1.10 Triangulating Valuation — Putting It All Together
No single valuation method is trusted on its own. Professionals line up each method’s range as bars on one chart, called a football field. It shows the trading range, the buyout analysis, the DCF, comparable companies and precedent deals. Where the standalone methods overlap is the most defensible value. Precedent deals show what a sale might fetch, and the buyout analysis gives a floor.
Why it matters: Several methods that agree give far more confidence than one precise number.
Summary: Professionals triangulate value by setting each method’s range side by side on a football field chart: trading range, LBO, DCF, comps and precedents. The overlap of the standalone methods is the most defensible value; precedents show a sale value, and an LBO a floor.
- A football field compares methods; a sensitivity table is a grid within one method that sets the DCF bar’s ends.
- Ridgeline’s standalone bars overlap at $18.98 to $21.00; precedents sit at $24.30 to $28.20.
- Delaware courts give heavy weight to deal prices from robust sale processes (DFC, Dell, Aruba).
- In Aruba, dissenters claiming $32.57 were awarded $19.10, below the $24.67 deal price they refused.
No professional valuation rests on a single method. The standard practice is to build a DCF, a comps analysis, and — where relevant — a precedent transactions analysis and an SOTP, then plot all of them on a single chart of valuation ranges, commonly called a football field. Where the ranges overlap is treated as the most defensible zone of value; where they diverge sharply, that divergence itself becomes the analytical story — is the market undervaluing intrinsic cash generation, or is the DCF’s growth assumption too aggressive?
A football field is not the same thing as a sensitivity table. The sensitivity table (1.3) is a grid inside one method, the DCF’s value at each combination of WACC and long-term growth; its corners set the two ends of the DCF bar. The football field sets that bar beside the bars from every other method, so you can see how the different ways of asking “what is it worth?” agree or disagree.
Each row is one horizontal bar, in dollars per share, for the illustrative Ridgeline Instruments (1.7, 1.8). Per share = (EV − $300m net debt) ÷ 50m shares.
| Method (question it answers) | Inputs that set the bar’s ends | Low | High |
|---|---|---|---|
| 52-week trading range (what has the market paid?) | Assumed low and high | $15.50 | $21.00 |
| LBO (what can a financial buyer pay? Part 2: M&A, Private Equity & LBOs) | 5.0× debt, 5% EBITDA growth, exit at 8.5×, $250m repaid, 20% to 25% target IRR | $16.39 | $18.06 |
| DCF (what are the cash flows worth? 1.3) | Next-year unlevered FCF $85m; WACC 9.5% to 8.5%; growth 2.0% to 3.0% | $16.67 | $24.91 |
| Trading comps (what do peers trade at? 1.7) | 8.33× to 9.38× LTM EBITDA of $150m | $18.98 | $22.13 |
| Precedents (what have buyers paid for control? 1.8) | 10.1× to 11.4× LTM EBITDA | $24.30 | $28.20 |
The DCF bar uses a one-stage shortcut, EV = FCF ÷ (WACC − g): at the low end $85m ÷ (0.095 − 0.020) = $1,133.3 million, or $16.67 a share; at the high end $85m ÷ (0.085 − 0.030) = $1,545.5 million, or $24.91. The LBO bar works backward from a 5-year exit: EBITDA $150m × 1.055 = $191.4 million, exit EV at 8.5× = $1,627.3 million, less $500 million of remaining debt = $1,127.3 million of equity. At a 20% IRR the sponsor can invest $1,127.3m ÷ 1.205 = $453.0 million of equity; adding $750 million of debt gives an EV of $1,203.0 million, or $18.06 a share.
Reading it. The standalone bars (trading range, DCF, comps) overlap between $18.98 and $21.00, so that is the most defensible standalone value, and the $19.00 price sits at its bottom. The precedent bar sits entirely above it: that gap of roughly $3 to $9 a share is the control premium a strategic buyer would have to fund from synergies. The LBO bar sits below the price, so a private equity bid is unlikely.
Courts face this question in appraisal cases, where shareholders who reject a merger ask a judge for “fair value”. Since 2017 the Delaware Supreme Court, in DFC Global and Dell, has held that the price produced by a robust sale process deserves heavy weight, and in Verition Partners v. Aruba Networks (April 16, 2019) it valued Aruba at the $24.67 deal price less $5.57 of synergies: $19.10 a share. The court’s ordering suits any analyst: a price from a competitive process beats a market price, which beats one expert’s model. When a banker presents these bars to public shareholders in a fairness opinion, FINRA Rule 5150 requires disclosure of the banker’s contingent fees, recent relationships with the parties, and whether it independently verified the company’s figures.
