IRR, NPV, and Incremental Analysis – Exam Prep Notes

Exam logistics and note-card guidelines

  • Exam One is on Monday; you may use an up-to-8x11 size note card on one side only (no formulas or illegible content on the other side). Plan ahead and bring a note card if you have one; extras may be available in the instructor’s office. Writing on ripped-from-a-book pages is not allowed.
  • If you don’t have a note card, the instructor may provide extras, but size and one-sided rule still apply.
  • Blackboard contains the exam guide, labeled Exam One Guide, which mirrors previous material but with end-of-chapter problems included for the chapters covered so far.

Key financial concepts: NPV and IRR

  • Net Present Value (NPV) basics
    • Definition: NPV=∑<em>t=0TCF</em>t(1+r)tNPV = \sum<em>{t=0}^{T} \frac{CF</em>t}{(1+r)^t} where $CF_t$ are cash flows at time $t$ and $r$ is the discount rate.
    • Decision rule: accept a project if $NPV > 0$ (positive value creation).
  • Internal Rate of Return (IRR)
    • Definition: IRR is the discount rate $r$ that makes NPV(r)=0.NPV(r) = 0.
    • Often described as the rate of return that equates inflows and outflows on a project. In practice, IRR is the rate that would lead to zero NPV.
    • Another intuition mentioned: IRR as an average rate of return, but with caveats discussed later.
  • General IRR-based rule (contextual): compare the IRR to the required rate of return $\text{(required return)}$ to decide acceptance or rejection. In standard cases, if the IRR exceeds the required return, the project is attractive; if it is below, reject.

Conventional vs borrowing cash flows and their NPV profiles

  • Conventional investing (costs before benefits)
    • Cash flow pattern: negative cash flow up front (cost) followed by positive inflows.
    • NPV profile: downward sloping with discount rate; one unique IRR exists.
    • Decision rule: accept if the required return is less than the IRR (i.e., $r < IRR$) or equivalently if $NPV > 0$ at the required rate.
  • Conventional borrowing (benefits up front, costs later effect)
    • Cash flow pattern: initial positive cash inflow (borrowing) followed by subsequent outflows (repayment).
    • NPV profile: upward sloping; IRR exists but interpretation differs due to the cash-flow signs.
    • Decision rule: accept if the required return is greater than the IRR (i.e., $r > IRR$), which corresponds to a positive NPV under the borrowing setup.

Issues with IRR as a sole decision criterion

  • Borrowing vs lending sign-flip issue
    • The same set of cash flows with signs flipped yields the same IRR, but the interpretation of the decision rule changes depending on whether you’re in a conventional investing or borrowing context.
  • Unconventional cash flows
    • Defined as cash-flow sign changes (e.g., positive, negative, positive, etc.).
    • Consequence: can yield no IRR or multiple IRRs; IRR becomes unreliable as a decision criterion.
  • Mutually exclusive investments
    • When choosing between two investments, IRR can be misleading due to scale and timing differences.
    • Two main problems:
    • Scale problem (different project sizes): a higher IRR may be from a smaller project with less total value; NPVs can tell the true value created.
    • Timing problem (cash-flow timing differences): even with similar lifetimes, the timing of cash inflows matters for NPV more than IRR alone.

Unconventional cash flows: example and implications

  • Typical pattern showing multiple IRRs
    • Example shape: cash flows with an outflow late in the project (e.g., shutdown costs or reinvestment needs) can produce two zero-crossings of the NPV curve, i.e., two IRRs.
    • Result: no unique IRR to base a decision on; IRR becomes unreliable for choosing between projects.
  • Practical takeaway
    • In cases with unconventional cash flows, prefer NPV analysis or incremental methods over relying on IRR alone.

Mutually exclusive investments: scale and timing in practice

  • Scale problem (size matters)
    • Example intuition: 50% return on $1 vs 10% return on $100.
    • Higher IRR on the smaller project can be misleading; the larger project may create more value in total (higher NPV).
    • Lesson: with mutually exclusive choices, do not use the highest IRR alone; compare NPVs to decide.
  • Timing problem (when cash flows occur matters)
    • Example: two projects with similar life cycles but different timing of big cash inflows.
    • At same discount rate, IRRs may differ; however, the one with larger NPV (better value today) should be preferred.
  • Consequence for practice
    • IRR is not reliable for mutually exclusive investments; NPVs and incremental analysis are preferred.

Remedies to IRR pitfalls for mutually exclusive investments

  • Three approaches to resolve scale and timing issues: 1) Compare NPVs directly: compute and compare NPV<em>A(r)NPV<em>A(r) and NPV</em>B(r)NPV</em>B(r) at the given discount rate and choose the higher. 2) Incremental NPV (and incremental IRR):
    • Compute incremental cash flow: CFinc<em>t=CFA</em>t−CFtBCF^{inc}<em>t = CF^A</em>t - CF^B_t for each period $t$ (often subtract the smaller project from the larger one so the first cash flow is negative, yielding conventional investing cash flows).
    • Then compute: NPVinc=∑<em>t=0TCFinc</em>t(1+r)tNPV^{inc} = \sum<em>{t=0}^{T} \frac{CF^{inc}</em>t}{(1+r)^t} and find its IRR.
    • Decision: accept the larger project if the incremental IRR exceeds the required rate; otherwise prefer the smaller project.
      3) Incremental IRR (alternative): use the IRR of the incremental cash flows directly as the criterion, similar to incremental NPV.

