Accumulated Value

Study guide: Exam FM-2

The accumulated value (AV) is the value at a future time $t$ of a cash flow (or series of cash flows), grown forward using the Accumulation Function $a(t)$ . It is the time- $t$ analogue of Present Value: where PV discounts back to time 0, AV accumulates forward to time $t$ :

$AV = PV \cdot a(t)$

Under Compound Interest at rate $i$ , $a(t) = (1+i)^t$ , so $AV = PV \cdot (1+i)^t$ .
The ratio $a(t)/a(s)$ for $t > s$ is the accumulation factor from $s$ to $t$ : the value at time $t$ of 1 deposited at time $s$ . Under compound interest it equals $(1+i)^{t-s}$ ; under a force of interest $\delta_r$ it equals $\exp\left(\int_s^t \delta_r\,dr\right)$ .
For a series of cash flows $C_{t_k}$ occurring at times $t_k \leq t$ — under compound interest, a force of interest, or any accumulation function a question gives other than simple interest — each cash flow is carried forward by its own accumulation factor:

$AV = \sum_k C_{t_k} \cdot \frac{a(t)}{a(t_k)}$

Under compound interest at rate $i$ this is:

$AV = \sum_k C_{t_k} \cdot (1+i)^{t - t_k}$

Simple Interest is the exception. When a question specifies simple interest, SOA measures $t$ in $a(t) = 1 + it$ from the moment each cash flow occurs, so each cash flow grows by $1 + i(t - t_k)$ — not by $a(t)/a(t_k) = (1+it)/(1+it_k)$ :

$AV = \sum_k C_{t_k} \bigl[1 + i\,(t - t_k)\bigr]$

Accumulated Value of a Savings Plan An investor deposits $ $500$ today and $ $800$ two years from now into an account earning $i = 5\%$ per year effective. What is the total accumulated value at the end of 4 years?

Answer Accumulate each deposit to time 4: $AV = 500 \times (1.05)^4 + 800 \times (1.05)^2$ $= 500 \times 1.21551 + 800 \times 1.10250$ $= 607.75 + 882.00 = \$1{,}489.75$

Deposits Earning Simple Interest An account credits simple interest at $10\%$ per year. A policyholder deposits $ $100$ at time 0 and another $ $100$ at time 1. What is the accumulated value at time 3?
Answer Each deposit earns simple interest from its own deposit date:
 $\begin{align*} AV &= 100\,(1 + 0.10 \times 3) + 100\,(1 + 0.10 \times 2) \\ &= 130 + 120 \\ &= \$250 \end{align*}$ 
Carrying the second deposit by the ratio $a(3)/a(1) = 1.3/1.1$ instead would give $100(1.3) + 100(1.3/1.1) = 248.18$ — not SOA's convention for simple interest.