Professor Jessie

From Cash Flows to Decisions

In Chapter 12, we learned how to estimate the cash flows a project will generate. Now we face the next question: Given those cash flows, should we accept or reject the project? This chapter introduces the decision rules — the statistical techniques that translate cash flow estimates into clear accept/reject signals.

A Shift in Perspective

Throughout our earlier time-value-of-money work, we assumed financial markets were perfectly competitive — securities traded at fair prices. Now we're evaluating real assets like factories, machines, and land. These trade in much less competitive markets, which means there can be genuine bargains (or overpriced traps) that our analysis must uncover.

The Six Capital Budgeting Techniques

Finance professionals use six major decision tools. Each measures project attractiveness differently — in dollars, in time, or as a rate of return — and each has strengths and weaknesses.

Notice that NPV is the only technique that works well across every dimension: it uses TVM, handles non-normal cash flows, and correctly ranks mutually exclusive projects. This is why NPV is considered the gold standard — and why the other techniques are best understood in relation to it.

How to Choose Among Techniques

Selecting which technique to use depends on five interconnected factors:

• Statistical format: Do you prefer seeing the answer in dollars, time, or as a rate?
• Benchmark: What do you compare the statistic against — $0, the cost of capital, or an externally imposed limit?
• TVM usage: Does the method account for the time value of money, or does it treat all dollars equally regardless of when they arrive?
• Cash flow pattern: Are your cash flows normal (one outflow followed by inflows) or non-normal (sign changes multiple times)?
• Project set: Are you evaluating independent projects (accept all that pass) or mutually exclusive projects (pick the best one)?

Independent vs. Mutually Exclusive Projects

Payback: How Long Until We Break Even?

The payback method answers an emotionally compelling question: How long will it take to get our money back? It's the simplest capital budgeting technique — you add up the project's cash inflows year by year until the cumulative total equals or exceeds the initial investment.

How It Works

If a project costs $50,000 and generates $20,000 per year, the payback period is 2.5 years. After year 2 you've recovered $40,000; the remaining $10,000 is half of year 3's $20,000 inflow.

The Decision Rule

Compare the payback period to a maximum allowable payback set by management. This benchmark is exogenous — it's not derived from any formula or market rate. It might reflect a bank loan deadline, a protective covenant, or simply management's comfort level with risk.

Discounted Payback: Adding Time Value

The discounted payback (DPB) method is payback's smarter cousin. It works the same way, but instead of adding up raw cash flows, it first discounts each year's cash flow back to present value using the cost of capital. The question becomes: How long to recoup the investment in present-value terms?

How It Works

If a $50,000 project generates $20,000/year at a 10% cost of capital, the present values are $18,182 (year 1), $16,529 (year 2), $15,026 (year 3)… Cumulative PV reaches $50,000 sometime in year 3 — later than the 2.5 years that regular payback suggested, because discounting shrinks each year's contribution.

Strengths and Weaknesses

Net Present Value: The Gold Standard

Net Present Value (NPV) is the most theoretically sound capital budgeting technique. It sums the present value of every cash flow — both inflows and outflows — discounted at the project's cost of capital. The result is measured in dollars, which makes it directly interpretable: "This project creates $X of value."

The Formula

This is the same present-value math we used for bonds and stocks — just applied to a project's cash flows instead of a security's. The difference is that project cash flows are estimates, not contractual promises.

The Decision Rule

• NPV > $0: Accept. The project earns more than its cost of capital — it creates value.
• NPV < $0: Reject. The project doesn't earn enough to cover its cost of capital — it destroys value.
• NPV = $0: Indifferent. The project exactly meets its required return.

NPV Strengths and Weaknesses

• Strength: Works equally well for independent projects and mutually exclusive projects — pick the highest NPV
• Strength: Handles non-normal cash flows without issue
• Weakness: Managers sometimes misinterpret NPV by comparing it to the project cost, even though cost is already built into the NPV

Internal Rate of Return: The Most Popular Rate-Based Method

The Internal Rate of Return (IRR) is the discount rate that makes the project's NPV equal to zero. In other words, it's the project's implicit expected rate of return — the geometric average return the project earns on its invested capital.

