Valuation of Free Cash Flows: Operating Forecasts, Reinvestment, Terminal Value, WACC, and APV

Free cash flow connects a business’s operating economics to its value. It asks how much cash the operations generate after taxes and necessary reinvestment, before deciding how debt and equity divide that cash. The discipline is to separate operating value from financing effects, then combine them consistently.

Learning goals and conventions

You should finish able to reconcile three FCF formulas, construct operating working capital, forecast reinvestment, distinguish liquidation from continuation, calculate terminal value, and match cash flows with either WACC or APV.

Here FCF means unlevered free cash flow to the firm, also called FCFF. Some companies report a different “free cash flow” measure, such as operating cash flow minus capital expenditure, potentially after interest. Read the definition rather than assuming every reported FCF equals the valuation cash flow used here.

Use annual year-end nominal cash flows in one currency. Assume the tax deductions in the examples can be used immediately, book and tax depreciation coincide, and no other accounting adjustments occur unless stated otherwise. Real forecasts may require cash-tax schedules, loss carryforwards, amortization adjustments, and other reconciliations.

Symbol	Meaning
$EBIT_t$	Operating earnings before interest and taxes
$\tau$	Relevant marginal tax rate in the example
$D_t$	Depreciation expense consistent with the stated tax assumptions
$CapEx_t$	Cash capital expenditure
$\Delta NWC_t$	Change in operating net working capital
$NA_t$	Net operating assets under a specified accounting definition
$k_A$	Unlevered operating-asset cost of capital
$TV_T$	Value at date $T$ of cash flows after $T$

1. Think of the operations as if all equity financed

Unlevered FCF estimates after-tax operating cash flows as if debt financing were absent. That means operating taxes are computed without interest deductions. It does not mean the actual company must have no debt.

Exclude interest, debt issuance, principal repayment, dividends, and share repurchases from this operating measure. Financing tax shields are handled later, through a consistent WACC or a separate APV adjustment.

Depreciation’s tax effect is different: it arises from operating asset investment and belongs in operating cash taxes. Excluding the interest tax shield does not exclude every tax deduction.

Three-step roadmap

1. Forecast incremental unlevered operating FCF.
2. Assess the risk and the appropriate discounting framework.
3. Value the operations and incorporate financing effects consistently.

A beautifully estimated discount rate cannot rescue missing working capital or an impossible terminal-growth assumption.

2. Derive the core FCF formula

Define unlevered after-tax operating profit, often called NOPAT:

$$NOPAT_t=EBIT_t(1-\tau).$$

Add back depreciation because it reduced accounting profit without consuming cash in that period. Then subtract actual capital spending and the cash tied up in additional working capital:

$$\boxed{FCF_t=EBIT_t(1-\tau)+D_t-CapEx_t-\Delta NWC_t.}$$

Depreciation is not a cash inflow received from a customer. Under the example’s usable-deduction assumptions, its benefit relative to no deduction is the tax saving $\tau D_t$. Adding back the full expense reverses its accounting subtraction; it does not create a second tax benefit.

Equivalent EBITD expression

The outline’s EBITD means $EBIT+D$, earnings before interest, tax, and depreciation. If no separate amortization adjustment is present:

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

This follows because

$$(EBITD-D)(1-\tau)+D=EBITD(1-\tau)+\tau D.$$

Do not add only $\tau D$ to after-depreciation EBIT, or add both $D$ and $\tau D$ to NOPAT. Both mix the two derivations. If using EBITDA and depreciation plus amortization, expand the definitions and cash-tax assumptions consistently.

Worked one-year calculation

Sales are 1,000, cash operating costs 650, depreciation 80, tax rate 25%, CapEx 100, and operating NWC increases by 30.

$$EBITD=350,\quad EBIT=270,\quad NOPAT=202.50.$$

Then

$$FCF=202.50+80-100-30=152.50.$$

The equivalent calculation is $350(0.75)+0.25(80)-100-30=152.50$. Any interest payments are deliberately absent.

3. Operating NWC is not every current account

The broad accounting identity $CurrentAssets-CurrentLiabilities$ is not automatically the operating NWC used in an unlevered valuation. A common definition is

$$NWC^{op}=Receivables+Inventory+OtherOperatingCurrentAssets -TradePayables-OtherOperatingCurrentLiabilities.$$

Exclude interest-bearing current debt and normally exclude excess financial cash. Required operating cash can be included under an explicit convention; then keep that convention consistent and do not add the same cash again as a separate nonoperating asset.

For receivables rising from 100 to 120, inventory from 80 to 95, and payables from 60 to 70:

$$NWC_0=120,\quad NWC_1=145,\quad\Delta NWC=25.$$

The increase reduces FCF by 25. The firm has recognized sales or acquired stock without receiving or releasing equivalent cash. Increased trade payables partly offset the requirement by providing operating financing.

