Joint lives

A JointLife combines two SingleLifes and a contingency. Independent() (also available as Frasier()) models independent future transitions. It is the only supported dependence assumption.

The lives keep their own issue ages and mortality selection. An observed dead life stays dead; a table ending with unknown survival does not establish death.

using LifeContingencies, MortalityTables

life = SingleLife(UltimateMortality([0.5, 0.5, 1.0]; start_age=60))
last_survivor = JointLife(lives=(life, life), contingency=LastSurvivor())
first_to_die = JointLife(lives=(life, life), contingency=FirstToDie())
@assert survival(last_survivor, 1) == 0.75
@assert survival(first_to_die, 1) == 0.25
(last_survivor_probability = survival(last_survivor, 1),
 first_to_die_probability = survival(first_to_die, 1))

(last_survivor_probability = 0.75, first_to_die_probability = 0.25)

A joint insurance pays once when its contingency ends; a joint annuity pays while that contingency remains in force. They use the same product constructors as single lives:

(
    last_death_insurance = present_value(Insurance(last_survivor, 0.05)),
    first_death_insurance = present_value(Insurance(first_to_die, 0.05)),
    last_survivor_annuity = present_value(AnnuityDue(last_survivor, 0.05)),
)

(last_death_insurance = 0.8994708994708994, first_death_insurance = 0.9383435914048157, last_survivor_annuity = 2.111111111111111)

What is known at the valuation date?

For last-survivor business, “the contract is still in force” does not tell you whether one or both lives are alive. Consider year one in the example above:

Conditional on at least one alive at year one, the probabilities of (alive, alive), (alive, dead) and (dead, alive) are each 1/3. The last answer is therefore $\frac13\frac34+\frac23\frac12=\frac7{12}$.

one_known_alive = survival(life, 1, 2)
both_known_alive = 1 - (1 - one_known_alive)^2
aggregate = survival(last_survivor, 1, 2)
@assert aggregate ≈ 7 / 12
@assert aggregate ≈ both_known_alive / 3 + 2one_known_alive / 3
(one_known_alive = one_known_alive, both_known_alive = both_known_alive,
 unknown_individual_statuses = aggregate)

(one_known_alive = 0.5, both_known_alive = 0.75, unknown_individual_statuses = 0.5833333333333334)

Two-time survival for JointLife conditions on its contingency, using its original statuses at issue. It does not assert that both individuals are alive at the starting date. The conditional joint distribution above is no longer a product of its marginals, even though subsequent transitions remain independent.

Reserves from known statuses

The default reserve uses aggregate in-force conditioning. Use state to supply the observed pair at the valuation date instead:

lc = LifeContingency(last_survivor, 0.05)
r = (
    unknown_individual_statuses = reserve_premium_net(lc, 1),
    both_alive = reserve_premium_net(lc, 1; state=(:alive, :alive)),
    only_first_alive = reserve_premium_net(lc, 1; state=(:alive, :dead)),
)
# Each in-force state has probability 1/3 at year one; the two lives are identical.
@assert r.unknown_individual_statuses ≈ r.both_alive / 3 + 2r.only_first_alive / 3
r

(unknown_individual_statuses = 0.26315789473684204, both_alive = 0.18796992481203012, only_first_alive = 0.300751879699248)

The state tuple follows the order of lives. Passing a state does not reprice the premium or change the original curve or mortality basis.

Horizons and assumptions

Last-survivor cover follows the longer life; first-to-die follows the shorter life. An extinct shorter life stays dead while the longer life continues. Incomplete tables still need enough support for the contingent payments. Unsupported dependence assumptions raise an ArgumentError rather than silently using independence.

The indicator identity $S_{\mathrm{first}}(t)+S_{\mathrm{last}}(t)=S_1(t)+S_2(t)$ counts the two individual alive indicators at each date, so it also holds for correlated starting distributions. Here $S_1$ and $S_2$ are the individual survival probabilities. It gives $S_{\mathrm{last}}=S_1+S_2-S_{\mathrm{first}}$ and provides a useful check of a joint model. With the same payment dates and discount factors, the identity also gives last-survivor life-annuity values from the single-life and first-to-die values. The engine guide shows how distributions and rows are supplied separately.