Private Equity puzzles, solved step by step

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

Showing 1–10 of 20 · filtered from 100Clear filters

006A sponsor invests 300 of equity in a buyout. Four years later it sells, and the equity is worth 750. What are the MOIC and the IRR?Mental paper LBOsWarm upWarburg PincusSan Francisco · 2014
Try it first
Which IRR is closest?
Show the worked solution
The MOIC is 2.5x and the IRR is about 25.7%. MOIC is money out over money in: 750 divided by 300. The IRR is the yearly rate that compounds 300 into 750 over four years, the fourth root of 2.5. Take the square root twice: 2.5 becomes 1.581, then 1.257. Check: 1.25 to the fourth is 2.44, just short of 2.5, so the rate sits a little above 25%.
Why work out the multiple before the rate?
Picture a plant that grew from 30 centimetres to 75 in four years. Asking how many times bigger it got is one division; asking how fast it grew each year needs a root. Getting the multiple first strips the problem down to one number, 2.5, so the only hard step left is a fourth root. In a paper LBO the multiple is also what the interviewer will ask for first, because it is the number a sponsor quotes when it talks about a deal.
Compounding at 25.7% a year takes 300 to 377, 474, 596 and then 750, the same x1.257 step each year, which is why a 2.5x multiple over four years is a 25.7% IRR and not the 37.5% a straight line suggests.
How do you take a fourth root in your head?
A fourth root is a square root of a square root. The square root of 2.5 is about 1.58, because 1.58 squared is about 2.5, and the square root of 1.58 is about 1.257, because 1.26 squared is 1.59. Then check from the other side with a round rate you can compound: 1.25 squared is 1.5625, and squared again is 2.44. That is a little short of 2.5, so the IRR is a little above 25%, which matches 25.7%.
The relationship
$\text{MOIC} = \frac{750}{300} = 2.5 \qquad \text{IRR} = 2.5^{1/4} - 1 \approx 25.7\%$
MOIC multiple on invested capital: equity out divided by equity in
1/4 the fourth root, because the money is invested for four years
What it says in wordsThe multiple tells you how much the money grew; the root of the multiple tells you how fast it grew each year.
What would change the IRR without changing the multiple?
Time and interim cash. The same 2.5x earned over five years is about 20%, and over three years about 36%, so a sponsor always quotes both numbers together. Say also that this calculation assumes one cash flow in and one out. A dividend paid in year two, or a second equity cheque in year one, means the IRR has to be solved over several cash flows, and the shortcut gives only an approximation.
Where candidates lose it
The fast wrong answer is 37.5%: a 150% gain split evenly over four years. It is the most common error in paper LBOs because the multiple and the rate get mixed up under time pressure.
The other loss is a correct number with no check. A fourth root guessed at 26% sounds lucky; 1.25 to the fourth equals 2.44, so slightly above 25% sounds reasoned.
What the interviewer asks next
What IRR does 2.5x give over five years?
The sponsor also took a 100 dividend in year two. Is the IRR higher or lower, and roughly by how much?
What exit equity value gives a 30% IRR over four years?
Asked at Warburg Pincus, Private Equity, San Francisco, 2014 (Wall Street Oasis): The interviews consisted of case studies and technical questions (paper LBO, basic returns, IRR, etc.)
010In your head, and out loud: what are 48 x 52, 97 x 103 and 499 x 501?Mental mathsWarm upGeneral AtlanticNew York · 2026
Try it first
What is 97 x 103?
Show the worked solution
2,496, 9,991 and 249,999. Each pair sits evenly either side of a round number, so the product is that number squared less the gap squared. 48 x 52 is 50 squared less 2 squared, 2,500 less 4. 97 x 103 is 10,000 less 9. 499 x 501 is 250,000 less 1. Say the rule first, then the three answers come in seconds.
What do the three pairs have in common?
Look at where each pair sits. 48 and 52 are both 2 away from 50; 97 and 103 are both 3 away from 100; 499 and 501 are both 1 away from 500. Two numbers placed symmetrically around a round number multiply to that number squared, less the gap squared. In symbols, (a minus b) times (a plus b) is a squared minus b squared. The round number is easy to square, the small gap is easy to square, and one subtraction finishes the job.
Cutting the 2 by 48 strip from a 50 by 50 square that is missing a 2 by 2 corner and laying it along the bottom gives a rectangle 48 wide and 52 tall, so 48 x 52 equals 2,500 less 4, which is 2,496.
Why is the product always a little less than the square?