A negative working-capital change releases cash. Forecasting permanent releases while the business expands can be unrealistic; test collection periods, inventory needs, and supplier terms rather than using the balance as an unexplained plug.

4. The net-assets formula needs a bridge

In a simple model with net PPE and operating NWC:

$$NA_t=NetPPE_t+NWC_t^{op}.$$

If there are no disposals, acquisitions, foreign-exchange movements, revaluations, or other noncash changes:

$$\Delta NetPPE_t=CapEx_t-D_t,$$

so

$$\Delta NA_t=CapEx_t-D_t+\Delta NWC_t.$$

Substitute into the core formula:

$$\boxed{FCF_t=NOPAT_t-\Delta NA_t.}$$

The outline’s $TotalAssets-CurrentLiabilities$ works only when the categories have been restricted so they describe the same operating asset base. Excess cash, goodwill changes, current debt, and nonoperating investments can otherwise contaminate the expression. The symbol NA is not a substitute for defining the accounts.

If assets are sold, include sale proceeds and taxes and reconcile the related book-value change separately. A mechanically computed balance-sheet difference can mix cash investment with noncash changes.

5. Worked three-year operating forecast

A project requires initial PPE of 600 and initial operating NWC of 60, so date-0 outlay is 660. Use tax rate 25%, annual depreciation 100, no interest in operating FCF, and the following hypothetical forecast:

Item	Year 1	Year 2	Year 3
Sales	500	550	600
Cash operating costs	300	330	360
EBITD	200	220	240
Depreciation	100	100	100
EBIT	100	120	140
NOPAT	75	90	105
CapEx	80	90	100
Ending operating NWC	66	72	80
Change in NWC from preceding year	6	6	8
FCF	89	94	97

Reconcile the asset method:

Asset bridge	Date 0	Year 1	Year 2	Year 3
Net PPE	600	580	570	570
Operating NWC	60	66	72	80
Net operating assets	660	646	642	650
Change in net operating assets	—	-14	-4	+8

FCF is $75-(-14)=89$, $90-(-4)=94$, and $105-8=97$. The two methods agree because their definitions and assumptions agree.

At this point, the forecast is incomplete for valuation: either the assets continue beyond year 3 or there are closure and sale cash flows. Ending the spreadsheet does not imply the business becomes worthless.

6. Forecast incremental cash flows

For a project, compare the firm with the project to the firm without it. For an acquisition, distinguish target stand-alone FCF from achievable incremental synergies and integration costs.

Item	Correct treatment
Unrecoverable research paid last year	Exclude from the current decision as a sunk cost
Land that could be sold today	Include foregone after-tax sale proceeds as opportunity cost
Sales lost from an existing product	Include lost incremental contribution and related cash effects
Allocated overhead that does not change	Exclude the allocation as such
Additional staff or support capacity required	Include actual incremental cash costs
Tax losses or deductions	Forecast usable cash effects and their timing

An initial opportunity cost can coexist with terminal resale proceeds; the project uses a resource and may later release it. Keep the with-versus-without alternatives consistent.

Negative FCF in a growth phase is not automatically bad: it may finance positive-NPV investment. Positive FCF from cutting essential investment is not automatically good. The quality of reinvestment and future cash flows matters.

7. Terminal liquidation: value the assets actually released

If the project ends at $T$, include final operating FCF and then the incremental closure proceeds and costs. For an asset sold at $S$ with remaining tax basis $BV_T$, under a simplified fully usable gain/loss tax treatment:

$$AfterTaxSalvage=S-\tau(S-BV_T)=S(1-\tau)+\tau BV_T.$$

Add realistically recoverable operating NWC and subtract closure obligations. The outline’s PPE term must mean remaining tax basis, not the asset’s market value or an unrelated accounting balance.

Worked liquidation of the forecast project

Suppose the Year-3 PPE sells for 500, tax basis equals the forecast book value 570, all 80 of operating NWC is recovered, and there are no closure costs. Assume the loss deduction can be used immediately:

$$AfterTaxSalvage=500-0.25(500-570)=517.50.$$

Liquidation proceeds are $517.50+80=597.50$. Total Year-3 cash flow is $97+597.50=694.50$.

At an assumed appropriate 10% discount rate:

$$NPV=-660+\frac{89}{1.10}+\frac{94}{1.10^2}+\frac{694.50}{1.10^3} \approx20.38.$$

The actual tax treatment and recovery may differ. Receivable defaults, inventory write-downs, environmental obligations, or delayed deductions can change the result. Do not assume every dollar of recorded NWC becomes a dollar of cash at closure.

