Force of Interest: Continuous Compounding for Actuaries

Definition

The force of interest delta(t) = a'(t)/a(t) is the instantaneous rate of growth of the accumulation function at time t. For constant force of interest delta, the accumulation function is a(t) = e^(delta*t). The force of interest is the limit of the nominal interest rate as the compounding frequency approaches infinity: delta = lim_{m->infinity} i^(m) = ln(1+i).

Conversely, i = e^delta - 1. For example, if delta = 0.05, then i = e^0.05 - 1 = 0.05127. Also, d = 1 - e^(-delta) and v = e^(-delta).

Variable Force of Interest

When the force of interest varies with time, the accumulation function from time 0 to time t is a(t) = exp(integral from 0 to t of delta(s) ds). The present value of 1 due at time t is 1/a(t) = exp(-integral from 0 to t of delta(s) ds). The accumulation from time t1 to time t2 is exp(integral from t1 to t2 of delta(s) ds).

Example: if delta(t) = 0.01*t, the accumulation from 0 to 5 is exp(integral from 0 to 5 of 0.01*t dt) = exp(0.01 * 25/2) = exp(0.125) = 1.1331.

Exam FM Applications

Exam FM tests the force of interest in two main ways: converting between delta and other rate measures (i, d, v, i^(m), d^(m)), and computing present or accumulated values when delta(t) is a given function of t. For the first type, memorize the chain of relationships: delta = ln(1+i) = -ln(v) = -ln(1-d). For the second type, set up and evaluate the integral. Common forms include delta(t) = a + bt (linear) and delta(t) = a/(b+t) (reciprocal).