term insurance function

Whole life pays whenever death comes; term insurance is narrower. It pays 1 only if the person dies within a fixed window — say the next 20 years — and pays nothing if they survive past the end of that window. It is the simplest, cheapest form of life cover: pure protection for a limited time, like a guardrail that protects you only while you are on a particular stretch of road. A 30-year-old buying coverage 'until the mortgage is paid off' is buying term insurance.

Its actuarial present value, written A^1_{x:n} (the small 1 sits over the x to mean 'pays on death of the life, but only if death comes before the n years are up'), is computed exactly like A_x except the sum stops at the end of the term. You add, over each of the first n years only, the discounted benefit weighted by the probability of dying in that year. Anything beyond year n contributes nothing. Because it covers only part of life, term insurance is always cheaper than the corresponding whole life insurance: A^1_{x:n} is less than A_x, and the gap is the value of the death cover you are giving up after the term ends.

Term insurance is the backbone of the protection market because it delivers the most death benefit per dollar of premium. In the mathematics it is also a key Lego brick: a whole life insurance equals a term insurance plus a deferred insurance covering the years after the term, and an endowment insurance equals a term insurance plus a pure endowment. A frequent confusion is reading the little 1 as a power or a duration; it is a status marker in international actuarial notation meaning the benefit is contingent on death happening first, before the term expires.