Enterprise value is EBITDA times a multiple, so when both move the change splits into three parts, not two, and the third is where bridges get argued.
Value EBITDA growth at the entry multiple and multiple expansion on the entry EBITDA, and a third piece is left over: the change in EBITDA times the change in multiple. On a deal that grows EBITDA from 14.0 to 24.5 and re-rates from 9.5× to 10.2×, that cross-term is 7.35, and depending on where you put it, multiple expansion is worth either 9.80 or 17.15. Show it on its own line.
Worked in full in The Private Equity Operating Partner by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
Every value creation bridge in a private equity deck splits the change in enterprise value into EBITDA growth and multiple expansion. Enterprise value is EBITDA times a multiple, and when both move, the change does not separate cleanly into two parts. It separates into three. The third is the one most bridges quietly allocate, and the allocation is a choice, not a fact.
| Input | Entry | Exit | Change |
|---|---|---|---|
| EBITDA | 14.0 | 24.5 | 10.5 |
| EV / EBITDA multiple | 9.5× | 10.2× | 0.7× |
| Enterprise value | 133.0 | 249.90 | 116.90 |
EBITDA rises 75.0 per cent over the hold. The multiple rises 7.4 per cent. Enterprise value rises by 116.90.
Write exit value as entry value plus the two changes and expand the product:
E1M1 − E0M0 = ΔE × M0 + ΔM × E0 + ΔE × ΔM
In Excel, with entry and exit EBITDA in B2:C2 and the multiples in B3:C3:
=(C2-B2)*B3, =(C3-B3)*B2 and =(C2-B2)*(C3-B3). Add a
fourth cell, =C2*C3-B2*B3-SUM(...), that must return zero.
The first two terms are unambiguous. Growth at an unchanged multiple belongs to growth; re-rating of an unchanged business belongs to the multiple. The cross-term is the extra value from the new EBITDA being priced at the new multiple. It exists only because both moved, so it belongs to both and to neither.
Most bridges never show the cross-term. They fold it into one of the two buckets, usually by the order in which the steps are computed. Here is what each convention reports for the same deal:
| Convention | EBITDA growth | Multiple expansion | Cross-term shown | Multiple share of change |
|---|---|---|---|---|
| Growth valued at the exit multiple | 107.10 | 9.80 | not shown | 8.4% |
| Re-rating applied to exit EBITDA | 99.75 | 17.15 | not shown | 14.7% |
| Midpoint split | 103.42 | 13.47 | not shown | 11.5% |
| Cross-term on its own line | 99.75 | 9.80 | 7.35 | 8.4% |
Moving 7.35 between buckets changes the growth figure by 7.4 per cent. It changes the multiple figure by 75.0 per cent. The cross-term is only 6.3 per cent of the value created, but it is almost as large as the pure re-rating, so whichever bucket receives it, the multiple line is the one that moves.
That asymmetry is why the choice is not neutral. A buyer discounts the multiple bucket, because the next owner cannot repeat a re-rating. A seller has every reason to value growth at the exit multiple and leave the multiple bucket at 9.80. Nothing in the arithmetic stops them, and the bridge still reconciles to the cent.
Hold EBITDA at 24.5 and move only the exit multiple. The pure growth term stays at 99.75 at every exit, because it is valued at the entry multiple. Everything else moves.
| Exit multiple | Change in EV | Pure re-rating | Cross-term | Growth if cross goes to growth | Multiple if cross goes to multiple |
|---|---|---|---|---|---|
| 8.5× | 75.25 | −14.00 | −10.50 | 89.25 | −24.50 |
| 9.0× | 87.50 | −7.00 | −5.25 | 94.50 | −12.25 |
| 9.5× | 99.75 | 0.00 | 0.00 | 99.75 | 0.00 |
| 10.2× | 116.90 | 9.80 | 7.35 | 107.10 | 17.15 |
| 11.0× | 136.50 | 21.00 | 15.75 | 115.50 | 36.75 |
| 12.0× | 161.00 | 35.00 | 26.25 | 126.00 | 61.25 |
Read the 8.5× row. The cross-term is now negative, and a bridge that values growth at the exit multiple reports operating growth of 89.25 instead of 99.75. The operating team is charged for a market move it did not make. The convention that flatters the growth bucket in a rising market penalises it in a falling one, and a fund that switches convention between the two has two bridges that cannot be compared.
The usual error is not arithmetic. It is computing the bridge sequentially, "first grow EBITDA, then re-rate", without noticing that the order decides the answer. Grow EBITDA first at the entry multiple (99.75), then re-rate on the exit EBITDA (17.15), and the cross-term lands in the multiple; re-rate first on the entry EBITDA (9.80), then value growth at the exit multiple (107.10), and it lands in growth. The two orders produce bridges that both reconcile, both look rigorous, and disagree by 7.35 on every line that matters. A model that hardcodes one order never reveals that a choice was made.
The second error is percentage arithmetic: adding 75.0 per cent EBITDA growth and 7.4 per cent multiple growth and expecting the sum to explain the change. It cannot, because 1.75 × 1.074 is 1.879, while adding the two rates gives 1.824. The missing product is the cross-term again.
The enterprise value bridge is only half the job. The equity bridge adds deleveraging, fees and add-on acquisitions, and the free workbook for this case builds all six buckets of a buyout with a residual of zero, at the companion page. For how entry leverage changes the operating share of that bridge, see does higher leverage dilute the operating partner's contribution, and for the full deal model, the LBO model template.
It is the change in EBITDA multiplied by the change in the multiple: the extra value from new EBITDA being priced at a new multiple. On EBITDA rising by 10.5 and the multiple by 0.7 turns it is 7.35. It exists only because both moved, so neither growth nor re-rating can claim it on arithmetic alone, and best practice is to show it separately.
At the entry multiple, if you want a figure that does not depend on the market. Valuing growth at the exit multiple silently adds the cross-term to growth: 107.10 instead of 99.75 in an illustrative deal re-rating to 10.2x. When the multiple compresses to 8.5x the same convention cuts reported growth to 89.25 and charges the operating team for the market.
Usually because they compute the steps in a different order. Growing EBITDA first at the entry multiple and re-rating second on exit EBITDA puts the cross-term in multiple expansion; re-rating first on entry EBITDA and valuing growth at the exit multiple puts it in growth. Both reconcile to the same total. On an illustrative deal the multiple bucket reads 9.80 one way and 17.15 the other, a difference of 75.0 per cent from one hidden choice.
Building the value creation bridge is the subject of chapter 13 of The Private Equity Operating Partner. The book takes the same case from first principles to the decision, chapter by chapter, and every figure it prints is a live formula in the free companion workbooks.
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