# NPV and IRR Concept Card: [Your course, case or project]

> A concept card for the capital budgeting rules every MBA corporate finance course tests and every investment
> committee uses: net present value (NPV), internal rate of return (IRR), modified IRR (MIRR), payback, discounted
> payback and the profitability index (PI). It is for MBA students in the core finance course, admitted students who
> want a head start, and managers who sign off on capital requests. It explains each rule in plain language, gives the
> formula with a notation table, shows where IRR misleads (multiple IRRs, scale and timing conflicts, unequal lives),
> flags the Excel `NPV` timing convention that catches out new users, and covers dated cash flows with `XNPV` and
> `XIRR`. Worked example A is fully solved, Worked example B is half solved and Exercise C is left for you, with a
> separate answer key. Source basis: Fisher (1930), Dean (1951), Lorie and Savage (1955), Hirshleifer (1958), Lin
> (1976) and Graham and Harvey (2001), written up in original words with fictional examples (Corvane Robotics and
> Northfield Outdoor Co.). Replace every [bracketed] value and every row marked *Example*. This is a learning aid, not
> investment advice.

<!--
Guidance: How to use this template
- Read "When to use it" and "The idea in plain language" first. If you can explain why NPV is in money and IRR is a
  rate, and why that difference matters, you understand most of the card.
- Rebuild Worked example A in a spreadsheet with the Excel formulas shown. Getting the same numbers to the third
  decimal is the point: cases and exams reward speed and accuracy with these functions.
- Complete Worked example B by filling every [ ] before you look at the answer key. Do Exercise C on paper first.
- Answer the self-test from memory, then check the answer key. Log every miss in mba/mba-error-log and repeat the
  self-test after two days and after a week (spacing).
- Every company and number is fictional and was computed in Python (numpy-financial); the calculation script is kept
  with the library sources.
- Related templates: mba/mba-corporate-finance-guide, mba/mba-capm-wacc-card, mba/mba-dcf-valuation-card,
  mba/mba-capital-structure-card, mba/mba-options-real-options-card, mba/mba-decision-trees-card,
  mba/pre-mba-finance-primer, mba/pre-mba-excel-modeling-drills, mba/mba-formula-sheet, mba/mba-error-log,
  mba/mba-concept-flashcards, finance/investment-appraisal, finance/capex-request, finance/post-investment-review,
  academic/finance-case-analysis.
-->

## Document control

<!-- Guidance: Record which course and which spreadsheet you used, so that you (or a study partner) can reproduce every number. Update the version when you replace the example with your own project. -->

| Field | Value |
|---|---|
| Course, case or project | [Core corporate finance, session 3 / capital request for project name] |
| Learner or analyst | [Your name] |
| Discount rate source | [mba/mba-capm-wacc-card worksheet / company hurdle rate memo, date] |
| Spreadsheet used | [Excel / Google Sheets / LibreOffice Calc] |
| Reviewed by | [Study partner, tutor or finance business partner] |
| Version | [0.1] |
| Last updated | YYYY-MM-DD |

## Contents

- [When to use it](#when-to-use-it)
- [The idea in plain language](#the-idea-in-plain-language)
- [Formulas and notation](#formulas-and-notation)
- [Assumptions and limits](#assumptions-and-limits)
- [How managers use these rules](#how-managers-use-these-rules)
- [Worked example A (fully solved): Corvane Robotics Atlas line](#worked-example-a-fully-solved-corvane-robotics-atlas-line)
- [When IRR misleads](#when-irr-misleads)
- [Excel conventions and dated cash flows](#excel-conventions-and-dated-cash-flows)
- [Worked example B (half solved): Northfield store refit](#worked-example-b-half-solved-northfield-store-refit)
- [Exercise C (unsolved): two automation options of different lives](#exercise-c-unsolved-two-automation-options-of-different-lives)
- [Decision diagram](#decision-diagram)
- [Connections to other cards](#connections-to-other-cards)
- [Common mistakes and misconceptions](#common-mistakes-and-misconceptions)
- [Self-test questions](#self-test-questions)
- [My notes and next steps](#my-notes-and-next-steps)
- [Sources and further reading](#sources-and-further-reading)
- [Answer key](#answer-key)
- [Revision history](#revision-history)

## When to use it

<!-- Guidance: Start with the one-sentence rule. Then check the table: if your decision is not about committing cash now for cash later, another card fits better. -->

**One-sentence rule:** accept a project when the present value of the cash it will generate, discounted at the
opportunity cost of capital for projects of that risk, exceeds the cash it needs; among mutually exclusive projects,
choose the one with the highest NPV.

