Capital Budgeting: NPV, Incremental Cash Flows, Project Risk, and Real Options

Capital budgeting decides which long-lived investments a firm should undertake. The central question is whether the future benefits are worth more today than the resources committed. Answering it requires the right incremental cash flows, discount rates that match their risks, and a comparison of all feasible alternatives—including waiting.

Learning goals and conventions

You should finish able to build incremental after-tax project cash flows, calculate NPV, choose a project-specific opportunity cost, explain conflicting IRR and PI rankings, and distinguish a positive immediate-investment NPV from an optimal exercise decision.

Use year-end nominal cash flows in one currency, annual effective rates, and no financing effects in the basic operating-project example. All future cash flows are forecasts unless stated otherwise. A constant discount rate is a simplifying assumption, not a requirement of NPV analysis.

Symbol	Meaning
$CF_t$	Incremental project cash flow at date $t$
$r$	Constant opportunity cost per year in simple examples
$r_{0,t}$	Annualized maturity-specific discount rate from today to $t$
$\tau$	Relevant marginal tax rate
$D_t$	Tax depreciation for the period
$CapEx_t$	Capital expenditure
$\Delta NWC_t$	Change in operating net working capital

1. The NPV rule: maximize value added today

Represent the project as dated cash flows:

$$(CF_0,CF_1,\ldots,CF_T).$$

With a constant rate:

$$NPV_0=CF_0+\sum_{t=1}^{T}\frac{\mathbb E_0[CF_t]}{(1+r)^t}.$$

The initial investment is usually negative. NPV is the value of future incremental cash flows minus the resources committed, measured at a common date. A project earning exactly its opportunity cost has NPV zero; it has not failed merely because it adds no value above that benchmark.

Under the standard value-maximization model, with correct cash flows and discounting:

• An independent project with positive NPV adds value; one with negative NPV reduces value. At zero NPV, the model gives financial indifference.
• Accept all positive-NPV independent projects if funding and other resources are unconstrained.
• For mutually exclusive alternatives on a consistent comparison basis, choose the feasible alternative with the highest positive NPV, including the option to do nothing.

Budget constraints, dependencies, different replacement opportunities, and timing options require comparing feasible combinations or policies rather than applying isolated accept/reject rules.

Discount rates across maturities versus across periods

The outline’s expression

$$NPV_0=CF_0+\sum_{t=1}^{T}\frac{\mathbb E_0[CF_t]}{(1+r_{0,t})^t}$$

uses an annualized rate from date 0 to each maturity $t$. The two-year rate applies to the whole two-year interval.

If instead $k_1,k_2,\ldots$ are appropriate successive one-period discount rates in a deterministic-rate framework, the date-$t$ denominator is

$$\prod_{s=1}^{t}(1+k_s),$$

rather than $(1+k_t)^t$. For example, 121 due after two years discounted at 10% in each year has PV 100. With first-year rate 10% and second-year rate 5%, its PV is $121/(1.10\times1.05)\approx104.76$.

Risk-adjusted rates must be consistent with the particular cash-flow risks. Changing risk or uncertain state-dependent discounting can require a richer model than a mechanically assigned rate for each project phase.

2. Incremental cash flow is a with-versus-without comparison

The relevant quantity is

$$CF_t^{\mathrm{incremental}}=CF_t^{\mathrm{firm\ with\ project}} -CF_t^{\mathrm{firm\ without\ project}}.$$

This is not always the same as revenue or profit recorded in a project’s internal budget. Include consequences elsewhere in the business when caused by the decision.