How to Find It

You can't solve for IRR algebraically — the equation is too complex. Instead, you use trial-and-error, a financial calculator, or spreadsheet software (Excel's =IRR() function). You're searching for the rate r where:

The Decision Rule

• IRR > Cost of Capital: Accept. The project earns more than it costs to finance.
• IRR < Cost of Capital: Reject. The project earns less than its financing cost.

Why IRR Is So Popular

Managers love rates of return. "This project earns 18%" is more intuitive to many executives than "This project has an NPV of $12,000." The IRR gives the same accept/reject decision as NPV when evaluating independent projects with normal cash flows — so in those straightforward cases, it's a perfectly valid alternative.

When IRR Goes Wrong

IRR is consistent with NPV only when two conditions hold: the project has normal cash flows, and you're evaluating it independently (not against competing projects). When either condition fails, IRR can produce misleading or contradictory results.

Problem 1: Non-Normal Cash Flows

If a project's cash flows change sign more than once (e.g., an initial outflow, then inflows, then a major cleanup outflow at the end), the math can produce multiple IRRs — each one technically valid. The Rule of Signs tells us: the number of positive IRRs can never exceed the number of sign changes in the cash flow stream. A project with two sign changes might have two IRRs, and you'd have no way to know which is "the" rate of return.

Problem 2: Mutually Exclusive Projects

When choosing between competing projects, IRR and NPV can disagree. A small project with a 25% IRR might have a lower NPV than a large project with a 15% IRR. IRR measures return per dollar invested but ignores the total dollars of value created. For mutually exclusive projects, always follow NPV.

Problem 3: The Reinvestment Rate Assumption

IRR implicitly assumes that interim cash flows are reinvested at the IRR itself — which is often unrealistically high. NPV assumes reinvestment at the cost of capital, which is far more realistic. A project with a 30% IRR is unlikely to have follow-on opportunities that also earn 30%; the cost of capital is a much more defensible reinvestment benchmark.

Modified IRR: Fixing the Reinvestment Problem

The Modified Internal Rate of Return (MIRR) directly addresses IRR's most criticized flaw: the unreasonable reinvestment rate assumption. MIRR modifies the project's cash flows before computing the rate of return, ensuring that interim cash flows are assumed to be reinvested at the cost of capital rather than at the IRR.

How It Works

• Compound all positive cash flows forward to the project's end at the cost of capital
• Discount all negative cash flows back to time zero at the cost of capital
• Find the rate that equates the present value of costs to the future value of inflows

What MIRR Fixes — and What It Doesn't

Profitability Index: Value per Dollar Invested

The Profitability Index (PI) converts the NPV into a rate-based metric by dividing the present value of a project's future cash flows by its initial investment. It answers: How much value does each dollar of investment create?

The Formula

The Decision Rule

• PI > 1: Accept. Each dollar invested returns more than $1 in present value — value is created.
• PI < 1: Reject. Each dollar invested returns less than $1 — value is destroyed.

Interpreting the Result

A PI of 1.23 means the project earns a 23% premium above and beyond the return necessary to repay the initial investment. Unlike IRR and MIRR, you don't compare PI to the cost of capital — the cost of capital is already embedded in the present value calculation.

Key Takeaways

• Six major techniques: PB, DPB, NPV, IRR, MIRR, PI — each measures project value differently
• NPV is the gold standard: it handles non-normal cash flows, works for mutually exclusive projects, and measures value in dollars
• Payback and discounted payback are intuitive but ignore cash flows after the payback cutoff
• IRR is popular but can produce multiple answers with non-normal cash flows and can misrank mutually exclusive projects
• IRR's reinvestment rate assumption (reinvest at IRR) is unrealistic; NPV's (reinvest at cost of capital) is more defensible
• MIRR fixes IRR's reinvestment rate and multiple-IRR problems but still struggles with mutually exclusive project ranking
• PI measures value per dollar invested — ideal for capital rationing, but can mislead on mutually exclusive projects of different sizes
• When in doubt, follow NPV — it's the most theoretically sound method and rarely leads you astray