Match the bars to the question: standalone value from the DCF, comps and trading range; a sale price from precedents and synergy capacity; a floor from the LBO. If at least three standalone bars overlap, present the overlap as the central range. If the DCF midpoint sits more than about 15% outside the comps range, reconcile the two (growth, margins, WACC) before presenting either. Never average the bars into one number. When a competitive sale process has produced a price, treat it as the strongest single piece of evidence.
Trusting one bar over the market evidence. In the Aruba appraisal, the dissenting shareholders turned down HP’s $24.67 a share and claimed a fair value of $32.57, a figure built on discounted cash flows. The Delaware Supreme Court awarded $19.10: deal price less synergies. Measured against the deal they refused, they gave up $5.57 a share, 22.6%, before statutory interest; against their own claim, the award was 41.4% lower. A DCF that sits far above every market-based bar is a hypothesis to test, not a value to bet on.
What is a football field in valuation?
It is a chart of horizontal bars, one per valuation method, each showing a low-to-high range of value per share or enterprise value, often with the current share price or offer price drawn as a vertical line. It is not a sensitivity table, which is one method’s grid of outcomes.
Which valuation method is most accurate?
None is reliable alone; each answers a different question. A DCF is the most complete but the most assumption-heavy; comps are current but inherit market moods; precedents include control premiums. Where a clean, competitive sale has happened, its price is usually the best evidence, which is why Delaware courts give it heavy weight.
How wide should a valuation range be?
Wide enough to reflect the real uncertainty in its inputs. Moving WACC by ±0.5 percentage point and growth by ±0.5 point moves Ridgeline’s DCF from $16.67 to $24.91, a spread of about 40% around $20.15. Narrower than the inputs justify is false precision.
A football field sets each method’s range side by side; it is not the DCF sensitivity table, which only sets the ends of the DCF bar. Read the overlap of standalone methods as the central value, precedents as sale value and the LBO as a floor, and give a competitive sale price the most weight, as Delaware courts do.
Four questions on this chapter. Decide on your answer first, then click “Reveal Answer.”
1. What does a valuation football field show?
- The stock price path over the last 52 weeks
- A company’s historical trading multiples over time
- Value ranges from several methods side by side
- DCF values across WACC and growth inputs
Reveal Answer
Answer: C. A football field compares methods; the WACC-by-growth grid is the DCF sensitivity table, which only sets the ends of the DCF bar.
2. Standalone bars run: DCF $30–$42, comps $34–$40, 52-week range $28–$38. Precedents run $45–$52. What is the central standalone range?
- $30 to $42
- $34 to $38
- $34 to $40
- $45 to $52
Reveal Answer
Answer: B. The overlap of the three standalone bars is $34 to $38; precedents answer a different question, the price of control.
3. In Verition Partners v. Aruba Networks (2019), what fair value per share did the Delaware Supreme Court award?
- $24.67
- $17.13
- $32.57
- $19.10
Reveal Answer
Answer: D. The court used the $24.67 deal price less $5.57 of synergies, $19.10, reversing the Court of Chancery’s $17.13 award; the dissenters had claimed $32.57.
4. A sponsor expects $600 million of equity at exit in 5 years, needs a 20% IRR and will borrow $400 million at entry. What is the most it can pay in enterprise value?
- About $641 million
- About $1,000 million
- About $241 million
- About $880 million
Reveal Answer
Answer: A. Entry equity = $600m ÷ 1.205 = $241.1m; add $400m of debt for an EV of about $641m. That is the LBO floor bar on the football field.
1.11 Capital Budgeting: NPV, IRR, Payback and the Profitability Index
Companies decide which long projects to fund by comparing the cash a project will bring in with the cash it costs, with future cash discounted at a rate that matches the project’s risk. Net present value, or NPV, is the deciding measure: it tells you how much value the project adds in today’s money. IRR, payback and the profitability index are helpful aids, but they can mislead, so they should not overrule NPV.
Why it matters: A project that looks attractive on payback can still destroy value.
Summary: Capital budgeting decides which long-term projects to fund by discounting each project’s incremental cash flows at a rate matched to its risk. Net present value is the deciding measure; IRR, payback and the profitability index are aids that can mislead.
