Private Equity puzzles, solved step by step

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

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002A deal returns 2.5x your money over five years. Without a calculator, what is the IRR, to the nearest percentage point?Returns mathsCoreMid-market buyout fundIndian mid-market PE
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Commit to a number first.
Show the worked solution
About 20%, and 20.1% exactly. Bracket it with round rates you can compound in your head. 1.2 to the fifth is 2.49 and 1.25 to the fifth is 3.05, so 2.5x sits right at the bottom of the bracket, a hair above 20%. Straight division, 150% over five years, gives 30% and overstates the return badly because it ignores compounding.
How do you compound 1.2 five times in your head?
Do it by squaring, the way you would fold a sheet of paper. 1.2 squared is 1.44. Square again to get the fourth power: 1.44 times 1.44 is about 2.07. One more 1.2 gives about 2.49. Chaining small multiplications you trust is safer than reaching for a formula you half remember, and it is exactly what the interviewer wants to watch. The same chain at 1.25 runs 1.56, then 2.44, then 3.05.
Compounding for five years turns 15% a year into 2.01x, 20% into 2.49x and 25% into 3.05x, so a 2.5x deal sits just past the 20% rung and its IRR is 20.1%.
Why is the answer not 30%?
Think of a savings account that pays interest on interest. If it pays 20% a year, the gain in year five is far larger than the gain in year one, because it is earned on a bigger balance. An IRR is a compound rate, so the five years of gains are not equal slices of the 150% total; the later years carry more of it. Dividing 150 by five assumes every year earns 30 on the original 100, which is simple interest. The compound rate needed is lower: 20.1%.
The relationship
$\text{IRR} = \text{MOIC}^{1/n} - 1 = 2.5^{1/5} - 1 \approx 20.1\%$
MOIC multiple on invested capital, money back divided by money in
n years the money is invested, here 5
What it says in wordsWith one cash flow in and one out, the IRR is the yearly growth rate that turns the money in into the money out.
What table is worth memorising before a buyout interview?
A small grid of multiples by holding period. Once you know that 2x in five years is about 15%, 2.5x is about 20% and 3x is about 25%, most return questions become a lookup with a small adjustment. If you need more precision, interpolate between rungs: 2.5 is 0.01 of the way along a 0.56 gap from 2.49 to 3.05, which adds about a tenth of a point to 20%, giving 20.1%. State that this assumes a single cash flow in and a single cash flow out; interim dividends would change it.
Where candidates lose it
The classic slip is 30%: dividing the total gain by the years. It sounds reasonable when said quickly and is wrong by ten points, which an interviewer in a returns-driven job notices immediately.
The second miss is guessing 25% because 2.5x feels like a strong result. Bracket it with two rates you can compound, then say which end of the bracket the deal sits at.
What the interviewer asks next
What multiple does 20% a year give over three years?
What IRR does 3x over seven years imply?
If half the proceeds come back in year three, is the IRR higher or lower than 20.1%, and why?
016A deal is on track to return 2x in three years. The exit slips to year six, but by then the equity value has doubled again, to 4x. Has the IRR improved?Returns mathsCoreMid-market buyout fund
Try it first
Compared with 2x in three years, the IRR on 4x in six years is:
Show the worked solution
No, the IRR is unchanged at about 26%. 2x in three years means the money doubles every three years. 4x in six years is two doublings in six years, the same pace. The cube root of 2 is 1.26, and the sixth root of 4 is the same number, so both IRRs are 26.0%. The multiple doubled; the annual rate did not move.
Why does a bigger multiple not mean a better IRR?
Think of two runners: one covers 2 km in 10 minutes, the other 4 km in 20 minutes. The second went further, but neither ran faster. IRR measures pace, and the multiple measures distance, so doubling the multiple over double the time leaves the pace exactly where it was. Three years to double, then another three years to double again, is the same doubling time throughout.
Compounding at 26.0% a year reaches 2x at year 3 and 4x at year 6, so both outcomes sit on the same curve with the same IRR, while 3x at year 6 would sit below it at 20.1%.
How do you show it in two lines?
Write both as roots. The first IRR is 2 to the power of one third, less one. The second is 4 to the power of one sixth, less one. Since 4 is 2 squared, 4 to the one sixth is 2 to the two sixths, which is 2 to the one third: the same number, 1.26. So both are 26.0%. If the slip had produced only 3x by year six, the IRR would fall to 20.1%, a real cost of the delay.
The relationship
$4^{1/6} = (2^2)^{1/6} = 2^{1/3} \approx 1.260$
2^{1/3} the yearly growth factor for doubling in three years
4^{1/6} the yearly growth factor for quadrupling in six years
What it says in wordsQuadrupling in six years is the same yearly pace as doubling in three.
So is the slipped exit just as good?
