endowment insurance function

Endowment insurance is the 'something either way' policy. It pays 1 if the person dies within a fixed period, and it also pays 1 if the person is still alive at the end of that period. Live or die, the benefit lands — either as a death benefit during the term, or as a maturity payment at the end. This makes it feel less like pure insurance and more like a disciplined savings plan with a death-benefit guarantee, which is exactly why it was historically popular for funding a future goal such as repaying a loan or building a retirement lump sum.

Because a payment is certain to be made (the only question is whether it is triggered by death or by survival to maturity), its value decomposes cleanly. The actuarial present value, written A_{x:n}, is simply the term insurance plus the pure endowment: A_{x:n} = A^1_{x:n} + nE_x, where A^1_{x:n} values the death side and nE_x values the survival side. For example, if the term insurance is worth 0.04 and the pure endowment is worth 0.74, the endowment insurance is worth 0.04 + 0.74 = 0.78. Notice the survival piece usually dominates for short, near-certain terms.

Endowment insurance shows the additive logic of life contingencies at its clearest: complex benefits are sums of simpler ones, and you value each piece separately then add. It also satisfies its own annuity identity, A_{x:n} = 1 - d times the temporary annuity-due ä_{x:n}, mirroring the whole life relationship. The honest caveat is cost: because a benefit is guaranteed either way, endowment insurance is expensive compared with term cover, and much of the premium is really savings rather than protection — a point modern buyers often overlook.