continuous vs discrete vs mthly functions

The same benefit can be timed in different ways, and the timing changes its value. A death benefit might be paid at the end of the policy year in which death occurs (the discrete case), or at the exact instant of death (the continuous case). A pension might be paid once a year, or in twelve monthly instalments (the mthly case). Actuaries distinguish all three because real products pay monthly or on death, not in tidy once-a-year lumps, and the differences, while small, add up over a large book.

Notation marks the timing. The plain discrete annual function is A_x (insurance at year-end of death) or ä_x (annuity once a year, in advance). A bar on top means continuous: Ā_x is the insurance paid at the moment of death, and ā_x is the annuity paid continuously, as a smooth stream. A small superscript (m) means paid m times a year: ä_x^{(m)} is an annuity of 1 per year split into m payments. The relationships are intuitive: paying a death benefit at the moment of death (rather than waiting until year-end) is worth slightly more because you get the money a little sooner, so Ā_x is a bit larger than A_x — roughly by a factor of i / delta (the interest rate over the force of interest). A monthly annuity is worth slightly less than an annual annuity-in-advance, because you receive each year's money a little later on average. Standard approximations (such as the so-called 'alpha-beta' formulas) convert between them without re-summing from scratch.

Getting the frequency right matters because it is real money. Annuities are usually paid monthly, so pricing them as annual would overstate their value; death benefits are effectively paid soon after death, closer to continuous than year-end. The honest point is that the continuous functions are a mathematical idealisation — nobody is literally paid in a continuous stream — but they are clean to work with and a good approximation to frequent payments. The common error is to mix frequencies inside one calculation, for example valuing premiums annually but benefits continuously without adjusting; consistency, or a deliberate documented adjustment, is essential.