accumulation and discount functions

Picture a single dollar dropped into an account and left alone. As time passes it grows according to some rule. The accumulation function, written a-of-t, is simply the recipe that tells you how much that one dollar has become after t periods. Run the clock backwards instead and ask what a future dollar is worth now, and you get its mirror image, the discount function. Together these two functions are the engine that moves any amount through time.

The accumulation function a-of-t gives the value at time t of one unit invested at time zero, with a-of-0 equal to 1 by definition. Under compound interest a-of-t equals (1 + i) to the power t; under simple interest it equals 1 plus i times t; in general it equals e raised to the integral of the force of interest up to time t. The discount function, written v-of-t, is its reciprocal, 1 divided by a-of-t — the present value today of one unit payable at time t. Under compound interest v-of-t equals v to the power t, where v is the one-period discount factor 1 divided by (1 + i).

These functions are the common abstraction that lets actuaries work without committing to any one interest pattern. By choosing the form of a-of-t you can model constant rates, simple interest, a smoothly varying force, or even a rate schedule that jumps. Every present value, accumulated value, annuity, and reserve in the discipline is ultimately built by multiplying cash flows by the appropriate value of a-of-t or v-of-t. They are the verbs of interest theory.