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Chapter 3 of 4 · 10 min

Probability-weighted value

A deal either closes or it breaks. Weight each outcome by its chance and you know what the stock is worth to you, and what odds the market is pricing.

By the end of this chapter you can
  • Calculate a stock's expected value from the close and break outcomes
  • Back out the probability of closing the market is pricing
  • Measure your edge against the market
  • Value the chance of a higher, competing bid
1

The intuition

A raffle ticket pays $100 if it wins and nothing if it loses, and you think it has an 80% chance of winning. It is worth $80 to you. If someone sells it for $70, they must think the chance is 70%. Whether to buy depends on whose number you trust.

A merger arb position is that ticket with two prizes: the offer if the deal closes and the break price if it fails. Weighting them by your probability gives the expected value. The probability that makes the expected value equal today's price is the implied probability: the market's number.

The key idea

Expected value = p × offer + (1 − p) × break price. Implied probability = (price today − break price) ÷ (offer − break price). Edge = your probability − implied probability.

2

Why it works

  • The conventions here: two outcomes, the offer or the break price, with no time value. The bump case adds a third outcome, a higher offer, whose probability comes out of the chance of closing, not out of the chance of a break.
  • The implied probability is how far the price has traveled from the break price toward the offer, as a share of the whole distance.
  • A wide spread is not a reason to trade; an edge is. If you think closing is less likely than the market does, the stock is worth less to you than its price.
  • The implied probability depends on the break price, which is an estimate: move it a few dollars and the implied probability moves several points.
  • A possible bump is worth its probability × the extra per share. A stock trading above the offer is the market already paying for one.
  • Hold your probability as a range, and ask whether the trade still works at the bottom of it.
Offer $60; break price $44; price today $56; your chance of closing 85%
Implied probability: (56 − 44) ÷ (60 − 44)75%
Expected value: 85% × 60 + 15% × 44$57.60
Expected return: 57.60 ÷ 56 − 12.9%
Edge: 85% − 75%10 points
A 20% chance of a 10% bump, out of the 85%: 20% × 66 + 65% × 60 + 15% × 44$58.80
The chance of a bump is worth: 58.80 − 57.60$1.20 a share

The same answer directly: a 20% chance of $6 more per share is worth $1.20.

3

The formulas

Expected value = p × offer + (1 − p) × break price

Weight each outcome by its chance.

Expected return = expected value ÷ price today − 1

What the position earns on your odds.

Implied probability = (price − break) ÷ (offer − break)

The market's chance of closing.

Edge = your probability − implied probability

The reason to trade, or not to.

With a bump: EV = p(bump) × bump + p(close) × offer + p(break) × break

Three outcomes.

4

Worked example

Weight the offer by the chance of closing and the break price by the rest, then compare with today's price.

Drawing the numbers…
5

See it move

Same company and the same break price. Change the offer's premium over the break price, the spread from today's price to the offer, your own odds, and the size and chance of a bump.

Drawing the numbers…
Try this
  • Raise your chance of closing. The expected value and your edge rise; the market-implied probability does not move.
  • Widen the spread. Today's price falls, the implied probability falls, and your edge grows.
  • Raise the offer's premium with the spread held. The implied probability rises.
  • Raise the size or the chance of a bump. The bump bar grows.
6

Run it backwards

Same company, reversed: from the offer, the break price and today's price, what chance of closing is the market pricing in?

Drawing the numbers…

Set today's price equal to the expected value and solve for the probability: how far the price has come from the break price, over the whole distance from the break price to the offer.

Compare it with your own number. The difference is your edge, or the lack of one, however wide the spread looks.

7

Traps

Treating the spread as the expected return.
The spread is what closing pays; the expected return also counts the loss if the deal breaks.
Measuring the implied probability from zero.
Measure from the break price: (price − break) ÷ (offer − break).
Taking a bump's probability out of the break.
A bump is one way the deal closes, so it comes out of the chance of closing.
Treating the implied probability as a fact.
It moves with your estimate of the break price.
Trading a positive expected value at any size.
A likely small gain and a rare large loss need sizing so a break is survivable.
8

Say it in the interview

The interviewer asks

“An arb says the market is pricing a deal at 80%. What does that mean?”

Say yours out loud first, then compare.
9

Check yourself

4 fresh questions, with new numbers. Answer each one correctly to finish the chapter. Get one wrong and you will see the full working, then you can try it again with new numbers.

Answers within 1% are marked right. Type the number; $, %, x and M are fine. First tries count toward Learned: the topic is Learned once every chapter is done and 75% of first tries were right.

0 of 4
Drawing your questions…
Remember
  • Expected value = p × offer + (1 − p) × break price.
  • Implied probability = (price − break) ÷ (offer − break).
  • Edge = your probability − the implied probability.
  • A bump is worth its chance × the extra per share.