The spread, its annualised return, and the completion probability the market price implies, for a cash deal and a hedged stock deal.
The merger arbitrage spread is the offer price less the target's current share price, plus any dividends due before closing, divided by the current price. A cash offer of 50.00 for a target trading at 47.50, with a 0.25 dividend, is a spread of 2.75, or 5.79 per cent, 11.91 per cent annualised over a six-month close. If the deal fails and the shares fall back to 36.00, the price implies the market gives completion a probability of about 80.36 per cent.
Worked in full in Mergers and Acquisitions by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
The spread is what the market charges for carrying deal risk: regulatory refusal, a failed shareholder vote, a financing gap, a material adverse change. Read the right way, it tells a target's board, an acquirer and an investor the same thing: how much doubt the market has about this deal closing on these terms, and by when.
| Input | Value |
|---|---|
| Offer price per share, cash | 50.00 |
| Target share price today | 47.50 |
| Unaffected price before the announcement | 38.00 |
| Estimated price if the deal fails | 36.00 |
| Dividend payable before closing | 0.25 |
| Expected time to closing | 6 months |
| Funding rate, for the refined probability | 4.0% |
The offer is a 31.58 per cent premium to the unaffected price. The fallback price is the hardest input: it is not the unaffected price, because the market has moved since, and the company's own news may have moved it further. Here it is set below the unaffected price on the view that peers have fallen.
1. Gross spread = 50.00 − 47.50 = 2.50, or 5.26 per cent of the price
2. Total spread with the dividend = 2.50 + 0.25 = 2.75, or 5.79 per cent
3. Annualised = (1 + 5.79%)12 ÷ 6 − 1 = 11.91 per cent; simple, 11.58 per cent
4. Loss if the deal fails = 47.50 − 36.00 − 0.25 = 11.25, or 23.68 per cent, 4.1 times the gain
5. Implied probability of completion = (47.50 − 0.25 − 36.00) ÷ (50.00 − 36.00) = 80.36 per cent
Excel: =(1+(Offer-Price+Div)/Price)^(12/Months)-1 and =(Price-Div-Fallback)/(Offer-Fallback)
The probability in step 5 is the one that makes today's price the expected value of the two outcomes: 80.36 per cent of 50.25 plus 19.64 per cent of 36.25 is 47.50. It ignores the cost of holding the position for six months. Carry the price forward at a 4.0 per cent funding rate, 47.50 becomes 48.45, and the implied probability rises to 87.14 per cent. The difference is the time value the arbitrageur also has to earn; the simple version understates the market's confidence.
| Months to close | Annualised return |
|---|---|
| 3 | 25.25% |
| 6 | 11.91% |
| 9 | 7.79% |
| 12 | 5.79% |
| 18 | 3.82% |
Timing moves the return as much as completion does. A second-request antitrust review that pushes the close from six months to eighteen cuts the annualised return from 11.91 to 3.82 per cent, below the funding rate, without the deal ever being in doubt. That is why spreads widen on a regulatory delay even when nobody expects a block.
| Fallback price | Implied probability | Loss / gain |
|---|---|---|
| 32.00 | 84.72% | 5.5x |
| 36.00 | 80.36% | 4.1x |
| 38.00 | 77.08% | 3.4x |
| 42.00 | 65.62% | 1.9x |
The implied probability is only as good as the fallback. Assume the shares would only drop to 42.00 and the market looks sceptical at 65.62 per cent; assume 32.00 and it looks relaxed at 84.72 per cent. Same price, same spread, a range of conclusions nearly twenty points wide.
When the consideration is shares, the arbitrageur buys the target and sells short the acquirer in the exchange ratio, so that a move in the acquirer's price cancels. At a ratio of 0.80 and an acquirer at 60.00, each target share will receive 48.00 of stock; with the target at 46.00 the spread is 2.00, or 4.35 per cent. If the acquirer falls to 54.00 the target share is worth 43.20 at closing, but the short of 0.80 shares has gained 4.80, and the 2.00 is still locked in. The dividends now go both ways: the position receives the target's 0.20 and owes 0.80 times the acquirer's 0.30, or 0.24, so the net spread is 1.96, 4.26 per cent. Fixed or floating ratios and collars change the hedge, as set out in how to calculate the exchange ratio.
The seven prices of one acquisition, and which of them the arithmetic supports, are set out in the free workbook for this case.
Subtract dividends and the fallback price from the current price and divide by the offer less the fallback. With a 50.00 offer, a 47.50 price, a 0.25 dividend and a 36.00 fallback, the market implies 80.36 per cent. Allowing six months of funding at 4.0 per cent raises it to 87.14 per cent.
Compound the total spread over the expected months to closing: (1 + spread) to the power 12 over months, minus one. A 5.79 per cent spread earned in six months is 11.91 per cent a year; if closing slips to eighteen months it is 3.82 per cent.
Buy the target and short the acquirer in the exchange ratio, so the spread is locked whatever the acquirer's price does. At 0.80 of a 60.00 acquirer share against a 46.00 target, the spread is 2.00, or 4.35 per cent; after the dividend owed on the short it is 1.96.
This article is one calculation from Mergers and Acquisitions. 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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