Think of a square garden 50 metres on each side. If you make it 2 metres longer and 2 metres narrower, the strip you gain along one side is 2 by 48, but the strip you lose is 2 by 50. You lose slightly more than you gain, and the shortfall is exactly the small 2 by 2 corner, which is why the answer is always the square less the gap squared. That is the picture worth having in your head when the interviewer asks why the trick works.
The relationship
$(a-b)(a+b) = a^2 - b^2 \qquad 499 \times 501 = 500^2 - 1^2 = 249{,}999$
a the round number in the middle, here 500
b the distance from it to each factor, here 1
What it says in wordsSquare the middle, square the gap, subtract.
Why would a growth equity interviewer bother with this?
Because deal conversations are full of quick products: 48 months of a 52 lakh run rate, a multiple of 9.7x on 10.3 crore of EBITDA. The test is not whether you know the identity but whether you look for structure before you start grinding through digits. If the numbers are not symmetric, make them so: 47 x 55 is 51 squared less 4 squared, 2,601 less 16, which is 2,585. Say the method aloud so the interviewer hears the route, not just the result.
Where candidates lose it
The common loss is grinding through long multiplication out loud, which works but takes a minute and invites a slip in the carries. The interviewer is looking for the moment you spot the symmetry.
The second trap is getting 10,009 for 97 x 103 by adding the gap squared. The product is always below the square, because the lost strip is bigger than the gained one.
What the interviewer asks next
What is 51 x 49?
What is 46 x 54, and what is 46 x 55?
Square 105 in your head.
Asked at General Atlantic, Generalist, New York, 2026 (Wall Street Oasis): The first round was behavioral with mental math at the end.
011Diligence flags two independent risks in a target: a 20% chance its revenue will need to be restated, and a 10% chance its largest customer leaves. What is the chance at least one of them happens?Probability and expected value in dealsWarm upMid-market buyout fund
Try it first
Fast answer?
Show the worked solution
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.
Splitting all outcomes 20 to 80 for the restatement and 10 to 90 for the customer gives four cells, 2%, 18%, 8% and 72%, so at least one problem is 28%, and adding 20% and 10% counts the 2% overlap twice.
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%.
The relationship
$P(\text{at least one}) = 1 - (1 - 0.2)(1 - 0.1) = 1 - 0.72 = 0.28$
0.2 chance of a revenue restatement
0.1 chance the largest customer leaves
0.72 chance neither happens, if the two are independent
What it says in wordsThe chance of at least one problem is one less the chance of no problem at all.
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.
Where candidates lose it
Saying 30% is the whole trap. It is quick, sounds right and is off by the overlap, which the interviewer chose small so that only careful candidates notice.
The second miss is treating independence as a given. Answer 28%, then add one sentence on why diligence risks are rarely independent.
What the interviewer asks next
What is the chance both happen?
With four independent 10% risks, what is the chance of at least one?
If the two risks are positively correlated, does the chance of at least one rise or fall?
012A company's enterprise value is 800. It has 500 of senior debt, 400 of subordinated notes and equity beneath both. In a restructuring, what does each class recover? What changes if enterprise value is 1,000 instead?Leverage and capital structureWarm upKKRNew York · 2025
Try it first
At an enterprise value of 800, what do the subordinated notes recover?
Show the worked solution
At 800, senior recovers 100%, the notes 75% and equity nothing. Value is paid strictly by priority. Senior takes its 500 first, leaving 300 for the 400 of notes, which is 75%. At 1,000, value covers all 900 of debt, so both classes recover in full and equity is left with 100. The 200 increase in value goes 100 to the notes and 100 to equity.
In what order does the value get paid out?
Think of a row of buckets under one tap, each bucket overflowing into the next. The first fills completely before a drop reaches the second. A capital structure works the same way: each class is paid in full before the class below it receives anything. With 800 of value, the senior bucket takes 500 and is full. The remaining 300 flows into the notes bucket, which holds 400, so it is 75% full. Nothing reaches equity.
At an enterprise value of 800 the senior debt of 500 is paid in full, the subordinated notes get the remaining 300 of their 400, a 75% recovery, and equity gets nothing; at 1,000 both debt classes are whole and equity keeps 100.
Which security does the interviewer care about most?