Use this card when a decision commits cash today (or over a few years) in exchange for uncertain cash flows later:
a new production line, a store refit, an IT system, a product launch, an acquisition of assets, a lease-or-buy
choice, or the replacement of old equipment. The same tools value a financing offer, a supplier discount or an
early-payment deal, because each of these is a stream of cash flows at different dates.

| Use when | Use another tool when |
|---|---|
| You can forecast incremental after-tax cash flows by period | You only have accounting profit (convert it to cash flow first, mba/mba-cash-flow-statement-card) |
| The risk of the project is similar to a known benchmark, so a discount rate can be set | The risk differs sharply from the firm's (estimate a project rate, mba/mba-capm-wacc-card) |
| The decision is now or never, or the timing is fixed | Waiting, expanding or abandoning has real value (mba/mba-options-real-options-card) |
| Projects are independent or a clear set of alternatives | You value a whole firm with a terminal value (mba/mba-dcf-valuation-card) |
| Capital is available at the cost of capital | Capital is truly rationed this period (use PI to rank, then check combinations) |

## The idea in plain language

<!-- Guidance: Be able to say each rule in one sentence without a formula. If you cannot, you will misuse it in a case discussion. -->

Money has a time value because it can be invested. A dollar received in three years is worth less than a dollar
today, because today's dollar could be invested in something of equal risk and grow. The **discount rate** is the
return investors give up by putting money into this project instead of a comparable investment in the capital
market. It is an opportunity cost, not the interest rate on whatever loan happens to fund the project (Fisher 1930;
Hirshleifer 1958).

- **NPV** turns every future cash flow into today's money and adds them up, including the initial outlay. A positive
  NPV is the amount by which the project is expected to increase the value of the firm today. It is in currency, so
  it answers "how much value?".
- **IRR** is the discount rate at which NPV equals zero. It answers "what rate of return does this cash-flow pattern
  imply?". For a normal project (outflow first, then inflows) a higher IRR than the cost of capital means a positive
  NPV, so the two rules agree on accept or reject.
- **MIRR** fixes one weakness of IRR: it compounds the inflows forward at a stated reinvestment rate (usually the cost
  of capital) and discounts the outflows at a finance rate, then solves for a single rate. It always has one answer
  (Lin 1976).
- **Payback** counts how many years it takes for undiscounted cash flows to repay the outlay. **Discounted payback**
  does the same with present values. Both ignore everything after the cut-off and neither measures value; they are
  crude liquidity and risk screens.
- **PI** divides the present value of future cash flows by the initial outlay. Above 1 means positive NPV. It is
  useful for ranking when capital is limited, because it measures value per unit of scarce capital.

The core point every MBA exam probes: **NPV is the rule that is consistent with maximising shareholder value**. The
other rules are useful summaries, and they agree with NPV for simple independent projects, but each can give the
wrong ranking in situations this card lists.

## Formulas and notation

<!-- Guidance: Learn the NPV formula and the IRR definition by heart. The others follow from them. Check that your spreadsheet uses the same timing (t = 0 for today). -->

$$
NPV = \sum_{t=0}^{T} \frac{CF_t}{(1+r)^t} = CF_0 + \frac{CF_1}{1+r} + \frac{CF_2}{(1+r)^2} + \dots + \frac{CF_T}{(1+r)^T}
$$

$$
\text{IRR is the rate } y \text{ that solves } \sum_{t=0}^{T} \frac{CF_t}{(1+y)^t} = 0
$$

$$
MIRR = \left( \frac{\sum_{t} CF_t^{+} (1+r_{re})^{T-t}}{\left| \sum_{t} CF_t^{-} / (1+r_{fin})^{t} \right|} \right)^{1/T} - 1
$$

$$
PI = \frac{\sum_{t=1}^{T} CF_t/(1+r)^t}{|CF_0|} \qquad EAA = \frac{NPV \cdot r}{1-(1+r)^{-T}}
$$

| Symbol | Meaning | Unit |
|---|---|---|
| $CF_t$ | Incremental after-tax cash flow at the end of period $t$ (negative for an outflow) | Currency |
| $CF_0$ | Cash flow today, usually the initial investment (negative) | Currency |
| $r$ | Opportunity cost of capital for this risk, per period | Rate per period |
| $T$ | Last period of the project | Periods |
| $y$ | Internal rate of return | Rate per period |
| $CF_t^{+}$, $CF_t^{-}$ | Positive and negative cash flows, treated separately in MIRR | Currency |
| $r_{re}$, $r_{fin}$ | Reinvestment rate and finance rate in MIRR (often both equal to $r$) | Rate per period |
| $PI$ | Profitability index | Ratio |
| $EAA$ | Equivalent annual annuity: the level annual amount with the same NPV over $T$ years | Currency per period |

Payback is the smallest $t$ at which cumulative cash flow turns non-negative, with linear interpolation inside the
year: payback $= (k-1) + \frac{\text{unrecovered amount at end of year } k-1}{CF_k}$, where $k$ is the year in which
the cumulative sum turns positive. Discounted payback uses $CF_t/(1+r)^t$ in place of $CF_t$.

