Variable Force of Interest

Study guide: Exam FM-2

A variable (time-varying) force of interest $\delta(t)$ allows the instantaneous rate of interest to change over time. The Accumulation Function from time $0$ to time $t$ is:

$a(t) = \exp\!\left(\int_0^t \delta(s)\,ds\right)$

The accumulation from time $s$ to time $t$ ( $s < t$ ) is:

$\frac{a(t)}{a(s)} = \exp\!\left(\int_s^t \delta(u)\,du\right)$

Solving Time Value of Money Equations under variable force of interest requires evaluating these integrals.

Accumulation Under Variable Force The force of interest at time $t$ is $\delta(t) = 0.04 + 0.002t$ . Find the accumulation of $1$ from $t=0$ to $t=3$ .

Answer $\int_0^3 (0.04 + 0.002t)\,dt = [0.04t + 0.001t^2]_0^3 = 0.12 + 0.009 = 0.129$ $a(3) = e^{0.129} \approx 1.1377$