The one the value line cuts through. That tranche is the fulcrum securityThe most senior class of claims that is not repaid in full, and so typically ends up owning the restructured company., here the subordinated notes at an EV of 800. Every change in enterprise value between 500 and 900 lands entirely on the notes, so they are the class whose price moves with the valuation argument while senior sits at par and equity sits at zero. In a restructuring the fulcrum holders usually swap their claim for the new equity, which is why distressed investors spend their time on it.
What changes if enterprise value is 1,000?
The line clears the debt. Senior is still paid 500, the notes are now paid their full 400, and the 100 left over belongs to equity. The 200 of extra value is split 100 to the notes and 100 to equity, and none of it reaches senior, which was already whole. That asymmetry is the lesson: a senior lender's upside is capped at par, while the junior classes capture the swing. Say the limitations too: real cases add administrative claims ahead of senior debt, and negotiated outcomes sometimes give junior classes a little value to secure their agreement.
Where candidates lose it
The usual slip is sharing value pro rata, giving every creditor 800 over 900, about 89%. That ignores priority, which is the whole point of having senior and subordinated claims.
The second miss is stopping at the recoveries. The follow-up about 1,000 is there to see whether you notice which tranche absorbs the change in value.
What the interviewer asks next
At what enterprise value does equity start to recover anything?
If the senior debt were secured on assets worth only 400, how would the answer change?
Why might a distressed fund buy the notes at 60 when they recover 75?
Asked at KKR, Distressed Debt, New York, 2025 (Wall Street Oasis): What are your weaknesses? A capital structure question with enterprise value.
020A sponsor buys a company at 10x EBITDA of 100, funded with 60% debt. EBITDA stays flat for five years and the exit is also at 10x. What is the MOIC if no debt is repaid, and what is it if 300 of debt is repaid over the five years?Leverage and capital structureWarm upMid-market buyout fund
Try it first
With 300 of debt repaid and nothing else changing, the MOIC is:
Show the worked solution
1.0x with no paydown and 1.75x with 300 repaid. The purchase price is 1,000, with 600 of debt and 400 of equity. With flat EBITDA and the same multiple, the exit value is again 1,000. If debt is still 600, equity is still 400: 1.0x. If 300 has been repaid, debt is 300 and equity is 700, which is 1.75x, an IRR of about 12% over five years.
Where does the return come from if nothing grows?
Think of buying a flat for 1 crore with a 60 lakh home loan, then renting it out and using the rent to pay down 30 lakh of the loan over five years. If the flat is still worth 1 crore, your share has gone from 40 lakh to 70 lakh, though the flat itself is unchanged. Debt paydown moves value from the lenders to the owner: the business is worth the same, but a larger slice of it belongs to the sponsor. The company's own cash flow is doing the repaying, so the sponsor's cheque never changes.
The business is worth 1,000 at entry and at exit, but repaying 300 of debt cuts the lenders' claim from 600 to 300 and lifts the sponsor's equity from 400 to 700, a 1.75x multiple and a 11.8% IRR with no growth at all.
What are the numbers, step by step?
Entry: 10 x 100 is 1,000. Sixty per cent debt is 600, so equity is 400. Exit with nothing repaid: 1,000 less 600 is 400 of equity, 1.0x, a zero return over five years. Exit with 300 repaid: 1,000 less 300 is 700 of equity, and 700 over 400 is 1.75x. Over five years, 1.75x is about 11.8% a year: 1.12 to the fifth is 1.76, so a shade under 12%.
The relationship
$\text{MOIC} = \frac{EV_{\text{exit}} - D_{\text{exit}}}{E_{\text{entry}}} = \frac{1000 - 300}{400} = 1.75x$
EV_exit exit enterprise value, 10x EBITDA of 100
D_exit debt left at exit, 600 less 300 repaid
E_entry the sponsor's equity cheque, 400
What it says in wordsThe sponsor's multiple is exit equity, enterprise value less remaining debt, over the equity it put in.
Is 300 of paydown realistic, and what is left out?
It means 60 a year of free cash flow after interest and tax on a business with EBITDA of 100, which is possible for a capital-light company and unlikely for a capital-hungry one. Paydown is the most reliable of the three return levers, alongside EBITDA growth and multiple change, because it depends on cash the business already generates rather than on the future. Say what the simple version ignores: transaction fees at entry and exit, any cash left on the balance sheet, and the risk that interest rates or a downturn absorb the cash meant for repayment.
Where candidates lose it