## Assumptions and limits

<!-- Guidance: Every rule rests on the cash-flow forecast. Before you debate NPV versus IRR, ask whether the cash flows are incremental, after tax and complete. -->

1. **Cash flows, not profits.** Use incremental after-tax cash flows: include capital spending, changes in working
   capital and the tax effect of depreciation; exclude sunk costs and allocated overheads that do not change.
2. **Incremental.** Include side effects: lost sales of existing products (cannibalisation), the opportunity cost of
   assets the firm already owns, and costs that only exist because of the project.
3. **Financing is in the rate, not the cash flows.** Do not subtract interest from project cash flows when you discount
   at the weighted average cost of capital; the WACC already prices the debt (mba/mba-capm-wacc-card). Adjusted
   present value (APV) is the alternative when financing is unusual (mba/mba-capital-structure-card).
4. **Consistent inflation.** Discount nominal cash flows at a nominal rate, or real cash flows at a real rate. Mixing
   them is a classic error.
5. **One rate for the whole life.** The standard formula assumes a flat term structure and constant risk. When risk
   changes over time (a pilot followed by a scale-up), value the stages separately.
6. **End-of-period timing.** The formula and Excel's `NPV` place each cash flow at the end of its period. Cash that
   arrives through the year is valued slightly too low; mid-year discounting or `XNPV` with actual dates fixes this.
7. **Flexibility is ignored.** Static NPV assumes the plan is followed whatever happens. Options to expand, delay or
   abandon add value that the formula misses (Myers 1977 coined "real options").

## How managers use these rules

<!-- Guidance: Know what a real investment committee looks at, so that your case recommendation speaks its language. -->

Surveys of chief financial officers find that NPV and IRR are the two most widely used techniques, with payback still
common, especially in smaller firms (Graham and Harvey 2001). In practice a capital request (see
finance/capex-request) shows NPV at the hurdle rate, IRR, payback and a sensitivity table, and the committee reads
them together:

- **NPV** answers the value question and sets the ranking among alternatives.
- **IRR** gives a quick margin of safety: how far the discount rate could rise before NPV turns negative. Managers like
  it because it is a percentage they can compare with a hurdle rate.
- **Payback** is a liquidity and risk check: a venture-backed company with 18 months of runway cares when the cash
  comes back, not only whether it does.
- **PI** ranks projects when a fixed budget must be allocated across many small projects.
- **Sensitivity and break-even** (which revenue level or margin gives NPV = 0) usually decide the debate, because the
  forecast, not the arithmetic, is where the risk sits.

A good recommendation states the NPV, the IRR and the break-even assumption in one sentence, for example: "The line
adds about USD 2.3m of value at a 10% cost of capital; it stays positive unless the discount rate exceeds about 17%."

## Worked example A (fully solved): Corvane Robotics Atlas line

<!-- Guidance: Rebuild this table in a spreadsheet, row by row. Then change the discount rate and watch NPV and the profile chart move. -->

**Situation (fictional).** Corvane Robotics, a growth-stage warehouse-robotics company, considers a dedicated
production line for its Atlas picking robot. The line costs USD 12.0m today. Finance forecasts incremental after-tax
cash flows of USD 3.0m, 3.6m, 4.2m, 4.4m and 4.0m in years 1 to 5, with no salvage value. The cost of capital for
projects of this risk is 10% (see the WACC estimate in mba/mba-capm-wacc-card, Worked example A).

| *Example* year $t$ | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| Cash flow (USD m) | -12.000 | 3.000 | 3.600 | 4.200 | 4.400 | 4.000 |
| Discount factor at 10%, $1/1.1^t$ | 1.0000 | 0.9091 | 0.8264 | 0.7513 | 0.6830 | 0.6209 |
| Present value (USD m) | -12.0000 | 2.7273 | 2.9752 | 3.1555 | 3.0053 | 2.4837 |
| Cumulative cash flow | -12.000 | -9.000 | -5.400 | -1.200 | 3.200 | 7.200 |
| Cumulative present value | -12.0000 | -9.2727 | -6.2975 | -3.1420 | -0.1367 | 2.3469 |

Lay the cash flows out in a row: B2 holds year 0 (-12.0), C2 to G2 hold years 1 to 5, and B1 holds the rate (10%).

| *Example* measure | Result | Excel formula | Reading |
|---|---|---|---|
| NPV at 10% | USD 2.3469m | `=NPV(B1, C2:G2) + B2` | Accept: the line adds about USD 2.35m of value |
| IRR | 16.96% | `=IRR(B2:G2)` | Above 10%, consistent with positive NPV |
| MIRR (finance and reinvestment rate 10%) | 14.00% | `=MIRR(B2:G2, B1, B1)` | Lower than IRR because inflows earn 10%, not 16.96% |
| PI | 1.1956 | `=NPV(B1, C2:G2) / -B2` | Each dollar invested returns about 1.20 dollars of present value |
| Payback | 3.27 years | 3 + 1.2 / 4.4 | Cumulative cash turns positive in year 4 |
| Discounted payback | 4.06 years | 4 + 0.1367 / 2.4837 | Present values recover the outlay early in year 5 |

**MIRR by hand.** Compound each inflow to year 5 at 10%: $3.0 \times 1.1^4 + 3.6 \times 1.1^3 + 4.2 \times 1.1^2 +
4.4 \times 1.1 + 4.0 = 23.1059$. Then $MIRR = (23.1059/12.0)^{1/5} - 1 = 14.00\%$.

