Lecture 5 — Investment Decisions & Alternative Selection¶
Session Info
Date: Sunday, 2026-06-07 · Time: 16:20–18:10 · Room: SY109 Instructor: Dr. Zhijiang Chen · Session 5 of 16 — End of Week 1
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Learning Objectives¶
By the end of this lecture, you should be able to:
- State the Net Present Value (NPV) definition, compute NPV from a cash flow stream at a given discount rate, and apply the accept-iff-NPV>0 decision rule.
- Define the Internal Rate of Return (IRR) as the rate at which NPV = 0, compute it (analytically for short streams, iteratively otherwise), and apply the IRR > MARR decision rule — including its multiple-root and reinvestment-rate caveats.
- Choose an appropriate MARR for a given organisational setting — cash-rich enterprise, venture-backed startup, government, AI/experimental — and explain why an engineer should never silently substitute their own.
- Compute simple payback and discounted payback, and recognise when payback is a useful screen and when it is a dangerous sole criterion.
- Compute the Profitability Index (PI) and use it to rank projects under capital rationing — the situation where you must choose a subset of positive-NPV projects under a fixed budget.
- Choose between mutually-exclusive alternatives correctly — picking by NPV (not IRR) when sizes or lives differ — and reconcile the rules with incremental IRR analysis.
- Resolve the Lecture 1 Elasticsearch-vs-SaaS puzzle end-to-end, including a sensitivity check that identifies which assumptions actually drive the verdict.
Lecture Map — Four Rules, One Verdict¶
By the end of today you should be able to look at any project's cash flow and reach a defensible accept/reject verdict using four decision rules — and know which rule to trust when they disagree. Today closes Week 1: every cost concept, every cash-flow shape, and every equivalence move from Lectures 1–4 funnels into a single answer.
| § | Topic | Minutes |
|---|---|---|
| I | Net Present Value (NPV) & the discount-rate question | 20 |
| II | Internal Rate of Return (IRR) & MARR | 20 |
| III | Payback period & Profitability Index (PI) | 15 |
| IV | Comparing mutually-exclusive alternatives | 15 |
| — | Discussion — a real decision from your team | 10 |
| V | Verdict on Lecture 1's Elasticsearch vs SaaS | 20 |
| — | HW 5, Week 1 recap, questions | 10 |
Prerequisites from Lectures 3–4
This lecture assumes you can price a single sum (\(P/F, F/P\)), a uniform series (\(P/A, A/P\)), an arithmetic gradient (\(P/G, A/G\)), and a geometric gradient. If any of those still feel slow, revisit Lecture 4 before today — every decision rule below is NPV underneath, and NPV is a sum of priced cash-flow shapes.
§ I. Net Present Value — The One Number That Says Yes or No¶
If you remember only one rule from this entire course, remember NPV. Every other rule we discuss today earns a footnote; NPV is the only one finance professors will defend without one.
Definition¶
The Net Present Value of a project is the sum of every cash flow, each discounted to the present at the firm's cost of capital \(i\):
where \(CF_t\) is the net cash flow at the end of period \(t\) (positive for inflows, negative for outflows), \(i\) is the discount rate (the firm's MARR — see § II), and \(n\) is the project horizon. Conventionally \(CF_0\) is the initial investment and is negative.
Decision rule¶
Accept the project iff NPV > 0. The positive NPV is the surplus value the project creates over and above the firm's required return.
NPV = 0 means the project earns exactly the required return — the firm is indifferent. NPV < 0 means the firm would be richer doing literally anything else that earns the MARR. There is no "small negative NPV is fine" tier; if the rate truly is your MARR, the rule is binary.