Item	Treatment	Reason
Research fee already paid and unrecoverable	Exclude from today’s decision	Sunk cost does not change between alternatives
Owned land that could be sold	Include foregone after-tax sale proceeds	The project consumes an alternative use of a resource
Sales lost from an existing product	Include lost incremental contribution	Cannibalization changes the firm’s total cash flows
Existing overhead allocation	Include only the change caused by the project	An allocation is not automatically an incremental cost
Extra support staff or maintenance	Include if required by the project	These are incremental outflows
Financing interest	Match the valuation method	Do not subtract it in unlevered cash flow and count financing again in WACC

Worked opportunity-cost example

A feasibility study cost 20 last month and cannot be recovered. The project would use a plot of land whose after-tax sale proceeds today would be 150. Even if the land’s historical accounting cost was 40, the relevant date-0 opportunity cost is 150. Exclude the sunk study fee and include the foregone sale value.

If selling the land later is part of the project, include its forecast after-tax terminal sale proceeds too. Including its initial opportunity cost and its later recovery is consistent; ignoring the initial alternative sale would overstate value.

3. From earnings to after-tax operating cash flow

Define cash operating surplus before depreciation as revenue minus cash operating costs:

$$EBITDA_t=Revenue_t-CashOperatingCosts_t, \qquad EBIT_t=EBITDA_t-D_t.$$

In a simplified setting where taxes equal $\tau EBIT_t$ and tax deductions can be used immediately:

$$OCF_t=EBIT_t(1-\tau)+D_t =EBITDA_t(1-\tau)+\tau D_t.$$

Unlevered free cash flow is then

$$FCF_t=EBIT_t(1-\tau)+D_t-CapEx_t-\Delta NWC_t,$$

or equivalently

$$FCF_t=EBITDA_t(1-\tau)+\tau D_t-CapEx_t-\Delta NWC_t.$$

This resolves an ambiguity in the outline: adding $\tau D_t$ is correct when “operating profit” means the cash surplus before depreciation. If operating profit means EBIT after depreciation, add back all of $D_t$, not just the tax shield. Do not mix the two forms.

Depreciation is not a cash payment when recorded, but deductible tax depreciation can reduce actual tax payments. Book depreciation and tax depreciation may differ. If the firm cannot immediately use a deduction, its tax benefit may be delayed or limited; it is not automatically worth $\tau D_t$ in current cash.

Working capital is a use of cash

Operating NWC commonly includes inventory and operating receivables minus operating payables. Investing in more inventory or waiting longer for customers to pay uses cash, so a positive $\Delta NWC_t$ reduces FCF. A release of working capital has a negative change and adds cash.

Recover only amounts realistically collectible or saleable at termination. Full recovery is an assumption, not a rule: obsolete inventory, bad debts, or continuing operations can prevent it. Cash and financing debt are handled consistently outside operating NWC in this framework.

Terminal assets and after-tax sale proceeds

Under a simplified tax treatment, for sale proceeds $S$ and remaining tax book value $BV$:

$$AfterTaxSalvage=S-\tau(S-BV).$$

A 160 sale with tax book value 100 and tax rate 25% yields 145 after tax. The treatment of gains, losses, and deduction usability must match the applicable assumptions; the formula is not a universal tax rule.

4. Complete worked project: machine, taxes, and working capital

A machine costs 1,000 today and requires initial NWC of 100. It operates for three years. Annual revenue is 700 and cash operating cost is 250. Assume tax depreciation of 300 each year, tax rate 25%, immediate use of deductions, no later CapEx, no further NWC changes during operation, and a 10% appropriate discount rate.

At year 3, sell the machine for 160. Its remaining tax book value is $1{,}000-3(300)=100$, giving after-tax proceeds 145. Recover the full 100 of NWC at that date.

Annual cash operating surplus is 450, EBIT is 150, tax is 37.50, and operating cash flow is $150(0.75)+300=412.50$.

