recursion relationships

Computing an insurance or annuity value the brute-force way means summing over every future year — a long, tedious calculation. Recursion relationships offer a shortcut by linking a value at one age to the same value at the next age, one step at a time. The idea is intuitive: a whole life insurance on a person aged x is 'what happens in the next year' plus 'whatever the policy is worth a year from now, if they survive'. Split off the first year, and the rest is the same kind of problem on an older life. Climb that ladder rung by rung and you can build a whole table of values cheaply.

The classic recursion for whole life insurance reads A_x = v times q_x + v times p_x times A_{x+1}, in words: discount one year, then either the person dies this year (probability q_x) and you pay the benefit of 1, or they survive (probability p_x) and you are left holding next year's insurance A_{x+1}. The annuity version is ä_x = 1 + v times p_x times ä_{x+1}: you definitely pay 1 now, then if they survive you owe next year's annuity, discounted. Reserves follow the same pattern: this year's reserve, plus this year's premium, accumulated with interest, must cover this year's expected death benefit and the survivors' next reserve. These step-by-step formulas are how spreadsheets and actuarial systems actually compute values.

Recursions are the computational backbone of life contingencies: they turn an infinite sum into a simple loop, they make 'what if mortality changes in one year' easy to test, and they reveal the year-by-year mechanics that a single closed-form symbol hides. The one thing to respect is direction and boundary conditions — most life recursions are solved backwards from a known endpoint (for example, a reserve is 0 at issue and the benefit at the final age), and starting from the wrong boundary or stepping the wrong way gives confident nonsense.