Why NPV is the gold standard¶
| Property | What it gives you |
|---|---|
| Uses every cash flow | Doesn't truncate at a payback horizon; doesn't ignore tails. |
| Respects time value of money | Each dollar lives at its real economic time. |
| Directly measures value created | The number is dollars in the shareholder's pocket today. |
| Additive across a portfolio | NPV(A + B) = NPV(A) + NPV(B). Lets you build budgets bottom-up. |
| Currency-denominated | "+$35K" tells you something "+17% IRR" never can. |
Worked example — NPV of an internal AI code-review tool¶
Your team proposes to build an internal AI-assisted code-review tool. The estimated cash flow is:
| Year | Cash flow |
|---|---|
| 0 | −$200,000 (build) |
| 1 | +$60,000 |
| 2 | +$80,000 |
| 3 | +$80,000 |
| 4 | +$80,000 |
At a MARR of 10%, compute NPV. The discount factors \(1/(1.10)^t\) are tabulated, so this is mechanical:
| Year | Cash flow | DF @ 10% | PV |
|---|---|---|---|
| 0 | −$200,000 | 1.0000 | −$200,000 |
| 1 | +$60,000 | 0.9091 | +$54,545 |
| 2 | +$80,000 | 0.8264 | +$66,116 |
| 3 | +$80,000 | 0.7513 | +$60,105 |
| 4 | +$80,000 | 0.6830 | +$54,641 |
| NPV | +$35,407 |
Verdict. NPV ≈ +$35K > 0 — the project creates value over the firm's 10% cost of capital. Accept.
Working principle
NPV converts a stream of dollars across years into a single dollar amount today. The single dollar amount is the only number that decides yes or no.
The discount-rate question — where NPV is fragile¶
NPV depends on the discount rate you choose. Halve it and the project looks great; double it and it looks terrible. At \(i = 5\%\) the AI tool above is worth about $53K; at \(i = 20\%\) it is worth about −$8K. Same cash flows, opposite verdicts. The rate is half the answer. This is why § II is about the discount rate, not the math.
§ II. Internal Rate of Return — The Rate Implied by the Cash Flows¶
NPV needs a rate as input. IRR outputs a rate from the cash flows themselves — and asks the firm whether that rate clears the bar.
Definition¶
The Internal Rate of Return is the discount rate \(i^*\) at which the project's NPV equals zero:
It is the project's implied return — what you would earn if you put your money in this project and reinvested all the returns at \(i^*\).
Decision rule¶
Accept iff IRR > MARR. If the project's implied return clears the firm's required return, accept.
How to compute IRR¶
| Stream | Method |
|---|---|
| 2 cash flows (one out, one in) | Solve algebraically: \(-P + F/(1+i)^n = 0 \Rightarrow i = (F/P)^{1/n} - 1\). |
| 3+ cash flows | Numerical: spreadsheet IRR() / XIRR(), Newton's method, or bisection. |
| Uniform-series projects | Solve \((P/A, i, n) = P / A\) for \(i\) — interpolate in a factor table. |
For the AI tool above, the IRR is ≈ 17.5% — well above the 10% MARR. Decision: accept. Consistent with the NPV verdict (they almost always agree on accept/reject for a single project — the disagreements appear when comparing two projects, in § IV).
Three honest limitations of IRR¶
- Multiple IRRs. Any cash flow with more than one sign change can have more than one IRR. A project that costs money, makes money, then costs decommissioning money has two roots — neither of which is "the" return. In practice, when sign changes are present, fall back to NPV.
- Reinvestment-rate assumption. IRR implicitly assumes intermediate cash flows are reinvested at the IRR itself. If your IRR is 35% but the rest of the firm earns 10%, that assumption is fiction. The Modified IRR (MIRR) fixes this by assuming reinvestment at the MARR.
- Scale blindness. IRR is a rate, so it ignores project size — a 30% return on $1K beats a 16% return on $1M on rate, but the firm cashes the dollars, not the rate. See § IV.
When NPV and IRR disagree, NPV wins
NPV measures value created in dollars. IRR measures rate of return. The firm is in business to maximise dollars — not rates. Whenever the two rules conflict on a project comparison, NPV is the rule that aligns with shareholder wealth.
MARR — the hurdle the firm sets¶
MARR (Minimum Attractive Rate of Return) is the lowest return the firm will accept for an investment of similar risk. It is set by leadership, not chosen by the engineer — and it varies enormously across settings:
| Setting | Typical MARR | Rationale |
|---|---|---|
| Cash-rich enterprise | 8–12% | Cost of capital + small risk premium |
| Venture-backed startup | 25–40% | Equity investors demand high return |
| Government / non-profit | 3–5% | Treasury-rate floor |
| AI / experimental product | 20–30% | Outcome uncertainty bumps the floor |
Three rules for engineers using a MARR:
- If the firm has stated a discount rate, use it — do not quietly substitute a number you prefer.
- If the firm has not stated a rate, ask finance. Most students do not realise they are allowed to.
- If you must default, 10% is the safe textbook number for a mid-cap technology company in a normal interest-rate environment. State the assumption explicitly.
