Lin receives two valuations for the same company: one says $9 a share and the other $17. Both analysts used discounted cash flow, and neither spreadsheet reports an error. Did someone use the wrong formula, or do their views of the future differ?
Discounted cash flow, or DCF, connects future cash, waiting time, and required returns. It makes disagreements inspectable without automatically resolving them. Take a small model apart and you can identify the assumption worth challenging instead of arguing over the final share value.
What you will be able to do
- Explain cash, timing, discount rates, and terminal value within a valuation.
- Match enterprise or equity cash flows with the appropriate discount rate and avoid deducting debt twice.
- Recalculate the model, change two key assumptions, and save sensitivities and evidence gaps.
Why future 100 is not today's 100
Begin with an assumed certain payment, expressed in generic money units. If money today can grow at a specified 5% annual rate, about 95.24 today grows to 100 in one year. Dividing 100 by 1.05 works backward from the future payment to its present value.
For payment in two years, divide by 1.05 twice, giving about 90.70 today. With other conditions unchanged, waiting longer reduces the present value. Amounts arriving in different years need to be brought to one date before being added.
This illustrates time value using certainty; it does not make a stock's future cash certain. A business may deliver less than expected or fail to deliver at all. A discount rate is the return requirement used to translate the specified future cash into present value. Risk and the character of the cash matter. Lowering the rate just to obtain a preferred share price does not provide evidence for that rate.
Ask whose cash it is
Free cash flow to the firm, or FCFF, concerns cash available to all capital providers after operating taxes and the reinvestment required for operations and growth. Free cash flow to equity, or FCFE, concerns common shareholders and incorporates financing effects such as interest and net borrowing.[1]
Discount FCFF at a capital cost matched to its debt-and-equity claim; discount FCFE at the required equity return. These routes can connect under consistent assumptions. Selecting the larger cash figure and the lower rate from different routes creates a mismatch rather than a conservative or optimistic scenario.
Nor does any reported “operating cash flow minus capital expenditure” automatically qualify as FCFF. Check how interest, taxes, and financing items enter the statements and adjust for the chosen definition. Net income is not distributable cash, and EBITDA has not deducted every investment needed to maintain and grow the business.
We will fully work through only the FCFF route. The operating cash forecasts exclude an additional nonoperating cash asset, which will enter later in the equity bridge. That separation prevents the asset from appearing once in operating value and again at the end.
Bring three cash flows to one valuation date
Lanchuan Equipment and all figures below are synthetic, independent teaching inputs, not a valuation of Costco or another real business. The valuation date is the end of year zero, with each future cash flow arriving at year-end. Cash is nominal U.S.-dollar cash, matched with a nominal discount rate; there is no additional personal-tax adjustment.
The model's cash amounts are in units of $10,000. A displayed 100 therefore means $1 million, not $100 or $100 million. Keep the displayed units through the calculations, then convert them when dividing by actual shares.
FCFF for years one, two, and three is given as 100, 110, and 120. These amounts already deduct operating taxes and the reinvestment supporting the forecasts. They do not deduct debt interest or principal. The matched capital cost is assumed to be 10%, a teaching input rather than a rate recommended for every business.
Year one's 100 divided by 1.10 has a present value of about 90.91. Year two's 110 divided by 1.10 squared also gives about 90.91. Year three's 120 divided by 1.10 cubed gives about 90.16. Their present values total approximately 271.98 in the same model units.
| Operating cash | Year-end amount, units of $10,000 | Present value at year zero, same units |
|---|---|---|
| Year 1 | 100 | 90.91 |
| Year 2 | 110 | 90.91 |
| Year 3 | 120 | 90.16 |
| Total | 330 | 271.98 |
The second payment is larger but arrives later, producing nearly the same present value as the first in this example. That coincidence is not a rule. Each amount has its own arrival time and discounting.
