Private Equity puzzles, solved step by step

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  1. 053A commercial due diligence study costs Rs 2 crore. There is a 25% chance it uncovers a deal-breaker that would otherwise cost the fund Rs 30 crore. Is the study worth buying?Probability and expected value in dealsCoreMid-market buyout fund

    Try it first

    What is the most the fund should be willing to pay for this study?

    Show the worked solution

    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.

    Diligence is worth what it can change, times the chance it changes it (Rs crore)Buy thestudy?NoYes, pay 225%: problem, lose 3075%: fine, lose 0Expected: 0.25 x -30= -7.525%: found, walk away, -275%: clean, proceed, -2Expected: the fee only= -2.0Saves 7.5, costs 2Worth +5.5 net
    Proceeding blind carries a 25% chance of a Rs 30 crore loss, an expected loss of Rs 7.5 crore; buying the study caps the cost at its Rs 2 crore fee, so the study is worth Rs 5.5 crore more than it costs.
    The relationship
    V=p×L−c=0.25×30−2=5.5V = p \times L - c = 0.25 \times 30 - 2 = 5.5
    pthe chance the deal-breaker exists and the study finds it, 25%
    Lthe loss avoided by walking away, Rs 30 crore
    cthe cost of the study, Rs 2 crore
    What it says in wordsThe net value of a test is the loss it prevents times the chance it prevents it, less what the test costs.

    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.

    Where candidates lose it

    Candidates compare the Rs 2 crore fee with the Rs 30 crore loss and say yes because 30 is bigger, which would also say yes at a fee of Rs 20 crore. The comparison that decides it is the fee against the expected saving of Rs 7.5 crore.

    The opposite slip is dismissing the study because the problem probably is not there. A 75% chance of a clean result is exactly why the study is cheap insurance, not why it is wasted.

    What the interviewer asks next

    • What if the study also raises a false alarm on 10% of clean deals, and you would walk away from a good deal worth Rs 12 crore?
    • At what fee would you be indifferent?
    • How would you put a value on the time the study takes in a competitive auction?
  2. 097Each deal in a fund has a 20% chance of returning 0x, a 50% chance of 2x and a 30% chance of 4x, independently. What is the expected money multiple per deal, and what is the chance that at least one of three deals is a zero?Probability and expected value in dealsCoreLarge-cap buyout fundSecondaries and fund of funds

    Try it first

    Chance that at least one of three deals returns zero?

    Show the worked solution

    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.

    A 2.2x average hides a near coin-flip chance of at least one zero20%0x50%2x30%4xmean 2.2xOne deal: chance of each multiple51.2%12.8%12.8%3.2%12.8%3.2%3.2%0.8%Three deals: green = no zero, red = a zeroNo zero: 51.2% At least one zero: 48.8%
    One deal returns 0x with 20% chance, 2x with 50% and 4x with 30%, an average of 2.2x; across three independent deals only the path with no zeros, 51.2%, avoids a loss, so the chance of at least one zero is 48.8%.
    The relationship
    E[M]=0.2(0)+0.5(2)+0.3(4)=2.2P(≥1 zero)=1−0.83=0.488E[M] = 0.2(0) + 0.5(2) + 0.3(4) = 2.2 \qquad P(\geq 1 \text{ zero}) = 1 - 0.8^3 = 0.488
    E[M]expected money multiple of one deal
    0.8the chance one deal is not a zero
    0.8^3the chance none of three deals is a zero
    What it says in wordsAverage the outcomes by their chances; for at least one, take one minus the chance of none.

    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.

    Where candidates lose it

    The fast wrong answer to the second half is 60%, adding the probabilities. Probabilities of overlapping events cannot simply be added; the complement avoids the trap.

    The other slip is treating the expected 2.2x as what each deal returns. Say plainly that the average is a property of many deals, not of one.

    What the interviewer asks next

    • What is the chance that exactly one of the three deals is a zero?
    • With ten deals, what is the chance of at least one zero?
    • If outcomes are not independent, for example all deals in one sector, what changes?
  3. 099A fund screens 40 deals a year. 25% reach investment committee, 30% of those are signed and 60% of signed deals close. How many deals close? How many must be screened to expect 5 closes?Probability and expected value in dealsCoreMid-market buyout fundIndian mid-market PE

    Try it first

    How many deals close from 40 screened?

    Show the worked solution

    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%.

    Conversion rates multiply: 25% x 30% x 60% = 4.5% of screened deals close40Screenedx 25%10Investment committeex 30%3Signedx 60%1.8ClosedWork it backwards25% x 30% x 60% = 4.5%of screened deals closeFor 5 closes:5 / 0.045 = 111deals screenedabout 2.8 times today's 40
    Forty screened deals narrow to 10 at investment committee, 3 signed and 1.8 closed, an overall conversion of 4.5%, so expecting 5 closes needs about 111 deals screened.
    The relationship
    40×0.25×0.30×0.60=1.850.045≈11140 \times 0.25 \times 0.30 \times 0.60 = 1.8 \qquad \frac{5}{0.045} \approx 111
    0.25, 0.30, 0.60the conversion rate at each stage
    0.045the overall share of screened deals that close
    What it says in wordsMultiply the stage rates to get the overall rate; divide the target by it to size the top of the funnel.

    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.

    Where candidates lose it

    The common slip is averaging or adding the rates, or applying each one to the original 40. Every rate applies only to the deals that survived the stage before.

    The second is answering 111 and calling it certain. It is the screening volume for an expected 5 closes; to be confident of 5, you need more.

    What the interviewer asks next

    • Which stage would you try to improve first, and why?
    • If closing takes six months, how many deals must be in the funnel at any time?
    • How would you estimate these conversion rates for a new fund with no history?