Cash-flow component	Date 0	Year 1	Year 2	Year 3
Machine purchase	-1,000	0	0	0
Operating cash flow	0	412.50	412.50	412.50
Operating NWC cash effect	-100	0	0	+100
After-tax machine sale	0	0	0	+145
Total FCF	-1,100	412.50	412.50	657.50

Date                0              1              2              3
                     |--------------|--------------|--------------|
Cash flow         -1,100          +412.50        +412.50        +657.50

$$NPV=-1{,}100+\frac{412.5}{1.10}+\frac{412.5}{1.10^2} +\frac{657.5}{1.10^3}\approx109.90.$$

Using accounting EBIT of 150 as annual cash flow would miss the depreciation add-back. Forgetting initial NWC would make the project look 100 more valuable; forgetting its recovery would understate terminal value. Neither error is fixed by refining the discount rate.

5. Project-specific risk and financing consistency

The opportunity cost is the expected return available on investments with comparable risk. A company-wide rate can misprice a safer expansion or a riskier new business.

Under CAPM, a project’s unlevered required return can be modeled as

$$r_A=r_f+\beta_A\bigl(\mathbb E[R_M]-r_f\bigr).$$

For risk-free return 3%, market premium 5%, and project asset beta 0.8, the asset cost is 7%. A beta-1.6 project instead has required return 11%. Their risks differ even if both are funded by the same parent company.

Estimate business risk from comparable operations and adjust for leverage as needed. Cost of equity belongs with shareholder cash flows; WACC belongs with appropriate free cash flow to the firm under consistent financing assumptions; adjusted present value separates unlevered project value from financing effects.

R&D, exploration, and production may expose cash flows to different risks. Do not automatically apply a high production-stage rate to a nearly certain future cost, or assume every uncertain early-stage expense should receive a higher discount rate. Higher rates reduce the PV magnitude of negative future cash flows, which can perversely make a risky-cost project appear more attractive. Model each exposure consistently.

Keep expected nominal cash flows with nominal discount rates and real cash flows with real rates. Avoid double counting risk by pessimistically reducing cash flows for a risk already charged again in the discount rate without a coherent model.

6. Alternative decision rules: useful summaries, incomplete rankings

Payback and discounted payback

Payback asks when cumulative cash inflows recover the initial outlay. It emphasizes liquidity, but ignores time value and all cash flows after recovery or a chosen cutoff. Discounted payback corrects time value but still ignores later value.

Consider two projects costing 100, with annual cash flows:

Project	Year 1	Year 2	Year 3	Ordinary payback
Quick	60	40	0	2 years
Lasting	0	50	100	3 years under year-end timing

At 10%, Quick has NPV approximately -12.40; Lasting has NPV approximately 16.45. A two-year payback cutoff favors the project that destroys value and rejects the one that adds it.

Some payback calculations interpolate within a year. That requires an assumption about receipt timing: with uniform year-3 receipts, Lasting’s interpolated ordinary payback would be 2.5 years, rather than three year-end dates. NPV should still use timing assumptions appropriate to the cash flows.

Profitability index and scale

For a conventional project with one initial investment $I_0=-CF_0>0$:

$$PI=\frac{PV(\text{future net cash flows})}{I_0}, \qquad NPV=I_0(PI-1).$$

A PI above one corresponds to positive NPV under this definition. Some sources instead define a net-value-per-dollar index, so identify the convention.

At 10%, project A costs 100 and pays 130 in one year: PI is about 1.1818 and NPV 18.18. Project B costs 1,000 and pays 1,200 in one year: PI is about 1.0909 and NPV 90.91. If mutually exclusive and affordable, B creates more value despite its lower ratio.

Capital rationing is a combination problem

Suppose the date-0 budget is 1,000 and projects are indivisible and independent:

Project	Initial cost	PV of future cash flows	NPV	PI
A	600	720	120	1.20
B	500	595	95	1.19
C	500	595	95	1.19

Choosing highest PI first selects A and leaves an unusable 400. B plus C fits the budget and creates NPV 190, exceeding A’s 120. PI ranking can help with divisible projects and one funding constraint, but does not generally solve indivisible, dependent, or multi-period budgeting problems.

