Private Equity puzzles, solved step by step

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

Showing 21–30 of 100

021A secondary buyer pays 85% of NAV for an LP interest with a NAV of 100 and 20 of unfunded commitment. Over three years the NAV realises at 1.3x, and the unfunded 20 is drawn and returns 1.5x. What is the buyer's MOIC?Fund economics numeracyHardSecondaries and fund of funds

Try it first

Commit before you work it: the buyer's multiple is closest to

Show the worked solution

About 1.52x. The buyer pays 85 for the NAV today and 20 more when the unfunded commitment is drawn, 105 in all. It receives 130 when the NAV realises at 1.3x and 30 when the drawn 20 returns 1.5x, 160 in all. 160 over 105 is 1.52x, roughly 15% a year if the money went out at the start and came back at year three. At full NAV the same deal would be 1.33x.

Why does the unfunded commitment belong in the sum at all?

Think of taking over a friend's gym membership halfway through. You pay him something for the months already paid up, but you also inherit the remaining monthly instalments, and in return you get every month of the membership that is left. Buying an LP interest transfers both the assets already in the fund and the obligation to fund the rest of the commitment, so the cheque you write today is only part of what the position costs. Here the buyer pays 85 now and 20 later, and the unfunded commitmentThe part of an LP commitment that the fund has not yet called. The buyer of the interest takes over the obligation to pay it when called. is money the buyer must find whether or not it wants to.

The buyer pays 85 plus 20 drawn, 105, and receives 130 plus 30, 160, so the multiple is 1.52x, compared with 1.33x if it had paid full NAV, and the unfunded 20 appears on both sides of the sum.

What are the numbers, line by line?

Money out: 85% of 100 is 85, plus 20 when it is drawn, so 105. Money in: the existing NAV of 100 grows 1.3x to 130, and the 20 of new capital earns 1.5x, which is 30, so 160. 160 over 105 is 1.52x, and the discount is what lifts it, because at full NAV the cost would be 120 and the multiple only 1.33x. Note that the 15-point discount is 15% of NAV but only 12.5% of the 120 of total exposure. Quoting the discount on NAV alone makes a secondary look cheaper than it is when a lot of the commitment is still to be drawn.

The relationship

\text{MOIC} = \frac{\text{NAV} \times m_1 + U \times m_2}{p \times \text{NAV} + U} = \frac{130 + 30}{85 + 20} \approx 1.52x

NAV	net asset value of the interest at purchase, 100
p	the price paid as a share of NAV, 0.85
U	the unfunded commitment, 20, drawn later
m_1, m_2	the multiples earned on the existing NAV and on the new capital, 1.3 and 1.5

What it says in wordsProceeds from the old assets and the new capital, divided by the price paid plus the capital still to be put in.

What does the simple version leave out?

Timing, mostly. The 20 is drawn after the purchase and the 160 arrives in pieces, so the 15% figure is only a placeholder for an IRR that depends on when each call and distribution lands. A later draw raises the IRR because less money is out for less time, which is one reason secondary buyers like interests with a large unfunded tail: the headline discount applies to a small NAV and the rest of the exposure is priced at cost. Say that out loud, then say what else is missing: the GP's management fee on the commitment, transfer costs and the fact that 1.3x and 1.5x are assumptions, not knowledge.

Where candidates lose it

The common loss is a half-count: including the 20 in the cost but forgetting that it also earns a return, which gives 1.24x, or counting the 30 of proceeds while pretending the 20 never had to be paid, which gives 1.88x. The unfunded commitment is either on both sides or on neither, and on neither is wrong.

The second loss is quoting the discount as a 15% bargain without noting that it applies only to the funded part. On total exposure of 120 it is about 12.5%, and the interviewer wants to hear that distinction.

What the interviewer asks next

If the unfunded 20 is never called, what is the MOIC?
The buyer is offered the same interest at 70% of NAV but the unfunded is 60 rather than 20. Is that a better deal?
Why do secondary buyers often price a young fund at a larger discount than an old one?

