Joint life and last survivor
Two lives, four statuses: either alone, both alive, at least one alive. Slide the ages and basis and watch the joint-life and last-survivor curves, annuities and assurances all move together through one identity.
Survival probabilities
Lives and basis
Mortality basis (Gompertz-Makeham, illustrative)
Annuities-due and assurances
ä for life x alone16.542
ä for life y alone17.888
ä joint life (until first death)14.781
ä last survivor (until second death)19.648
A whole life, x alone0.3638
A joint life (first death)0.4315
A last survivor (second death)0.2443
Independent lives, so the joint survival is the product of the singles and every last-survivor value is “sum of the singles minus the joint” — the identity that solves most joint-life exam questions. The joint annuity is always the smallest (it stops at the first death) and the last-survivor annuity the largest; assurances order the other way round. Push the ages apart and watch the joint values track the older life.
One identity does all the work
With independent lives the joint-life status survives while both do, so its survival probability is the product of the individual ones, and its mortality is the first death. The last-survivor status fails at the second death, and the inclusion-exclusion identity — last survivor equals sum of singles minus joint — turns every last-survivor annuity or assurance into three quantities you already know. That is the entire trick behind most joint-life exam parts: value the two single lives, value the joint status, add and subtract. Pension scheme spouse benefits and reversionary annuities are the same algebra with a benefit attached.
Make it stick. Single-life values on the same basis live in the life contingencies calculator, and the notation in the notation cheat sheet. Memori is a flashcard app built by actuarial students, with a ready-made CM1 set in the shop. Join the beta.
For education only: an illustrative Gompertz-Makeham law with independent lives, not any published table, and not financial advice.