The same project can be accepted at one MARR and rejected at another. That is not a flaw; it is the point — the MARR encodes the firm's risk appetite, capital cost, and opportunity set. A startup that accepts only IRR ≥ 30% is making the right call for its investors even if a large bank would happily accept the same project at IRR = 9%.
§ III. Payback Period & Profitability Index — The Rough-and-Ready Rules¶
NPV and IRR answer "is it worth it?" Payback and PI answer two follow-up questions managers actually ask: "when do we get our money back?" and "what's the bang for the buck?"
Payback period¶
The payback period is the number of periods until the cumulative cash flow turns from negative to positive. For the AI tool above:
| Year | Cash flow | Cumulative |
|---|---|---|
| 0 | −$200,000 | −$200,000 |
| 1 | +$60,000 | −$140,000 |
| 2 | +$80,000 | −$60,000 |
| 3 | +$80,000 | +$20,000 ← crosses zero |
| 4 | +$80,000 | +$100,000 |
Cumulative cash crosses zero partway through year 3. Interpolating: $60K still owed at start of year 3 ÷ $80K incoming that year = 0.75 years, so payback ≈ 2.75 years.
Decision rule: Accept iff payback < some firm-set ceiling (commonly 2–3 years for IT projects, longer for infrastructure).
Strengths and weaknesses¶
| Strengths | Weaknesses |
|---|---|
| Easy to compute, easy to explain | Ignores all cash flows after payback |
| Prioritises liquidity — protects against running out of cash | Ignores the time value of money (unless discounted) |
| Reasonable proxy for risk in unstable or fast-changing environments | Biases against long-horizon projects (R&D, infrastructure) |
| Useful as a screen before deeper analysis | Says nothing about value created |
Discounted payback¶
A halfway-house: discount each year's cash flow first, then find when cumulative discounted cash flow turns positive. This fixes the time-value-of-money flaw but still ignores post-payback flows. For the AI tool at 10%, discounted payback comes in at ≈ 3.4 years — meaningfully longer than the undiscounted 2.75.
Payback is a screen, never a verdict
Treat payback as a secondary criterion, never the sole decision rule. A high-NPV project with a 4-year payback is still a high-NPV project. Conversely, a 1-year payback on a project that also has negative NPV is still a value-destroyer. The classic management error is rejecting a great long-tailed project because its payback exceeds an arbitrary ceiling.
Profitability Index¶
The Profitability Index (PI) measures bang-for-the-buck — present-value benefit per dollar of capital invested:
For our AI tool: PV(future) = $54,545 + $66,116 + $60,105 + $54,641 = $235,407. Divide by initial $200,000:
Interpretation. Every $1 of capital deployed returns $1.18 of present-value benefit — i.e., $0.18 of NPV per $1 invested.
Decision rule: PI > 1 ⟺ NPV > 0. So as a single-project rule, PI gives the same accept/reject as NPV.
Where PI earns its keep — capital rationing¶
The real power of PI is in capital rationing — when you have several positive-NPV projects but a fixed total budget. Ranking by NPV alone would have you pick the biggest projects regardless of efficiency; ranking by PI picks the most efficient use of each scarce dollar.
| Project | Investment | NPV | PI |
|---|---|---|---|
| α | $100K | $40K | 1.40 |
| β | $200K | $60K | 1.30 |
| γ | $80K | $24K | 1.30 |
| δ | $120K | $24K | 1.20 |
If the budget is $300K, taking the single highest-NPV project (β) captures $60K of NPV. Ranking by PI lets you assemble the combination α + γ + δ (total cost = $300K) for total NPV of $88K — $28K better than β alone. PI is the right ranking when capital is the binding constraint.
§ IV. Mutually-Exclusive Alternatives — When "Accept All Positives" Is the Wrong Rule¶
So far we have evaluated projects one at a time. Many real engineering decisions are choose-one: build vs. buy, vendor A vs. vendor B, architecture α vs. architecture β. The rule changes.
For mutually-exclusive alternatives, pick the one with the highest NPV — not the highest IRR.
Why IRR misleads here¶
Consider two ways of solving the same internal-tooling problem:
| Project | Investment | NPV @ 10% | IRR |
|---|---|---|---|
| A (small) | $10,000 | $3,500 | 30% |
| B (large) | $80,000 | $15,000 | 16% |
A wins on IRR; B wins on NPV. Which should you choose?
