temporary life annuity function

A whole life annuity pays for as long as you live; a temporary life annuity caps that at a fixed number of years. It pays 1 at the start of each year, but only while the person is both alive and still within the first n years — payments stop either at death or at the end of the n-year window, whichever comes first. Think of an income bridge: a payout that carries someone from early retirement at 60 until their state pension kicks in at 67, and no longer.

Its actuarial present value, written ä_{x:n} (the subscript x:n means 'while the life aged x survives, for at most n years'), is computed just like the whole life annuity but the sum stops after n terms. You add, for each of the first n years, the discount factor times the survival probability to that year, and ignore all years beyond n. Because it pays for a limited time, the temporary annuity is always worth less than the whole life annuity: ä_{x:n} is less than ä_x, and the difference is exactly an n-year deferred annuity (the value of the payments you would have received after year n). It is also the natural denominator in the equivalence principle whenever premiums are paid only for a limited period — divide the benefit's EPV by ä_{x:n} to get the level premium for an n-year payment plan.

Temporary annuities are everywhere in practice: limited-premium-payment policies, fixed-period pension top-ups, child's education income, and the streams used inside reserve and premium formulas. The caveat to keep straight is the double contingency: a temporary annuity payment requires both survival and being within the term, so two things can stop it. It is also easy to confuse a temporary life annuity (which stops on death) with a period-certain annuity-certain (which pays for n years regardless of survival) — the life version is cheaper precisely because death can end it early.