**NPV profile.** The chart plots NPV against the discount rate. Where the line crosses zero is the IRR (about 17%).
The slope tells you how sensitive the value is to the cost of capital: long-lived projects with late cash flows have
steeper profiles.

```vega-lite
{
  "$schema": "https://vega.github.io/schema/vega-lite/v5.json",
  "description": "NPV profile of the Corvane Robotics Atlas line (fictional data, USD m)",
  "width": 420,
  "height": 220,
  "data": {"values": [
    {"rate": 0, "npv": 7.2}, {"rate": 2, "npv": 6.047}, {"rate": 4, "npv": 4.996}, {"rate": 6, "npv": 4.035},
    {"rate": 8, "npv": 3.155}, {"rate": 10, "npv": 2.347}, {"rate": 12, "npv": 1.604}, {"rate": 14, "npv": 0.919},
    {"rate": 16, "npv": 0.287}, {"rate": 18, "npv": -0.298}, {"rate": 20, "npv": -0.84}, {"rate": 22, "npv": -1.343},
    {"rate": 24, "npv": -1.811}, {"rate": 26, "npv": -2.247}, {"rate": 28, "npv": -2.653}, {"rate": 30, "npv": -3.033}
  ]},
  "layer": [
    {"mark": {"type": "line", "point": true},
     "encoding": {
       "x": {"field": "rate", "type": "quantitative", "title": "Discount rate (%)"},
       "y": {"field": "npv", "type": "quantitative", "title": "NPV (USD m)"}
     }},
    {"data": {"values": [{"y": 0}]}, "mark": {"type": "rule", "strokeDash": [4, 4]},
     "encoding": {"y": {"field": "y", "type": "quantitative"}}}
  ]
}
```

**Conclusion.** At a 10% cost of capital the Atlas line has an NPV of USD 2.35m and should be accepted if the
forecast is credible. The IRR of 16.96% gives about 7 percentage points of headroom on the discount rate. Payback of
3.27 years matters if Corvane's investors set a cash-recovery limit; the value case does not depend on it.

## When IRR misleads

<!-- Guidance: These are the exam favourites. For each problem, be able to show the numbers and to say what to do instead. -->

### 1. Multiple IRRs (non-normal cash flows)

When the sign of the cash flows changes more than once, the IRR equation can have more than one solution (Lorie and
Savage 1955). *Example:* Corvane signs a site-services contract that costs USD 100k to set up, earns USD 230k in year 1
and then requires USD 132k of site restoration in year 2. The cash flows (-100, +230, -132, in USD k) have two IRRs,
**10% and 20%**. NPV is negative at 0% (-2.0), zero at 10%, positive between (0.189 at 15%), zero again at 20% and
negative above (-0.48 at 25%). An "IRR" output here is meaningless on its own. Use NPV at the cost of capital, or MIRR,
which has a single answer. Excel returns whichever root is nearest its starting guess: `=IRR(B2:D2, 0.05)` and
`=IRR(B2:D2, 0.25)` give different answers, which is a warning sign, not a feature.

### 2. Scale (mutually exclusive projects of different size)

IRR is a rate, so it ignores how much money is at stake. *Example:* Corvane can either retrofit an old cell (S) or
rebuild it fully (L); it cannot do both. Cost of capital 10%.

| *Example* project | $CF_0$ | $CF_1$ | $CF_2$ | $CF_3$ | NPV at 10% | IRR | PI |
|---|---|---|---|---|---|---|---|
| S, retrofit | -1.0 | 0.6 | 0.5 | 0.4 | 0.2592 | 25.35% | 1.2592 |
| L, rebuild | -4.0 | 1.6 | 1.8 | 2.0 | 0.4448 | 15.98% | 1.1112 |
| L minus S (incremental) | -3.0 | 1.0 | 1.3 | 1.6 | 0.1856 | 13.22% | |

