Private Equity puzzles, solved step by step

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

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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?
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?
043An asset earns 9% a year unlevered and debt costs 12%. What is the return on equity at 0%, 40% and 70% debt funding? When does adding leverage lower the equity return?Leverage and capital structureHardLarge-cap buyout fundPrivate credit
Try it first
At 70% debt, roughly what does equity earn?
Show the worked solution
9%, 7% and 2%: here every turn of debt lowers the equity return, because debt costs more than the asset earns. On 100 of assets earning 9, debt of 40 costs 4.8 and leaves 4.2 on equity of 60, which is 7%. Debt of 70 costs 8.4 and leaves 0.6 on equity of 30, which is 2%. Leverage helps only when the asset earns more than the debt costs; when it earns less, leverage works in reverse.
How does leverage change the equity return?
Borrow at 12% to buy a flat that earns 9% in rent and you lose 3% a year on every borrowed rupee, and the more you borrow, the bigger the loss as a share of your own money. Equity keeps whatever the asset earns after the debt is paid, so each unit of debt adds the gap between the asset return and the debt cost, times the debt-to-equity ratio. With a 9% asset and 12% debt that gap is minus 3 points.
The relationship
$r_E = r_A + (r_A - r_D)\frac{D}{E} = 9\% + (9\% - 12\%) \times \frac{70}{30} = 2\%$
r_A the unlevered asset return, 9%
r_D the cost of debt, 12%
D/E debt over equity, 70/30 at 70% debt
What it says in wordsEquity earns the asset return plus the asset-debt gap, scaled up by how much debt sits on each unit of equity.
With an asset earning 9%, debt at 12% drags the equity return down to 7% at 40% debt and 2% at 70%, while debt at 6% lifts it to 11% and 16%, because leverage magnifies whatever the gap between the two rates is.
Why does a buyout investor worry about this?
A business bought at a high multiple has a low unlevered return: at 12x EBITDA, pre-tax cash yield on the purchase price is in single digits. If borrowing costs rise above that yield, the leverage that used to lift returns now pulls them down, and the deal depends entirely on growth and multiple expansion. This is called negative leverage, and it is why higher rates squeeze what sponsors can pay.
Debt share Equity return, debt at 12% Equity return, debt at 6%
0% 9% 9%
40% 7% 11%
70% 2% 16%
The same three capital structures give falling returns when debt costs more than the asset earns and rising returns when it costs less.
Say the limitations. The sums ignore tax: interest is usually deductible, which lowers the after-tax cost of debt and moves the break-even. And they treat the asset return as fixed. A growing business can out-earn expensive debt later even if it does not today, which is the bet some sponsors knowingly make.
Where candidates lose it
The common loss is the reflex that leverage always increases equity returns. It magnifies the spread, and when the spread is negative it magnifies losses of return. Interviewers ask this exactly to catch that reflex.
The second loss is working in percentages of the asset instead of the equity. Compute earnings, subtract interest, and divide by the equity cheque: 0.6 on 30 is 2%, not 0.6%.
What the interviewer asks next
At what cost of debt does leverage stop mattering for this asset?
How does the interest tax shield change the 70% debt answer at a 25% tax rate?
If the asset return grows 1 point a year, how long before 70% debt at 12% starts to help?
052A company carries debt of 600 against EBITDA of 100, so it is levered 6.0x. It repays 100 of debt every year and EBITDA grows 10% a year. What is leverage after three years?Leverage and capital structureCoreMid-market buyout fundIndian mid-market PE
Try it first
Pick the closest figure for leverage at the end of year 3.
Show the worked solution
About 2.25x. Three repayments take debt from 600 to 300, and three years of 10% growth take EBITDA from 100 to 133.1. Leverage is 300 over 133.1, or 2.25x, down from 6.0x. Halving the debt alone would have left 3.0x; growth removes most of another turn, and that matters to a lender reading the covenant.
Why does leverage fall faster than the debt?
Think of a family loan measured against the family's income. Paying down the loan helps, and so does a pay rise, because the bank looks at how many years of income the loan represents. Leverage is debt divided by EBITDA, so it falls when the top shrinks and when the bottom grows, and here both are happening at once. Track both lines year by year rather than jumping to the end.
Debt falls from 600 to 300 while EBITDA rises from 100 to 133.1, so leverage drops from 6.0x to 2.25x; repayment alone would leave 3.0x and growth alone 4.51x.
Year Debt EBITDA Debt / EBITDA
0 600 100.0 6.00x
1 500 110.0 4.55x
2 400 121.0 3.31x
3 300 133.1 2.25x
Each year debt falls by a flat 100 and EBITDA rises by 10%, so the ratio drops from 6.00x to 4.55x, 3.31x and finally 2.25x.
How much of the drop came from growth?
