Capital budgeting decision rules

Capital budgeting decisions shape a firm for years: which plants to build, which products to launch, which machines to replace, which acquisitions to pursue. Lesson 1 argued that the goal is to maximize the value of the firm, and lessons 3 to 6 built the tools for valuing cash flows. This lesson asks which decision rule turns those tools into the right choices. BUS-110 lesson 10 introduced net present value, the internal rate of return, payback and the profitability index for a single delivery van. Here the focus is on why NPV is the benchmark, the situations in which the other rules give wrong answers, and how to handle common complications: mutually exclusive projects, projects with different lives and limited capital. Lessons 8 and 9 deal with estimating the cash flows and their risk.

Net present value

A project's net present value is the present value of its expected cash flows, discounted at the opportunity cost of capital for projects of that risk, minus its initial cost. If NPV is positive, the project is expected to return more than investors could earn elsewhere at the same risk, so accepting it adds value to the firm equal to the NPV. If NPV is negative, the firm's owners would be better off if the money were returned to them. The rule is simple: accept projects with positive NPV, and among mutually exclusive alternatives choose the one with the highest NPV.

NPV has properties no other rule shares. It uses all the cash flows, discounts them at the market's opportunity cost of capital, and measures value in dollars. NPVs add: the value of two projects together equals the sum of their separate NPVs, so a bad project cannot be disguised by bundling it with a good one, and the value of the firm is the sum of the values of its projects. And NPV depends only on the project's cash flows and the cost of capital, not on managers' preferences, accounting choices or the firm's other activities.

The internal rate of return

The internal rate of return (IRR) is the discount rate at which a project's NPV is zero. The IRR rule accepts projects whose IRR exceeds the cost of capital. For a conventional project, one with an initial outflow followed by inflows, NPV falls as the discount rate rises, so IRR and NPV give the same accept-or-reject answer: if the IRR is above the cost of capital, the NPV at the cost of capital is positive. Managers like the IRR because a percentage is easy to communicate and compare with a hurdle rate. But it has four well-known pitfalls.

First, lending or borrowing. A project that brings in cash first and pays out later, such as a customer prepayment or a loan received, has an NPV that rises with the discount rate. A high IRR is then bad, not good, because it is the rate you are paying. Second, multiple IRRs. If cash flows change sign more than once, as in a mine that requires a large clean-up cost at the end or a project needing a major overhaul midway, there can be two or more IRRs, or none. Third, mutually exclusive projects. A small project can have a higher IRR but a lower NPV than a large one, and a project with quick payoffs can beat a project with distant payoffs on IRR while losing on NPV. Fourth, a term structure: when the cost of capital differs by year, there is no single hurdle rate to compare an IRR with.

# NPV profile of the mining project and a scan for internal rates of return
flows = [-4, 25, -25]
def npv(rate, cfs):
    return sum(cf / (1 + rate) ** t for t, cf in enumerate(cfs))

for r in (0.0, 0.10, 0.25, 0.60, 1.00, 2.00, 4.00, 6.00):
    print(f"rate {r:5.0%}: NPV {npv(r, flows):7.3f}")
roots, prev = [], npv(0.0, flows)
for k in range(1, 6001):
    r = k / 1000
    cur = npv(r, flows)
    if prev * cur <= 0:
        roots.append(round(r, 3))
    prev = cur
print("IRRs found:", roots)

rate    0%: NPV  -4.000
rate   10%: NPV  -1.934
rate   25%: NPV   0.000
rate   60%: NPV   1.859
rate  100%: NPV   2.250
rate  200%: NPV   1.556
rate  400%: NPV   0.000
rate  600%: NPV  -0.939
IRRs found: [0.25, 0.251, 4.0, 4.001]

Mutually exclusive projects

When a firm must choose between projects, because they use the same site, the same budget or serve the same need, ranking by IRR can mislead. The reason is that IRR is a rate, not an amount of value, and it implicitly assumes intermediate cash flows can be reinvested at the IRR itself. NPV assumes reinvestment at the cost of capital, which is the market's actual opportunity rate. When the two rankings conflict, NPV is right. One way to use IRR correctly is to look at the incremental cash flows, the difference between the larger and smaller project: if the IRR of the increment exceeds the cost of capital, the larger project is better.

The figure shows the timing version of the problem. A project whose cash flows arrive late has a steep NPV profile, because distant cash flows are hit hard by higher discount rates. Below the crossover rate, the rate at which the two NPVs are equal, the late project has the higher NPV; above it, the early project wins. The IRR ranking corresponds to the comparison at high discount rates, which is irrelevant if the firm's cost of capital is below the crossover.

The modified internal rate of return

The modified IRR (MIRR) addresses the reinvestment and multiple-root problems by compounding all inflows forward to the end of the project at the cost of capital, discounting all outflows back to today at the cost of capital, and then finding the single rate that links the two. It always gives a unique answer and ranks independent projects consistently with NPV, though it can still rank projects of different size differently. Many firms report MIRR alongside NPV for communication; it adds no information that NPV does not already contain.

Payback and discounted payback

The payback period is the time it takes for a project's cash inflows to recover its initial cost. It is simple and widely used as a screen, particularly by smaller firms and for risky projects in volatile markets, because it favors projects that return cash quickly. But it ignores the time value of money within the payback period (discounted payback fixes that), ignores all cash flows after the cutoff, and the cutoff itself is arbitrary. A project that pays back in three years and then stops can beat one that pays back in four years and then produces large cash flows for twenty more. Payback is best treated as a rough indicator of liquidity and risk, not as a decision rule.

Capital rationing and the profitability index

In theory, a firm that can raise money in competitive markets should take every positive-NPV project. In practice, many firms impose a budget limit, either because senior managers want to control the growth of divisions and discipline optimistic forecasts (soft rationing), or because the firm genuinely cannot raise more funds on reasonable terms (hard rationing). When capital is limited, the goal becomes maximizing the total NPV from the projects that fit within the budget. The profitability index, the present value of future cash flows divided by the initial investment, or equivalently NPV per dollar invested, ranks projects by value created per unit of the scarce resource.

Projects with different lives

Choosing between machines that do the same job but last different lengths of time requires care, because a machine that lasts longer saves the firm from replacing it sooner. If the machine will be replaced repeatedly, compare the equivalent annual cost: the level annual payment with the same present value as the machine's costs over its life. It converts each machine's lifetime cost into a cost per year that can be compared directly, assuming each can be replaced on the same terms.

What firms actually use

Surveys of chief financial officers in North America, beginning with Graham and Harvey's study published in 2001 and repeated in later work, find that most large firms use NPV and IRR, often together, and many also look at payback, especially smaller firms. That is sensible if the rules are used together intelligently: NPV for the decision, IRR for communication, payback as a check on how long capital is at risk. The danger is in relying on IRR or payback alone in exactly the situations where they fail. Whatever the rule, the quality of the decision depends mostly on the quality of the cash flow forecasts and the discount rate, the subjects of the next lessons.