Corporate finance: rank projects by value, not by a percentage alone
A project ranking requires a decision set, dated incremental cash flows and a discount rule. An attractive internal rate of return can belong to a smaller project that creates less present value. This route makes those comparisons explicit before adding uncertainty, financing or market evidence.
Course topic: Corporate Finance
Study route and exercise: University Reading List editorial project. How we curate · Sources checked:
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Scope and responsibility
For learners comfortable with algebra, powers and the difference between profit and cash. All projects below are invented, certain classroom cash flows; rates are stipulated inputs. The worksheet teaches a decision rule, not an investment recommendation or a claim about real required returns.
Choose your starting point
I need a free valuation spine
Use Welch’s fifth edition for present value, capital-budgeting rules and the economic cash-flow question. Reconstruct the timeline before using a formula.
I want one organizing principle
Use Berk and DeMarzo’s law-of-one-price approach to connect valuation rules across topics. Separate a pricing argument from a numerical discount-rate assumption.
I need wider managerial context
Use the selected Brealey and colleagues release for investment, risk and financing connections. Keep the cash-flow comparison small before selecting broader applications.
Compare the three books
Corporate Finance
Ivo Welch · Edition: 5 · 2022
Free conceptual and numerical spine
- Choose it when
- You can manipulate discounted sums and want to examine why a capital-budgeting rule works.
- Use another route when
- You expect free access to mean unrestricted reuse of the fifth edition or access to instructor-only material.
- Selected reading
- Fifth edition, 2022: present value, capital-budgeting rules and pitfalls, then economic cash flows by title.
- Limits and trade-offs
- The author provides free reading under stated conditions. The fourth edition’s open-source arrangement must not be transferred to the copyrighted fifth edition.
Try before choosing
Put A and B on separate three-date timelines and calculate their difference B−A.
Check your work: The difference is an incremental project comparison; do not average IRRs or add mutually exclusive project values.
Corporate Finance
Jonathan Berk; Peter DeMarzo · Edition: 6 · 2023
Law-of-one-price framework
- Choose it when
- You want a consistent valuation argument and a structured commercial course sequence.
- Use another route when
- You still need introductory algebra or want a freely licensed full text.
- Selected reading
- Sixth edition: valuation principles, investment decision rules and fundamentals of capital budgeting, selected by topic.
- Limits and trade-offs
- The chosen print ISBN 9780137845026 was published in 2023, while the product header carries ©2024. Digital and bundled products have other identifiers.
Try before choosing
Explain why both projects must use the same stipulated rate in this packet before computing either NPV.
Check your work: Changing a rate separately to favor a project changes the comparison; real project-specific risk would need a separate model.
Principles of Corporate Finance
Richard A. Brealey; Stewart C. Myers; Franklin Allen; Alex Edmans · Edition: 2025 Release · 2025
Broad investment-and-financing reference
- Choose it when
- You need to relate investment appraisal to risk and corporate financial choices.
- Use another route when
- You want to complete the entire reference before attempting one cash-flow decision.
- Selected reading
- Selected 2025 Release: present-value and investment-criteria discussions, then risk or financing only when the course needs them.
- Limits and trade-offs
- The selected release includes Edmans with Brealey, Myers and Allen. Match its hardcopy identifier to the assigned object rather than substituting a different edition or digital bundle.
Try before choosing
State the outside option and identify which cash flows are incremental, certain and dated.
Check your work: A positive IRR ranking is not itself a decision among mutually exclusive projects, and a negative best project can lose to doing neither.
Check your preparation
What does NPV measure?
The stipulated dated cash flows converted to time-zero value, including the initial outlay. Here a zero-value outside option is feasible.
What does IRR solve?
A rate that makes a project’s NPV zero. These conventional cash flows have one nonnegative IRR, but the rate does not measure the scale of value created.
What is held fixed?
Certainty, incremental flows, common constant discounting and mutual exclusivity. Taxes, financing effects, real risk estimates and cash-flow revisions are outside the supplied packet.
A six-step study plan
Steps completed: 0 / 6
These are your own study marks, not an assessment. They are not sent to this site.
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Build the timeline
Selected reading: Welch present-value reading.
What to do: Create columns for now, year 1 and year 2, with signed cash flows for both projects.
Check your work: An outlay now is negative and is not discounted another period.
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Value each alternative
Selected reading: Berk and DeMarzo investment criteria.
What to do: Calculate both NPVs at 10% and include the outside option.
Check your work: Choose the highest feasible time-zero value, not the largest year-2 payment alone.
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Audit the rate ranking
Selected reading: The same investment-criteria discussion.
What to do: Solve each IRR equation and compare its ranking with NPV.