The cash path already assumes particular operating results. In real research, connect it to revenue, margins, taxes, equipment, and working capital. Filling 100, 110, and 120 into consecutive cells creates a smooth forecast; it does not establish that the business can generate it.
The company does not disappear after year three
If operations continue, three years are not the whole valuation. To make the later period calculable, assume FCFF grows at 2% annually from year four onward, after reinvestment needed to support that growth. The risk and capital cost remain at the assumed 10%.
Year-four cash is 120 times 1.02, or 122.40. In the perpetual-growth model, terminal value at the end of year three is 122.40 divided by the difference between 10% and 2%, giving 1,530. The numerator is next year's cash; the resulting value is located at year three, not today.
Discount 1,530 by 1.10 cubed to obtain a terminal present value of about 1,149.51. Add the first three years' present value of 271.98 to obtain operating value of approximately 1,421.49. Retain year three's own 120 cash flow, but do not add the undiscounted 1,530 to today's values.
The terminal component represents about 80.87% of operating value. Roughly $81 of every $100 estimated today depends on cash after year three. That does not automatically invalidate the model. It changes the investigation: near-term delivery matters, yet long-term competition, returns, and reinvestment support most of the calculated value.
Stable growth is not a button that makes cash accelerate forever without investment. Growth, reinvestment, risk, and mature operating conditions must be consistent, a constraint emphasized in Damodaran's terminal-value discussion.[2]
Our teaching case supplies cash growth after reinvestment to isolate the arithmetic. A real company needs evidence that growth and cash availability can coexist. If the model only reaches an appealing result after repeatedly increasing growth, return to operating assumptions rather than keep turning the formula.
Bridge operating value to one common share
At the valuation date, the company also has nonoperating cash of 50, excluded from the operating forecasts, and debt with market value of 300. Both remain in units of $10,000. There is one common-share class with one million shares and no preferred stock, noncontrolling interests, leases, other required claim adjustments, or potential dilution.
Operating value of 1,421.49 plus cash of 50 minus debt of 300 gives common-equity value of approximately 1,171.49 model units. Multiply by $10,000 per unit before dividing by one million shares. The result is about $11.71 per share. Dividing a displayed model number by a share count without carrying the currency scale loses the meaning of the result.
The debt deduction is the debt's value at the valuation date, not next year's interest bill. We did not subtract interest from FCFF. First value operating cash available to capital providers, then bridge to the relevant equity claim. Do not deduct the entire debt here and also subtract future principal repayments from the same FCFF forecast as though they were an additional operating burden.
With an FCFE route, the discounted value already concerns equity. Reusing this bridge to subtract all debt again would count the claim twice. Actual work also needs restrictions on cash, debt valuation, other capital claims, and dilution. The example's short bridge is not a complete capital-structure checklist for every company.[1]
How one formula can approach $9 or $17
Keep the three forecast cash flows, cash of 50, debt of 300, and one million shares unchanged. Change only capital cost and long-term growth to produce this per-share sensitivity table.
| Capital cost | Growth 1% | Growth 2% | Growth 3% |
|---|---|---|---|
| 8% | $14.07 | $16.52 | $19.95 |
| 10% | $10.34 | $11.71 | $13.49 |
| 12% | $7.97 | $8.84 | $9.90 |
This is a mechanical sensitivity exercise, not nine equally probable futures or a guaranteed range for the share price. Real assumptions may interact: risk can change the required return, and growth can alter reinvestment. Every item held fixed needs justification outside this controlled exercise.
Different parameters can make two otherwise consistent reports approach the opening $9 and $17 figures. Compare the evidence behind their cash forecasts, transition to stable growth, and return requirements. Choosing the more optimistic answer does not settle those questions.
The perpetual model also requires capital cost above stable growth. Equality makes the denominator zero; growth above the rate invalidates this interpretation. A negative terminal number from such inputs is not evidence that growth destroys the business. Stop using the violated model premise.
