Private Equity puzzles, solved step by step

Puzzles: 100
Traced to a firm: 22
Topics: 12
Hard: 30

Showing 1–4 of 4 · filtered from 100Clear filters

035Without a calculator, estimate the square root of 2.5. Then use it: an investment returns 2.5x in 2 years. What is the annual return?Mental mathsCoreGrowth equity
Try it first
What annual return turns 1 into 2.5 over two years?
Show the worked solution
The square root of 2.5 is about 1.58, so 2.5x over two years is about 58% a year. 1.5 squared is 2.25 and 1.6 squared is 2.56, so the root lies between them and much nearer 1.6: 2.5 is 25 of the 31 hundredths along, giving about 1.581. One Newton step from 1.6 gives 1.5813, against a true 1.5811.
How do you find a square root you have not memorised?
A train leaves one station at 6:00 and reaches the next at 9:00. A signal four fifths of the way along is passed at about 8:24: you slide between the two times you know. Find the two squares you know that bracket the number, then slide between them in proportion: the root sits about as far along as the number sits between the squares. 1.5 squared is 2.25 and 1.6 squared is 2.56. The target 2.5 is 0.25 above the lower square out of a gap of 0.31, about 0.81 of the way, so the root is about 1.5 plus 0.081, or 1.581.
Since 1.5 squared is 2.25 and 1.6 squared is 2.56, the root of 2.5 sits about 25/31 of the way from 1.5 to 1.6, close to 1.581, which makes 2.5x over two years about 58.1% a year rather than 75%.
How do you sharpen the estimate and check it?
Use one Newton step: take a guess, divide the number by it, and average the two. From 1.6, 2.5 divided by 1.6 is 1.5625, and the average of 1.6 and 1.5625 is 1.58125. One averaging step from a good guess gets you to three decimals, close enough for any interview. Check by squaring: 1.58 squared is 2.4964, just under 2.5, so the root is a hair above 1.58.
The relationship
$x_{1} = \tfrac{1}{2}\left(x_0 + \frac{N}{x_0}\right) = \tfrac{1}{2}\left(1.6 + \frac{2.5}{1.6}\right) = 1.58125$
x_0 the starting guess, 1.6
N the number whose root you want, 2.5
x_1 the improved guess
What it says in wordsAverage your guess with the number divided by your guess, and you land much closer to the root.
Now the investing use. A growth deal returning 2.5x in two years earns the square root of 2.5, less one, each year: about 58%. The fast answer, 150% over two years so 75% a year, ignores compounding: 1.75 squared is 3.06, not 2.5. Say the limitation: short holds produce eye-catching IRRs on modest money, so investors read the multiple alongside it.
Where candidates lose it
The common loss is splitting the gain evenly: 150% over two years, 75% a year. That treats the return as simple interest. Over two years the factor is squared, so you need a square root, about 1.581.
The second loss is freezing on the root itself. Interviewers are not testing whether you have 1.5811 memorised. They want to see you bracket it with 1.5 and 1.6 and slide.
What the interviewer asks next
Estimate the cube root of 3, and use it for 3x over three years.
What annual return turns 1 into 2 over two years?
A deal returns 2.5x in two years and another 3x in four. Which would an LP prefer, and why might they disagree?
048Without a calculator, what is 1.1 to the power 7, to two decimal places?Mental mathsCoreLarge-cap buyout fund
Try it first
Which is right to two decimals?
Show the worked solution
About 1.95. Multiply up one step at a time: 1.21, 1.331, 1.4641, 1.6105, 1.7716, then 1.7716 plus a tenth of itself is 1.9487. Each step is the last number plus a tenth of it, which is easy to do in your head. Check with the rule of 72: at 10%, money doubles in about 7.2 years, so after 7 years it should be just under 2.
What is the cleanest way to do it in your head?
Adding 10% is the friendliest step there is: move the decimal one place and add. A bill of 1,771 plus 10% is 1,771 plus 177.1, which is 1,948.1. Multiplying by 1.1 is the same as adding a tenth of the number to itself, so seven careful additions beat any half-remembered formula. Keep four decimals until the last step, then round.