The common error is saying flat EBITDA and a flat multiple mean no return. That ignores the capital structure, which is exactly what the interviewer is testing.
The second slip is computing the return on enterprise value, 1,000 to 1,000, rather than on equity. The sponsor owns the equity; say 400 in and 700 out.
What the interviewer asks next
What exit multiple would give 2.0x with the same paydown?
If EBITDA grows to 120 as well, what is the MOIC?
Why do lenders accept higher leverage for businesses with stable cash flow?
027A company trades at 10x EV/EBITDA and 2x EV/Sales. What is its EBITDA margin?Valuation riddlesWarm upMid-market buyout fund
Try it first
Answer before you write anything down.
Show the worked solution
The EBITDA margin is 20%. Both multiples share the same enterprise value on top. Divide EV/Sales by EV/EBITDA and EV cancels, leaving EBITDA divided by sales: 2 over 10, which is 20%. Check it with any EV you like: at 1,000, sales are 500 and EBITDA is 100, and 100 is 20% of 500.
Why does the enterprise value not matter?
A cricket bat costs as much as 10 balls, and as much as 2 sets of pads. How many balls is a set of pads worth? Five, and you never needed the price of the bat. When two ratios share the same top line, dividing one by the other cancels it and leaves the ratio of the two bottom lines. Here the shared top line is enterprise value, and the bottom lines are sales and EBITDA.
The relationship
$\frac{EV/\text{Sales}}{EV/\text{EBITDA}} = \frac{\text{EBITDA}}{\text{Sales}} = \frac{2}{10} = 20\%$
EV/Sales enterprise value over revenue, 2x
EV/EBITDA enterprise value over EBITDA, 10x
What it says in wordsThe sales multiple divided by the EBITDA multiple is the EBITDA margin.
Picking any enterprise value, say 1,000, gives sales of 500 at 2x and EBITDA of 100 at 10x, and 100 is 20% of 500; the EV cancels, so the margin is the ratio of the two multiples.
How do you check the direction of the division?
A margin must be smaller than 100%, and EBITDA is a slice of sales, so EBITDA has to be the smaller number. The higher multiple sits on the smaller number, so the margin is the low multiple over the high multiple, never the other way. Dividing 10 by 2 gives 5, which would be a 500% margin and is impossible. Dividing 2 by 10 gives 0.2.
A buyout investor uses this the other way round all the time. If comparable companies trade at 2x sales and the target earns a 10% margin, then 2x sales is 20x EBITDA for this target, which is expensive. Sales multiples hide margin differences; converting to EBITDA puts them back. The limitation: EBITDA multiples carry their own blind spots, such as heavy capital spending that EBITDA leaves out.
Where candidates lose it
Candidates freeze because no enterprise value is given and assume the question is missing data. It is not. Saying out loud that EV appears in both ratios and cancels is the whole answer.
The second loss is dividing the wrong way and saying 5, then not noticing that a margin cannot be 500%. A two-second sense check catches it.
What the interviewer asks next
The company also trades at 25x earnings. What do you learn, and what do you still need?
Peers trade at 2x sales with 30% margins. Is this company cheap or expensive on EBITDA?
When would you prefer a sales multiple to an EBITDA multiple?
039A company sells two products with gross margins of 60% and 20%. Sales shift from 50/50 to 40/60, towards the low-margin product. What happens to the blended gross margin?Operating levers and margin mathsWarm upPortfolio operations team
Try it first
Where does the blended margin go?
Show the worked solution
The blended margin falls from 40% to 36%, with no change in either product's economics. Before, 0.5 x 60% + 0.5 x 20% = 40%. After, 0.4 x 60% + 0.6 x 20% = 24% + 12% = 36%. Moving 10 points of the mix from a 60% product to a 20% product costs 10% of the 40-point gap between them, which is 4 points of blended margin.
How can margin fall when nothing got worse?
A cafe earns 70% on coffee and 25% on sandwiches. If more customers come at lunch and order sandwiches, the cafe's overall margin drops even though neither the coffee nor the sandwich got less profitable. A blended margin is a sales-weighted average, so shifting weight towards the lower-margin product pulls the average down by itself. Each 1 point of mix moved costs 1% of the margin gap, here 0.4 points.
With product margins fixed at 60% and 20%, a 50/50 mix earns 30 plus 10 on revenue of 100, a 40% blend, while a 40/60 mix earns 24 plus 12, a 36% blend, so mix alone takes 4 points off the margin.
The relationship
$m = w_A m_A + w_B m_B = 0.4 \times 60\% + 0.6 \times 20\% = 36\%$
w_A, w_B each product's share of sales
m_A, m_B each product's gross margin
What it says in wordsThe blended margin is each product's margin weighted by its share of sales.