S has the higher IRR and PI, L the higher NPV. Choose **L**: the extra USD 3.0m invested in L earns 13.22%, above the
10% cost of capital, so it adds USD 0.19m of value. The incremental IRR (`=IRR` on the L minus S row) is also the
**crossover rate**: below 13.22% L has the higher NPV, above it S does. The chart shows the two profiles crossing.

```vega-lite
{
  "$schema": "https://vega.github.io/schema/vega-lite/v5.json",
  "description": "NPV profiles of two mutually exclusive projects, S and L (fictional data, USD m)",
  "width": 420,
  "height": 220,
  "data": {"values": [
    {"rate": 0, "project": "S retrofit", "npv": 0.5}, {"rate": 4, "project": "S retrofit", "npv": 0.3948},
    {"rate": 8, "project": "S retrofit", "npv": 0.3018}, {"rate": 12, "project": "S retrofit", "npv": 0.219},
    {"rate": 16, "project": "S retrofit", "npv": 0.1451}, {"rate": 20, "project": "S retrofit", "npv": 0.0787},
    {"rate": 24, "project": "S retrofit", "npv": 0.0188}, {"rate": 28, "project": "S retrofit", "npv": -0.0353},
    {"rate": 0, "project": "L rebuild", "npv": 1.4}, {"rate": 4, "project": "L rebuild", "npv": 0.9807},
    {"rate": 8, "project": "L rebuild", "npv": 0.6124}, {"rate": 12, "project": "L rebuild", "npv": 0.2871},
    {"rate": 16, "project": "L rebuild", "npv": -0.0017}, {"rate": 20, "project": "L rebuild", "npv": -0.2593},
    {"rate": 24, "project": "L rebuild", "npv": -0.49}, {"rate": 28, "project": "L rebuild", "npv": -0.6977}
  ]},
  "mark": {"type": "line", "point": true},
  "encoding": {
    "x": {"field": "rate", "type": "quantitative", "title": "Discount rate (%)"},
    "y": {"field": "npv", "type": "quantitative", "title": "NPV (USD m)"},
    "color": {"field": "project", "type": "nominal", "title": "Project"}
  }
}
```

### 3. Timing (same outlay, different cash-flow shapes)

A project that returns cash early has a higher IRR than one that returns more cash later, even when the later one is
worth more at the firm's cost of capital. *Example:* two USD 3.0m software projects at Corvane. Early returns 2.0, 1.2,
0.6, 0.3; Late returns 0.2, 0.6, 1.5, 2.8 (USD m, years 1 to 4).

| *Example* project | NPV at 10% | IRR |
|---|---|---|
| Early | 0.4656 | 19.80% |
| Late | 0.7171 | 17.48% |
| Crossover rate (IRR of Late minus Early) | | 15.05% |

At 10% Late is worth more; only above 15.05% would Early win. IRR implicitly assumes interim cash can be reinvested at
the project's own IRR, which flatters early cash flows; NPV assumes reinvestment at the cost of capital, which is the
market's opportunity rate (Hirshleifer 1958).

### 4. Unequal lives

When alternatives last different lengths of time and the activity will continue, compare them over a common horizon
(replacement chain) or by their **equivalent annual annuity**: $EAA = NPV \times r / (1 - (1+r)^{-T})$, Excel
`=PMT(r, T, -NPV)`. Exercise C practises this.

### 5. Borrowing-type cash flows

If the first cash flow is positive and later ones negative (a customer prepayment, a loan you receive), a high IRR is
**bad**: it is the rate you pay. Read the sign pattern before you read the IRR.

## Excel conventions and dated cash flows

<!-- Guidance: Most wrong answers in spreadsheet exams come from function conventions, not from finance. Test each function on a case where you know the answer. -->

| Function | Convention | Pitfall and fix |
|---|---|---|
| `=NPV(rate, values)` | Discounts the **first** value one full period: it treats the range as years 1, 2, 3 and so on | Put only years 1 to T in the range and add year 0 outside: `=NPV(B1, C2:G2) + B2`. Including year 0 in the range divides the true NPV by $(1+r)$ |
| `=IRR(values, [guess])` | Needs at least one negative and one positive value; equally spaced periods | Returns one root only; with several sign changes, check the NPV profile |
| `=MIRR(values, finance_rate, reinvest_rate)` | Outflows discounted at the finance rate, inflows compounded at the reinvestment rate | State both rates in your write-up |
| `=XNPV(rate, values, dates)` | Actual dates, annual rate, day count actual/365, first value **not** discounted | Dates must be real date values; the first date is the valuation date |
| `=XIRR(values, dates, [guess])` | Annual rate consistent with `XNPV` | Use for venture, private equity and lease cash flows on irregular dates |
| `=PMT(rate, nper, pv)` | Returns a negative number for a positive `pv` | Use `=PMT(r, T, -NPV)` for an EAA in positive form |

Microsoft's documentation of `NPV` states that the investment begins one period before the first value and that an
initial investment made at the start should be added outside the function (checked October 2026 on
support.microsoft.com; check the current page for your version).

*Example of the timing pitfall.* For the Atlas line, `=NPV(10%, B2:G2)` returns USD 2.1336m instead of 2.3469m. The
difference is exactly the factor 1.1: $2.3469 / 1.1 = 2.1336$. Under exam pressure this error looks plausible, which is
why you should always check one discount factor by hand.

