Private Equity puzzles, solved step by step

28%. The easy route is through the opposite event. The chance there is no restatement is 80% and the chance the customer stays is 90%; because the risks are independent, the chance of neither is 0.8 x 0.9, which is 72%. At least one problem is everything else, 100% less 72%, or 28%. Adding 20% and 10% gives 30% and double counts the 2% chance of both.

Why not just add the two chances?

Picture two friends who each might be late for dinner. If you add their chances of being late, the evening where both are late gets counted once for each of them. Adding probabilities works only when the events cannot happen together; when they can, the overlap is counted twice. Here both problems happen together 0.2 x 0.1, or 2%, of the time, so 20 plus 10 overshoots by exactly that: 30 less 2 is 28.

Why is one minus none the safest route?

Because the opposite of at least one is a single clean case: nothing goes wrong. At least one problem is everything except the case where neither happens, so you multiply the two chances of no problem and subtract from one. It scales without effort. With five independent risks of 10% each, the chance none happens is 0.9 to the fifth, about 59%, so at least one is about 41%, a number that adding would put at 50%.

Is independence a fair assumption in diligence?

Usually not, and saying so earns credit. A company that needs a revenue restatement may have weak controls or stretched customer relationships, so the two risks tend to move together, which raises the chance of both and lowers the chance of at least one below 28%. If the risks were perfectly linked, so the customer leaves only when revenue is also restated, the answer would fall to 20%. The 28% is the answer the question asks for; the correlation point is the judgement the interviewer is listening for.

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.

Yes: the study is worth Rs 7.5 crore in expectation and costs Rs 2 crore, a net gain of Rs 5.5 crore. Without it, the fund carries a 25% chance of losing Rs 30 crore, an expected loss of Rs 7.5 crore. With it, the fund walks away in the bad case and pays only the fee. The study stays worth buying as long as the chance of a problem is above 2 over 30, about 6.7%.

What exactly is the fund buying?

Think of paying a mechanic to inspect a second-hand car before you buy it. The inspection is worth nothing if the car is sound and a great deal if the gearbox is about to fail. A diligence study is worth the loss it can prevent, multiplied by the chance that the loss is really there. It is not worth the loss itself, and it is not worth zero because the problem is unlikely. Draw both choices as branches and the answer is visible.

What assumptions is that answer resting on?

Two, and you should say both. First, the study is perfect: it always finds the problem when it is there and never raises a false alarm. If the study catches the problem only 60% of the time, its value falls to 0.25 x 0.6 x 30, Rs 4.5 crore, still above the fee. Second, walking away costs nothing beyond the fee. In a live auction, pausing for a study can lose the deal to a faster bidder, and that cost belongs on the yes branch.

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.

The expected multiple is 2.2x, and the chance of at least one zero in three deals is 48.8%. Expected value is 20% x 0 plus 50% x 2 plus 30% x 4, which is 2.2. For the zero, take the complement: each deal avoids a zero 80% of the time, so all three do 0.8 cubed, 51.2% of the time. A good average hides a near coin-flip chance of a loss.

How do you get the expected multiple?

Weight each outcome by its chance and add. It is the same as asking what a hundred identical deals would return on average: twenty return nothing, fifty return 2x and thirty return 4x, which is 0 plus 100 plus 120, or 220 on 100 put in. The expected multiple is 2.2x, but no single deal ever returns 2.2x: each one returns 0, 2 or 4.

Why does at least one zero use the complement?

There are many ways to get at least one zero: the first deal, the second, the third, or any two, or all three. There is only one way to get no zero, so count that and subtract it from one. Adding 20% three times gives 60% because it counts the two-zero and three-zero cases more than once. The chance all three are zeros is 0.2 cubed, 0.8%.

The fund-level point is the one to close on: a manager with a 2.2x average still writes off a deal in about half of all three-deal stretches. That is why investors judge a manager over many deals, and why a single write-off early in a fund says little on its own.

About 1.8 deals close, and about 111 must be screened to expect 5. Forty screened becomes 10 at investment committee, 3 signed and 1.8 closed. The stage rates multiply to 4.5%, so expecting 5 closes takes 5 divided by 0.045, about 111 deals screened. Conversion rates multiply, which is why sourcing volume matters so much.

Why do the stage rates multiply?

Each rate applies only to what survived the stage before, like a cricket trial where a quarter of players make the camp, a third of those make the squad and most of the squad play. A funnel's overall conversion is the product of its stage rates, so three reasonable-looking rates combine into a small number: 25% x 30% x 60% is 4.5%.

What would you do with this as a deal team?

You can lift closes by screening more deals or by improving any one stage, and improving a stage is often cheaper. Raising the committee rate from 25% to 35%, by screening sharper, takes closes from 1.8 to 2.5 without a single extra deal. One honest caveat: 1.8 is an expected number. In a year with 40 screens the fund could close zero or four, so a target of 5 closes should be planned with a margin above 111.

Interviewers in Indian mid-market funds often ask this to see whether you understand that origination is a numbers game. A candidate who says a team needs to see more than a hundred opportunities to close five has understood why associates spend so much time on screening.