6A.7 Merger arbitrage · M06A Events, sectors, factors

Motivation

After a merger is announced, the target trades below the value the deal will deliver. The gap, the spread, pays whoever bears the risk that the deal fails. Pricing that risk is a probability problem, and its payoff has the shape of a short option.

6A.7.1Implied completion probability#

Setup

A deal pays $D$ per target share at completion, $T$ years away. For cash deals $D$ is the offer price. For stock deals with exchange ratio $R$ (acquirer shares per target share), the arbitrageur buys one target share and shorts $R$ acquirer shares, so $D = R\,P_A$ is locked in at today's acquirer price. If the deal fails, the target falls to a fallback value $F < D$ , and in a stock deal the short is bought back at whatever the acquirer then trades at. Financing costs $r$ per year (simple), and net carry $c$ (target dividends received minus $R$ times the acquirer dividends owed on the short, plus any short rebate) accrues per target share.

Proposition 6A.10implied completion probability

If the target price makes the hedged position earn the financing rate in expectation, then

P_T(1 + rT) = p\,D + (1 - p)F + c \quad\Longrightarrow\quad p = \frac{P_T(1 + rT) - c - F}{D - F}. \tag{6A.12}

The annualised return if the deal closes is $\big[(D + c)/P_T\big]^{1/T} - 1$ (compound) or $\big[(D + c)/P_T - 1\big]/T$ (simple).

Proof

At $T$ the position pays $D + c$ with probability $p$ and $F + c$ otherwise; its expected value equals the financed cost $P_T(1 + rT)$ . Solve the linear equation for $p$ .∎

The formula is risk-neutral in spirit: it ignores any premium for bearing deal risk. Because arbitrage capital is limited and failure correlates with bad markets, the true completion probability is usually higher than (6A.12); Baker and Savaşoğlu (2002) found that US merger arbitrage earned excess returns consistent with limited arbitrage. A result above 1 means the inputs are inconsistent: usually the fallback $F$ is set too low, or the spread does not even cover financing.

6A.7.2The short-put payoff#

Proposition 6A.11the payoff is a short put

Suppose the acquirer completes the deal only if the target's standalone value $S$ at the decision date is at least $D$ , and otherwise walks away (a material-adverse-change clause, or a re-cut that the target rejects), leaving the target at $S$ . Then the payoff of the long target position is

\min(S, D) = D - (D - S)^+, \tag{6A.13}

a riskless amount $D$ minus a put on the target's standalone value struck at $D$ .

Proof

If $S \ge D$ the deal completes and pays $D = \min(S, D)$ . If $S < D$ it fails and the holder keeps $S = \min(S, D)$ . And $\min(S, D) = D - \max(D - S, 0)$ .∎

Real failure rules are messier (regulatory refusals that are unrelated to $S$ , competing bids that pay more than $D$ ), but the shape survives: small steady gains, rare large losses, and losses concentrated in falling markets when standalone values fall. Mitchell and Pulvino (2001), with 4,750 US deals over 1963–1998, found risk-arbitrage returns nearly uncorrelated with the market in flat and rising markets and strongly positively correlated in severe declines: like selling index puts. Size with M03, not with the headline annualised spread.

6A.7.3A cash deal and Indian practice#

Worked example

A cash offer at ₹500, target at ₹470, fallback ₹380, expected close in six months, financing 6.5% a year, no carry. If it closes, the position earns $500/470 - 1 = 6.38\%$ , or 13.2% annualised (compound). By (6A.12), $p = (470 \times 1.0325 - 380)/120 = 0.877$ . The trade is a bet that the true probability is above 87.7%; at 95% the expected six-month return is $0.95 \times 500 + 0.05 \times 380 = 494$ , i.e. 5.1% before financing. The Kelly fraction for a binary bet that wins 6.4% or loses 19.1% with probability 0.95 is $0.95/0.191 - 0.05/0.0638 = 4.2$ (more than fully invested), which shows how sensitive Kelly is to $p$ : at $p = 0.191/(0.191 + 0.064) = 0.75$ , where the bet has zero expected return before financing, it is zero (M03 · 3.5 on estimation risk). The financing-adjusted break-even is the 0.877 of (6A.12).

Indian specifics

Approvals: CCI (competition), sector regulators (RBI for banks and NBFCs, IRDAI for insurers), stock exchanges and SEBI on the scheme, shareholders and creditors, and the NCLT. Timelines of 9–18 months are common, so $T$ is long and financing dominates.
The short leg: single-stock futures (if the acquirer is in F&O), whose basis embeds the financing and expected dividends, or SLB. Lot sizes force rounding of the hedge ratio $R$ .
Index events: when a merged entity's weight changes, passive funds trade on the effective date; this can move the spread in the final days.