- A $10m production line has NPV $2.552m, IRR 17.0%, payback 3.45 years, discounted payback 4.21 years and PI 1.23 at 9%.
- 100% bonus depreciation for property acquired after January 19, 2025 adds $0.293m of NPV in the example.
- Cash flows that change sign twice can have two IRRs (10% and 20%); MIRR or NPV resolves them.
- Choosing a smaller project for its higher IRR forgoes $1.368m of NPV below the 15.0% crossover rate.
- Under a fixed budget, maximize total NPV; ranking by PI can leave money idle.
Capital budgeting is how a company decides which long-term projects to fund. It discounts (1.2) a project’s incremental cash flows, the cash that changes only because the project happens, at a rate matched to the project’s risk; the company’s WACC (1.4) fits only projects like its existing business. Four measures dominate:
- Net present value (NPV): the present value of all project cash flows, outlay included; a positive NPV is value added above the cost of capital.
- Internal rate of return (IRR): the discount rate at which NPV equals zero; accept if it beats the hurdle rate.
- Payback period and discounted payback: the years until cumulative cash flows, undiscounted or discounted, recover the outlay.
- Profitability index (PI): present value of future cash flows ÷ initial investment; above 1.0 means a positive NPV.
Of 392 CFOs surveyed by Graham and Harvey (2001), 75.7% always or almost always used IRR, 74.9% NPV, 56.7% payback and 29.5% discounted payback.
Inputs (USD millions): equipment $10.0, straight-line to zero over 5 years ($2.0 a year); working capital $1.0, recovered in year 5; revenue $8.0 and cash costs $4.5 a year; tax 21%; resale $1.0 in year 5 ($0.79 after tax); discount rate 9%.
Operating cash flow = (Revenue − Cash costs − Depreciation) × (1 − t) + Depreciation = (8.0 − 4.5 − 2.0) × 0.79 + 2.0 = 1.185 + 2.0 = $3.185 a year. Year 5 adds $1.0 of working capital and $0.79 of salvage: $4.975.
| Year | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| Cash flow | −11.000 | 3.185 | 3.185 | 3.185 | 3.185 | 4.975 |
| Cumulative cash flow | −11.000 | −7.815 | −4.630 | −1.445 | 1.740 | 6.715 |
| PV at 9% (CF ÷ 1.09n) | −11.000 | 2.922 | 2.681 | 2.459 | 2.256 | 3.233 |
| Cumulative PV | −11.000 | −8.078 | −5.397 | −2.938 | −0.681 | 2.552 |
NPV = sum of PVs = $2.552 million: accept. IRR = the rate that sets NPV to zero = 17.0%. Payback = 3 + 1.445 ÷ 3.185 = 3.45 years; discounted payback = 4 + 0.681 ÷ 3.233 = 4.21 years. PI = 13.552 ÷ 11.000 = 1.23.
Tax timing matters: equipment acquired after January 19, 2025 qualifies for a permanent 100% first-year deduction. Deducting all $10.0 in year 1 lifts that year’s tax saving from $0.42 to $2.10 and removes it in years 2 to 5: NPV rises $0.293 million to $2.845 million and IRR to 18.9%.
IRR misleads in two recurring cases:
- Multiple IRRs. When cash flows change sign more than once, say −$100, +$230, then −$132 for a site clean-up, NPV is zero at both 10% and 20% and positive only between them; at 9% it is −$0.09. The modified internal rate of return (MIRR) discounts outflows and compounds inflows at the cost of capital: (230 × 1.09 ÷ (100 + 132 ÷ 1.092))1/2 − 1 = (250.70 ÷ 211.10)1/2 − 1 = 8.98%, below 9%: reject, as NPV says.
- Mutually exclusive projects of different scale. Project B costs $4.0 million and returns $1.6 million a year for 4 years: IRR 21.9%, NPV $1.184 million, PI 1.30. The production line has the lower IRR (17.0%) but over twice the NPV. The crossover rate, the IRR of the difference in their cash flows, is 15.0%: below it the line wins on NPV, so at 9% choose the line.
Capital rationing applies when a budget caps spending. With $20 million for five independent projects: the line (cost $11.0m, NPV $2.55m, PI 1.23), C ($8.0m, $1.60m, 1.20), D ($6.0m, $1.38m, 1.23), E ($5.0m, $0.85m, 1.17) and F ($9.0m, $1.62m, 1.18). Ranking by PI picks the line and D ($17.0m, NPV $3.93m), leaving $3.0m idle. The line plus F uses all $20.0m for NPV $4.17m. PI ranking works for divisible projects; with lumpy ones, test combinations.