Not necessarily, and the judgement is the point of the question. Whether 4x in six years is as good depends on what the money could have done in years four to six: if the fund could have redeployed it at 26%, the two are equal; if not, the larger multiple is better. For the GP, 4x means more carry in absolute terms. For the LPs, the delay holds capital in an older fund for longer. And three extra years of exposure carry three more years of risk that the second doubling does not happen.
Where candidates lose it
The trap is answering that the IRR improved because the multiple doubled. Candidates who think in multiples alone fall into it every time.
The second loss is answering correctly and stopping. The follow-on judgement, about redeployment, carry and risk, is what separates a calculator from an investor.
What the interviewer asks next
What multiple would the deal need at year six to beat 26%?
Why do LPs track both net IRR and net multiple?
If you could redeploy at 15%, which outcome would you prefer?
028A sponsor invests 100. In year 2 it takes a dividend recap of 60, and in year 5 it exits for 180. What is the money multiple, and is the IRR higher or lower than a deal that simply turns 100 into 240 at year 5?Returns mathsCoreMid-market buyout fund
Try it first
Both deals return 2.4x. Which has the higher IRR, and by roughly how much?
Show the worked solution
The multiple is 2.4x in both cases, but the recap deal's IRR is about 24.1% against 19.1%. Money back is 60 plus 180, which is 240 on 100. The single exit's IRR is 2.4 to the power one fifth, less one. The recap returns part of the money in year 2, and cash that comes back sooner lifts the IRR even though the total is unchanged.
Why does the same multiple give a different IRR?
Lend a friend Rs 100 and get Rs 240 back. If Rs 60 of it comes back after two years, you have that money in hand for three years while waiting for the rest. The multiple counts how much money comes back; the IRR counts how fast it comes back, so pulling cash forward raises the IRR without touching the multiple. The two deals below differ only in timing.
Both deals turn 100 into 240 over five years, but the deal that returns 60 in year 2 earns an IRR of 24.1% while the single exit at year 5 earns 19.1%, because early cash shortens the average time the money is out.
How do you get the recap IRR without a spreadsheet?
The single exit is a one-line calculation: 2.4 to the power 0.2 is about 1.191, so 19.1%. The recap needs a guess and a check, because there are two inflows. Try 24%: 60 divided by 1.24 squared is about 39.0, and 180 divided by 1.24 to the fifth is about 61.4. Together that is 100.4, just above the 100 invested, so the true rate is a touch higher. Guess, discount each cash flow, and nudge the rate until the present values add back to the cheque. The answer is 24.1%.
The relationship
$100 = \frac{60}{(1+r)^2} + \frac{180}{(1+r)^5} \quad\Rightarrow\quad r \approx 24.1\%$
r the IRR, the rate that makes the discounted inflows equal the investment
60 the recap dividend in year 2
180 the exit proceeds in year 5
What it says in wordsThe IRR is the single rate at which the discounted cash coming back exactly repays the cash put in.
Say what the recap costs, because an interviewer will push. The dividend was paid with new debt, which is why the exit cheque is 180 and not 240. The company carried more leverage for three years, so the higher IRR came with higher risk of a covenant problem. That is the honest trade: an LP sees a better IRR and early cash back, and the business sees a thinner cushion.
Where candidates lose it
The fast wrong answer is that the IRRs are equal because the multiples are equal. Candidates who think of IRR as a multiple spread over years miss that IRR weights early cash more heavily.
The second loss is getting stuck on a two-cash-flow IRR. You do not need the exact figure in your head; bracket it with one trial rate, say it is a little above 24%, and explain why.
What the interviewer asks next
Why might a GP favour a dividend recap late in a fund's life?
If the recap had been 100 in year 2 and exit 140 in year 5, what happens to the IRR and to the multiple?
What does an LP look at besides IRR to judge whether the recap added value?
031On a 5-year hold, how much more exit equity do you need to lift the IRR from 20% to 25%?Returns mathsCoreLarge-cap buyout fund
Try it first
Pick the closest answer.
Show the worked solution
About 22.6% more exit equity: from 2.49x the cheque to 3.05x. 1.2 to the fifth is 2.49 and 1.25 to the fifth is 3.05. Their ratio is (1.25 / 1.2) to the fifth, about 1.226. Five points of IRR compound every year, so over five years they add up to almost a quarter more money at exit, and more again over longer holds.
Why is the answer not five per cent?
Two runners leave together, one about 4% faster. After one lap the gap is small. After five laps it is five small gaps stacked, each measured on a longer distance. An extra five points of IRR is an extra growth factor of 1.25 / 1.20, about 1.042, applied every year, so over five years the exit has to be 1.042 to the fifth, about 1.226 times larger. That is a 22.6% increase in exit equity.
The relationship
$\frac{(1.25)^5}{(1.20)^5} = \left(\frac{1.25}{1.20}\right)^5 = 1.0417^5 \approx 1.226$
1.25^5 the exit multiple at 25% a year for five years, 3.05x
1.20^5 the exit multiple at 20% a year, 2.49x