**Dated cash flows.** *Example:* Corvane pays USD 5.0m for a fleet of demonstration robots on 15 January 2027 and
expects lease income of USD 1.5m on 30 September 2027, 2.2m on 30 June 2028 and 2.6m on 31 March 2029 (258, 532 and 806
days after the start).

| *Example* method | Result | Excel formula |
|---|---|---|
| XNPV at 10% (actual dates) | USD 0.4235m | `=XNPV(10%, B2:E2, B3:E3)` |
| XIRR | 15.94% | `=XIRR(B2:E2, B3:E3)` |
| NPV if the flows were wrongly treated as one year apart | USD 0.1352m | `=NPV(10%, C2:E2) + B2` |
| IRR on the same wrong annual spacing | 11.40% | `=IRR(B2:E2)` |

Treating irregular dates as annual periods understates the value here because the cash actually arrives sooner. Use
dates whenever cash flows do not fall on equal intervals.

## Worked example B (half solved): Northfield store refit

<!-- Guidance: Fill each [ ] in order, using the Excel formula given. Compare with the answer key only after you have a number in every gap. -->

**Situation (fictional).** Northfield Outdoor Co. considers refitting a flagship store for USD 2.4m. Incremental
after-tax cash flows are USD 0.7m, 0.8m, 0.9m and 0.9m in years 1 to 4; the refit has no value after year 4. The cost
of capital is 9%.

| *Example* year | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| Cash flow (USD m) | -2.400 | 0.700 | 0.800 | 0.900 | 0.900 |
| Discount factor at 9% | 1.0000 | 0.9174 | 0.8417 | [ ] | 0.7084 |
| Present value (USD m) | -2.4000 | 0.6422 | [ ] | 0.6950 | 0.6376 |

1. NPV at 9% = [ ] (`=NPV(9%, C2:F2) + B2`).
2. IRR = [ ] (`=IRR(B2:F2)`). Does it agree with the NPV decision?
3. MIRR with finance and reinvestment rate 9% = 11.71% (`=MIRR(B2:F2, 9%, 9%)`). Explain in one sentence why it is
   below the IRR.
4. PI = [ ].
5. Payback = [ ] years. Discounted payback = 3.61 years.
6. Northfield's board uses a three-year payback rule. Does the refit pass? Would you recommend it, and what would you
   ask the board to reconsider?

## Exercise C (unsolved): two automation options of different lives

<!-- Guidance: Work on paper first, then confirm in a spreadsheet. The trap is to compare the two NPVs directly. -->

**Situation (fictional).** Corvane must automate the packing area of its own plant and will keep doing so
indefinitely. Two mutually exclusive systems are available; the cost of capital is 12%.

| *Example* option | Year 0 | Year 1 | Year 2 | Year 3 | Year 4 |
|---|---|---|---|---|---|
| P, conveyor system (3-year life) | -6.0 | 2.8 | 2.8 | 2.8 | |
| Q, robotic cell (4-year life) | -9.0 | 1.2 | 3.4 | 4.2 | 4.8 |

1. Compute NPV and IRR for each option (Excel formulas included).
2. Which has the higher NPV? Which has the higher IRR?
3. Compute each option's equivalent annual annuity with `=PMT(12%, T, -NPV)`.
4. Assuming each system can be replaced on identical terms, which do you choose, and why does the direct NPV comparison
   mislead?
5. Check your answer with a replacement chain over 12 years (P four times, Q three times).

## Decision diagram

<!-- Guidance: Use this flow to decide which rule to trust in a case. It mirrors the sections above. -->

Start from the cash flows and the rate, then let the structure of the decision pick the rule. Every path ends with
sensitivity analysis, because the forecast is where most of the risk sits.

```mermaid
flowchart TD
    A["Forecast incremental after-tax cash flows"] --> B["Set discount rate for this risk"]
    B --> C{"Independent project?"}
    C -->|Yes| D{"Cash-flow signs change more than once?"}
    D -->|No| E["NPV > 0 and IRR > r agree: accept"]
    D -->|Yes| F["Use NPV or MIRR, ignore raw IRR"]
    C -->|No, mutually exclusive| G{"Same life?"}
    G -->|Yes| H["Pick highest NPV, check with incremental IRR"]
    G -->|No| I["Compare EAA or a replacement chain"]
    E --> J{"Capital rationed this period?"}
    H --> J
    I --> J
    F --> J
    J -->|Yes| K["Rank by PI, then test combinations"]
    J -->|No| L["Run sensitivity and break-even, then recommend"]
    K --> L
```

## Connections to other cards

<!-- Guidance: Each link answers a question this card leaves open. Follow the one that matches your next case. -->

- **Where the discount rate comes from:** mba/mba-capm-wacc-card (CAPM cost of equity, after-tax cost of debt, target
  weights, project versus company rate).
- **Valuing a whole business:** mba/mba-dcf-valuation-card adds free cash flow forecasting and a terminal value; the
  NPV logic is the same.
- **Financing side effects:** mba/mba-capital-structure-card covers APV, tax shields and when the WACC shortcut breaks.
- **Flexibility:** mba/mba-options-real-options-card and mba/mba-decision-trees-card value the options static NPV
  ignores.
- **From profit to cash:** mba/mba-cash-flow-statement-card and mba/mba-three-statements-card.
- **Course map:** mba/mba-corporate-finance-guide places this card in session 3 of the roadmap.
- **Practice formats:** finance/investment-appraisal and finance/capex-request are working documents built on the same
  rules; finance/post-investment-review checks the forecast after the fact.