Accept independent projects with positive NPV at a risk-matched rate. Between mutually exclusive projects, take the higher NPV whatever the IRRs say. Quote IRR only when cash flows change sign once; otherwise use MIRR or NPV. Payback is a liquidity screen, not a measure of value. Under a hard budget, pick the affordable combination with the highest total NPV.
Ranking mutually exclusive projects by IRR. Choosing Project B (IRR 21.9%) over the production line (17.0%) adds $1.184 million of value instead of $2.552 million, forgoing $1.368 million, 53.6% of what was available, at any cost of capital below the 15.0% crossover. Rank exclusive choices by NPV.
What is the difference between NPV and IRR?
NPV is a dollar amount: the value a project adds after paying for its capital. IRR is the discount rate at which that value would be zero. They agree on accepting a single conventional project but can rank projects of different size or timing differently, so a higher IRR is not always better; follow NPV.
What discount rate should a company use?
The return investors require for the project’s own risk: WACC for an ordinary expansion, more for riskier ventures, based on the betas of companies doing that kind of work. US federal agencies use a separate convention: since April 2025 OMB again applies its 1992 Circular A-94.
Why do companies still use the payback period?
It is simple and answers a liquidity question: how long the money is at risk. But it ignores the time value of money (discounted payback fixes that) and all cash after the cutoff, so it should screen projects, not choose them.
Capital budgeting discounts a project’s incremental cash flows at a risk-matched rate and accepts positive NPV. IRR, payback and the profitability index summarize a project but can mislead with multiple sign changes, mutually exclusive projects of different scale, or lumpy budgets, where the rule is to maximize total NPV.
Four questions on this chapter. Decide on your answer first, then click “Reveal Answer.”
1. A project costs $1,000 and returns $400 at the end of each of the next 3 years. At a 10% cost of capital, what is its NPV?
- −$5.3
- +$9.7
- +$94.7
- +$200.0
Reveal Answer
Answer: A. PV = $400 × (1 − 1.10−3) ÷ 0.10 = $994.7; NPV = $994.7 − $1,000 = −$5.3. The IRR is 9.7%, just below the 10% hurdle.
2. Mutually exclusive projects at a 9% cost of capital: A has NPV $2.6 million and IRR 17%; B has NPV $1.2 million and IRR 22%. Which should the company choose?
- B, because its IRR is higher
- A, because it adds more value
- B, because it pays back sooner
- Neither, since the IRRs disagree
Reveal Answer
Answer: B. Between exclusive projects, take the higher NPV. B only wins at discount rates above the crossover rate.
3. A project’s cash flows are −$100, +$230 and −$132, and the cost of capital is 9%. What should you do?
- Accept it: the higher IRR is 20%
- Accept it: both IRRs exceed 9%
- Reject it: NPV at 9% is slightly negative
- Reject it: a project cannot have two IRRs
Reveal Answer
Answer: C. NPV is zero at 10% and 20% and positive only between them; at 9% it is −$0.09, and MIRR is 8.98%.
4. With a $10 million budget, X costs $6m (NPV $1.5m), Y costs $5m (NPV $1.2m), Z costs $5m (NPV $1.1m). Which choice maximizes value?
- X and Y, total NPV $2.7m
- X alone, total NPV $1.5m
- X and Z, total NPV $2.6m
- Y and Z, total NPV $2.3m
Reveal Answer
Answer: D. X and Y or X and Z would cost $11m, over budget. Ranking by PI picks X first and leaves $4m idle; Y and Z use the full $10m for the highest feasible NPV.
- Delaware Supreme Court, Verition Partners Master Fund v. Aruba Networks (April 16, 2019) — Deal price less synergies $19.10 vs $24.67 deal price
- FINRA Rule 5150, Fairness Opinions — Fairness opinion disclosures
- IRS, guidance on the additional first-year depreciation deduction amended by the One, Big, Beautiful Bill — Permanent 100% bonus depreciation after January 19, 2025
- OMB Memorandum M-25-23, Rescission and Reinstatement of Circular No. A-94 (April 8, 2025) — Federal discount-rate guidance
- 26 U.S.C. §11 (corporate tax rate) — 21% federal rate