deferred insurance function

Deferred insurance is cover that switches on only after a waiting period. For the first m years there is no death benefit at all; from year m onward, the usual death benefit applies. Picture an employer who promises life cover to staff, but only once they have served five years — the protection is real, but it sits dormant until the deferral period is over. The earlier you might die, the more this 'switch-on later' design saves the insurer.

Its value, written m|A_x (the bar after m means 'deferred m years'), is the actuarial present value of a whole life insurance that begins paying only after m years. You can compute it directly by summing discounted, mortality-weighted death benefits but only for years m+1 onward, ignoring the first m years entirely. More elegantly, it is a subtraction: m|A_x = A_x minus A^1_{x:m}, the whole life value minus the term insurance that would have covered the first m years. It can also be written as the m-year pure endowment times the whole life insurance on the older life: m|A_x = mE_x times A_{x+m}, which reads as 'survive the deferral, then buy whole life at the attained age'.

Deferred insurance is the natural complement to term insurance — term covers the early window, deferred covers everything after it, and together they rebuild whole life. It appears whenever benefits have a waiting period, vesting condition or postponed start, and the same deferral idea carries straight over to deferred annuities. The misconception to avoid is thinking 'deferred' means 'cheaper because it pays later' in a vague sense; the precise reason it is cheaper is that it drops all the early-year death payments, which for younger lives are heavily weighted and lightly discounted.