Note: Day 1 this week is the Midterm MC Exam. This pack covers the Day 2 session, which introduces Chapter 9 — Net Present Value and Other Investment Criteria.
For the first half of this course, you've been building a toolkit for valuing cash flows across time — computing present values, pricing bonds, valuing stocks. All of that machinery was preparing you for the question at the heart of corporate finance: Should a firm invest in this project?
Recall from Week 1 that capital budgeting is the process of planning and managing a firm's long-term investments. A firm's managers constantly face decisions like these: Should we build a new factory? Should we launch a new product line? Should we replace aging equipment? Each of these decisions involves spending money now in exchange for uncertain cash flows in the future — and each one should only be pursued if it creates value for shareholders.
The question is deceptively simple: does the project generate more value than it costs? But answering it requires comparing cash flows that occur at different points in time, which is exactly what discounting does. This week, we'll study four criteria that firms use to evaluate investment projects. One of them — net present value — is the gold standard. The others have intuitive appeal but real limitations. Understanding both the strengths and weaknesses of each method is essential for making good capital budgeting decisions.
The net present value (NPV) of a project is the difference between the present value of its future cash flows and the cost of the investment. In plain terms, NPV answers the question: "How much richer does this project make the firm's shareholders, in today's dollars?"
The logic is straightforward. You already know how to discount future cash flows back to the present. If the present value of everything the project will generate exceeds what you have to spend to get it, the project creates value. If it doesn't, the project destroys value.
The NPV rule is simple: accept a project if its NPV is positive; reject it if its NPV is negative. A positive NPV means the project earns more than the required return — it creates value beyond what shareholders could earn by investing elsewhere at the same level of risk. A negative NPV means the project earns less than the required return — the firm would be better off returning the money to shareholders and letting them invest it on their own.
What about a project with NPV exactly equal to zero? Technically, shareholders break even — the project earns exactly the required return. In practice, NPV = 0 projects are rare, and most firms would be indifferent about them. We'll treat NPV = 0 as marginal: not a clear accept, not a clear reject.
The discount rate r deserves a moment of attention. It represents the required return — the minimum rate the project must earn to justify the investment. This rate reflects the riskiness of the project's cash flows. We'll formalize how to determine the appropriate discount rate when we study the cost of capital in Weeks 9 and 10. For now, take the discount rate as given.
When a project generates the same cash flow every year, the NPV calculation simplifies. The stream of equal payments is an annuity, so you can use the annuity present value formula from Week 2 to find the PV of the future cash flows in one step, then subtract the initial investment.
Notice what this result tells us. Red Acre's shareholders are $20,358.22 richer (in present-value terms) if the firm buys this machine than if it doesn't. The project more than compensates for the time value of money and the risk involved.
Many real projects don't generate the same cash flow every year. A new product might ramp up slowly, peak, then decline. A branch office might take time to build a client base. When cash flows vary across periods, you have to discount each one individually — there's no annuity shortcut.
A project's NPV is sensitive to the discount rate you use. Higher discount rates shrink the present value of future cash flows, making it harder for a project to have a positive NPV. Lower discount rates do the opposite. This matters because the discount rate is often estimated — and reasonable people can disagree about the right number. When you're evaluating a project, it's good practice to ask: "How much would the discount rate have to change before my decision flips?"
This example illustrates a general principle: for any project with positive cash flows in the future, NPV is a decreasing function of the discount rate. As r rises, future cash flows are worth less in today's terms, and eventually the NPV turns negative. The rate at which NPV crosses zero has a name — it's the internal rate of return, and it's our next topic.
The internal rate of return (IRR) is the discount rate that makes a project's NPV exactly equal to zero. Put differently, the IRR is the rate of return the project actually earns on the invested capital. If a project costs $100 and generates cash flows whose present value is exactly $100 when discounted at 15%, then the IRR is 15%.
The IRR rule says: accept a project if its IRR exceeds the required return; reject it if the IRR falls below the required return. The intuition is appealing — if the project earns a higher rate of return than shareholders require, it creates value.
For projects with conventional cash flows (an initial outflow followed by a series of inflows), the IRR rule and the NPV rule will always give the same accept/reject decision for independent projects. This is because, as we saw in Example 3, NPV declines as the discount rate increases. If the IRR exceeds the required return, then evaluating NPV at the required return must yield a positive number.
We'll see in Week 7 that IRR can be misleading when comparing mutually exclusive projects or when cash flows change sign more than once. For now, we'll focus on computing the IRR and applying it to straightforward cases.
Finding the IRR means solving for the unknown rate in the NPV equation — which generally can't be done with algebra alone. For projects with equal annual cash flows, you can narrow it down using annuity tables or a financial calculator. For uneven cash flows, you'll need a financial calculator, spreadsheet (Excel's IRR() function), or trial-and-error. In this course, exam questions will either give you equal cash flows (where you can solve for the annuity factor) or provide enough information to identify the IRR from answer choices.
Notice the consistency: in Example 1, we found the NPV was positive at a 12% discount rate. Here we find the IRR is 15.24% — above 12%. Both methods agree. They always will for conventional, independent projects.
Again, this is consistent with Example 2, where we found a positive NPV of $69,632.60 at a 10% discount rate.
The NPV profile for the Red Acre stamping machine (Examples 1 and 4). NPV declines as the discount rate rises, crossing zero at the IRR of 15.24%. At any required return below the IRR, the project has positive NPV and should be accepted.
