NPV as the Decision Rule

Computing NPV

For a project with an initial outlay $C_0$ (negative cash flow) and a stream of future cash flows $C_1, C_2, \ldots, C_T$ discounted at rate $r$:

$$\text{NPV} \;=\; C_0 \;+\; \sum_{t=1}^{T} \frac{C_t}{(1 + r)^t}.$$

The decision rule:

• If NPV > 0: take the project. You are creating value.
• If NPV < 0: don’t take it. You are destroying value.
• If NPV = 0: indifferent. The project earns exactly its cost of capital.

Equivalently and importantly: among mutually exclusive projects (you can only pick one), choose the one with the highest NPV.

Try a few

Initial outflow CF₀: $-1000Base annual CF: $250Growth g: 3.0%Discount rate r: 8.0%Horizon T (years): 8

NPV = $578, IRR ≈ 21.2%

Cashflows by year

NPV sensitivity to discount rate

The interactive lets you set initial outflow, base annual cash flow, growth, discount rate, and horizon. Three things to confirm:

1. NPV falls monotonically as the discount rate rises. (More impatience → future cash flows worth less today.)
2. The IRR is whatever discount rate makes NPV equal to zero. Read it off the right-panel chart.
3. A small change in growth has a larger effect at long horizons. (Compounding asymmetries.)

Why NPV beats the competition

Three rival rules are still taught, and still misused. Each has a specific failure mode:

IRR (Internal Rate of Return)

IRR is the rate at which NPV = 0. Rule: take the project if IRR exceeds the cost of capital.

When it works: a conventional project (one negative cash flow up front, all-positive cash flows afterward).

When it breaks:

• Non-conventional cash flows (cash flow changes sign more than once, e.g. mining projects with closure costs at the end). Multiple IRRs exist; the rule is ambiguous.
• Mutually exclusive projects of different scale. Project A: invest $1, get $2 back. IRR = 100%. Project B: invest $1M, get $1.2M back. IRR = 20%. NPV at 10% cost of capital: A = $0.82, B = $90,000. IRR ranks A higher; NPV (correctly) ranks B higher.
• Mutually exclusive projects of different timing. IRR favors short, early-payoff projects; NPV correctly accounts for the size of long-term value creation.

Payback period

Rule: take the project if you recoup the initial investment within some cutoff (often 3 years). Easy to compute, easy to communicate.

Problems:

• Ignores time value of money. A dollar in year 3 and a dollar in year 1 count equally.
• Ignores cash flows after the cutoff. A 30-year solar farm with a 4-year payback fails a 3-year payback test, even though its NPV is enormous.

A discounted-payback variant fixes problem 1 but not problem 2. Use payback only as a rough liquidity screen, never as a decision rule.

Accounting Rate of Return (ARR)

Rule: take the project if the average accounting net income divided by average book value of assets exceeds a hurdle.

Problems:

• Uses accounting income (which includes depreciation and accruals), not cash flows.
• Ignores the timing of returns.
• The denominator is sensitive to depreciation policy, which has nothing to do with the project’s economic value.

Useful for performance evaluation of existing units; useless for project selection.

The one place NPV needs help

NPV requires a single discount rate. For projects financed with a mix of debt and equity, choosing the right rate is the topic of Unit 4 (cost of capital) and Unit 5 (WACC, APV, FTE). Until then, treat $r$ as given and focus on the mechanics.

A worked decision

Your firm is considering two mutually exclusive logistics-warehouse investments. Cost of capital = 10%.

Project A: C_0 = -\500,k$, then$$150,k$ per year for 5 years.

NPV(A) = -500 + 150 × PV-annuity(10%, 5) = -500 + 150 × 3.791 ≈ -500 + 568.65 = +$68.65k.

Project B: C_0 = -\500,k$, then$$250,k$in year 1 and 2,$$0$ thereafter (a shorter contract).

NPV(B) = -500 + 250 / 1.10 + 250 / 1.21 = -500 + 227.27 + 206.61 = -$66.12k.

Take A. B has a faster payback (under 2 years) and a higher IRR (~12%) but destroys value because the contract ends just as the investment has been recouped. NPV picks the value-creating project; the competing rules don’t.

Calibrating your intuition

Once you’ve internalized NPV, every decision you’ll encounter in this course becomes a special case: which discount rate is right (cost of capital), which cash flows to count (capital-budgeting unit), how to handle risk and leverage (units 4 and 5). The framework is settled. The interesting work is in the specifics.