022A business earns a 20% return on capital, grows 5% a year forever and has a 10% cost of capital. What share of its profit must it reinvest, and what P/E does that imply? Now redo it with a 10% return on capital.Valuation riddlesHardLarge-cap buyout fund
Try it first
At a 10% return on capital, with the same 5% growth, the P/E is
Show the worked solution
Reinvest 25% and the P/E is 15x; at a 10% return on capital it falls to 10x. To grow 5% a year on a 20% return, the business reinvests 5 over 20, a quarter of profit, and pays out 75 of every 100. 75 over 10% less 5% is 1,500, fifteen times profit. At a 10% return it must reinvest half, pays out 50, and 50 over 5% is 1,000, ten times profit, the same value it would have with no growth at all.
Why does growth cost anything in the first place?
Picture a tiffin service that wants 5% more customers next year. It needs more tiffin boxes and a bigger kitchen, and that comes out of this year's profit before the owner can take anything home. Growth has to be funded, and the amount of profit that must be ploughed back is the growth rate divided by the return the business earns on new capital. A business earning 20% on capital needs to reinvest 5 over 20, a quarter of profit, to add 5% a year. One earning 10% needs 5 over 10, half its profit, for the same growth. The owner of the second business gives up twice as much to get the same thing.
At a 20% return on capital the business reinvests 25 of every 100 of profit and pays out 75, worth 1,500 or 15x profit, while at a 10% return it reinvests 50 and pays out 50, worth 1,000 or 10x, exactly the no-growth value.
How does the reinvestment rate turn into a multiple?
Value is the cash the owner actually receives, growing at 5%, discounted at 10%. With 20% returns the owner receives 75 of every 100, so value is 75 over 5%, which is 1,500 and a P/E of 15x; with 10% returns the owner receives only 50, and 50 over 5% is 1,000, a P/E of 10x. Compare that with no growth at all: the owner receives the whole 100, value is 100 over 10%, which is also 1,000 and 10x. So growth at a 10% return on capital has moved the multiple from 10x to exactly 10x. The 5% growth consumed exactly as much as it created, because each rupee reinvested earned precisely the rate the owner demanded.
The relationship
$\frac{P}{E} = \frac{1 - g/\text{ROIC}}{r - g} \qquad \frac{1 - 0.05/0.20}{0.10 - 0.05} = 15x \qquad \frac{1 - 0.05/0.10}{0.05} = 10x$
g the growth rate of profit, 5% forever
ROIC the return earned on each unit of profit reinvested, 20% or 10%
r the cost of capital the owner demands, 10%
What it says in wordsThe multiple is the share of profit paid out, divided by the discount rate less the growth rate, and the share paid out is one minus growth over return on capital.
What is the lesson for a buyer, and where does the formula break?
That a growth story is only worth paying for when the growth is earned at a return above the cost of capital. Below 10% in this example, growth destroys value: the business would reinvest more than it earns on the reinvestment, and the multiple would fall below the no-growth 10x. That is why a buyout investor asks what return the incremental capital earns before asking how fast revenue grows. The limits are real: the formula assumes the growth and the return last forever, that profit is a fair proxy for cash, and that the business can keep finding projects at the same return, which it usually cannot as it grows.
Where candidates lose it
The common error is reaching for one over (r less g), which gives 20x, as though the owner could keep all the profit and still grow. That multiple belongs to a business that grows for free, which does not exist.
The second loss is treating the 10% case as a trick with no meaning. Say the sentence that matters: when return on capital equals the cost of capital, growth is worth nothing, and below it growth is worth less than nothing.
What the interviewer asks next
What P/E does a 5% return on capital imply with the same 5% growth, and why is the answer uncomfortable?
At a 20% return on capital, what growth rate would push the P/E to 20x?
Why do sponsors pay high multiples for capital-light businesses even when their growth is modest?
023Two businesses each earn EBITDA of 100 on revenue of 500. A has fixed costs of 50; B has fixed costs of 300. Revenue falls 10% at both. What is each one's EBITDA, and what happens to 4x leverage?Operating levers and margin mathsHardOaktree Capital ManagementLos Angeles · 2024
Try it first
After the 10% revenue fall, B's EBITDA is
Show the worked solution