Pick B. The firm exists to maximise wealth, not rate. A's "extra" $70K of capital (not invested in A) can presumably earn the MARR somewhere else — but no better than that, so B's larger absolute value wins. A 30% return on $10K is $3K of value; a 16% return on $80K is $15K of value. The firm cashes the $15K, not the rate.
This is the scale problem: IRR doesn't know how big the project is. A 100% IRR on a $1 investment is meaningless if there's a 12% IRR on a $1M investment on the table.
Incremental IRR analysis — reconciling NPV and IRR¶
If your CFO insists on rate-based language, there is a way to use IRR consistently with NPV: compute the IRR of the incremental cash flow (B − A). The question becomes: "Is the extra investment in B (beyond what A would cost) worth it at our MARR?"
For our two projects, the increment is: invest $70K more today to gain $11,500 more NPV at 10%. Solving for the rate that zeroes the incremental NPV gives an incremental IRR of ≈ 13.7%. Since 13.7% > 10% MARR, the additional capital deployed in B beats the MARR — so choose B. Consistent with the NPV verdict.
| Approach | Says |
|---|---|
| Pick higher NPV | B ($15K > $3.5K) |
| Pick higher IRR | A (30% > 16%) — wrong |
| Incremental IRR | B (13.7% > 10% MARR) |
When to use which
For mutually-exclusive alternatives, default to NPV — it never lies. Use incremental IRR only when you need to speak rate to a rate-speaking audience. Never use plain IRR ranking for mutually-exclusive alternatives of different sizes.
Different lives — the LCM trick¶
A second wrinkle: what if alternative A has a 4-year life and B has a 6-year life? Comparing their NPVs directly is unfair — B has 50% more time to generate value. Two clean fixes:
- Repeat each to the least common multiple of lives (here, 12 years), then compare NPVs of the replicated streams. Crude but transparent.
- Convert each to its Equivalent Annual Annuity (EAA) using \(A = \text{NPV} \cdot (A/P, i, n)\). The EAA is the project's "value per year" and is directly comparable across lives.
We will return to the EAA approach in Lecture 7 (mutually-exclusive decisions with unequal lives).
Class Discussion — A Real Decision From Your Team¶
Format. Pairs, 4 minutes each, then 6 minutes whole-room. Total ~10 minutes.
Prompt: Bring me a decision your team (or someone you've worked with) actually made — and tell me whether it would survive an NPV check.
Step 1 — Pair (4 min). Each partner describes one real decision: a tooling switch, a hire, a contract, a build-vs-buy, a refactor. Estimate the cash flows on paper: initial cost, annual benefits or savings, horizon. Use 10% as the discount rate unless your organisation has a stated MARR.
Step 2 — Compute (4 min). Rough NPV. Two columns are fine: cumulative cost, cumulative discounted benefit. Doesn't need to be precise — just defensible.
Step 3 — Whole room (6 min). Two pairs share. We'll surface one decision where NPV would have endorsed the call and one where it would have rejected it.
Questions to wrestle with while you compute:
| Question | Why it matters |
|---|---|
| What discount rate did you assume — and why? | Half the answer lives in the rate. |
| Which assumption is your NPV most sensitive to? | Sensitivity tells you where to investigate. |
| Did the decision look smart in NPV terms — or only with hindsight? | Hindsight bias inflates ex-post judgements. |
| What cash flows did you exclude — and why? | Sunk vs. opportunity vs. common — see Lecture 2. |
On hindsight bias
A decision that turned out well is not the same as a decision that was well-made. Conversely, a good decision can produce a bad outcome (the world is uncertain). NPV asks about the quality of the decision at the time it was made, with the information available then — not about the eventual outcome. That is exactly why it survives as a discipline.
§ V. Verdict — Elasticsearch vs SaaS, The Answer At Last¶
In Lecture 1 we posed a build-vs-buy puzzle and asked you to vote on instinct. Today we settle it with numbers.
The setup, recalled¶
| Alternative | Year 0 cost | Annual operating cost | Life |
|---|---|---|---|
| A. Build in-house (Elasticsearch) | $120,000 | $30,000 | 5 years |
| B. Subscribe to SaaS search | $20,000 | $80,000 | 5 years |
Same functionality, same horizon. Discount rate: 10%. Which alternative is economically preferred?
A quick undiscounted glance: A totals $120K + 5 × $30K = $270K; B totals $20K + 5 × $80K = $420K. A "wins" by $150K on raw totals — but the timing matters, and discounting will sharpen the picture.
