Fund Accumulation

Study guide: Exam FM-2

Fund accumulation is the growth of a fund's balance over time from an initial deposit and/or a stream of periodic deposits or withdrawals, each earning interest at the fund's rate. It generalizes the single-sum Accumulation Function to a fund that also receives ongoing cash flows.

$AV_n = F_0(1+i)^n + \sum_t C_t(1+i)^{n-t}$

$F_0$ is the initial balance, $C_t$ is the net deposit (or withdrawal, if negative) at time $t$ , and $i$ is the fund's effective interest rate.
When the deposits are a level $C$ at the end of each period — times $t = 1, \ldots, n$ — $\sum_t C_t(1+i)^{n-t}$ reduces to $C\cdot s_{\overline{n}|i}$ , the annuity-immediate accumulated value. Level deposits at the start of each period ( $t = 0, \ldots, n-1$ ) give $C\cdot \ddot{s}_{\overline{n}|i}$ instead (an Annuity Due).
A common Exam FM scenario: amounts withdrawn from one fund are immediately deposited into and accumulate within a second fund earning a different rate — the accumulated value in the second fund uses its own rate, even though the deposits originated from the first fund's activity.
Directly related to Accumulated Value and Future Value.

Fund with an Initial Deposit and Level Additions $ $2{,}000$ is deposited into a fund today, and an additional $ $300$ is deposited at the end of each year for 6 years. The fund earns an annual effective interest rate of $5\%$ . Find the accumulated value of the fund at the end of year 6.

Answer $F_0(1.05)^6 = 2{,}000(1.34010) \approx \$2{,}680.19$ $300\cdot s_{\overline{6}|0.05} = 300\left(\frac{1.34010-1}{0.05}\right) = 300(6.80191) \approx \$2{,}040.57$ $AV_6 \approx 2{,}680.19 + 2{,}040.57 = \$4{,}720.76$

Reinvesting Interest From One Fund Into Another $ $500$ is deposited into Fund A, which earns an annual effective rate of $5\%$ . At the end of each year, the interest earned by Fund A is withdrawn and deposited into Fund B, which earns $8\%$ . Find the accumulated value of Fund B at the end of year 4.

Answer Since the $ $500$ principal stays in Fund A, it earns level interest of $500(0.05) = \$25$ per year, deposited into Fund B. $s_{\overline{4}|0.08} = \frac{(1.08)^4 - 1}{0.08} = \frac{1.36049-1}{0.08} \approx 4.50611$ $AV_B = 25 \times 4.50611 \approx \$112.65$