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Why do a DCF and a transaction multiple disagree by 26.8 per cent?

The three methods are not three opinions about the business. They are one model, run three times, at three settings of a single unobservable parameter.

Julian R. Sterling · from Business Valuation

A 26.8 per cent spread between a discounted cash flow and a transaction multiple is 1.96 points of perpetual growth, and nothing else. Put each value back into the same model — same forecast, same 9.0 per cent cost of capital — and solve for the growth rate that produces it. The trading comparables at 8.20x imply 1.44 per cent. The discounted cash flow uses 2.00 per cent. The precedent transactions at 10.40x imply 3.41 per cent.

The translation, in four steps

A multiple and a present value are answers in the same currency to questions posed in different languages. No amount of staring at them side by side will translate one into the other. The move is to stop comparing outputs and start comparing inputs: take the discounted cash flow exactly as it was built, feed in the value some other method produced, and solve for the single assumption that would make the model return that value.

The natural candidate is the perpetual growth rate. It has the largest leverage on the answer, because it drives the terminal value, and on Cawdrey Instruments the terminal value is 53,176,478 of a 73,060,585 enterprise value — 72.78 per cent of the total. It is the input least constrained by evidence, since nobody can observe a perpetuity. And it is an input a human being can argue about without a spreadsheet, which is the whole point.

• Hold the explicit forecast fixed. The discounted value of years one to five is 19,884,106, and that number does not move.
• Subtract it from the target enterprise value. What remains is the present value the terminal value must carry.
• Un-discount that remainder back to the end of year five, by multiplying by 1.09 to the fifth power — 1.538624.
• Solve the terminal value formula for growth. There is no iteration and no solver. It is one division.

Run it on the trading comparables. The peer median of 8.20x on EBITDA of 8,400,000 gives 68,880,000. Subtract 19,884,106 and 48,995,894 remains; multiply by 1.538624 and the implied terminal value is 75,386,256. Then g = (0.09 × 75,386,256 − 5,615,002) ÷ (75,386,256 + 5,615,002), which is 1.44 per cent. On the precedent transactions the deal median of 10.40x gives 87,360,000, leaving 67,475,894, which grosses up to 103,820,027 and solves to 3.41 per cent. The discounted cash flow itself returns 2.00 per cent, as it must, since that is what was fed into it. That control is worth running every time, because a reverse solve built wrongly fails it.

Three methods on one company, restated as one parameter.

Method	Value	Implied perpetual growth
Trading comparables, 8.20x	68,880,000	1.44 per cent
Discounted cash flow	73,060,585	2.00 per cent
Precedent transactions, 10.40x	87,360,000	3.41 per cent

What the gap actually is

The spread between the lowest and the highest of the three values is 18,480,000, which is 26.8 per cent of the lowest. That is the gap that fills the range slide in every pitch and gets described in committee as the honest width of a difficult valuation. It is 1.96 points of perpetual growth.

It is not a disagreement about the order book, the customer concentration, the quality of the plant or the durability of the margins. It is not a disagreement about the forecast, which is identical in all three columns because the reverse solve holds it fixed by construction. It is not a disagreement about the cost of capital, which is 9.0 per cent in all three. One parameter, three settings, and out the other end comes an 18,480,000 range that people treat as a fact about the company.

The value of the restatement is what it lets a room do next. Before the solve, the available move is to weight: sixty per cent to the discounted cash flow, forty to the comparables, or a third each. A weighting is arithmetically unimpeachable and intellectually empty, because it asserts nothing about the world and therefore cannot be attacked. After the solve, the available move is to argue about 1.44 versus 3.41 — whether a specialist manufacturer grows at inflation or above it, for ever, once its plan runs out. That is a claim with evidence attached to it.

The cell the range chart does not show

Restating the spread as 1.96 points of growth makes it look small: a rounding error in a perpetuity, magnified by discounting into apparent disagreement. The opposite is true. Hold growth at 2.0 per cent, change nothing about the company, and move only the cost of capital across two hundred basis points. The value runs from 85,393,469 to 63,814,891 — a movement of 21,578,578, or 29.5 per cent of the base case, and 1.17 times the entire spread between the three methods.

One cell moved, forecast untouched.

Cost of capital	Enterprise value	Implied EBITDA multiple	Change from base
8.0 per cent	85,393,469	10.17x	12,332,884
8.5 per cent	78,752,060	9.38x	5,691,475
9.0 per cent	73,060,585	8.70x	—
9.5 per cent	68,129,031	8.11x	(4,931,554)
10.0 per cent	63,814,891	7.60x	(9,245,694)

The multiple column carries the same warning. At an 8.0 per cent cost of capital the discounted cash flow says Cawdrey is worth 10.17x, short of the precedent median of 10.40x by less than a quarter of a turn. At 10.0 per cent it says 7.60x, below every peer in the trading set but one. The discounted cash flow does not sit between the two multiple-based methods as a matter of principle. It sits wherever the discount rate puts it, and the discount rate can put it anywhere on the page. A reader who takes the shaded band on the chart seriously will treat 68,880,000 as a floor; a defensible half-point revision to the cost of capital puts the value below it with no change of view about the business whatsoever.

What to do with it

Nothing about the technique is specific to growth. The move is to fix everything that is not in dispute and solve for the thing that is, choosing the input the people in the room are competent to argue about. Solve for the cost of capital when the buyer claims the business is safer in their hands. Solve for the steady-state margin when the price requires nineteen or twenty against Cawdrey's 17.50 per cent, which is an operational claim somebody either owns or does not. Solve for the forecast itself: a price that requires 6,900,000 of year-five free cash flow when the plan delivers 5,615,002 is not a valuation difference, it is a different plan.

Then change what the range is a range of. Rank the drivers by how far each moves the answer and publish the rank order, state the span each is varied over and defend it, and put the driver's name next to the number — not “high case 85,393,469” but 85,393,469 at an 8.0 per cent cost of capital and 2.0 per cent growth. A value without its assumption attached is an orphan, and orphans get adopted by whoever wants them most.

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