Net Present Value (NPV)

The formula

NPV = −C₀ + Σₜ₌₁ⁿ CFₜ / (1 + r)ᵗ

where C₀ is the initial investment (a positive number; the minus sign treats it as a cash outflow), CFₜ is the expected cash flow in period t, n is the project life, and r is the discount rate — the project's risk-adjusted opportunity cost of capital.

Each term CFₜ / (1 + r)ᵗ is the present value of a future dollar: a payment of $100 in 5 years at a 10% discount rate is worth $100 / 1.10⁵ = $62.09 today. Summing across all years and netting against the initial cost gives the dollar value created by undertaking the project.

Why discount future cash flows?

Three reasons together justify the discount factor:

1. Time preference. A dollar today buys things today. A dollar in five years buys things in five years. People prefer earlier consumption — they require compensation for waiting.
2. Inflation. $100 in 2031 buys less than $100 in 2026 if prices rise. Discount rates in nominal terms include expected inflation; real discount rates strip it out.
3. Risk. Future cash flows are uncertain. A risk-free Treasury yields, say, 4%. A risky project must offer more — typically the risk-free rate plus a CAPM-style risk premium.

Putting these together, the discount rate is the risk-adjusted opportunity cost — the return investors could earn on a different investment of equivalent risk. Discounting at that rate normalizes future cash flows to today's purchasing-power-and-risk-equivalent dollars.

Worked example: a 5-year manufacturing project

A company is considering investing $10M today to build a production line that will generate cash flows over five years. The project's risk profile matches the firm's overall business, so the firm's WACC of 10% is the right discount rate.

Year	Cash flow	Discount factor (1.10⁻ᵗ)	PV
0	−$10,000,000	1.0000	−$10,000,000
1	$3,000,000	0.9091	$2,727,273
2	$3,500,000	0.8264	$2,892,562
3	$3,500,000	0.7513	$2,629,602
4	$2,500,000	0.6830	$1,707,532
5	$1,500,000	0.6209	$931,382
Total			NPV = $888,351

NPV = +$888,351 → accept the project. Building the line is expected to create roughly $888K of shareholder value, expressed in today's dollars.

Notice the cash flow in year 1 is undiscounted only by 9.1%, while year 5 is discounted by 38%. Time value compounds: the further out a cash flow lies, the more aggressively it gets shrunk. This is why early-cash projects beat back-loaded ones at the same total dollar payoff.

NPV vs IRR vs other criteria

Criterion	What it measures	Decision rule	Strength	Weakness
NPV	Dollars of value created	Accept if > 0	Theoretically pure; additive across projects	Doesn't show return per dollar invested
IRR	Discount rate that sets NPV to zero	Accept if > cost of capital	Intuitive (a percentage); works without specifying r	Multiple solutions for non-conventional cash flows; misranks scale-different projects; assumes reinvestment at IRR
Payback Period	Years to recover initial outlay	Accept if < threshold	Simple; useful liquidity proxy	Ignores time value; ignores cash flows after payback
Discounted Payback	Years until cumulative discounted CF = 0	Accept if < threshold	Fixes time value	Still ignores post-payback flows
Profitability Index (PI)	PV of inflows ÷ PV of outflows	Accept if > 1.0	Useful for capital rationing	Same NPV-rank-misalignment risks as IRR
Modified IRR (MIRR)	IRR with explicit reinvestment-rate assumption	Accept if > cost of capital	Fixes IRR's reinvestment problem	Two extra inputs (financing rate, reinvestment rate)
EVA / Residual Income	Period profit minus capital charge	Accept if positive	Aligns with NPV under right assumptions	Period-by-period, not lifecycle

Brealey, Myers and Allen — the standard corporate finance textbook — devote a chapter to demonstrating why NPV is the only criterion that consistently maximizes shareholder wealth. IRR is the most popular alternative because finance professionals find percentages more intuitive than dollar amounts, but it produces wrong answers in three documented cases:

• Multiple sign changes. A project with cash flows −100, +250, −150 has two IRRs (around 0% and 50%) — both mathematically valid roots of the polynomial.
• Mutually exclusive projects of different scales. Project A: $1K invested, IRR 100%, NPV $500. Project B: $100K invested, IRR 20%, NPV $15K. IRR ranks A first; NPV ranks B first; B is the wealth-maximizing choice.
• Reinvestment assumption. IRR implicitly assumes interim cash flows are reinvested at the IRR itself — typically unrealistic when IRR is far from market rates.

