recursive reserve formulas

Instead of recomputing a reserve from scratch each year by summing up all future cash flows, you can grow it one step at a time, like updating a bank balance: take last year's reserve, add this year's premium, let it earn interest, then pay out for those who died and split what is left among the survivors. Recursive reserve formulas are these step-by-step rules that turn the reserve at the start of a year into the reserve at the end.

Precisely, the fundamental recursion for a fully discrete policy says: (reserve at start of year t + net premium for year t) accumulated one year at interest (1+i) must cover the death benefit paid to those who die during the year plus the reserve needed for the survivors. In symbols, (t-V + P)(1+i) = q_(x+t) * b + p_(x+t) * (t+1)-V, where q is the death probability, p = 1 - q the survival probability, and b the benefit. This says the money carried forward, grown with interest, is exactly enough to pay claims and re-establish next year's reserve. Rearranged, it cleanly separates the 'cost of insurance' for the year from the 'savings' that build the reserve.

These recursions are how reserves are actually computed in production software — they are fast, numerically stable, and reveal the inner mechanics of a policy year (interest credited, mortality charge, reserve build-up). They also generalize to multiple-state and multiple-decrement models. The continuous-time limit of this same balance is Thiele's differential equation. The one thing to keep straight is the timing convention (premium at start of year, benefit at end), since getting the order of interest, premium, and death wrong is the classic source of off-by-one reserve errors.