Why does a buyout team care, and what should it check next?
When a target's margin falls, the first question is whether the products got worse or the mix moved. A mix-driven margin fall is a different problem from a pricing or cost problem, and it can come with rising profit. If the low-margin product is growing fast, total gross profit may still rise even as the percentage falls: a business at a 36% margin on revenue of 150 earns more than one at 40% on 100.
Say the limitation. Gross margin ignores the overheads each product uses. A low-margin product that needs little selling effort can be more attractive after overheads than it looks here, so ask for contribution by product before judging the mix.
Where candidates lose it
The common loss is saying the margin stays at 40% because neither product changed. The interviewer is checking whether you know that a blend is a weighted average and moves with its weights.
The second loss is reading a falling margin as bad news without asking about volume. Margin percentage and profit in rupees can move in opposite directions.
What the interviewer asks next
What mix would bring the blended margin down to 30%?
If revenue grows from 100 to 130 with the new mix, does gross profit rise or fall?
How would you separate price, cost and mix effects in a margin bridge?
040A company has floating-rate debt of 500 priced at a 5% base rate plus a 4% margin, and EBITDA of 110. The base rate rises by 200 basis points. What happens to interest cover?Credit and PIK mathsWarm upPrivate creditIndian mid-market PE
Try it first
Where does EBITDA interest cover go?
Show the worked solution
Interest cover falls from 2.44x to 2.0x. The all-in rate is 5% plus 4%, so 9% on 500 is 45 of interest, and 110 / 45 is 2.44x. After a 200 basis point rise the rate is 11%, interest is 55, and 110 / 55 is 2.0x. A two-point rise in the base rate lifts the interest bill by 22% and cuts cash left after interest from 65 to 55.
Why does a two-point rate rise do so much damage?
A household with a floating-rate home loan feels every rate rise in the next EMI, while one with a fixed rate does not. On floating-rate debt the borrower carries the rate risk, so a rise in the base rate goes straight into the interest bill. Two points on 500 is 10 more a year, which is 22% more than the 45 being paid today, with no change in how the business is running.
With EBITDA fixed at 110, a 200 basis point rise in the base rate lifts interest on 500 of floating debt from 45 to 55, so cover falls from 2.44x to 2.0x and the cash left after interest falls from 65 to 55.
The relationship
$\text{Cover} = \frac{EBITDA}{D \times (b + m)} = \frac{110}{500 \times 11\%} = 2.0x$
D floating debt, 500
b the base rate, now 7%
m the lender's margin, 4%
What it says in wordsInterest cover is EBITDA over the interest bill, and on floating debt the bill moves with the base rate.
What does a sponsor do about it?
Most leveraged loans are floating, so sponsors usually hedge part of the debt. An interest rate swap fixes the rate on a portion of the loan, and a cap limits how high it can go in return for an upfront premium. Lenders often require a minimum hedged share in the loan agreement. The trade-off is cost and lost upside: a swap set before rates fall locks the borrower into the higher rate.
Say what a credit analyst would look at next. EBITDA interest cover ignores capital spending and tax, so cash cover is tighter than 2.0x. A cover covenant set at, say, 2.0x would now be right at its limit, which turns a market move into a negotiation with lenders.
Where candidates lose it
The common loss is saying cover is unchanged because EBITDA is unchanged. Candidates forget that the coupon on floating debt resets with the base rate.
The second loss is adding the 200 basis points to the margin and calling it a 2% rise in interest. The rate goes from 9% to 11%, which is a 22% rise in the bill. Say the percentage change in the bill, not in the rate.
What the interviewer asks next
How much EBITDA growth would restore 2.44x cover after the rate rise?
If 60% of the debt is swapped to a fixed 7.5%, what is cover after the rise?
Why do lenders often insist on hedging, and who benefits when rates fall?
042Which makes more money on the same cheque: a 25% IRR for 3 years, or a 20% IRR for 5 years?Returns mathsWarm upMid-market buyout fund
Try it first
Which ends with more money?
Show the worked solution
The 20% for 5 years makes more money: 2.49x against 1.95x. 1.25 cubed is about 1.95 and 1.2 to the fifth is about 2.49. On a cheque of 100 that is a profit of 149 against 95. The 25% deal is faster, but the money is back after three years, and it only catches up if it can be reinvested at about 13% for the remaining two.
Why does the higher IRR make less money?