When a precise share value is the wrong output
An unclear cash path, rapidly changing capital structure, or doubtful survival may leave this stable-growth framework unsupported. Additional evidence, liquidation analysis, or a different staged model may be relevant. Every company does not need a target-price cell filled today.
C1's real-company work also keeps its information boundaries. Its two frozen observation packets can use only information public at their respective dates. The Costco fiscal-2025 10-K used elsewhere as C1-R retrospective material was published after both dates. It cannot become an early forecast input merely because it is convenient now.
This chapter gives Costco no target price and uses no later share price to identify supposedly correct assumptions. The $11.71 is a result for synthetic conditions. Model uncertainty, market price, and personal funding constraints still stand between a valuation calculation and an investment action.
Optional: the capital cost is not a universal constant
The weighted average cost of capital, WACC, combines relevant debt and equity financing costs using capital-structure weights. Its estimate can involve tax, currency, and risk assumptions. The essential point here is matching it to FCFF, not deriving it from scratch or combining book debt weights with an arbitrary equity return.
For a real business, record market-value weights, the sources for debt and equity costs, the valuation date, and the applicable assumptions. A range may be more defensible than a point estimate. Extra calculator precision cannot strengthen unsupported inputs.
Explain the model, then recalculate it
Why can terminal value of 1,530 not be added directly to 271.98? An analyst discounts equity cash flow to obtain common-equity value, then subtracts debt of 300 “to be conservative.” What is wrong? Write one reason for each before checking the discussion.
The terminal amount belongs to year three and must be discounted to the valuation date. The second calculation mixes an FCFE valuation with an FCFF bridge. Deducting a claim twice is a mismatch, not an additional margin of safety.
Now keep forecast cash and every bridge item unchanged, with long-term growth still 2%, but raise the matched capital cost to 12%. Recalculate year-four cash, terminal value at year three, operating value, and value per share. Can you retain $11.71?
Reference answers · reveal
Answer guide: Year-four cash remains 122.40. Dividing it by 12% minus 2% gives terminal value of 1,224. The first three years' present value is approximately 262.39, and terminal present value is about 871.22. Operating value is 1,133.61. Add 50, subtract 300, convert model units to dollars, and divide by one million shares: about $8.84.
This tests whether a rate change reaches both the terminal denominator and each year's discounting. It does not prove that 12% is more reasonable than 10%. If real evidence also changes cash forecasts, save a revised full model rather than treating “everything else unchanged” as an eternal business fact.
Save an assumptions and sensitivity record
Record valuation date, currency, nominal or real basis, the recipients of the cash, each year's cash and reinvestment assumptions, discount-rate support, terminal method, stable growth, bridge items, share count, and potential dilution. Identify each input as disclosed fact, working estimate, or supplied teaching assumption.
Recalculate at least two parameter changes. Include terminal present value's share of operating value and one long-term assumption most in need of evidence. Check common valuation dates, matched cash and rates, and single treatment of debt and cash. Preserve gaps rather than bury them under more decimals.
Make the valuation lead back to its assumptions
Identify whose cash is being valued, then discount forecast and terminal amounts to the same date. Keep the equity bridge consistent with that claim. Correct arithmetic establishes the result under the inputs, not the truth of the inputs.
Investigate long-term cash, reinvestment, and required returns, using sensitivities to expose dependence. If evidence does not support stability or continued operations, pause the point valuation. The useful deliverable is an assumption chain another person can question and update, not merely a target price.
References
- CFA Institute: Free Cash Flow Valuation, 2026 public introduction and summary. FCFF/FCFE, matched discounting, and claims; no paid curriculum reviewed.
- Aswath Damodaran, NYU Stern: Terminal Value. Stable growth, reinvestment, and terminal sensitivity. The model inputs and dates are original synthetic teaching assumptions.
Sources inspected October 30, 2026. AI-assisted draft; independent AI full-text and bilingual review is complete; human financial review and target-reader testing remain pending.