A keeps 85 and B keeps 60; leverage goes to 4.7x and 6.7x. A's variable costs are 350, or 70% of revenue, so at 450 they are 315 and EBITDA is 450 less 315 less 50, which is 85. B's variable costs are 100, or 20%, so at 450 they are 90 and EBITDA is 450 less 90 less 300, which is 60. Debt of 400 was 4.0x at both; it is now 4.7x at A and 6.7x at B.
Why does the same revenue fall hit the two businesses so differently?
Think of two auto drivers. One rents his vehicle by the day and pays a fixed 300 whatever happens; the other owns his and pays mostly for fuel. On a slow day the owner-driver still goes home with something, while the renter may go home with nothing. Fixed costs do not shrink when revenue shrinks, so the whole of a revenue fall lands on profit after only the variable costs have been saved. At A, 70 of every 100 of lost revenue was variable cost that disappears with it, so profit falls by 30 on 50 of lost revenue. At B only 20 of every 100 was variable, so profit falls by 40 on the same 50.
A 10% revenue fall from 500 to 450 cuts A's EBITDA from 100 to 85 because its fixed costs are only 50, but cuts B's from 100 to 60 because 300 of its costs do not move, so 400 of debt goes from 4.0x to 4.7x at A and 6.7x at B.
What are the numbers, and what do they do to the lenders?
A: revenue 450, variable costs 70% of that is 315, fixed 50, EBITDA 85, down 15%. B: revenue 450, variable 20% of that is 90, fixed 300, EBITDA 60, down 40%. Debt of 400 was 4.0x EBITDA at both companies, and after one bad year it is 4.7x at A and 6.7x at B, which is the difference between a conversation with the lender and a covenant breach. The operating leverageThe ratio of the percentage change in profit to the percentage change in revenue. High fixed costs mean high operating leverage. here is 1.5 at A and 4.0 at B: each 1% of revenue lost costs B 4% of EBITDA. Lenders feel it first because their claim is fixed and sits ahead of the equity; the equity feels it hardest because it is what is left.
The relationship
$\text{EBITDA}_1 = R_1(1 - v) - F \qquad B: 450 \times (1 - 0.20) - 300 = 60 \qquad \frac{400}{60} = 6.7x$
R_1 revenue after the fall, 450
v variable costs as a share of revenue, 0.70 at A and 0.20 at B
F fixed costs, 50 at A and 300 at B
What it says in wordsProfit after the fall is the new revenue less the costs that move with it, less the costs that do not.
How should a lender and a sponsor use this?
By sizing the debt to the cost structure, not to the EBITDA alone. Two businesses with identical EBITDA can carry very different debt safely, because the one with high fixed costs needs far less of a downturn to stop covering its interest. B breaks even at revenue of 375, a fall of only 25%, while A breaks even at 167. A lender to B wants lower leverage, a wider cushion in the covenant and a close look at whether any of the 300 can be made variable. The limits: the split of costs into fixed and variable is never clean, fixed costs do move over a long enough horizon, and the same leverage cuts the other way in an upswing, where B's EBITDA would rise 40% on a 10% revenue gain.
Where candidates lose it
The common slip is to cut EBITDA by 10% along with revenue, giving 90 at both companies. That treats every cost as variable and misses the whole point of the question.
The second loss is stopping at 85 and 60. The question says 4x leverage for a reason: convert both numbers into leverage and name which one has become a lender's problem, and say that it is the debt holders who feel operating leverage first because their claim does not shrink.
What the interviewer asks next
What revenue fall would take B to zero EBITDA?
Revenue rises 10% instead. What is each EBITDA, and which company would you rather own the equity of?
How would you check what share of a target's costs is really fixed during diligence?
Asked at Oaktree Capital Management, Credit, Los Angeles, 2024 (Wall Street Oasis): How does operating leverage affect debt vs. equity holders
024A company has a 500 toggle note that pays either 10% in cash or 11% in kind. It toggles to PIK for three years. What is the note worth at the end, and what is total leverage if there is 300 of other debt and EBITDA stays flat at 100?Credit and PIK mathsCoreAres ManagementLos Angeles · 2026
Try it first
After three years of PIK at 11%, the note is
Show the worked solution
About 684, and leverage of about 9.8x. Each year 11% is added to the balance rather than paid: 500 grows to 555, then 616, then 683.8. Add the 300 of other debt and total debt is 984 against EBITDA of 100, so 9.8x, up from 8.0x today. The company has kept 150 of cash over the three years and taken on 184 of extra debt to do it.
What does toggling to PIK actually do?