What it says in wordsThe extra exit value needed is the ratio of the two yearly growth factors, raised to the number of years.
Lifting the IRR from 20% to 25% over five years needs exit equity of 3.05x the cheque instead of 2.49x, 22.6% more, and the same five points need 33.1% more over a seven-year hold.
How much more enterprise value does that take?
Less than 22.6%, if the deal carries debt at exit. Say exit EV is 1,000 with 400 of debt, so equity of 600 earns 20%. Equity needs to rise to about 736, and with debt unchanged that needs EV of about 1136. Leverage works in your favour here: a 13.6% rise in enterprise value delivers the 22.6% rise in equity. That is also why sponsors stress-test exit multiples so carefully, since small EV moves swing equity hard either way.
Say the limitation: the sums assume a single cheque in and a single exit. With interim dividends or staged investments, the required change in exit value differs, and the IRR needs a full cash flow schedule.
Where candidates lose it
The common loss is thinking linearly: five more points sounds like a small tweak, so candidates say exit value needs to rise by 5% or so. Compounding turns five points a year into about 23% over five years.
The second loss is dividing 25 by 20 and answering 25%. That compares the rates, not the money multiples. Raise the ratio of the growth factors, 1.25 over 1.20, to the number of years.
What the interviewer asks next
What exit multiple of the cheque does 30% over five years need?
Over a three-year hold, how much more exit equity does the same move need?
If exit equity can only rise 10%, how much shorter must the hold be to reach 25%?
095A deal returned 3x the money at a 20% IRR. Roughly how long was the hold?Returns mathsCoreMid-market buyout fundIndian mid-market PE
Try it first
Answer inside ten seconds.
Show the worked solution
About 6 years. 1.2 to the sixth power is 2.99, so six years of 20% a year turns 1 into very nearly 3. Build it from anchors: 1.2 cubed is 1.728, and squaring that gives 2.99. The exact answer, the log of 3 over the log of 1.2, is 6.03 years.
How do you find the hold without logarithms?
Climb the powers of 1.2 until you pass 3. It is like counting how many 20% pay rises it takes to triple a salary: each rise is on the new salary, so they compound. 1.2 cubed is 1.728, and 1.728 squared is about 2.99, so six steps of 20% take you to 3. Seven would be 3.58, well past.
Raising 1.2 to the powers 1 to 7 gives 1.20, 1.44, 1.73, 2.07, 2.49, 2.99 and 3.58, so a 3x return at 20% a year needs about six years, 6.03 exactly.
Why is this worth memorising for buyout interviews?
Interviewers ask for any one of the three numbers, multiple, IRR or hold, given the other two. Knowing a few powers of 1.2 and 1.25 answers most of them instantly: 2x at 20% is about 4 years, 3x is about 6, and 3x at 25% is about 5. The rule of 72 gives a cross-check: doubling at 20% takes about 3.6 years, and tripling takes about 1.6 times as long as doubling, near 5.8.
The relationship
$n = \frac{\ln 3}{\ln 1.2} = \frac{1.0986}{0.1823} \approx 6.03$
n years in the hold
ln 3 natural log of the money multiple
ln 1.2 natural log of one plus the IRR
What it says in wordsThe hold is the log of the multiple divided by the log of one plus the yearly rate.
One caveat worth saying: this assumes all the money went in at the start and came out at the end. If proceeds came back in stages, the same 3x and 20% would imply a longer total hold, because early returns lift the IRR.
Where candidates lose it
The slip is simple interest: 3x means gaining 200%, and at 20% a year that looks like ten years. Compounding gets there in six.
The other loss is fumbling the powers under time pressure. Learn 1.44, 1.73, 2.07, 2.49 and 2.99 and the question takes five seconds.
What the interviewer asks next
A deal returns 2.5x in 4 years. Roughly what IRR?
Same 3x, but over 4 years. What IRR?
Why do sponsors often prefer a 2.5x in 4 years to a 3x in 6?

MOIC	multiple on invested capital, money back divided by money in
n	years the money is invested, here 5

2^{1/3}	the yearly growth factor for doubling in three years
4^{1/6}	the yearly growth factor for quadrupling in six years

r	the IRR, the rate that makes the discounted inflows equal the investment
60	the recap dividend in year 2
180	the exit proceeds in year 5

1.25^5	the exit multiple at 25% a year for five years, 3.05x
1.20^5	the exit multiple at 20% a year, 2.49x

Try it first

How do you compound 1.2 five times in your head?

Why is the answer not 30%?

What table is worth memorising before a buyout interview?

Where candidates lose it

What the interviewer asks next

Try it first

Why does a bigger multiple not mean a better IRR?

How do you show it in two lines?

So is the slipped exit just as good?

Where candidates lose it

What the interviewer asks next

Try it first

Why does the same multiple give a different IRR?

How do you get the recap IRR without a spreadsheet?

Where candidates lose it

What the interviewer asks next

Try it first

Why is the answer not five per cent?

How much more enterprise value does that take?

Where candidates lose it

What the interviewer asks next

Try it first

How do you find the hold without logarithms?

Why is this worth memorising for buyout interviews?

Where candidates lose it

What the interviewer asks next