2.2 NPV, IRR, and Decision Rules

NPV as the Primary Decision Rule

Once cash flows are expressed in today's dollars, net present value (NPV) asks a single question: after paying for the investment, does the project increase the value of the firm? CFI's Corporate Finance Fundamentals and the later DCF courses all use the same definition:

NPV = Σ CF_t / (1 + r)^t − C0

CF_t is the incremental cash flow at date t, C0 is the time-0 outlay (written as a positive setup cost in this form, or as CF_0 inside the sum if you keep the sign negative), and r is the opportunity cost of capital — on the FMVA core, usually WACC for unlevered project cash flows. The decision rule for an independent project (you may accept it without rejecting another) is:

• Accept if NPV > 0
• Reject if NPV < 0
• Indifferent if NPV = 0 (the project earns exactly the required return)

Positive NPV means the present value of inflows exceeds the present value of outflows. The firm is better off in market-value terms by that dollar amount. That is why NPV, not accounting profit and not IRR, is the ranking rule when two projects compete for the same slot.

Worked NPV Example

Project Atlas requires $100,000 at t = 0 and is expected to return $40,000, $45,000, and $50,000 at the ends of years 1, 2, and 3. WACC is 10%.

NPV = −100,000 + 40,000/1.10 + 45,000/1.21 + 50,000/1.331 = $11,119.46

Atlas creates about $11,119 of value at a 10% cost of capital, so an independent Atlas is accepted. If WACC rose to 16%, the same cash flows have NPV ≈ −$42 — essentially zero — and the project is a wash. Above 16% it destroys value.

IRR: The Rate That Sets NPV to Zero

The internal rate of return (IRR) is the discount rate r* that solves NPV = 0:

0 = Σ CF_t / (1 + r)^t − C0*

For a conventional cash-flow pattern (one initial outflow, then inflows), there is one positive IRR. The paired decision rule is:

• Accept if IRR > hurdle rate / WACC
• Reject if IRR < hurdle rate / WACC

For Atlas, NPV is +$11,119 at 10% and about zero at 16%, so IRR ≈ 16.0%. Because 16% > 10% WACC, IRR agrees with NPV: accept. On conventional independent projects the two rules cannot conflict. IRR is still a diagnostic, not a substitute for NPV. A 16% IRR on a $100,000 project is not "better" than an 11% IRR on a $5 million project until you look at dollar NPV.

Excel: =IRR({-100000, 40000, 45000, 50000}) returns approximately 16.0%. IRR iterates from a guess (default 10%). If you supply a wild guess on a messy cash-flow sheet, IRR can converge to a different root or return #NUM!.

Mutually Exclusive Projects: NPV Wins

Mutually exclusive means accepting one project requires rejecting the other (one plant site, one software vendor, one capital slot). IRR can rank the pair backwards for two well-known reasons: scale and cash-flow timing.

Scale conflict. Project Small costs $10,000 and returns $15,000 in one year. Project Large costs $100,000 and returns $125,000 in one year. WACC is 10%.

IRR prefers Small (50% > 25%). NPV prefers Large ($13,636 > $3,636). You cannot do both. Ranking by IRR would leave $10,000 of value on the table. The incremental IRR of Large minus Small is ($110,000 / $90,000) − 1 = 22.2%, which still exceeds 10%, so the extra capital in Large is worth taking. That incremental test agrees with NPV. NPV is the ranking rule.

Timing conflict. Two $5,000 outlays, WACC 10%:

Near's cash arrives early, so its IRR is higher. Far's large year-3 inflow is still worth more in present-value dollars at a 10% WACC ($826 > $665). Choose Far. IRR overweighting early cash is the same phenomenon as a high coupon bond looking "higher yield" than a long zero when you forget that reinvestment and scale live in the NPV number, not in the percentage.

Multiple IRRs, Excel Traps, and Secondary Screens

The Atlas NPV profile in the chart crosses zero once, at about 16%. That single crossing is why a conventional project has one IRR and why IRR and NPV agree on accept/reject. Change the sign pattern and the profile can cross zero more than once.

