Case 012Buy-side portfolio judgementHard

Merger arbitrage: Kaldrin Power Systems offers Rs 450 cash for Pravena Cables, which trades at Rs 420, with closing expected in six months and a fall-back price of Rs 330 if the deal breaks. What probability is priced in, and is the spread worth it if you think 85%?

AQR Capital ManagementGreenwich · 2021

The situation

Kaldrin Power Systems has agreed to buy Pravena Cables for Rs 450 a share in cash. Pravena trades at Rs 420. The deal needs competition approval and a shareholder vote, and is expected to close in six months. Before the offer Pravena traded at Rs 300; since then its sector has risen, and you estimate it would trade at Rs 330 if the deal broke.

Your read of the competition review puts the chance of closing at 85%. The fund's cost of money is 7% a year.

Your task

What probability of closing does Rs 420 imply, what is your expected return at 85%, and is the trade worth taking?

Quick check

What chance of closing is the market pricing?

Worked solution

Try it on paper, then open one step at a time.

30-second answerThe answer to give first

The market prices about a 75% chance of closing; at your 85%, the expected return is about 2.9% over six months, 5.8% a year, which does not clear a 7% cost of money. The spread offers Rs 30 of upside against Rs 90 of downside. You need about 87% confidence to beat the cost of money, so your edge is real but too thin. A small position at most, or a wider spread.

Step 1How does a price turn into a probability?

A ticket that pays Rs 100 if your team wins and Rs 20 if it loses, selling at Rs 80, says the crowd gives your team three chances in four: 20 plus three quarters of the 80 gap is 80. A target's share price in a cash deal is a weighted average of the offer and the break price, so the weights are the market's probability. For Pravena, (420 - 330) / (450 - 330) is 75%. That spreadThe gap between the offer price and the current price of the target. In merger arbitrage it is the reward for bearing the risk that the deal fails. is the market's doubt, priced in rupees.

The price sits where the market thinks the odds areMarket price Rs 420Deal breaks: Rs 330Deal closes: Rs 45090 of the 120 gap30Implied probability = (420 - 330) / (450 - 330) = 90 / 120 = 75%Upside +30 (7.1%) against downside -90 (21.4%)
Pravena at Rs 420 sits 90 rupees above its Rs 330 break price and 30 below the Rs 450 offer, three quarters of the way up the gap, so the market prices a 75% chance that the deal closes.
Step 2What is the trade worth at 85%?

Expected value is 85% of Rs 450 plus 15% of Rs 330, which is Rs 432. Against Rs 420 paid, that is a 2.86% expected gain over six months, about 5.8% a year. The fund's money costs 7% a year, and to earn that the deal needs about an 87% chance of closing. Your 85% beats the market's 75% but not the 87% hurdle, so the trade adds risk without covering its cost.

The relationship
pimplied=P−BO−B=420−330450−330=75%E=p O+(1−p) B=432p_{\text{implied}} = \frac{P - B}{O - B} = \frac{420 - 330}{450 - 330} = 75\% \qquad E = p\,O + (1-p)\,B = 432
Pcurrent price of the target, Rs 420
Bestimated break price, Rs 330
Ooffer price, Rs 450
pprobability the deal closes
What it says in wordsThe implied probability is how far the price has climbed from the break price towards the offer; your expected value weights the two outcomes by your own probability.
Your edge against the market, and against the cost of money-10%0%+10%7% cost of moneyMarket: 75%, 0%You: 85%, 5.8% a yearNeeded: 87%Certain: 14.8%60%70%80%90%100%Your probability that the deal closes; y axis is annualised expected return
Buying Pravena at Rs 420 returns nothing a year at the market's 75% probability, about 5.8% at your 85% and 14.8% if the deal is certain; beating a 7% cost of money needs about 87% confidence.
Step 3What would change the answer?

The break price is the weakest input. If Pravena would fall to Rs 300 rather than Rs 330 on a break, the market is implying 80% rather than 75%, and your edge shrinks. Time matters too: if approval takes twelve months instead of six, the annual return halves. Hedging can help: shorting shares of the sector would protect the break price against a market fall, which is exactly what the Rs 330 estimate assumes has not happened. Say that the payoff is lopsided, a small gain most of the time and a large loss occasionally, so position size should allow for the loss, not the average.

Where candidates lose it

The fast wrong answer divides 420 by 450 and calls it 93%. That treats a failed deal as worth nothing, and the shares do not go to zero; they fall back to a break price, and the probability must be measured from there.

The second loss is saying the trade is good because 85% beats 75%. Beating the market's probability is not enough; the expected return must also beat the cost of the money tied up for six months.

What the interviewer asks next

  • How would you estimate the break price more carefully?
  • Kaldrin pays in its own shares instead of cash. How would you set up the trade?
  • Competition approval slips by six months. What is your annualised return now?
  • Why do merger arbitrage returns look like selling insurance?

Asked at AQR Capital Management, Quantitative Research, Greenwich, 2021 (Wall Street Oasis): Questions about merger arbitrage strategies. Hedging. Python programming. Data analysis and regression.

Company names and figures are illustrative.