# Corporate finance: rank projects by value, not by a percentage alone — Study worksheet [Study guides](https://unireadinglist.com/guides/corporate-finance/) [Course topic: Corporate Finance](https://unireadinglist.com/subjects/business/#course-corporate-finance) Sources checked: 2026-11-04 Study route and exercise: University Reading List editorial project. 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 · 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. Your notes: ________________________________ [Official free reading](https://corpfin.ivo-welch.org/read/) [Book record and access options](https://unireadinglist.com/subjects/business/corporate-finance-welch/) ### Corporate Finance Jonathan Berk; Peter DeMarzo · 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. Your notes: ________________________________ ISBN: 9780137845026 [Book record and access options](https://unireadinglist.com/subjects/business/corporate-finance-berk-demarzo/) ### Principles of Corporate Finance Richard A. Brealey; Stewart C. Myers; Franklin Allen; Alex Edmans · 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. Your notes: ________________________________ ISBN: 9781266586156 [Book record and access options](https://unireadinglist.com/subjects/business/principles-of-corporate-finance/) ## Check your preparation ### What does NPV measure? Your notes: ________________________________ ### What does IRR solve? Your notes: ________________________________ ### What is held fixed? Your notes: ________________________________ ## A six-step study plan - [ ] 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. Your notes: ________________________________ - [ ] 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. Your notes: ________________________________ - [ ] 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. Your notes: ________________________________ - [ ] 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). Your notes: ________________________________ - [ ] 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. Your notes: ________________________________ - [ ] 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. Your notes: ________________________________ ## 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. Your notes: ________________________________ ## 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. Your notes: ________________________________ ## 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. ## Answer and interpretation ### 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. ### Original mutually exclusive projects: the higher IRR does not win at 10% 1. 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. 2. 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%. 3. 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. 4. 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. 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 1. 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. 2. 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%. 3. 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. 4. 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. 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. ## Sources and edition checks - [Author: free fifth edition 2022 and conditions](https://corpfin.ivo-welch.org/read/) The author provides the free fifth-edition reading route and topic sequence. - [Author: distinct fifth-edition access and fourth-edition licence](https://corpfin.ivo-welch.org/home/) Fifth-edition conditions and fourth-edition open-source status are kept distinct. - [Pearson: selected sixth-edition 2023 print](https://www.pearson.com/en-us/subject-catalog/p/corporate-finance/P200000009791/9780137845026) The exact sixth-edition print and copyright-year distinction is preserved. - [McGraw Hill: selected 2025 release and format products](https://www.mheducation.com/highered/product/principles-of-corporate-finance-brealey.html) The selected 2025 Release and its author/format identity are retained; all project figures are original. 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