Commutation Function Calculator

About this calculator

Commutation functions compress the two moving parts of a life-insurance calculation — survivorship and interest discounting — into single tabulated values so premiums and reserves can be found with simple ratios instead of re-summing a whole mortality table each time. The building blocks are D_x = v^x · l_x, the discounted number of survivors to age x, and N_x, the sum of D_x over every age from x to the end of the table, representing the discounted value of a lifetime annuity of 1 paid each year survival occurs.

Actuaries combine these with the related C_x and M_x functions (built the same way from deaths instead of survivors) to price whole life insurance and annuities: a net single premium for a whole life policy is simply M_x / D_x, and a life annuity-due is N_x / D_x. Commutation tables were standard practice long before spreadsheets and are still taught because they make the mechanics of actuarial present value transparent — this calculator reproduces D_x and N_x for a chosen age under constant mortality and interest assumptions.