## Common mistakes and misconceptions

<!-- Guidance: Read the list before every exam. Tick the ones you have made and log them in mba/mba-error-log. -->

1. **Putting year 0 inside Excel's `NPV`.** The result is too small by the factor $(1+r)$. Add $CF_0$ outside.
2. **Choosing between exclusive projects by IRR.** Scale and timing differences make the higher-IRR project the
   lower-value one. Use NPV or the incremental IRR.
3. **Discounting at the cost of the loan that funds the project.** The discount rate reflects the project's risk, not
   the source of this year's cash. A safe-looking 6% bank loan does not make a risky robot line a 6% project.
4. **Including interest in the cash flows and then discounting at WACC.** That counts financing twice.
5. **Counting sunk costs.** Last year's feasibility study is spent whatever you decide; it is not part of the decision.
6. **Ignoring working capital.** Inventory and receivables tie up cash at the start and are released at the end; leaving
   them out overstates NPV.
7. **Mixing nominal cash flows with a real rate** (or the reverse). Keep both on the same basis.
8. **Treating payback as a value measure.** A project can pay back in two years and still destroy value, or pay back in
   six and create a lot.
9. **Reading a high IRR on financing-type flows as good.** For a loan you receive, the IRR is your cost.
10. **Trusting a single IRR when signs change twice.** Plot the profile; there may be two IRRs or none.
11. **Comparing NPVs of projects with different lives that will be repeated.** Use EAA or a common horizon.
12. **Believing IRR "assumes reinvestment at the IRR" is the reason to reject IRR.** The deeper issue is that IRR is a
    rate, not a measure of value; the reinvestment point explains why rankings differ, not which rule creates value.

## Self-test questions

<!-- Guidance: Answer in writing, from memory, before scrolling to the answer key. Repeat after two days and after a week. -->

1. In one sentence each, define NPV and IRR, and say what unit each is in.
2. A project's NPV at 11% is USD 0.8m. Without computing anything, is its IRR above or below 11%? What assumption do you
   need?
3. A colleague's spreadsheet shows `=NPV(8%, B5:H5)` where B5 is the initial outlay. The result is USD 4.20m. What is
   the correct NPV?
4. Why can a project have two IRRs? Give a business situation that produces this pattern.
5. Project X: NPV 1.2m, IRR 28%. Project Y: NPV 2.0m, IRR 18%. They are mutually exclusive and the cost of capital is
   10%. Which do you choose and what number would convince a sceptic who prefers X?
6. What does MIRR assume about interim cash flows, and why is it always single-valued?
7. When is the profitability index the right ranking tool, and when can it mislead?
8. A machine with a 3-year life has NPV 0.9m; an alternative with a 5-year life has NPV 1.2m; cost of capital 10%.
   Both will be replaced indefinitely. Which is better? Show the EAA calculation.
9. Cash flows occur on irregular dates over 26 months. Which Excel functions do you use and what is the first date?
10. Interleaving: for each, say which tool you use. (a) Comparing two leases with monthly payments on fixed dates.
    (b) Deciding whether to delay a launch by a year to learn about demand. (c) Valuing a whole company.

## My notes and next steps

<!-- Guidance: This section is yours. Rate your confidence honestly and plan the next review. -->

| Item | My entry |
|---|---|
| Confidence with NPV, IRR and the Excel functions (1 = lost, 5 = could teach it) | [ ] |
| The pitfall I am most likely to fall into | [ ] |
| A real capital decision I can apply this to (work, past job, a case) | [ ] |
| Mistakes logged in mba/mba-error-log | [Date and short description] |
| Next review date (spacing: +2 days, +1 week, +1 month) | YYYY-MM-DD |
| Next card to study | [mba/mba-capm-wacc-card / mba/mba-dcf-valuation-card] |

## Sources and further reading

<!-- Guidance: The primary works are listed for attribution and depth. Textbooks are named only; read them for more practice. -->

- Fisher, I. (1930). *The Theory of Interest*. Macmillan.
- Dean, J. (1951). *Capital Budgeting*. Columbia University Press.
- Lorie, J. H., and Savage, L. J. (1955). Three problems in rationing capital. *The Journal of Business*, 28(4), 229-239.
- Hirshleifer, J. (1958). On the theory of optimal investment decision. *Journal of Political Economy*, 66(4), 329-352.
- Lin, S. A. Y. (1976). The modified internal rate of return and investment criterion. *The Engineering Economist*,
  21(4), 237-247.
- Myers, S. C. (1977). Determinants of corporate borrowing. *Journal of Financial Economics*, 5(2), 147-175.
- Graham, J. R., and Harvey, C. R. (2001). The theory and practice of corporate finance: evidence from the field.
  *Journal of Financial Economics*, 60(2-3), 187-243.
- Microsoft Support. NPV function, XNPV function, IRR function, XIRR function, MIRR function (support.microsoft.com,
  checked 2026-10-28).
- Textbooks (by name only): Berk, J., and DeMarzo, P., *Corporate Finance*; Brealey, R. A., Myers, S. C., and Allen, F.,
  *Principles of Corporate Finance*; Ross, S. A., Westerfield, R. W., and Jaffe, J., *Corporate Finance*.
- Open resource: OpenStax, *Principles of Finance* (CC BY-NC-SA 4.0), chapters on time value of money and capital budgeting.

