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. ${}_tV^{(0)}$ is the reserve while both lives survive; ${}_tV^{(1)}$ and ${}_tV^{(2)}$ 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.

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

Common mistakes

• Setting ${}_tV^{(1)} = 0$ 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: ${}_tV = \text{EPV future benefits} - \text{EPV future premiums}$, computed in EACH non-absorbing state (a 4-state contract carries one reserve per non-absorbing state).
• Thiele recursion: $\frac{d}{dt}{}_tV = \delta \cdot {}_tV + P_t - B_t - \sum_j (S_t^{j} - {}_tV) \mu_t^{0j}$, 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: ${}_tp_{xy} = {}_tp_x \cdot {}_tp_y$; under the Common Shock model ${}_tp_{xy} = {}_tp_x \cdot {}_tp_y \cdot e^{-\lambda t}$, 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 $\frac{d}{dt}{}_tp^{00}_{xy} = -{}_tp^{00}_{xy}(\mu^{01} + \mu^{02})$, ${}_0p^{00}_{xy}=1$.

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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