t-B_t-{\sum }_j(S_t^j-{}_{t}V){\mu }_t^{0j}$, one ODE per state; integrate with Euler $V_{t+h}\approx V_t+h\cdot \text{RHS}$, where step size h controls accuracy.
• Independent lives: ${}_t{p}_{xy}={}_t{p}_x\cdot {}_t{p}_y$; under the Common Shock model ${}_t{p}_{xy}={}_t{p}_x\cdot {}_t{p}_y\cdot e^{-\lambda t}$, adding $\lambda$ to every transition into both-dead.
• Independence test: confirm ${\mu }^{01}={\mu }^{23}$ and ${\mu }^{02}={\mu }^{13}$; one failure proves dependence, so build survival from Kolmogorov $\frac{d}{dt}{}_t{p}_{xy}^{00}=-{}_t{p}_{xy}^{00}({\mu }^{01}+{\mu }^{02})$, ${}_0{p}_{xy}^{00}=1$.

Exam shortcut

Decompose every joint benefit into single-life and joint pieces before plugging numbers. $A_{\overline{xy}}$ and ${\ddot{a}}_{\overline{xy}}$ almost never need direct computation: build them from $A_x,A_y,A_{xy}$. DECISION: Constant benefits with standard annuity factors. Use prospective EPV directly. Variable benefits or non-Markov shocks. Set up Thiele and solve backward. "JLS" check for any joint reserve: identify the Joint-alive piece, the single-Life piece per survivor, and the Second-death liability if last-survivor.

The full lesson (about 9,459 words, 63 min read) adds 8 worked examples, all 16 common mistakes, a self-check, free in the app.

Learning objectives

• 3d

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