Private Equity puzzles, solved step by step

About Rs 49.6 crore off the price, but a cut is the weaker fix. Expected damages are 30% x 200 = Rs 60 crore, payable in two years, so 60 / 1.1^2 is Rs 49.6 crore today. A price cut is fair on average yet leaves you Rs 200 crore exposed if the case is lost. A specific indemnity from the seller backed by an escrow moves the bad branch to the seller instead.

How do you put a number on something that may never happen?

A shopkeeper who knows that 3 in 10 customers return a gadget within the warranty sets aside money for returns before counting profit. A contingent liability is priced as chance times cost, then discounted for time: here 0.3 x 200 = Rs 60 crore, and two years at 10% brings it to Rs 49.6 crore. That is the average cost across all the ways the case could end, in today's money.

Why is a price cut not the best answer?

Because a buyer does not live the average; it lives one branch. Cut the price by 49.6 and lose the case, and you have paid for a Rs 200 crore liability with a Rs 49.6 crore discount. A specific indemnityA promise by the seller to reimburse the buyer for one named liability, here the lawsuit, if it costs money after completion. puts the bad branch back with the seller, who knows the case best. Backing it with an escrow, part of the price held back until the case ends, makes sure the money is there to collect.

Say the trade-offs. Sellers dislike escrows because cash is locked up, so expect to give something back, perhaps a smaller cut or a cap on the indemnity. Warranty and indemnity insurance usually excludes known issues like this one, so check the policy before relying on it. And the 30% is itself an estimate: ask the legal team how they reached it, because the answer moves one for one with it.

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.

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.