IRR: a rate, not value added

IRR solves

$$0=CF_0+\sum_{t=1}^{T}\frac{CF_t}{(1+IRR)^t}.$$

For a conventional investment with an initial outflow followed by nonnegative inflows, a unique economically relevant IRR can support an accept/reject comparison with a suitable constant hurdle rate. It does not automatically rank mutually exclusive projects.

In the scale example, A’s IRR is 30% and B’s is 20%, yet B has higher NPV at 10%. Timing differences can also cause ranking conflicts. Use incremental cash flows and NPV at the appropriate opportunity cost to resolve them.

With nonconventional signs, IRR may be multiple or nonexistent. For $(-100,+230,-132)$:

$$-100+\frac{230}{1+x}-\frac{132}{(1+x)^2}=0$$

has IRRs of 10% and 20%. At a 15% hurdle, NPV is about 0.19. There is no single IRR to compare mechanically with 15%; calculate NPV directly.

7. Project interactions and unequal lives

A positive stand-alone NPV can disappear when another project changes its cash flows. Include shared capacity, prerequisite systems, cannibalization, and synergies in the feasible combinations.

If project A has NPV 20, B has NPV 15, and doing both creates an additional interaction PV of -25, their combined NPV is 10. Choosing A alone is better than taking both despite both individual NPVs being positive. If the interaction were +10, the combined value would be 45.

For mutually exclusive equipment with different lives, a single-cycle NPV comparison may omit future replacement opportunities. Equivalent annual value can help when repeated replacement is genuinely feasible under comparable conditions:

$$EAV=\frac{NPV}{[1-(1+r)^{-T}]/r}.$$

For cost-only machines use consistent annualized cost comparisons. Do not assume projects can be repeated unchanged merely to justify the method; technological change, capacity, and a finite planning horizon can require an explicit replacement schedule instead.

8. Timing and real options: positive NPV need not mean invest now

The right to defer an irreversible investment can have value. Compare investing now with the best feasible wait-and-decide policy, including lost interim revenues, expiry, competition, and changing costs. The rule remains maximize NPV across policies, not just across immediate projects.

A transparent one-period option example

Suppose a project can be obtained by paying 100 now or at date 1. It produces no interim cash flow. At date 1 its operating-asset value will be 160 or 60. Assume market-consistent risk-neutral state weights of 0.5 each, risk-free return 5%, and that these values and discounting come from a valid spanning model.

Investing now commits to both outcomes. Its NPV is

$$NPV_{\mathrm{now}}=\frac{0.5(160)+0.5(60)}{1.05}-100 \approx4.76.$$

Waiting lets the firm invest only in the favorable state. At date 1, the exercise payoffs are $\max(160-100,0)=60$ and $\max(60-100,0)=0$:

$$V_{\mathrm{wait}}=\frac{0.5(60)+0.5(0)}{1.05}\approx28.57.$$

Waiting is preferable here despite positive immediate NPV. The difference is about 23.81. This is an intentionally simplified option model: no loss from deferral, no competitive erosion, no expiry before the decision, and a fixed exercise cost. Different assumptions can reverse the result.

The state weights are pricing probabilities, not subjective odds chosen and then discounted risk-free. When real projects cannot be spanned by traded assets, risk adjustment and option valuation require additional assumptions. The example does not imply that every uncertain positive-NPV project should be delayed.

9. Audit the decision, not just the spreadsheet

Check the with-versus-without comparison, payment dates, taxes, working capital, terminal value, and the cash-flow/discount-rate match. Vary revenue, costs, salvage, project life, and risk assumptions; use coherent scenarios when variables move together.

A sensitivity calculation that changes sales while holding capacity, working capital, and operating costs unrealistically fixed can mislead. Avoid treating a detailed spreadsheet as evidence that uncertain inputs are accurate. Follow the chosen decision with a review of realized outcomes and the assumptions that drove the original valuation.

