the insurance-annuity relationship

Insurance pays a lump sum at death; an annuity pays a stream while alive. They look like opposites, but they are two sides of the same coin, connected by one of the most elegant identities in the whole subject: A_x = 1 - d times ä_x, where d is the annual rate of discount. Rearranged, ä_x = (1 - A_x) / d. Know one and you instantly know the other — no separate calculation needed.

Where does this come from? Imagine investing 1 today and promising to keep the interest it earns flowing out each year while the person lives, then handing over the leftover principal at their death. The yearly interest payments form an annuity, and the principal returned at death forms an insurance. Together they must reconstitute the original 1 you invested. Working through the discounting turns that conservation-of-value statement into exactly A_x = 1 - d times ä_x. The same identity holds in temporary form for endowment insurance, A_{x:n} = 1 - d times ä_{x:n}, and in continuous form with the force of interest replacing d.

This relationship is enormously useful in practice. Life tables and software often tabulate annuity values, and the identity lets you read off the matching insurance value (and vice versa) for free; it also lets you rewrite premium and reserve formulas in whichever form is cleaner. The one thing to get right is which interest quantity to use: the discrete identity uses d, the rate of discount (d = i / (1 + i)), not the interest rate i and not the force of interest. Plug in the wrong rate — a beginner's classic error — and the elegant identity quietly gives a wrong answer.