NPV and IRR

NPV and Decision Rules

Discounting converts future cash into today's units: $PV = \frac{CF_t}{(1+r)^t}$, where $r$ is the discount rate (its opportunity cost).

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

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

Accept if NPV > 0: the project creates value beyond alternatives.

Worked example. Invest 100k€ now, receive 60k€ in year 1 and 60k€ in year 2, with $r = 10\%$:

$$NPV = -100 + \frac{60}{1.1} + \frac{60}{1.21} = 4.1\text{k EUR} > 0$$

Internal Rate of Return asks: what discount rate makes $NPV = 0$? Accept when $IRR > r$.

Pitfall: A positive NPV means the project beats the alternative investment, not just "makes money." A project earning 3% when $r = 10\%$ makes money but still destroys value.

The NPV Profile

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

Why it slopes down: a higher $r$ shrinks every future cash flow, hitting distant cash hardest via $(1+r)^t$.

Reading the curve

• The height at your discount rate gives the NPV value verdict.
• The horizontal crossing point is the IRR, where NPV hits zero.

Where rankings flip. Compare two mutually exclusive projects: one pays early, the other pays late and big. The late-cash project has a steeper profile, so the two profiles cross. At low discount rates, the big-late project wins on NPV; at high rates, the early-cash project wins.

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. Always let NPV measure value created.