NPV and IRR

Money Has a Time Price

A euro today is worth more than a euro next year — today's euro can be invested and grow. Discounting converts future cash into today's units:

$$PV = \frac{CF_t}{(1+r)^t}$$

where $r$ is the discount rate — the return the money could earn elsewhere (its opportunity cost).

Net Present Value sums a project's entire cash-flow life in today's euros:

$$NPV = -C_0 + \sum_{t=1}^{T} \frac{CF_t}{(1+r)^t}$$

The decision rule is beautifully blunt: accept if $NPV > 0$ — the project creates value beyond what the money would earn elsewhere; reject if negative.

Worked example. Invest 100k€ now, receive 60k€ after one year and 60k€ after two, with $r = 10\%$:

$$NPV = -100 + \frac{60}{1.1} + \frac{60}{1.21} = -100 + 54.5 + 49.6 = 4.1\ \text{k EUR} > 0 \; \Rightarrow \; \text{accept}$$

Internal Rate of Return asks the reverse question: what discount rate would make this project's NPV exactly zero?

$$NPV(IRR) = 0$$

Accept when $IRR > r$: the project out-earns the alternative. NPV and IRR usually agree — but when they conflict (unconventional cash flows, mutually exclusive projects of different scale), NPV is the safer master, because it measures value created rather than a percentage.

Two rules, one verdict (usually)

Tip: The discount rate is an opportunity cost — the return the money would earn in its best alternative use. Pick it before looking at the project's numbers, not after.

Common pitfall: Reading a positive NPV as "the project makes money." It means more than that: the project beats the alternative investment. A project earning 3% when $r = 10\%$ makes money and still destroys value.

The NPV Profile: One Curve, Both Rules

Plot a project's NPV against the discount rate and you get its NPV profile — a single downward-sloping curve that contains both decision rules at once.

$$NPV(r) = -C_0 + \sum_{t=1}^{T} \frac{CF_t}{(1+r)^t}$$

Why it slopes down: a higher $r$ shrinks every future cash flow — and hits distant cash hardest, since $(1+r)^t$ compounds the punishment with $t$.

Reading the curve

• The height at your discount rate is the project's NPV — the value verdict.
• The horizontal crossing point is the IRR — where the curve hits zero.
• If your rate sits left of the crossing, NPV is positive and both rules say accept.

Where the ranking flips. Compare two projects: one pays early cash flows, the other pays late but bigger. The late-cash project's profile is steeper (long-dated cash suffers more from discounting), so the two profiles cross. At low discount rates the big-late project wins on NPV; at high rates the early-cash project wins — even though each project's IRR never moved. This is why ranking mutually exclusive projects by IRR can mislead: IRR ignores how much value is created at your actual cost of capital. The profile makes the flip visible — and NPV at your true rate settles the argument.

Tip: One curve, two readings: the height at your discount rate is the NPV verdict; the horizontal crossing is the IRR. If your rate is left of the crossing, both rules agree — accept.

Common pitfall: Ranking mutually exclusive projects by IRR. When profiles cross, the project with the higher IRR can have the lower NPV at your actual discount rate — and NPV, not IRR, measures value created.