Exam FMCash flow analysisFree to read

Net present value

Discount every cash flow to today at the required rate and add them up; positive NPV means the project beats the alternative.

The formulas

NPV: $NP V (i) = t \geq 0 \sum C_{t} v^{t}$
Decision rule: $NP V (i) > 0 \Rightarrow accept at required return i$
NPV is decreasing: $\frac{d}{d i} NP V (i) < 0$
for a conventional project

Where it comes from

Each cash flow is worth $C_{t} v^{t}$ today, and value is additive, so the project is worth the sum.
Later inflows are discounted more heavily, so raising $i$ lowers the NPV of any project whose inflows come after its outflows — which is why the NPV curve crosses zero exactly once at the IRR.

Worked example

A project costs 50,000 now and returns 18,000, 22,000 and 25,000 at the ends of years 1 to 3. Find the NPV at a required return of 9%.

$NP V = - 50, 000 + 18, 000 v + 22, 000 v^{2} + 25, 000 v^{3}$ at $i = 0.09$ .
$18, 000 (0.917431) = 16, 513.76$ ; $22, 000 (0.841680) = 18, 516.96$ ; $25, 000 (0.772183) = 19, 304.58$ .
$NP V = - 50, 000 + 54, 335.31 = 4, 335.31$ .
Positive, so the project clears the 9% hurdle.

Answer: 4,335.31

This answer is recomputed from the site’s own interest-theory and probability functions every time the test suite runs, so the page and the mathematics cannot drift apart.

Memory hooks

NPV answers 'how much richer, today'. IRR answers 'at what rate does that become zero'.
Time 0 flows are NOT discounted — v⁰ = 1.

Traps

Discounting the initial outlay by one period.
Comparing projects of different lengths by NPV without a common horizon.

Drill this: the Exam FM question bank has original questions on this topic, and today’s free round is open to everyone.

Net present value

The formulas

Where it comes from

Worked example

Memory hooks

Traps

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