Nonconventional Cash Flows and Multiple IRRs

A nonconventional stream has more than one sign change — for example, a mine with an environmental cleanup outflow at the end, or a project that requires a mid-life overhaul. Descartes' rule of signs says the number of positive IRRs can be as large as the number of sign changes.

Worked example. Cash flows: −$1,000, +$3,000, −$2,200 (outflow, inflow, outflow — two sign changes).

NPV(r) = −1,000 + 3,000/(1+r) − 2,200/(1+r)^2

Set NPV = 0, let x = 1 + r, and divide by −1,000:

x^2 − 3x + 2.2 = 0

Discriminant = 9 − 8.8 = 0.2, so x = [3 ± √0.2] / 2

The two roots are x = 1.2764 and x = 1.7236, which means r ≈ 27.6% and r ≈ 72.4%**. Excel's IRR will return whichever root sits near your guess. Neither number is a unique hurdle test. Plot NPV at the WACC and accept only if that NPV is positive. This is also why CFI-style models prefer a full NPV profile (or a data table on WACC) over a single IRR cell when cash-flow signs are messy.

Excel NPV Is Not "Native" NPV

This is one of the most expensive Excel traps on the FMVA final, because case studies are taken in Microsoft Excel.

Excel's NPV(rate, value1, value2, ...) assumes the first argument is a cash flow at the end of period 1, the second at the end of period 2, and so on. It does not treat the first number as t = 0.

Correct Atlas formula, with CF0 in C0 and CF1:CF3 in C1:C3:

=NPV(0.10, C1:C3) + C0 → $11,119.46

Incorrect:

=NPV(0.10, C0:C3) → $10,108.60

The wrong version discounts the $100,000 outlay by one extra year (−100,000/1.10 = −90,909) and understates NPV by about $1,011. You will sometimes see a "fix" of =NPV(WACC, C0:C3)*(1+WACC), which mathematically undoes the extra period on every cash flow, including CF0. That happens to restore Atlas, but it is fragile: if C0 is blank, or if a mid-year convention is later layered on, the multiplier fights you. CFI-style practice is add CF0 outside Excel NPV.

IRR does include the first cell as t = 0, so =IRR(C0:C3) is aligned with the economics even while =NPV(rate, C0:C3) is not. That inconsistency is the trap.

XNPV and XIRR for Irregular Dates

Annual year-end models are a teaching convenience. Live deals close on 15 March, pay capex on 1 June, and harvest cash daily. XNPV and XIRR take a parallel date range:

=XNPV(rate, values, dates)

=XIRR(values, dates)

XNPV discounts each cash flow by (1 + rate)^(days/365) from the first date in the date range (Excel's actual day-count is 365-day). If the first date is the valuation date and CF0 sits on that date, XNPV does not extra-discount CF0 — unlike NPV. Use XNPV/XIRR when the case gives actual dates; use NPV + CF0 when the case is a clean annual model. Do not mix them on the same timeline.

Payback and Discounted Payback Are Secondary Screens

Payback is the years until undiscounted inflows recover C0. Discounted payback uses present values of those inflows. For Atlas at 10%:

A treasurer who must not tie capital up beyond three years might use these as a liquidity screen. They are not FMVA primary decision rules. Payback ignores cash after the cutoff (Atlas's year-3 $50,000 is only partly counted) and undiscounted payback ignores TVM entirely. A project can pay back in 2.3 years and still have negative NPV if late cash is weak and WACC is high. A project can pay back slowly and still have a large NPV if later cash is huge. When the exam asks which project to choose, answer with NPV, use IRR as a rate diagnostic on conventional independent projects, and mention payback only if the question is explicitly about liquidity or recovery speed.

Putting the Rules Together

1. Build incremental cash flows (Section 2.3).
2. Discount at the correct r (WACC for unlevered project cash flows).
3. NPV > 0 → accept an independent project.
4. Mutually exclusive → highest NPV (not highest IRR).
5. Check the sign pattern; if it is nonconventional, do not trust a single IRR.
6. In Excel, NPV(rate, CF1:CFn) + CF0, or XNPV with real dates.
7. Payback is a screen, not a substitute for steps 3–4.