The payback period is the amount of time it takes for a project's cumulative cash flows to recover the initial investment. If you spend $300,000 on a project and it generates $100,000 per year, the payback period is 3 years.
The appeal of payback is obvious: it's simple, it's intuitive, and it answers a question that managers naturally care about — "How long until I get my money back?" Many firms use a payback rule that sets a cutoff: accept projects that pay back within a specified number of years, reject those that take longer.
Payback is widely used in practice, but it has serious flaws that you need to understand. There are three big ones:
1. Payback ignores the time value of money. A dollar received in Year 1 counts the same as a dollar received in Year 4. We've spent weeks establishing that this is wrong — a dollar today is worth more than a dollar tomorrow. Payback treats them identically.
2. Payback ignores cash flows after the cutoff. A project that pays back in 2 years and then generates $10 million in Year 3 looks the same as a project that pays back in 2 years and generates nothing afterward. By focusing only on how quickly the initial investment is recovered, the payback rule can reject highly valuable projects that happen to have a slow start.
3. The cutoff is arbitrary. There's no economic theory behind choosing a 3-year cutoff versus a 4-year cutoff. Different firms use different cutoffs, and none of them are derived from the project's risk or the time value of money.
The next example illustrates why payback can lead you astray.
This is the core lesson about payback: it can give you a quick sanity check on liquidity — how fast you'll recover your cash — but it should never be the primary basis for investment decisions. NPV is the superior criterion because it accounts for the time value of money, considers all cash flows, and directly measures value creation.
The profitability index (PI) measures the value created per dollar invested. It's calculated as the ratio of the present value of future cash flows to the initial investment.
The PI rule is closely related to NPV. If the PV of future cash flows exceeds the initial investment, PI > 1, and NPV is positive. If PI < 1, NPV is negative. For a single project, PI and NPV will always agree on the accept/reject decision.
Where PI adds value is when a firm faces capital rationing — a situation where the firm has more positive-NPV projects than it can fund. In that case, PI helps rank projects by their efficiency: how much value each project generates per dollar of scarce capital. We'll briefly touch on this idea in Week 7 when we discuss conflicts between investment criteria.
Notice the link to NPV: the NPV we computed in Example 3 was $39,599.05 at the 8% rate. Dividing that by the $180,000 investment gives $0.22 per dollar — the same net value per dollar that PI tells us. PI is just NPV rescaled to a per-dollar basis.
You now have four tools for evaluating investment projects. Here's how they stack up:
NPV is the gold standard. It directly measures value creation in dollar terms, accounts for the time value of money, and considers all cash flows. Use it as your primary criterion.
IRR gives you the project's rate of return, which is intuitive and easy to communicate ("This project earns 15%"). For conventional, independent projects, IRR and NPV always agree. But IRR has limitations with non-conventional cash flows and mutually exclusive projects — we'll study these in Week 7.
Payback tells you how quickly you'll recover your investment, which is useful as a liquidity measure. But it ignores the time value of money, ignores cash flows after the cutoff, and uses an arbitrary benchmark. Don't rely on it as a primary decision tool.
PI tells you the value per dollar invested, which is useful when capital is scarce. For independent projects, it always agrees with NPV on accept/reject decisions.
In practice, most well-managed firms compute NPV as their primary criterion and use IRR and payback as supplementary information. You should do the same on exams: when asked to evaluate a project, lead with NPV unless the question specifically asks for another criterion.
| Criterion | Formula | Decision Rule |
|---|---|---|
| Net Present Value | NPV = −C₀ + Σ Cₜ ÷ (1+r)ᵗ | Accept if NPV > 0 |
| Internal Rate of Return | 0 = −C₀ + Σ Cₜ ÷ (1+IRR)ᵗ | Accept if IRR > required return |
| Payback Period | Years to recover C₀ from cumulative CFs | Accept if payback < cutoff |
| Profitability Index | PI = PV of future CFs ÷ C₀ | Accept if PI > 1 |
White Acre Technologies is considering a server upgrade costing $320,000. The upgrade is expected to generate net cash flows of $95,000 per year for 5 years. White Acre's required return is 11%. Calculate the NPV and make an accept/reject recommendation.
A project requires an initial investment of $150,000 and is expected to produce the following cash flows: Year 1: $45,000; Year 2: $55,000; Year 3: $65,000; Year 4: $40,000. If the required return is 9%, compute the IRR (using a financial calculator or trial-and-error) and determine whether the project should be accepted.
A project costs $200,000 and produces net cash flows of $60,000, $70,000, $80,000, and $50,000 over four years. What is the payback period? If the firm's payback cutoff is 3 years, should the project be accepted under the payback rule?
A project costs $275,000 and will generate net cash flows of $80,000 per year for 5 years. The discount rate is 10%. Calculate the profitability index and the NPV. Verify that both criteria give the same accept/reject decision.
Your firm is choosing between two independent projects, each costing $500,000. Project X generates cash flows of $200,000, $200,000, $150,000, and $100,000 over four years. Project Y generates cash flows of $50,000, $100,000, $200,000, $300,000, and $200,000 over five years. The required return is 12%. (a) Compute the payback period for each project. (b) Compute the NPV of each project. (c) If the firm uses a 3-year payback cutoff, which project(s) would it accept? Does NPV agree? Explain any discrepancy.
A colleague argues that payback is a perfectly good investment criterion because "if a project pays for itself quickly, it must be a good investment." Identify and explain three specific weaknesses of the payback rule that undermine this argument. For each weakness, give a brief example of how payback could lead to a poor decision.