Check your work: Explain how differing scale and timing permit a ranking conflict.
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Use incremental cash flows
Selected reading: Welch capital-budgeting pitfalls.
What to do: Value B−A and connect its sign to the original ranking.
Check your work: The incremental comparison must reproduce NPV(B)−NPV(A).
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Change the assumption
Selected reading: The solved follow-up.
What to do: Recompute at 20% and solve the crossover rate algebraically.
Check your work: State the nonnegative-rate domain and retain the outside option.
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Write a bounded recommendation
Selected reading: Selected broader investment discussion.
What to do: Write a short classroom decision memo with the rule, result and missing real-world inputs.
Check your work: Do not turn a stipulated discount rate into a claim about a real asset’s required return.
Worked exercise
Original mutually exclusive projects: the higher IRR does not win at 10%
Choose at most one of two invented projects, or neither. A pays (−90,60,60) at dates (0,1,2 years); B pays (−140,0,200). At a common annual rate of 10%, calculate NPV and each nonnegative IRR. Which ranking answers this decision?
Assumptions
All amounts use one fictional money unit. Flows are certain, incremental, end-of-year where applicable, and complete; there are no other costs, taxes, financing effects or constraints. Choosing neither project has value 0. Annual discounting uses the same constant r for both alternatives.
- At r=0.10, NPV(A)=−90+60/1.1+60/1.1²=14.1322314. NPV(B)=−140+200/1.1²=25.2892562. B exceeds A by 11.1570248 and both exceed 0.
- Let z=1+r. For A, −90 z²+60 z+60=0 gives IRR(A)=(√7−2)/3≈21.5250%. For B, −140+200/z²=0 gives IRR(B)=√(10/7)−1≈19.5229%.
- IRR ranks A above B, but NPV at the stipulated opportunity-cost rate ranks B above A. IRR is a break-even percentage, not an amount of value created.
- The incremental flow B−A=(−50,−60,140) has NPV 11.1570248 at 10%, exactly the gain from replacing A with B. Mutual exclusivity makes that comparison relevant.
Answer and interpretation
At 10%, choose B within this classroom packet: NPV 25.2893 versus A 14.1322 and 0 for choosing neither. A’s higher IRR does not reverse the value comparison.
Test the result
Substitute each IRR into its own NPV equation and verify the incremental NPV identity. This result does not estimate any real project’s cash flows or appropriate rate.
Change an assumption
Changed rate: identify the crossover and preserve the outside option
Keep every cash flow and constraint unchanged but set the common rate to 20%. Find the crossover rate between A and B for r≥0, and explain why a relative ranking alone is insufficient at very high rates.
- At 20%, NPV(A)=−90+60/1.2+60/1.2²=1.6666667; NPV(B)=−140+200/1.2²=−1.1111111. Choose A rather than B or neither.
- For equality, −50−60/z+140/z²=0, so 50 z²+60 z−140=0. The positive root is z=(√79−3)/5, hence r≈0.177638883, or 17.7639%.
- For r≥0 below the crossover B has the larger NPV; above it A has the larger NPV. At equality they tie in value. The sign follows the incremental expression, not an IRR average.
- Once r exceeds A’s approximately 21.5250% IRR, both NPVs are negative and neither wins. At that IRR, A and neither tie at 0. The outside option prevents a forced choice of the less negative project.
Answer and interpretation
At 20%, choose A with NPV 1.6667. The pairwise crossover is about 17.7639%; a project must also beat the feasible zero-value outside option.
Test the result
Evaluate the incremental NPV on both sides of the crossover and compare each project with 0. The ranking statements apply only to these fixed flows and r≥0.
Common mistakes
IRR treated as value
A percentage omits the scale and timing needed for a mutually exclusive value comparison.
Outside option omitted
The relatively better project can still destroy value under the stipulated rule.
Classroom rate promoted to evidence
A supplied rate is an assumption, not an empirical cost-of-capital estimate.
Before moving on
- I can reproduce both dated NPV sums and the incremental identity.
- I can solve the IRRs and crossover without confusing their questions.
- I can state the decision set, outside option and limits of the cash-flow packet.
Sources and edition checks
Sources checked:
- Author: free fifth edition 2022 and conditions
The author provides the free fifth-edition reading route and topic sequence.
- Author: distinct fifth-edition access and fourth-edition licence
Fifth-edition conditions and fourth-edition open-source status are kept distinct.
- Pearson: selected sixth-edition 2023 print
The exact sixth-edition print and copyright-year distinction is preserved.
- McGraw Hill: selected 2025 release and format products
The selected 2025 Release and its author/format identity are retained; all project figures are original.
These are proposed study activities, not official course assignments or a claim that every book has been read cover to cover.