8. Going concern: terminal value is at the forecast horizon

If the business continues, value post-horizon FCF at date $T$:

$$TV_T=\frac{FCF_{T+1}}{k}\quad\text{for a flat perpetual FCF},$$

or

$$TV_T=\frac{FCF_{T+1}}{k-g},\qquad k>g,$$

for a stable growing perpetuity. Match $k$ to the valuation method and risks. Discount $TV_T$ from date $T$ back to today, alongside separately forecast cash flows through $T$.

Date              0             1             2             T             T+1
                   |-------------|-------------|-------------|--------------|
Explicit FCF     initial        FCF1          FCF2          FCFT          first continuation FCF
                                                             TVT values cash flows from T+1 onward

Do not include $FCF_T$ both in the explicit forecast and again as the first cash flow inside terminal value. Do not add full liquidation proceeds to an operating continuation value unless the combined scenario genuinely contains both without double counting assets.

No growth still requires maintenance

A flat perpetuity assumes sustainable FCF, not that CapEx is zero forever. Maintaining productive capacity generally requires replacement investment. In a simple steady state, CapEx may approximately equal depreciation and NWC may remain constant, but accounting depreciation is not automatically a reliable estimate of economic maintenance spending.

9. Growth must be funded by reinvestment

Under the net-assets bridge, assume that after date $T$, NOPAT and the operating asset base grow at the same constant rate $g$, with stable operating economics:

$$NOPAT_{T+1}=(1+g)NOPAT_T,\qquad \Delta NA_{T+1}=gNA_T.$$

Then

$$FCF_{T+1}=(1+g)EBIT_T(1-\tau)-gNA_T,$$

and

$$TV_T=\frac{(1+g)EBIT_T(1-\tau)-gNA_T}{k-g}.$$

The subtraction is essential. Forecasting rising profits without the assets needed to support them overstates distributable cash.

Worked terminal-growth calculation

In a separate hypothetical mature business, date-$T$ NOPAT is 100, net operating assets are 500, long-run growth is 3%, and the appropriate terminal rate is 8%:

$$NOPAT_{T+1}=103,\quad Reinvestment=0.03(500)=15,$$ $$FCF_{T+1}=88,\qquad TV_T=88/(0.08-0.03)=1{,}760.$$

Using $103/(0.08-0.03)=2{,}060$ would omit necessary reinvestment. Using 88/0.08 would incorrectly remove the assumed future growth.

At fixed $g=3\%$ and FCF 88, terminal rates of 7%, 8%, and 9% give values of 2,200, 1,760, and 1,466.67. Small changes in the spread matter greatly. When varying growth, recompute reinvestment too rather than assuming growth changes at no cost.

An equivalent steady-state expression uses next-year return on beginning invested capital:

$$ROIC=\frac{NOPAT_{T+1}}{NA_T},\qquad ReinvestmentRate=\frac{g}{ROIC},$$ $$FCF_{T+1}=NOPAT_{T+1}\left(1-\frac{g}{ROIC}\right).$$

Here $ROIC=103/500=20.6\%$, and reinvestment rate is about 14.5631%. The implied economics must be plausible and the relevant return on future investment must be distinguished from historical average returns. Growth creates value only when the investment opportunities justify the capital committed relative to risk.

10. Choose WACC or APV consistently

WACC approach

For a suitable financing policy, such as a consistent target market-value leverage ratio under standard assumptions:

$$WACC=\frac{E}{D+E}r_E+\frac{D}{D+E}r_D(1-\tau).$$

Discount unlevered FCF using the appropriately estimated WACC. Interest tax effects are reflected in the discount-rate framework; do not add a second interest shield to FCF or value.

APV approach

First value unlevered operations at $k_A$:

$$V_U=PV_{k_A}(FCF).$$

Then add financing effects separately:

$$V_L=V_U+PV(InterestTaxShields)-PV(FinancingCosts) +PV(OtherFinancingEffects).$$

Shield cash flows have their own timing and risk. The perpetual shortcut $\tau D$ requires particular debt and deduction assumptions and should not be applied to every forecast financing schedule.

Worked consistent comparison

Suppose sustainable unlevered FCF is 100 forever, $k_A=10\%$, debt is permanently 400 at 5%, tax rate is 25%, all deductions are usable, and the shield is discounted at 5%. Ignore other financing effects.

$$V_U=100/0.10=1{,}000,\quad PV(TS)=0.25(400)=100,$$

so APV gives $V_L=1{,}100$. Under the matching fixed-debt assumptions, equity is 700, its cost is approximately 12.1429%, and WACC is approximately 9.0909%. Thus $100/WACC=1{,}100$ too.

The agreement follows from matched assumptions, not from an arbitrary choice of two discount rates. Adding 100 of shield to the already WACC-derived 1,100 would incorrectly report 1,200. WACC and APV are alternative accounting frameworks for the same underlying effects when implemented consistently.

