Accumulation Function

Study guide: Exam FM-2

The accumulation function $a(t)$ gives the accumulated value at time $t \geq 0$ of 1 unit invested at time 0. It must satisfy $a(0) = 1$ and be non-decreasing.

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

Four standard forms arise from different interest assumptions:

Regime	$a(t)$
Compound interest (rate $i$ )	$(1+i)^t$
Simple interest (rate $i$ )	$1 + it$
Constant Force of Interest $\delta$	$e^{\delta t}$
Time-varying force $\delta(t)$	$\exp\!\left(\displaystyle\int_0^t \delta(s)\,ds\right)$

The Force of Interest at any time is recovered from $a(t)$ by $\delta(t) = a'(t)/a(t) = \frac{d}{dt}\ln a(t)$ .
The amount function $A(t) = k \cdot a(t)$ gives the accumulated value of an initial investment of $k$ .

Accumulated Value Under Simple vs. Compound Interest An investor deposits $ $1{,}000$ at time 0. Compare the accumulated values at $t = 5$ years under (a) simple interest at $8\%$ and (b) compound interest at $8\%$ .

Answer (a) Simple interest: $A(5) = 1{,}000 \times (1 + 0.08 \times 5) = 1{,}000 \times 1.40 = \$1{,}400.00$

(b) Compound interest: $A(5) = 1{,}000 \times (1.08)^5 = 1{,}000 \times 1.46933 \approx \$1{,}469.33$

Compound interest produces a higher accumulated value beyond $t = 1$ because interest itself earns interest.