A car doing 100 km an hour for three hours covers 300 km; one doing 80 km an hour for five hours covers 400. Speed and distance are different questions. IRR measures how fast money grows, the multiple measures how much money you end with, and a longer hold at a lower speed can end further ahead. Here 1.25 cubed is 1.95 and 1.2 to the fifth is 2.49.
Compounding at 25% for three years ends at 1.95x, while 20% for five years ends at 2.49x, so the higher IRR makes less money unless its proceeds can be reinvested at about 12.9% for the two years it is not running.
So which would an LP prefer?
It depends on what happens to the money after year three. If the LP can redeploy A's proceeds at more than about 12.9% a year, A ends ahead; below that, B wins. Reinvested at 20%, A reaches 2.81x by year five; parked at 8%, only 2.28x. That is why LPs read IRR and the money multiple together, and why GPs who sell early to protect a high IRR are sometimes accused of leaving money on the table.
The relationship
$1.25^3 = 1.95 \qquad 1.20^5 = 2.49 \qquad (1+r)^2 = \frac{2.49}{1.95} \Rightarrow r \approx 12.9\%$
1.25^3 A's money multiple after three years
1.20^5 B's money multiple after five years
r the reinvestment rate at which A catches B by year five
What it says in wordsCompare the two multiples, then ask what rate A's money must earn in the gap years to catch up.
Say the limitation. Both deals here are a single cheque in and out. A fund's IRR also depends on when capital is called and returned, and a high IRR on a small, quick deal can flatter a fund that made little money overall.
Where candidates lose it
The common loss is picking the higher IRR on reflex. The interviewer is checking whether you know that IRR is a rate and says nothing on its own about how much money comes back.
The second loss is giving the right answer without the reinvestment point. The full answer is that B makes more money, and A wins only if its proceeds can be redeployed at about 13% or better.
What the interviewer asks next
What IRR over 3 years would match 2.49x?
Why might a GP sell a winner early even though holding would make more money?
How does the timing of capital calls affect a fund's IRR but not its multiple?
049EBITDA grows from 100 to 250 over 6 years. What is the compound annual growth rate?Compounding and time valueWarm upMid-market buyout fundIndian mid-market PE
Try it first
Pick the CAGR.
Show the worked solution
About 16.5% a year. CAGR is the ratio of end to start, 2.5, raised to one over the number of years, less one. 1.15 to the sixth is about 2.31 and 1.17 to the sixth about 2.57, so the rate sits about three quarters of the way between them, near 16.5%. Dividing the 150% gain by 6 gives 25%, which would compound to 381.
Why is 25% wrong?
A savings account that pays interest on interest grows faster each year, so it needs a lower rate than you might think to reach a target. CAGR is the single rate that, compounded every year, turns the start into the end, so it is the sixth root of 2.5, not 150% divided by six. 25% compounded for six years would reach 381, far past 250.
Compounding at 16.5% a year takes EBITDA from 100 to 250 in six years, while the 25% from dividing 150% by six would compound to 381; bracketing between 1.15 and 1.17 to the sixth finds the rate.
The relationship
$CAGR = \left(\frac{250}{100}\right)^{1/6} - 1 \approx 16.5\%$
250 / 100 the ratio of end value to start value, 2.5
1/6 one over the number of years
What it says in wordsThe compound growth rate is the ratio of end to start, rooted by the number of years, less one.
How do you find a sixth root in your head?
Bracket it with rates whose sixth powers you can build. 1.15 squared is 1.3225, cubed that is about 2.31. 1.17 squared is 1.3689, cubed that is about 2.57. 2.5 sits about 0.74 of the way from 2.31 to 2.57, so the rate is about 15% plus 0.74 of 2 points, near 16.5%. A cross-check with the rule of 72: at 16.5% money doubles in about 4.4 years, and 2.5x in 6 years is a little more than one doubling, which fits.
Say what the number hides. A CAGR smooths the path: a business could have been flat for four years and then jumped, and the CAGR would be the same. A buyout investor asks for the yearly figures before trusting the rate, and checks whether the 250 includes acquisitions.
Where candidates lose it
The common loss is dividing the total growth by the years and saying 25%. That is the average simple growth, and it overstates the compound rate badly over six years.
The second loss is freezing on the sixth root. You do not need logarithms; bracket the rate between two you can compute and slide.
What the interviewer asks next
What CAGR turns 100 into 300 over 5 years?
If EBITDA grew 40% in year one and was flat after, what is the CAGR over six years?
How would you strip acquired EBITDA out of the growth rate?