## Answer key

<!-- Guidance: Check only after you have written your own answers. For each miss, write down why your first answer was wrong. -->

**Worked example B.** Discount factor year 3 = $1/1.09^3 = 0.7722$; present value year 2 = $0.8 \times 0.8417 =
0.6733$. (1) NPV = USD 0.2481m. (2) IRR = 13.42%, above 9%, so both rules accept. (3) MIRR assumes inflows are
reinvested at 9% rather than at 13.42%, so the compounded terminal value is smaller. (4) PI = $(0.2481 + 2.4)/2.4 =
1.1034$. (5) Payback = exactly 3.00 years ($0.7 + 0.8 + 0.9 = 2.4$); discounted payback 3.61 years. (6) It passes a
three-year undiscounted payback rule only just, and fails it on a discounted basis. Recommend the refit on NPV (positive
at 9%), and ask the board whether the payback rule reflects a real liquidity constraint; if not, NPV should decide.

**Exercise C.** (1) P: NPV at 12% = USD 0.7251m (`=NPV(12%, C2:E2) + B2`), IRR = 18.91%. Q: NPV = USD 0.8219m, IRR =
15.55%. (2) Q has the higher NPV, P the higher IRR. (3) EAA P = $0.7251 \times 0.12 / (1 - 1.12^{-3}) = 0.3019$
per year; EAA Q = $0.8219 \times 0.12 / (1 - 1.12^{-4}) = 0.2706$ per year (`=PMT(12%, 3, -0.7251)` and
`=PMT(12%, 4, -0.8219)`). (4) Choose P: it creates more value per year of service. The direct comparison favours Q only
because Q's NPV covers four years and P's covers three. (5) Over 12 years, P repeated four times has NPV USD 1.8701m and
Q repeated three times has NPV USD 1.6761m, confirming P.

**Self-test.**

1. NPV is the present value of all incremental cash flows at the cost of capital, in currency; IRR is the discount rate
   that sets NPV to zero, a percentage per period.
2. Above 11%, provided the project has normal cash flows (one outflow, then inflows), so the NPV profile slopes down.
3. The formula discounted the outlay one period too many, so the true NPV is $4.20 \times 1.08 = 4.536$, about USD 4.54m.
4. With more than one sign change the NPV polynomial can have more than one root. Examples: a mine or plant with a
   large closure or clean-up cost at the end; a contract with an upfront fee, inflows and a final restoration payment.
5. Choose Y, the higher NPV. Show the incremental cash flows (Y minus X): their IRR is above 10% (equivalently, Y adds
   0.8m more value at 10%), so the extra capital in Y earns more than its cost.
6. Inflows are compounded at the stated reinvestment rate and outflows discounted at the finance rate; after this one
   outflow and one terminal value remain, so there is exactly one solution.
7. When capital is rationed in a single period and projects are independent and divisible, PI ranks value per unit
   of capital. It misleads for mutually exclusive projects of different size, and with rationing over several
   periods (then use integer programming or test combinations).
8. EAA 3-year = $0.9 \times 0.10/(1-1.1^{-3}) = 0.3619$; EAA 5-year = $1.2 \times 0.10/(1-1.1^{-5}) = 0.3166$. The
   3-year machine is better.
9. `XNPV` and `XIRR`; the first date is the valuation date and its cash flow is not discounted.
10. (a) `XNPV` or monthly `NPV` with a monthly rate; (b) real options or a decision tree; (c) DCF valuation with a
    terminal value.

## Revision history

<!-- Guidance: Add a row each time you change the example, the rate or the conclusions. -->

| Version | Date | Author | Change summary |
|---|---|---|---|
| 0.1 | YYYY-MM-DD | [Your name] | First pass through the card |
| 0.2 | YYYY-MM-DD | [Your name] | *Example:* Replaced Atlas example with own capital request, added sensitivity |

## Standard and licence note

Based on the standard capital budgeting theory in the primary works listed above (Fisher 1930, Dean 1951, Lorie and
Savage 1955, Hirshleifer 1958, Lin 1976) and on Microsoft's published documentation of the spreadsheet functions,
checked in October 2026. The text, examples, numbers and questions are original; no textbook text is copied or
paraphrased, and Corvane Robotics and Northfield Outdoor Co. are fictional. The template is independent and not
affiliated with or endorsed by any business school, publisher or software vendor. Check the current documentation of
your spreadsheet before you rely on a function convention.
Template licence: MIT.