Computing NPV in Excel

=NPV(rate, value1, value2, ...)        // PV of period 1+ cash flows only
=NPV(rate, B2:B6) + B1                  // add initial outflow at period 0 separately

Excel's NPV function is famously misnamed: it computes the present value of cash flows starting in period 1, not period 0. Forgetting this is one of the most common spreadsheet errors in finance — an outflow entered in column B1 alongside other periods will be discounted by one period instead of treated as already in present-value terms.

The cleanest pattern: put the initial investment in cell B1 (period 0), period-1-onward flows in B2:B6, and write =NPV(0.10, B2:B6) + B1. The trailing + B1 adds the already-present-value initial outflow.

Sensitivity analysis

NPV depends on three inputs that are all uncertain: cash flow amounts, discount rate, and project life. Standard practice is a tornado chart varying each input ±10-30% around the base case:

Parameter	Low (−10%)	Base	High (+10%)	NPV swing
Year-1 revenue	$2.7M	$3.0M	$3.3M	±$273K
Margin	27%	30%	33%	±$890K
Discount rate	9%	10%	11%	±$420K
Salvage value	$0.9M	$1.0M	$1.1M	±$62K

The widest bars (margin, discount rate) are the parameters whose uncertainty matters most. Tighten estimates of those before refining the others.

When NPV undershoots: real options

Standard NPV assumes management commits to a single forecast at t = 0 and rides out whatever happens. Real projects offer flexibility: expand if demand exceeds expectations, abandon if it doesn't, defer if uncertainty resolves over time. These options have value that NPV understates by treating cash flows as deterministic.

Three classic real options that NPV misses:

• Option to expand. A first plant whose NPV is slightly negative may justify acceptance because it grants the option to build a second plant if market conditions improve.
• Option to abandon. The salvage value of equipment if the project fails caps downside and raises true expected value.
• Option to defer. Waiting until uncertainty resolves can be more valuable than committing now, even when "now" has positive NPV.

Real options valuation uses Black-Scholes-style techniques to quantify these. For commodity, R&D, and infrastructure projects with long lead times and high uncertainty, real options can swing the decision from "reject" to "accept."

Capital rationing and the profitability index

NPV ranks projects when capital is unlimited. When budget is constrained — say, $50M to spend on a slate of $20M projects — the right metric is NPV per dollar of investment, the profitability index:

PI = PV of future cash flows / Initial investment
   = (NPV + C₀) / C₀
   = 1 + NPV/C₀

Rank projects by PI and accept top-down until the budget is exhausted. The full integer-programming solution (with project indivisibility) can produce different rankings, but PI is a strong heuristic.

Common pitfalls

• Wrong discount rate. Using firm WACC for a project with different risk understates risk for risky projects and overstates it for safe ones. Use a project-specific cost of capital based on comparable assets.
• Ignoring inflation consistency. Nominal cash flows must be discounted at nominal rates; real cash flows at real rates. Mixing produces a 200-300 bp error.
• Including sunk costs. Past spending is irrelevant. Only incremental cash flows from this point forward matter.
• Excluding opportunity costs. Using existing factory floor space "for free" understates project cost — that space could be rented or used for something else.
• Forgetting working capital. Growth in receivables and inventory ties up cash. Subtract ΔWC from cash flows; recover it at project end.
• Excluding salvage value. Equipment doesn't drop to zero at project end. Include after-tax salvage value as a year-N inflow.
• Excel NPV function pitfall. The function discounts the first cell as period 1, not period 0 — always add the initial outflow separately.
• Optimistic forecasts. Bias toward managers who pitched the project. Stress-test by asking: "What's the NPV if revenue is 70% of base case?"