How to compute incremental cash flows and apply the three remedies

  • Steps to incremental analysis (example framework)
    • Determine which project is larger in total value or cost; compute incremental cash flows by subtracting the smaller project’s cash flows from the larger project’s cash flows at each period.
    • Ensure the first period cash flow of the incremental sequence is negative (to reflect the additional investment up front). This helps align with the conventional investing cash-flow pattern for the IRR rule.
    • Compute the incremental NPV: NPVinc=∑<em>t=0TCFinc</em>t(1+r)tNPV^{inc} = \sum<em>{t=0}^{T} \frac{CF^{inc}</em>t}{(1+r)^t}.
    • Compute the incremental IRR: the rate $r$ that makes NPVinc(r)=0.NPV^{inc}(r) = 0. If $IRR^{inc} > r_{required}$, adopt the larger project; otherwise adopt the smaller.
    • For the crossover-rate analysis: identify the rate at which the two projects have equal NPV, i.e., solve for $r$ in NPV<em>A(r)=NPV</em>B(r)NPV<em>A(r) = NPV</em>B(r). This rate is the crossover rate and equals the IRR of the incremental cash flows $CF^{inc} = CFB - CFA$ (up to sign conventions).
  • Practical note
    • If incremental IRR is computed with inverted sign (e.g., B minus A vs A minus B), the numerical IRR should be the same; the interpretation of “which project is better” depends on the sign convention and which project is treated as the incremental one.

Crossover rate concept and interpretation

  • Definition: The rate at which the NPVs of two projects are equal; above/below that rate, the preferred project can switch.
  • Relationship to incremental IRR: the crossover rate is also the IRR of the incremental cash flows (e.g., B − A).
  • Interpretation with rates:
    • If the required rate of return $r$ is below the crossover rate, one project (often the one with later larger payoffs) may be preferred; if $r$ is above the crossover rate, the other project (often the one with earlier payoffs) may be preferred.
  • Example numbers (from lecture): crossover rate around $10.55\%$; analysis often compares $r$ to this value to decide which project dominates at that $r$.

Practical examples discussed in class (conceptual summaries)

  • Example 1: Conventional scale issue (high IRR vs higher absolute value)
    • A: 50% return on $1$ vs B: 10% return on $100$.
    • IRR favors A due to a higher percentage, but A’s total value (NPV) is typically smaller due to the tiny scale; B may yield higher NPV.
    • Takeaway: do not rely on IRR when projects are mutually exclusive; compare NPVs.
  • Example 2: Mutually exclusive projects with conflicting IRRs but higher total value for one project
    • A costs $500, B costs $400; IRRA = 19.43%, IRRB = 22.17%; but NPVs show A provides higher value at the chosen rate.
    • Conclusion: with mutually exclusive projects, NPVs trump IRR in decision-making when lives are similar; IRR alone can be misleading.
  • Example 3: Incremental and crossover concepts in practice
    • Use incremental analysis to resolve which project adds more value by examining differences in cash flows across periods and their NPV/IRR.
    • Crossover rate helps identify when one project becomes preferable as the required rate changes.

Communication, interpretation, and real-world relevance

  • Emphasis on soft skills and clear communication
    • Employers value the ability to explain finance concepts in plain language to clients/managers without heavy jargon.
    • Use evidence from research papers to support arguments when presenting to non-finance stakeholders.
    • Illustrative explanations (e.g., comparing projects to everyday analogies) can improve understanding and credibility.
  • Use of evidence and research references
    • Example points drawn from Arnold and Nixon: early cash is reinvested, tying to the idea that money now is more valuable due to reinvestment opportunities.
    • Discuss Modified IRR (MIRR) as a more practical alternative that accounts for financing and reinvestment rate assumptions.
  • Practical takeaway for preparing explanations
    • Connect theory to real-world decision making with simple, layman terms.
    • Cite research where relevant to strengthen arguments.

Short case assignment and in-class activity: mining project in Excel

  • Overview of the short case (Seth, mining company)
    • Open a mine now, operate for 9 years, with costs to open and shut the mine at different times.
    • Expected cash inflows during the operation period.
    • Task: construct a spreadsheet and perform analysis for the period: IRR, Modified IRR, Profitability Index, NPV.
    • Decision rule: based on the analysis, decide whether to open the mine.
  • How the assignment is delivered
    • An Excel file is provided on Blackboard as a starting point.
    • Students must include their name, course code (e.g., 9437), and section in the file name or header.
    • Tab labeled “Review case analysis” provides the case text and cash flows; another tab or area for the executive summary (your yes/no recommendation).
    • The executive summary should be concise and not exceed the designated area; if more space is needed, you’re likely over-elaborating.
  • Collaboration and group work
    • You will work in groups to learn from each other, but the assignment itself is individual in final assessment.
    • At least one group member should be proficient with Excel functions and logical statements to assist others.
  • Timeline and logistics
    • The instructor plans to work through this case today and Friday; groups may be adjusted to balance; keep communicating with the instructor if you need changes.

Quick study and exam-prep tips

  • Plan ahead for the note card; ensure it contains essential formulas and concepts you can reference quickly (no pasted long-form content).
  • Be able to explain IRR, NPV, MIRR, and profitability index in your own words, with simple examples.
  • Be able to identify when to use NPV versus IRR (especially for mutually exclusive investments).
  • Practice incremental analysis: compute incremental cash flows, incremental NPV, and incremental IRR for pairs of projects.
  • Understand crossover rate and how to interpret it across different discount rates.
  • Be prepared to discuss both the theory (definitions, rules) and the practical application (Excel models, case analysis).