NPV and IRR Concept Card Template (MBA Capital Budgeting)

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.

Document control

Contents

• When to use it
• The idea in plain language
• Formulas and notation
• Assumptions and limits
• How managers use these rules
• Worked example A (fully solved): Corvane Robotics Atlas line
• When IRR misleads
• Excel conventions and dated cash flows
• Worked example B (half solved): Northfield store refit
• Exercise C (unsolved): two automation options of different lives
• Decision diagram
• Connections to other cards
• Common mistakes and misconceptions
• Self-test questions
• My notes and next steps
• Sources and further reading
• Answer key
• Revision history

When to use it

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.

The idea in plain language

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

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

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

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

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).

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%).

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.

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

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%.

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.

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).

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

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).

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

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%.

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

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%.

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

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.

Connections to other cards

• 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

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

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

Sources and further reading

• 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

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

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.