Think of a credit card where you can skip the monthly payment and let the interest be added to the balance. Nothing leaves your account this month, but next month's interest is charged on a bigger number. A toggle noteA bond or loan whose issuer can choose each period to pay interest in cash or to add it to the principal, usually at a higher rate for the in-kind option. lets the borrower choose that every period, and the price of choosing it is a higher rate charged on a balance that compounds. Here the cash option costs 50 a year. The PIK option costs 55 in year one, and because that 55 is added to the note, 61.05 in year two and 67.77 in year three.
With 300 of other debt and EBITDA flat at 100, three years of 11% PIK on the 500 note take total debt from 800 to 984 and leverage from 8.0x to 8.55x, 9.16x and 9.84x, while the 150 of cash interest avoided is less than the 184 of debt added.
How does the leverage climb, year by year?
Start at 800 of debt on 100 of EBITDA, 8.0x. After year one the note is 555 and total debt 855, so 8.55x. After year two the note is 616.05 and leverage 9.16x. After year three it is 683.82 and leverage 9.84x. Because EBITDA is flat, every unit the note accretes is a unit of leverage with no new earnings to carry it, and the ratio rises by almost two turns in three years without the business doing anything. That is the whole point of the question: PIK is deferred cash, not free cash, and it is deferred at a compounding rate.
The relationship
$N_3 = 500 \times 1.11^3 \approx 683.8 \qquad \text{Leverage} = \frac{683.8 + 300}{100} \approx 9.8x$
N_3 the note balance after three years of in-kind interest
1.11 one plus the PIK rate, applied to the growing balance each year
300 the other debt, which does not accrete
What it says in wordsThe note grows at the in-kind rate compounded, and leverage is all the debt divided by an EBITDA that has not moved.
When is toggling sensible, and what should you add?
When the cash has a better use than paying interest: a short period of heavy investment, or a downturn the company expects to come out of. The toggle buys time at a known price, and the question for the lender and the sponsor is whether the EBITDA that eventually has to carry 984 of debt will be there when the time runs out. Say the things the simple version leaves out: the 1-point premium over the cash coupon is what the lender charges for the option, PIK interest is often still a tax deduction even though no cash moves, and a 9.8x company usually needs an equity cure, a sale or growth, because refinancing at that level is hard.
Where candidates lose it
The common error is simple interest, three years of 55, which gives 665. PIK compounds on the accreted balance, so the answer is about 684, and the gap is the interest on interest.
The second loss is to say 684 and stop. The interviewer asked about leverage because that is what the toggle changes: convert to total debt over EBITDA, show the climb from 8.0x to 9.8x, and say that flat EBITDA is what makes the climb dangerous.
What the interviewer asks next
If EBITDA instead grows 10% a year, what is leverage after three years?
What would the lender want in exchange for the toggle option, beyond the extra 1%?
How does three years of PIK change the recovery for the 300 of other debt if the company is worth 7x EBITDA at the end?
Asked at Ares Management, Credit, Los Angeles, 2026 (Wall Street Oasis): First 1v1 they said was mainly behavioral had PIK question
025At a fork in the road stand two guides. One always lies and one always tells the truth, and you cannot tell which is which. You may ask one guide a single question. What do you ask to find the road to the deal site?Logic and brainteasersCoreMid-market buyout fund
Try it first
Which single question works?
Show the worked solution
Ask either guide: which road would the other guide say leads to the site? Then take the other road. If you asked the truth-teller, he truthfully reports the liar's wrong answer. If you asked the liar, he lies about the truth-teller's right answer, which is also the wrong road. Either way the answer is the wrong road, so the right road is the one not named. The question works because it passes through both guides, so exactly one lie is applied.
Why does a direct question fail?
Imagine asking two shopkeepers whether a price is fair, knowing one of them always exaggerates, but not which. One answer is honest and one is not, and with no way to tell them apart you have learned nothing you can act on. A direct question about the road gives the truth from one guide and a lie from the other, and since you do not know which guide you asked, the answer carries no usable information. Asking a guide whether he is the truth-teller is worse: both say yes, the honest one because it is true and the liar because it is false. The trick has to work the same way whichever guide you happen to be facing.
