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): 8Reset

NPV = $578, IRR ≈ 21.2%

Cashflows by year

NPV sensitivity to discount rate

Current parameters: CF₀ = $-1000, first annual cash flow = $250, growth = 3.0%, r = 8.0%, and T = 8 years. Cash flows grow from the year-1 base and are discounted at the same rate shown in the sensitivity chart.

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 today, get $2 back in one year. IRR = 100%. Project B: invest $1M today, get $1.2M back in one year. IRR = 20%. NPV at a 10% cost of capital: A = $0.82, B ≈ $90,900. 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$$325,k$in year 1 and 2,$$0$ thereafter (a shorter contract).

NPV(B) = -500 + 325 / 1.10 + 325 / 1.21 = -500 + 295.45 + 268.60 = +$64.05k.

Take A. Both projects create value, but the competing rules pick B: payback favors B (about 1.5 years vs. 3.3), and so does IRR (B’s is about 19.4%, A’s about 15.2%, since B returns cash sooner). NPV alone sees that A’s longer contract creates more total value. With mutually exclusive projects, rank by NPV, not by IRR or payback.

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.