Current Value

Study guide: Exam FM-2

The current value of a cash flow stream is its value at a specified reference date $t$ — not necessarily today (present) or at the end (future). It is computed by accumulating the cash flows that occur before $t$ and discounting those that occur on or after $t$ to the reference date, using the applicable Interest Rate or Accumulation Function. When every cash flow is discounted it is the Present Value; when every one is accumulated, the Future Value.

$\text{CV}_t = \sum_{k} C_k \cdot \frac{a(t)}{a(t_k)}$

$a(t)$ is the Accumulation Function and $C_k$ is the cash flow at time $t_k$ ; the ratio accumulates a cash flow with $t_k < t$ and discounts one with $t_k > t$ .
This form holds under Compound Interest, a force of interest, or any accumulation function a question gives other than simple interest. Under compound interest at rate $i$ :

$\text{CV}_t = \sum_{k} C_k\,(1+i)^{t - t_k}$

Simple Interest is the exception. When a question specifies simple interest, SOA starts each cash flow's simple interest on its own date, so a cash flow before $t$ grows by $1 + i(t - t_k)$ and one on or after $t$ is divided by $1 + i(t_k - t)$ — it is not moved by $a(t)/a(t_k)$ with $a(t) = 1 + it$ :

$\begin{aligned} \text{CV}_t = {} & \sum_{t_k < t} C_k \bigl[1 + i(t - t_k)\bigr] \\ & + \sum_{t_k \geq t} \frac{C_k}{1 + i(t_k - t)} \end{aligned}$

Value at an Intermediate Date Payments of $1000$ at time 0 and $1000$ at time 4 are made. Find the current value at time 2 using an effective annual rate of 5%.

Answer Accumulate the time-0 payment forward 2 years and discount the time-4 payment back 2 years: $\text{CV}_2 = 1000(1.05)^2 + 1000(1.05)^{-2} = 1102.50 + 907.03 = 2009.53$