Whichever guide you ask and whichever road is correct, the answer to the question of what the other guide would say is always the wrong road, so taking the road not named is right in all four cases.
How does routing the question through the other guide fix it?
By making sure the answer passes through exactly one liar. Ask the truth-teller what the liar would say and you get an honest report of a lie; ask the liar what the truth-teller would say and you get a lie about the truth; both contain one lie, so both point to the wrong road. The table in the figure checks all four cases: truth-teller or liar, left road or right road correct, and in every cell the named road is the wrong one. Because the answer is wrong by construction, you do not need to know which guide you asked. You simply take the other road.
You ask What he does Road he names What you do
The truth-teller Reports the liar's answer honestly The wrong road Take the other
The liar Lies about the truth-teller's answer The wrong road Take the other
Both guides name the wrong road when asked what the other would say, so the rule is the same whichever one you met.
Why does a deal desk ask a puzzle with no numbers in it?
Because it tests whether you can design a question whose answer is useful regardless of what you do not know. The puzzle is a small model of diligence: a management team, a seller's banker and a reference customer each have a reason to shade the truth, and the skill is to ask questions whose answers you can act on without first knowing who is being straight with you. Asking a supplier how the target compares with its other customers, rather than asking management how the supplier feels about them, is the same move: route the question so the bias cancels. The limit is that the puzzle assumes a guide who lies perfectly and consistently; a real counterparty who is merely selective needs several questions, not one.
Where candidates lose it
The common loss is panicking and asking the direct question, or asking a guide about himself, which both guides answer identically. Candidates also sometimes find the right question but then take the road it names, forgetting the final flip.
The second loss is giving the answer as a memorised line without showing it works in every case. Walk the four cases out loud, or draw the table, so the interviewer sees that you checked rather than recalled.
What the interviewer asks next
What question works if there are three roads instead of two?
Suppose the guides answer only yes or no. Rephrase the question so it still works.
What question would tell you which guide is the liar, and why is that less useful than finding the road?
026A portfolio company has debt of 450, EBITDA of 100 and a maximum leverage covenant of 5.0x net debt to EBITDA. How far can EBITDA fall before the covenant is breached? EBITDA then drops to 85. How much debt must be repaid so that EBITDA could fall another 20% before breach?Leverage and capital structureHardPrivate creditMid-market buyout fund
Try it first
Leverage is 4.5x against a 5.0x limit. How much can EBITDA fall before breach?
Show the worked solution
EBITDA can fall only 10%, from 100 to 90, before breach; after the drop to 85, about 110 of debt must be repaid. Breach comes at 450 divided by 5.0, which is 90. At 85 leverage is 5.29x, already in breach. For EBITDA to be able to fall 20% from 85, to 68, debt must be no more than 5.0 x 68 = 340. That is a repayment of 110.
Why measure headroom in EBITDA rather than in turns of leverage?
Think of a lift rated for 500 kg carrying 450 kg. The spare 50 kg sounds like a tenth of the rating, and it is: one more passenger and the alarm goes. A covenant works the same way, except that the load is fixed and the rating shrinks. Debt does not move between test dates, EBITDA does, so the only honest measure of headroom is how far EBITDA can fall before the ratio hits the limit. Here that point is 450 divided by 5.0, which is 90, a fall of 10% from today.
Half a turn of spare leverage, 4.5x against 5.0x, feels like comfort. It is not much. A business that misses budget by one bad quarter can lose 10% of its trailing EBITDA. The trailing twelve monthsThe covenant usually tests EBITDA over the last four quarters, so one weak quarter feeds straight into the ratio. test makes this worse, because a weak quarter enters the ratio in full and stays there for a year.