The math — both alternatives, both PVs¶
Both alternatives have a one-time Year-0 outflow plus a 5-year uniform annual operating cost. At \(i = 10\%\), the uniform-series PV factor is \((P/A, 10\%, 5) = 3.7908\).
| Alt. | Build / Subscribe | Year 0 | Years 1–5 stream | PV total cost |
|---|---|---|---|---|
| A | Elasticsearch | $120,000 | $30,000 × 3.7908 = $113,723 | $233,723 |
| B | SaaS | $20,000 | $80,000 × 3.7908 = $303,261 | $323,261 |
Verdict. At a 10% discount rate, build in-house has a PV of total cost $89,538 lower than the SaaS option. Pick A — build the Elasticsearch cluster.
The discipline behind the verdict
Same conclusion as the back-of-envelope total — but now with discipline. The math protects us when the gap is narrower; it lets us say exactly how big the advantage is; and it gives us a starting point for sensitivity analysis.
How robust is this verdict? — A sensitivity preview¶
A verdict is only as good as the assumptions behind it. The right post-NPV question is always: what would have to change for the verdict to flip?
| Assumption | Change tested | New verdict |
|---|---|---|
| Discount rate up to 25% | from 10% → 25% | A still wins, by ≈ $43K |
| Discount rate down to 5% | from 10% → 5% | A still wins, by ≈ $129K |
| SaaS price falls 30% | annual $80K → $56K | B wins by ≈ $13K |
| Build cost overruns 50% | $120K → $180K | A still wins, by ≈ $30K |
| Useful life shrinks to 3 years | 5 → 3 years | B wins by ≈ $25K |
| Build needs a 2nd hire ($50K/yr op) | annual op $30K → $50K | A still wins, by ≈ $14K |
Two findings worth carrying away.
- The verdict is insensitive to the discount rate across any plausible MARR for a mid-cap firm. Even at 25% (well above any realistic MARR), A wins. This is the kind of robustness that earns a CFO's trust.
- The verdict is sensitive to useful life and SaaS pricing trajectory. If the team is unsure they will still be using this search system in 3 years — or if there is a credible chance SaaS prices fall 30% (a real possibility in a competitive market) — the verdict flips.
The meta-lesson. NPV gives you an answer. Sensitivity tells you which assumptions you should investigate before signing. Today's homework includes a sensitivity exercise; tomorrow's lecture (Lecture 6) makes sensitivity formal with break-even analysis, tornado diagrams, and scenario tables.
Bridge to Lecture 6¶
You now know how to reach a verdict. Tomorrow you will learn how to stress-test that verdict, find the assumptions that matter, and present a confidence range — not a single number — to a decision-maker. NPV is a number; engineering judgement is what tells the decision-maker whether to trust it.
Homework 5 — Investment Decisions in Practice¶
Due: Tuesday 2026-06-09, before Lecture 7. Late: −10% per day.
Part A — Six problems (40 points)¶
Solve and show working. Use 10% MARR unless the problem states otherwise.
- NPV. A project costs $150K today and returns $45K/yr for 5 years. Compute NPV at \(i = 8\%\), \(12\%\), and \(18\%\). At which rate does it break even (approximately)?
- IRR. Compute the IRR of: −$200K, +$60K, +$80K, +$80K, +$80K. (Use spreadsheet or trial-and-error to ±0.1%.) Compare with the NPV-based decision at MARR = 10%.
- Payback & discounted payback. For the project in Q1, compute simple payback and discounted payback at 10%. Comment on the difference.
- PI under capital rationing. Given four projects with (investment, NPV) of ($80K, $24K), ($120K, $30K), ($60K, $21K), ($100K, $22K), and a budget of $240K, choose the optimal portfolio using PI ranking. Show the PI for each and the total NPV captured.
- Mutually exclusive — same life. Two designs: D1 costs $50K and saves $18K/yr for 5 years; D2 costs $120K and saves $36K/yr for 5 years. At MARR = 10%, compute NPV and IRR of each. Which do you pick? Then compute the incremental IRR (D2 − D1) and confirm.
- Mutually exclusive — different lives. Alternative X has NPV $60K over 4 years; Alternative Y has NPV $80K over 6 years. At MARR = 10%, compute the Equivalent Annual Annuity (EAA) of each. Which is preferred per year?
