Net Present Value (NPV)

What is NPV?

Net present value measures the value of a project today by discounting all future cash flows and subtracting the initial cost. A positive NPV means the project creates value; negative means it destroys value.

The formula

$NPV = \sum_{t=0}^{T} \dfrac{CF_t}{(1 + r)^t}$

where $CF_t$ is the cash flow at time $t$ (initial outlay $CF_0$ is negative), and $r$ is the discount rate (the investor's required return or cost of capital).

Step-by-step solution

Setup: $CF_0 = -50{,}000$; $CF_1 = CF_2 = \ldots = CF_5 = +15{,}000$; $r = 0.10$.

Step 1 — Discount each cash flow:

• Year 1: $15{,}000 / 1.10 \approx 13{,}636.36$
• Year 2: $15{,}000 / 1.21 \approx 12{,}396.69$
• Year 3: $15{,}000 / 1.331 \approx 11{,}269.72$
• Year 4: $15{,}000 / 1.4641 \approx 10{,}245.20$
• Year 5: $15{,}000 / 1.61051 \approx 9{,}313.82$

Step 2 — Sum the discounted inflows: $\text{PV of inflows} \approx 56{,}861.79$

Step 3 — Subtract initial outlay: $NPV = -50{,}000 + 56{,}861.79 = \boxed{6{,}861.79}$

Decision: NPV > 0 → accept the project. It earns more than the 10% required return.

Shortcut: annuity-style PV

Since years 1–5 are an equal-payment annuity: $PV = PMT \cdot \dfrac{1 - (1+r)^{-n}}{r} = 15{,}000 \cdot \dfrac{1 - 1.10^{-5}}{0.10} \approx 15{,}000 \cdot 3.79079 \approx 56{,}861.79$

Same answer, one multiplication — use the annuity factor when cash flows are uniform.

Interpreting NPV

• NPV > 0: project adds value above the hurdle rate. Accept.
• NPV = 0: project earns exactly the discount rate (break-even in DCF terms).
• NPV < 0: project fails to clear the hurdle. Reject.

The magnitude of NPV is the dollar value added in today's dollars.

Sensitivity to discount rate

At higher $r$, future cash flows shrink faster and NPV drops.

• $r = 5\%$: NPV $\approx +14{,}940$
• $r = 10\%$: NPV $\approx +6{,}862$
• $r = 15\%$: NPV $\approx +275$
• $r = 15.24\%$: NPV $\approx 0$ ← internal rate of return
• $r = 20\%$: NPV $\approx -5{,}141$

This sensitivity is central to valuation — small rate changes can flip a project's verdict.

Common mistakes

• Forgetting to include the initial outlay. $NPV$ must include $CF_0$, which is usually negative.
• Using nominal cash flows with a real discount rate (or vice versa). Keep inflation assumptions consistent on both sides.
• Double-counting costs. Sunk costs (already spent, unrecoverable) should not appear in NPV — only incremental cash flows from the decision.
• Ignoring tax and reinvestment effects. Real NPV usually uses after-tax cash flows.

Try it in the visualization

Bars above the time axis represent inflows; below, outflows. Each bar shrinks as the discount rate increases. Sum of signed heights = NPV, plotted as a function of rate with a zero-crossing marker at the IRR.