With debt of 450 and a 5.0x covenant, breach comes at EBITDA of 90, only 10% below today's 100, and at 85 the company is in breach at 5.29x. Debt has to fall to 340, a repayment of 110, before EBITDA can absorb another 20% fall.
How do you size the repayment that restores a 20% cushion?
Define the cushion the same way you measured headroom: the share by which EBITDA could fall before breach. A 20% cushion on EBITDA of 85 means the covenant must still hold at 85 x 0.8 = 68, so debt can be at most 5.0 x 68 = 340. Debt is 450, so 110 has to be repaid. Leverage after the repayment is 340 / 85 = 4.0x.
The relationship
$D_{max} = m \times E \times (1-c) = 5.0 \times 85 \times 0.8 = 340$
m the covenant maximum, 5.0x
E current EBITDA, 85
c the cushion, a 20% fall EBITDA must survive
D_max the most debt the business can carry with that cushion
What it says in wordsThe debt a business can carry with a cushion is the covenant multiple times the EBITDA it must survive falling to.
Say where the money comes from, because the interviewer will ask. A business that just lost 15% of its EBITDA rarely has 110 of spare cash, so this usually means an equity cureA sponsor injection of new equity, allowed by many loan agreements, used to repay debt or to count towards EBITDA for a covenant test. The terms vary by document., an asset sale, or a negotiated reset with lenders. Which of these the loan agreement allows is a question for the document, not for arithmetic.
Where candidates lose it
The common loss is reading headroom off the leverage numbers: half a turn spare out of 5.0, so 10% of the limit, or worse, the idea that half a turn is plenty. Candidates who never convert to an EBITDA fall miss how quickly a single bad quarter uses it up.
The second loss is an undefined cushion. Measured as EBITDA sitting 20% above the breach level, the answer is 354 of debt and a repayment of 96, not 110. Both are defensible; switching between them halfway is not. Say which definition you are using before you calculate.
What the interviewer asks next
If the covenant steps down to 4.5x next year, how much EBITDA headroom is left today?
Would you rather cure with equity used to repay debt, or with equity counted as EBITDA, and why does the lender care?
What does a lender gain from setting covenants tight, and what does a sponsor give up by accepting them?
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?
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?
029A sponsor buys a platform at 11x EBITDA of 100, then adds three bolt-ons with EBITDA of 10, 15 and 25, bought at 6x, 7x and 5x. What is the blended entry multiple for the whole group?Valuation riddlesCoreMid-market buyout fundIndian mid-market PE
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Which is closest to the blended multiple?
Show the worked solution
About 9.27x: total price of 1,390 over total EBITDA of 150. The platform costs 1,100. The bolt-ons cost 60, 105 and 125, which is 290 for 50 of EBITDA, a 5.8x average of their own. Add prices, add EBITDA, divide. The simple average of the four multiples, 7.25x, is wrong because it weights a 10 of EBITDA deal the same as a 100.
Why can you not average the multiples?
Buy 10 kg of rice at Rs 60 a kg and 1 kg of saffron rice at Rs 600 a kg. The average price per kg is not Rs 330; it is Rs 1,200 over 11 kg, about Rs 109. A multiple is a price per unit of EBITDA, so blending multiples means total price over total EBITDA, which weights each deal by its size. The platform holds two thirds of the group's EBITDA, so it pulls the blend towards 11x.
Drawn with width equal to EBITDA and height equal to the multiple, each deal's area is its price; the four blocks hold 1,390 of price over 150 of EBITDA, a blended multiple of 9.27x, well above the 7.25x simple average.
What does the blend tell you about the buy-and-build story?
The sponsor has bought 150 of EBITDA for 9.27x. If the larger group is worth the platform's 11x at exit, the same EBITDA is worth 1,650, against 1,390 paid. That gap of 260 is the multiple arbitrage that buy-and-build plans are built on, and it exists only if the market values the combined group at the platform's multiple.
Block EBITDA Multiple Price
Platform 100 11x 1,100
Bolt-on 1 10 6x 60
Bolt-on 2 15 7x 105
Bolt-on 3 25 5x 125
Group 150 9.27x 1,390
Adding prices and EBITDA separately gives 1,390 over 150, a blended 9.27x for the group.
Say the limitation. Bolt-on EBITDA is often bought on the seller's figures, before integration costs, lost customers and the management time each deal eats. Small businesses sell for lower multiples partly because they are riskier. The blend is a fair measure of what was paid; whether the group deserves 11x at exit is a separate judgement.