Seven 10% steps take 1 to 1.949, with each step adding a tenth of the bar before it, which lands just under the 2.0 that the rule of 72 predicts for seven years at 10%.
How do you check it a second way?
Two checks. First, the rule of 72: 72 divided by 10 is 7.2 years to double, so seven years should give a little under 2. Second, expand the bracket. (1 + 0.1) to the 7th is 1 + 0.7 + 0.21 + 0.035 + 0.0035 + smaller terms, which sums to about 1.9487. The terms shrink fast, so the first five get you there.
The relationship
$(1+0.1)^7 = 1 + 7(0.1) + 21(0.1)^2 + 35(0.1)^3 + 35(0.1)^4 + \dots \approx 1.9487$
7, 21, 35, 35 the binomial coefficients for the seventh power
0.1 the 10% growth rate
What it says in wordsExpanding the bracket term by term gives 1.9487, with each term much smaller than the one before.
Why a buyout interviewer asks this: a business growing 10% a year for seven years almost doubles. That is the sort of number you need to sanity-check a model's exit EBITDA in two seconds. The limitation of the rule of 72 is that it drifts at high rates; at 10% it is close, at 50% it is not.
Where candidates lose it
The common loss is adding seven lots of 10% to get 1.70, the simple-interest answer. The interviewer is checking whether compounding is automatic for you.
The second loss is rounding hard at every step, 1.2, 1.3, 1.5 and so on, which drifts well below the true value. Carry the decimals and round only once.
What the interviewer asks next
What is 1.1 to the 10th, roughly?
Use the same method for 1.08 to the 5th.
EBITDA of 100 grows 10% a year for 7 years. What is the exit EBITDA, and what if growth is 12%?
057Without a calculator: what is 1,000 divided by 7, to three decimal places? And what is 22 divided by 0.35?Mental mathsCoreSoftware buyout
Try it first
Which first step makes 22 / 0.35 easy?
Show the worked solution
1,000 / 7 is 142.857 and 22 / 0.35 is about 62.86. Seven goes into 1,000 142 times with 6 left, and six sevenths is 0.857142, repeating. For the second, multiply both numbers by 100 to get 2,200 / 35, divide both by 5 to get 440 / 7, and that is 62 with 6 left: 62 and six sevenths, about 62.86. Both answers end in the same sevenths.
Why is one seventh worth memorising?
Think of a clock face with six numbers on it instead of twelve. Every seventh, whatever it is, lands on the same circle of six digits, 1, 4, 2, 8, 5, 7, and only the starting point changes. Once you know 1/7 = 0.142857, every other seventh is a rotation of the same six digits: 2/7 starts at the 2, 3/7 at the 4, and 6/7 at the 8. That turns both of these divisions into one fact you already hold.
Dividing 1,000 by 7 leaves remainders 6, 4, 5, 1, 3 and 2 before 6 returns, so the digits 857142 repeat and the answer is 142.857; 22 / 0.35 rescales to 440 / 7, which is 62 and six sevenths, about 62.86.
How do you show the long division out loud?
Seven into 1,000 is 142, because 7 x 142 is 994, leaving 6. Bring down a zero: 60, seven goes 8 times, 4 left. 40 gives 5, with 5 left. 50 gives 7, with 1 left. The moment a remainder you have seen before comes back, the digits start repeating, so you can stop dividing and write the cycle. Three decimals is 142.857; the next digit is a 1, so no rounding is needed.
The relationship
$\frac{22}{0.35} = \frac{2200}{35} = \frac{440}{7} = 62\tfrac{6}{7} \approx 62.857$
x100 multiplying top and bottom by the same number leaves the value unchanged
440/7 after dividing 2,200 and 35 by 5
6/7 the leftover, read off the 142857 cycle as 0.857
What it says in wordsRescale the divisor to a whole number, cancel what you can, then divide.
Interviewers ask these at desks that live in spreadsheets because quick division is how you sanity check a multiple or a margin in a meeting. A 22 crore profit on a 0.35 crore unit is about 63 units. Saying the rescaling step out loud matters more than speed: it shows a method you would trust on any number.
Where candidates lose it
The usual slip on 22 / 0.35 is moving the decimal the wrong way and answering 6.29 or 628.6. Rescale out loud, 2,200 over 35, and the size of the answer is obvious before you start.