Part B — Mini-case (40 points)¶
Propose two implementation alternatives for a real product (yours, your team's, or a hypothetical you sketch in one paragraph). The alternatives must be genuinely mutually exclusive — e.g., build vs. buy, two cloud architectures, two vendor contracts.
- Cash flows. Build a 5-year cash flow table for each alternative. Identify every line: initial cost, recurring operating cost, expected benefits or savings, retirement/migration costs. Use Lecture 2's TCO discipline.
- All four rules. Compute NPV, IRR, simple payback, and PI for each alternative at MARR = 10%.
- Verdict. Pick a winner and defend it in one page. State explicitly: which rule did you rely on, and why? Did any rules disagree?
- Sensitivity (≥ 3 variables). For your chosen alternative, test sensitivity to (at least) the discount rate, the largest cost line, and the useful life. Present a small table like the Elasticsearch sensitivity table above. Which assumption(s) actually drive the verdict?
Part C — Reflection (20 points)¶
Half-page maximum. Would you have made the same recommendation before running the numbers? Why or why not? If the answer is "yes, the numbers just confirmed what I already thought" — be honest about whether that means the analysis was redundant, or that your intuition was well-calibrated. If the answer is "no" — what cost or benefit had you been ignoring?
Submission. A single PDF (your problems + spreadsheet export + mini-case writeup + reflection) committed to the course repository at submissions/HW5/<your-name>.pdf.
Office hours. Monday 2026-06-08, 10:00–11:30, before Lecture 6.
Week 1 — What Today Bought You¶
Five lines, each a load-bearing claim the rest of the course will build on.
- Engineers make economic decisions whether they admit it or not. Refusing to engage with the money does not exempt you — it just means someone else makes your decisions for you.
- Every cost lives in a fixed/variable × direct/indirect × non-recurring/recurring cell, across five lifecycle phases, and only future, differential costs are decision-relevant.
- Money has time value — opportunity cost, risk, inflation — and the discount rate \(i\) is how we encode all three in one number.
- Cash-flow streams have closed-form factors: \(P/A\), \(F/A\), \(P/G\), geometric. Any messy real-world stream decomposes into a sum of these shapes, and PV is linear.
- The decision rule is NPV > 0. Everything else — IRR, payback, PI — is a useful footnote on the conversation NPV is having with shareholder wealth.
From here forward, every lecture is built on these five lines.
Key Vocabulary — Quick Reference¶
| Term | Plain-language meaning |
|---|---|
| NPV | Sum of every cash flow, each discounted to today at rate \(i\). |
| IRR | The rate that makes NPV = 0. Project's implied return. |
| MARR | Minimum Attractive Rate of Return — the firm's hurdle rate. |
| Payback period | Time until cumulative cash flow turns positive. |
| Discounted payback | Same, but on discounted cash flows. |
| Profitability Index (PI) | PV(future) ÷ |initial investment|. PI > 1 ⟺ NPV > 0. |
| Capital rationing | Choosing a subset of positive-NPV projects under a fixed budget — rank by PI. |
| Mutually exclusive | Choose at most one of several alternatives — pick highest NPV. |
| Incremental IRR | IRR of (B − A) cash flow; lets IRR language agree with NPV. |
| Equivalent Annual Annuity (EAA) | NPV converted to a per-year amount via \(A/P\) — for comparing different-life alternatives. |
| MIRR | Modified IRR — assumes reinvestment at the MARR, not the IRR. |
| Sensitivity analysis | Re-running the verdict under perturbed assumptions to see what flips it. |
Further Reading¶
- Park, C. S. Contemporary Engineering Economics — chapters on NPV, IRR, payback, PI, and incremental analysis (the primary text for this lecture).
- Boehm, B. W. (1981). Software Engineering Economics. Prentice Hall — Chapter 5 on present-worth analysis and incremental comparison.
- Brealey, R., Myers, S., & Allen, F. Principles of Corporate Finance — capital budgeting chapters; the canonical treatment of NPV vs. IRR.
- Ross, S., Westerfield, R., & Jaffe, J. Corporate Finance — clean treatment of the multiple-IRR problem and MIRR.
- (Forward pointer) Lecture 6 — Break-Even & Sensitivity Analysis formalises the sensitivity check you saw in § V; Lecture 7 — Mutually-Exclusive Alternatives treats unequal lives and the EAA in depth.
Prepared by Dr. Zhijiang Chen — Frostburg State University, Summer 2026.