Where candidates lose it
The average of the four multiples, 7.25x, is the trap, and it is tempting because the question lists four multiples side by side. The interviewer is checking whether you treat a multiple as a price per unit and weight it by size.
The second loss is stopping at the number. A buyout interviewer wants the next sentence: the blend is below the platform multiple, and that gap is the paper value the strategy is counting on.
What the interviewer asks next
What exit multiple on the group would leave the sponsor no better off than the entry blend?
Why do smaller businesses trade at lower multiples, and is that gap sure to close once they are inside a platform?
How would integration costs change the blended multiple you report to the investment committee?
030A sponsor funds a buyout with 400 of preference shares compounding at 12% a year and 45 of ordinary shares for 90% of the ordinary. Management pays 5, at the same price per share, for the other 10%. After 5 years the equity is sold for 1,000. What does management receive?Fund economics numeracyHardMid-market buyout fundIndian mid-market PE
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Roughly what multiple does management make on its 5?
Show the worked solution
Management receives about 29.5, roughly 5.9x its money. The preference compounds to 400 x 1.12^5 = 704.9 and is paid first. That leaves 295.1 for the ordinary shares, and management's 10% is 29.5. The sponsor gets 970.5 on 445, about 2.18x. The structure gears management's small cheque on the ordinary hard.
Why does management do so much better than the sponsor on the same deal?
Imagine two friends buy a flat for Rs 50 lakh. One lends Rs 45 lakh at a fixed rate; both put a little cash in for the ownership. When the flat sells, the loan and its interest are repaid first and whatever is left belongs to the owners. If the price rises, the owners' small stake multiplies; the lender just gets the fixed rate. The preference share is the lender here: it takes a fixed 12% a year first, so all of the upside above that sits on the thin layer of ordinary equity where management's 10% lives. This is called sweet equityOrdinary shares sold to management at the same price as the sponsor, made valuable because most of the sponsor money sits in a senior preference instrument..
The relationship
$\text{Mgmt} = 10\% \times \big(1000 - 400 \times 1.12^5\big) = 10\% \times 295.1 = 29.5$
400 x 1.12^5 the preference with five years of compounding, 704.9
1000 exit equity
10% management's share of the ordinary equity
What it says in wordsManagement gets its share of whatever exit equity is left after the compounded preference.
Of 1,000 of exit equity the compounded preference takes 704.9 and the ordinary shares split 295.1, so management's 10% is 29.5, about 5.9x its 5; below 705 of exit equity management gets nothing.
What happens to management if the deal goes less well?
Run it at different exits. Because the preference keeps compounding whether the business grows or not, management's payout swings from nothing to many times its money over a narrow range of exit values. The table shows management's cheque and multiple at four exit values.
Exit equity Left for ordinary Management gets Multiple on 5
700 0.0 0.0 0.0x
800 95.1 9.5 1.9x
1,000 295.1 29.5 5.9x
1,200 495.1 49.5 9.9x
A 30% fall in exit equity, from 1,000 to 700, takes management from 5.9x its money to nothing, while the sponsor still recovers most of its cheque.
Sponsors measure this with the envy ratioThe price per 1% of ordinary equity paid by the sponsor, counting all its money, divided by the price per 1% paid by management.. The sponsor pays 445 for 90%, about 4.94 per point; management pays 5 for 10%, 0.5 per point, an envy ratio of about 9.9x. The limitation: the sums assume no leaver clauses, ratchets or management loan notes, all of which change who gets what in a real deal.
Where candidates lose it
The common loss is giving management 10% of the whole 1,000, which is 100, or 20x its money. That forgets that the preference sits ahead of the ordinary and has been compounding for five years.
The second loss is compounding the preference with simple interest: 400 plus 5 years of 48 is 640, not 704.9. That error hands management an extra 6.5 and makes the structure look safer than it is.
What the interviewer asks next
At what exit equity does management make 3x its money?
If the preference rate were 8% instead of 12%, how much would management get at 1,000?
Why do sponsors want the envy ratio high, and what stops them pushing it further?