On 1,000 / 7, candidates stop at 142.8 or round to 142.86 when asked for three decimals. Keep going until a remainder repeats: the cycle gives you every digit you need and shows you know why.
What the interviewer asks next
What is 5/7 as a decimal, without dividing?
Divide 1 by 13 to six decimals. What is the repeating cycle?
A company earns 38 on capital of 0.45 crore units. Quickly, what is the ratio?
078A value rises 20%, then falls 25%, then rises 10%. What is the net change from where it started?Mental mathsCoreMid-market buyout fundIndian mid-market PE
Try it first
Answer before you work it.
Show the worked solution
Down 1%. Start from 100. A 20% rise takes it to 120, a 25% fall takes away 30 to leave 90, and a 10% rise adds 9 to reach 99. In one line, 1.2 x 0.75 x 1.1 = 0.99. Percentage changes multiply; adding them gives plus 5%, which never happened.
Why can you not just add the percentages?
A shop marks a Rs 100 kurta up 20% to Rs 120, then runs a 25% sale, then nudges the price up 10%. The sale takes 25% of 120, which is 30, not 25. Every percentage is measured against the level it starts from, so each change has a different base and the changes multiply rather than add. Adding them treats every change as if it were applied to the original 100.
A value of 100 rises to 120, falls to 90 and rises to 99, because each change acts on the level before it; adding the three percentages suggests 105, while multiplying 1.20 by 0.75 by 1.10 gives 0.99, a net fall of 1%.
The relationship
$1.20 \times 0.75 \times 1.10 = 0.99 \quad\Rightarrow\quad -1\%$
1.20 a 20% rise as a multiplier
0.75 a 25% fall as a multiplier
1.10 a 10% rise as a multiplier
What it says in wordsTurn each change into a multiplier, multiply them, and the gap from 1 is the net change.
How do you do 1.2 x 0.75 x 1.1 fast?
Pick the pair that cancels nicely. 1.2 times 0.75 is exactly 0.9, because three quarters of 1.2 is 0.9, and 0.9 times 1.1 is 0.99. Saying the intermediate 0.9 out loud lets the interviewer follow you and catch a slip early. The order does not matter: multiplication gives the same answer whichever change comes first.
In private equity this is the same arithmetic as a multiple that expands 20% at one point and contracts later, or a business that grows, shrinks and recovers. Revenue up 20% and back down 20% does not return to the start: it ends 4% lower.
Where candidates lose it
The fast answer is plus 5%, and it comes from adding signed percentages as though they shared one base. The interviewer asked three changes in a row precisely to tempt that shortcut.
A second loss is getting 99 but saying it as 99% without naming the change. Say minus 1% and show 0.9 then 0.99 so the method is audible.
What the interviewer asks next
A value rises 50% then falls 50%. Where does it end?
What single yearly rate gives the same result as these three changes over three years?
Does the order of the three changes matter? Prove it.

x_0	the starting guess, 1.6
N	the number whose root you want, 2.5
x_1	the improved guess

7, 21, 35, 35	the binomial coefficients for the seventh power
0.1	the 10% growth rate

x100	multiplying top and bottom by the same number leaves the value unchanged
440/7	after dividing 2,200 and 35 by 5
6/7	the leftover, read off the 142857 cycle as 0.857

1.20	a 20% rise as a multiplier
0.75	a 25% fall as a multiplier
1.10	a 10% rise as a multiplier

Try it first

How do you find a square root you have not memorised?

How do you sharpen the estimate and check it?

Where candidates lose it

What the interviewer asks next

Try it first

What is the cleanest way to do it in your head?

How do you check it a second way?

Where candidates lose it

What the interviewer asks next

Try it first

Why is one seventh worth memorising?

How do you show the long division out loud?

Where candidates lose it

What the interviewer asks next

Try it first

Why can you not just add the percentages?

How do you do 1.2 x 0.75 x 1.1 fast?

Where candidates lose it

What the interviewer asks next