10. Practice questions

1. A project costs 500 and pays expected 300 at each of the next two year-ends. At 8%, calculate NPV.
2. A project has revenue 400, cash operating costs 180, tax depreciation 80, tax rate 25%, CapEx 30, and an NWC increase of 10 in one year. Find EBIT, operating cash flow, and unlevered FCF.
3. An unrecoverable study cost 15 last year. The project uses a warehouse that could be sold today for after-tax proceeds of 120. Which amount belongs in today’s opportunity-cost calculation?
4. Equipment sells for 90 with remaining tax book value 50. At a 25% tax rate, calculate after-tax sale proceeds under the simple gain-tax assumption.
5. A project asset beta is 1.3, risk-free return is 3%, and market premium is 5%. Find the asset cost of capital. Is the firm’s average equity beta necessarily appropriate instead?
6. Projects A and B are mutually exclusive. A costs 100 and has PI 1.4; B costs 500 and has PI 1.15. Both are affordable. Find NPVs and choose using NPV.
7. Check both IRRs for $(-100,230,-132)$. Why does comparing an arbitrary one of them with the hurdle rate fail?
8. Immediate investment has NPV 12. Waiting produces an option value of 18 today under a consistent model, with all deferral costs included. Which policy should be chosen, and does this contradict NPV?

11. Answers and reasoning

1. $NPV=-500+300/1.08+300/1.08^2\approx34.98$.
2. EBITDA is 220, EBIT is 140, taxes are 35, and OCF is $140(0.75)+80=185$. FCF is $185-30-10=145$. Equivalently use $220(0.75)+0.25(80)-30-10$.
3. Include 120 as foregone after-tax sale proceeds. Exclude the sunk study fee from today’s decision; its past payment is unchanged by the choice.
4. $90-0.25(90-50)=80$.
5. $r_A=3\%+1.3(5\%)=9.5\%$. The relevant beta belongs to the project exposure; the parent’s average levered equity beta can represent different business and financing risks.
6. A has NPV $100(1.4-1)=40$. B has NPV $500(1.15-1)=75$. Choose B on the stated mutually exclusive, unconstrained comparison basis.
7. At 10%, $-100+230/1.1-132/1.1^2=0$; at 20%, $-100+230/1.2-132/1.2^2=0$. The nonconventional signs create multiple roots, so use NPV at the appropriate discount rate rather than choosing a convenient IRR.
8. Choose waiting because 18 exceeds 12. This applies the NPV principle to the full feasible policy set, including flexibility, rather than rejecting the principle.

Further reading

• Andrew W. Lo, MIT OpenCourseWare, Capital Budgeting lectures: project cash flows, discount rates, interactions, and decision rules.
• MIT OpenCourseWare, Capital Budgeting slides, Lectures 18–20: the source lecture framework.
• MIT OpenCourseWare, The CAPM and APT: opportunity-cost and systematic-risk foundations.
• MIT OpenCourseWare, Options: replication and risk-neutral valuation concepts that motivate real-option analysis.

Feynman study lab

Explain it simply

A project adds value when its incremental future cash is worth more than its incremental cost.

Concrete example. Paying 100 today for 121 in two years has NPV zero at a 10% annual discount rate.

Teach it back without looking

Close the lesson and explain the idea aloud to a beginner in two minutes. Define each technical term you use, walk through the example, and state one assumption or limitation. Then answer:

Why should money already spent on an unrecoverable study usually be excluded from the accept-or-reject calculation?

Check your explanation after you try

It is a sunk cost that does not change between the alternatives. Include future cash flows that actually differ, including opportunity costs.

Find the gap and try again

If you hesitated, relied on a formula you could not explain, or missed the self-check, write down the exact gap. Revisit the relevant section of this lesson and work through its example. Close the notes, explain it again in simpler language, then change one number or assumption and predict what changes. Recalculate or check the lesson to test your prediction. Try once more tomorrow from memory.