MOIC	multiple on invested capital: equity out divided by equity in
1/4	the fourth root, because the money is invested for four years

a	the round number in the middle, here 500
b	the distance from it to each factor, here 1

0.2	chance of a revenue restatement
0.1	chance the largest customer leaves
0.72	chance neither happens, if the two are independent

EV_exit	exit enterprise value, 10x EBITDA of 100
D_exit	debt left at exit, 600 less 300 repaid
E_entry	the sponsor's equity cheque, 400

Try it first

Why work out the multiple before the rate?

How do you take a fourth root in your head?

What would change the IRR without changing the multiple?

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What do the three pairs have in common?

Why is the product always a little less than the square?

Why would a growth equity interviewer bother with this?

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Why not just add the two chances?

Why is one minus none the safest route?

Is independence a fair assumption in diligence?

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In what order does the value get paid out?

Which security does the interviewer care about most?

What changes if enterprise value is 1,000?

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Where does the return come from if nothing grows?

What are the numbers, step by step?

Is 300 of paydown realistic, and what is left out?

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Why does the enterprise value not matter?

How do you check the direction of the division?

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How can margin fall when nothing got worse?

Why does a buyout team care, and what should it check next?

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Why does a two-point rate rise do so much damage?

What does a sponsor do about it?

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Why does the higher IRR make less money?

So which would an LP prefer?

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Why is 25% wrong?

How do you find a sixth root in your head?

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