whole life insurance function (A_x)

Suppose an insurer promises to pay 1 the moment a person dies, no matter when that happens — next year or fifty years from now. What is that promise worth today? The death is certain to occur eventually, so the benefit will surely be paid; the only questions are when, and therefore how heavily to discount it for interest. A death soon means a payment soon, lightly discounted and so worth a lot; a death far off means a heavily discounted payment worth little. The whole life insurance function, written A_x, is the actuarial present value of that 'pay 1 at death, whenever it comes' promise on a life now aged x.

To compute it, you sweep over every future year, take the benefit of 1, discount it back from the year of death to today, and weight it by the probability that death falls in that particular year. Summed over all years, that is A_x. In words: A_x equals the sum over each future year k of (discount factor to that year) times (probability of dying in that year). Because death is certain, A_x is a kind of weighted-average discount factor for the unknown moment of death. A typical value for a healthy adult at moderate interest might be around 0.2 to 0.4 per unit — well below 1, since the payment is, on average, many years away.

A_x is the single most important insurance function in the subject; almost every other death-benefit value is a variation on it (term, deferred, endowment). It interlocks beautifully with annuities through the relationship A_x = 1 - d times the whole life annuity-due, where d is the rate of discount — pay attention to that identity, it ties the whole field together. A common misconception is that a higher A_x means 'more risk'; in fact A_x rises with age and falls as interest rates rise, simply reflecting that the death payment is, in present-value terms, closer or more cheaply financed.