Private Equity puzzles, solved step by step

About 1,133, so the winner overpays by about 133 on average. Each estimate is right on average, but the auction picks the highest of five. Five draws spread evenly between 800 and 1,200 sit on average at the sixths of the range, and the top one averages 800 plus five sixths of 400, about 1,133. To avoid overpaying, each bidder should bid below its own estimate, and by more as the number of bidders grows.

If every estimate is unbiased, how can the winner be wrong?

Picture five friends guessing the number of sweets in a jar. On average they are right, but the one who guesses highest is almost certainly too high. Each estimate is unbiased on its own, but the auction does not pick a random estimate; it picks the highest, and the highest of several noisy guesses is biased upwards. That is the {term('winner’s curse', 'The tendency for the winner of an auction, chosen because its estimate was highest, to have overestimated the value of what it bought.')}: winning is itself evidence that you overestimated.

Why does the top estimate average five sixths of the range?

Five points dropped at random on a line split it, on average, into six equal gaps. So the expected sorted estimates sit at one sixth, two sixths and so on up to five sixths of the way from 800 to 1,200: 867, 933, 1,000, 1,067 and 1,133. The top one is 400 x 5/6, or 333, above 800, which is 1,133. With n bidders the top estimate averages n over n plus 1 of the range, so ten bidders push it to about 1,164.

What does a disciplined bidder do with this?

Shade the bid. A bidder that wants to break even when it wins must bid as though its estimate is the highest of five, which in this setup means bidding about 133 below its estimate, and more in a crowded auction. That is the case for walking away from processes with many bidders and for building value on what the buyer can change rather than on a higher estimate of the same cash flows. Say the limits: real bidders have different information and different synergies, so not all of the gap is error, and a bidder with genuine private information is less exposed.

The earnout is worth about 15.9 to you, so the price in value terms is about 115.9. A 40% chance of 50 is 20 on average, payable in two years, and 20 / 1.12^2 is 15.9. The headline price is 150, but you are paying about 116 in value. A seller who thinks the chance is 80% values the same earnout at 31.9, which is why earnouts close price gaps.

How do you value a payment that may not happen?

A shopkeeper who offers a Rs 500 voucher to anyone who spends Rs 5,000 next month books its cost by how many customers she expects to claim it, not by the face value. An earnout is a contingent payment, so you value it as chance times amount, discounted for the wait: 0.4 x 50 / 1.12^2 is about 15.9. That number, not 50, is what the earnout adds to your price.

Why does an earnout make a deal possible?

Because both sides can be right in their own terms. The seller, confident in the plan, values the package at about 132; the buyer, more cautious, values it at about 116; and the contract lets each side bet on its own view. If the target is hit, the buyer pays more but has a business that performed. If not, the buyer has paid 100 for a business that did not.

Say the risks. Earnouts are a common source of disputes: how EBITDA is defined, which costs the buyer loads onto the business after closing, and whether the seller's management stays to run it. A buyer controls the business that decides the payout, so the contract needs a tight EBITDA definition and agreed accounting policies.

About 40%. Picture 1,000 targets: 100 have a problem and the review flags 90 of them. Of the 900 clean targets, it wrongly flags 15%, which is 135. So 225 targets are flagged and only 90 of those are real problems, 90 divided by 225. A good test on a rare problem still throws up more false alarms than true ones.

Why is the answer not 90%?

Think of a smoke alarm that goes off for almost every real fire, but also for burnt toast. In a house where fires are rare and toast is daily, most alarms are toast. The 90% is the chance of a flag when there is a problem; the question asks the reverse, the chance of a problem when there is a flag, and the two differ because clean targets are far more common. Counting people instead of using percentages makes the reversal visible.

What should a deal team do with a 40% flag?

Treat the flag as a reason to dig, not a reason to walk away. A flag lifts the chance of a problem from 10% to 40%, four times the starting level, but it still leaves the target more likely clean than not. A second, independent check changes the picture: if a forensic review with the same hit and false alarm rates also flags it, the chance rises to about 80%. The limit to state: that assumes the two reviews make independent mistakes, which two teams looking at the same ledgers may not.