g	the growth rate of profit, 5% forever
ROIC	the return earned on each unit of profit reinvested, 20% or 10%
r	the cost of capital the owner demands, 10%

R_1	revenue after the fall, 450
v	variable costs as a share of revenue, 0.70 at A and 0.20 at B
F	fixed costs, 50 at A and 300 at B

You ask	What he does	Road he names	What you do
The truth-teller	Reports the liar's answer honestly	The wrong road	Take the other
The liar	Lies about the truth-teller's answer	The wrong road	Take the other

m	the covenant maximum, 5.0x
E	current EBITDA, 85
c	the cushion, a 20% fall EBITDA must survive
D_max	the most debt the business can carry with that cushion

Try it first

Why does the unfunded commitment belong in the sum at all?

What are the numbers, line by line?

What does the simple version leave out?

Where candidates lose it

What the interviewer asks next

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Why does growth cost anything in the first place?

How does the reinvestment rate turn into a multiple?

What is the lesson for a buyer, and where does the formula break?

Where candidates lose it

What the interviewer asks next

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Why does the same revenue fall hit the two businesses so differently?

What are the numbers, and what do they do to the lenders?

How should a lender and a sponsor use this?

Where candidates lose it

What the interviewer asks next

Try it first

What does toggling to PIK actually do?

How does the leverage climb, year by year?

When is toggling sensible, and what should you add?

Where candidates lose it

What the interviewer asks next

Try it first

Why does a direct question fail?

How does routing the question through the other guide fix it?

Why does a deal desk ask a puzzle with no numbers in it?

Where candidates lose it

What the interviewer asks next

Try it first

Why measure headroom in EBITDA rather than in turns of leverage?

How do you size the repayment that restores a 20% cushion?

Where candidates lose it

What the interviewer asks next

Try it first

Why does the enterprise value not matter?

How do you check the direction of the division?

Where candidates lose it

What the interviewer asks next

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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 can you not average the multiples?

What does the blend tell you about the buy-and-build story?

Where candidates lose it

What the interviewer asks next

Try it first

Why does management do so much better than the sponsor on the same deal?

What happens to management if the deal goes less well?

Where candidates lose it

What the interviewer asks next