11. From operating value to equity value

Discounted FCFF values operations available to capital providers. A simplified bridge is

$$EquityValue=OperatingEnterpriseValue+ExcessCash+OtherNonoperatingAssets -Debt-OtherSeniorClaims.$$

Use consistent market or settlement values for relevant claims. Required operating cash already included in NWC is not automatically excess cash. Include minority interests or other claims when the operating cash flows being valued include the corresponding businesses.

If operating value is 1,100, truly excess cash is 80, debt value is 400, and other senior claims are 20, common-equity value is 760. With 100 consistently defined shares, value per share is 7.60 before any further dilution adjustments. Do not add cash or subtract a claim twice.

Project NPV is a related but different question: subtract the incremental acquisition or investment cost from the value of its future incremental cash flows. A business’s positive operating value alone does not establish that paying any acquisition price creates value.

12. Final consistency checks

Check each line’s sign and date, working-capital definition, asset bridge, terminal reinvestment, cash taxes, and financing method. Match nominal flows with nominal rates and real flows with real rates. Historical-cost depreciation need not inflate with sales, and specific prices or costs can grow differently from general inflation.

Stress operating margins, asset intensity, working-capital timing, long-run growth, ROIC, and discount rates in coherent scenarios. A valuation dominated by terminal value needs particularly strong scrutiny of mature operating economics. Use sensitivity ranges to reveal dependence on assumptions rather than presenting a precise answer as certainty.

13. Practice questions

1. EBIT is 200, depreciation 40, tax rate 25%, CapEx 60, and operating NWC increases by 15. Calculate FCF using both the EBIT and EBITD forms.
2. Receivables rise from 70 to 90, inventory from 50 to 55, and payables from 30 to 38. Find the NWC increase and its FCF effect.
3. NOPAT is 90, CapEx 50, depreciation 30, and NWC increases by 10. Find reinvestment, FCF, and the net-operating-asset change under the simple bridge.
4. An asset sells for 120 with tax basis 80. At 25% tax, find after-tax proceeds. Recoverable NWC is 30 and closure costs, already after tax, are 5. Find total liquidation proceeds excluding final operating FCF.
5. Date-$T$ NOPAT is 80, net operating assets 400, stable growth 2%, and terminal rate 8%. Find next-year reinvestment, FCF, and terminal value.
6. A terminal value of 1,000 is measured at year 3. At a 10% annual rate, find its PV today. Should year-3 operating FCF be included separately if TV covers year 4 onward?
7. Unlevered operations are worth 800, shield PV is 50, and financing-cost PV is 10. Find APV. If a consistently implemented WACC method already gives that levered value, should 50 be added again?
8. Operating enterprise value is 900, excess cash 40, debt value 250, and other senior claims 15. Find common-equity value and explain why operating cash already included in the forecast is not automatically added again.

14. Answers and reasoning

1. $FCF=200(0.75)+40-60-15=115$. EBITD is 240; $240(0.75)+0.25(40)-60-15=115$.
2. Beginning NWC is 90 and ending NWC is 107. The increase is 17, reducing FCF by 17 under the operating definitions.
3. Reinvestment is $50-30+10=30$. FCF is $90-30=60$. Net operating assets rise by 30 if no other movements occur.
4. After-tax sale proceeds are $120-0.25(120-80)=110$. Liquidation proceeds are $110+30-5=135$.
5. NOPAT next year is 81.60 and reinvestment is $0.02(400)=8$. FCF is 73.60, so $TV_T=73.60/(0.08-0.02)\approx1{,}226.67$.
6. $PV=1{,}000/1.10^3\approx751.31$. Include year-3 operating FCF separately because the terminal value covers later flows.
7. $APV=800+50-10=840$. Do not add the shield again if it is already reflected in the consistent alternative valuation.
8. Equity is $900+40-250-15=675$. Required operating cash and its future changes are already represented in the operating valuation; only separately excluded excess cash belongs in the nonoperating bridge.

Further reading

• Dirk Jenter, MIT OpenCourseWare, Valuation of Free Cash Flows, Lecture 12 Section A: operating cash-flow construction and financing separation.
• Katharina Lewellen, MIT OpenCourseWare, Valuation of Free Cash Flows, Lecture 12 Section B: alternative expressions, forecasting, and terminal-value context.
• MIT OpenCourseWare, Finance Theory II lecture notes: the course progression from FCF to WACC, APV, and company valuation.

Feynman study lab

Explain it simply

Free cash flow measures cash left after operating needs and reinvestment for the relevant claimholders.

Concrete example. With after-tax operating profit 100, depreciation 10, capital expenditure 30 and an increase in working capital of 5, firm free cash flow is 75.

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 add depreciation back but subtract capital expenditure?

Check your explanation after you try

Depreciation is a non-cash expense already deducted in profit; capital expenditure uses cash. Keep firm and equity cash-flow definitions consistent.

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.