Split the move multiplicatively, which is exact for a ratio. Halving the debt divides leverage by 2.0; growing EBITDA by 33.1% divides it by 1.331. Six divided by 2.0 and then by 1.331 is 2.25x. Measured that way, repayment does about 71% of the work and growth about 29%, so growth is worth roughly three quarters of a turn here. Say the limit too: the 100 a year of repayment is an assumption, and in a real deal the cash for it comes from the same EBITDA that is growing.
Where candidates lose it
The common slip is answering 3.0x: halving the debt and stopping. It treats leverage as a debt number when it is a ratio, and it misses the 33% rise in EBITDA that does a large share of the work.
The other slip is compounding the debt reduction, as if debt fell 100 then a percentage. The repayment is a flat 100 each year; only EBITDA compounds. Keep the two lines apart and the ratio falls out.
What the interviewer asks next
What leverage would a lender see at year 3 if EBITDA had fallen 10% a year instead?
The covenant steps down to 2.5x at year 3. Is there headroom?
Where does the 100 a year of repayment actually come from, and is it realistic at 6.0x?
066Two companies each have EBITDA of 100 and trade at 8x. One carries debt of 2x EBITDA, the other 6x. EBITDA falls 25% and the multiple stays at 8x. By how much does the equity value fall in each?Leverage and capital structureHardLarge-cap buyout fundMid-market buyout fund
Try it first
The business value falls 25% in both. What happens to the equity?
Show the worked solution
Company A's equity falls 33%; Company B's is wiped out. Both businesses fall from 800 to 600 of enterprise value, a loss of 200. The debt does not fall with them, so the loss lands entirely on the equity. A had 600 of equity and keeps 400. B had only 200 and keeps nothing: its lenders are now owed exactly what the business is worth.
Why does the equity take the whole fall?
Picture two people who each own a Rs 80 lakh flat. One owes Rs 20 lakh on it, the other Rs 60 lakh. If flat prices drop by Rs 20 lakh, both still owe the bank the same amount, so the first owner's stake falls from 60 to 40 and the second's from 20 to nothing. Debt is a fixed claim, so any change in the value of the business falls first and entirely on the equity. The more debt, the thinner the equity cushion that absorbs it.
Both companies lose 200 of enterprise value when EBITDA falls 25% at 8x; with debt of 200, Company A's equity drops from 600 to 400, 33%, while with debt of 600, Company B's equity drops from 200 to zero.
The relationship
$\%\Delta E = \%\Delta EV \times \frac{EV}{E} \qquad 25\% \times \tfrac{800}{600} = 33\% \qquad 25\% \times \tfrac{800}{200} = 100\%$
%Delta EV the fall in enterprise value, 25%
EV/E enterprise value over equity, the leverage multiplier
E equity value before the fall
What it says in wordsThe equity's percentage fall is the business's percentage fall multiplied by how many times larger the business is than the equity.
What does a buyout investor take from this?
Leverage works in both directions with the same multiplier. At 6x debt on 8x value, the equity is a quarter of the business, so every 1% move in enterprise value is a 4% move in the equity. That is what makes high leverage attractive on the way up and fatal on the way down. The limit worth saying: in practice B's equity rarely goes straight to zero on paper. It may keep some option value while the debt has years to run, and the lenders would usually step in through a restructuring before the equity holders lost control.
Where candidates lose it
The fast wrong answer is 25% for both, carrying the business's fall straight across to the equity. It ignores that the debt holds its value while the business shrinks.
The second slip is on Company B: saying 75% by scaling the 25% by three. Work in money, not percentages: 200 lost against 200 of equity is everything.
What the interviewer asks next
How far can EBITDA fall before Company A's equity is wiped out?
If EBITDA rises 25% instead, what are the two equity gains?
Why might B's equity still trade above zero after the fall?
088A business has EBITDA of 100 and debt of 5x EBITDA at 10%. Capex is 20 and tax is ignored. What is interest cover, and how far can EBITDA fall before free cash flow after interest reaches zero?Leverage and capital structureCorePrivate creditMid-market buyout fund
Try it first
How far can EBITDA fall before free cash flow hits zero?
Show the worked solution
Interest cover is 2.0x, and free cash flow reaches zero at EBITDA of 70, a 30% fall. Debt of 500 at 10% costs 50. EBITDA of 100 less capex of 20 and interest of 50 leaves 30 of free cash. That 30 is the cushion. At EBITDA of 70 the cover ratio still reads 1.4x, which looks fine just as the cash runs out.
Why is cover of 2.0x not the cushion it looks like?
A household earning Rs 1 lakh a month with a Rs 50,000 loan payment has income twice its instalment. But if it also spends Rs 20,000 on things it cannot skip, school fees and rent, only Rs 30,000 is truly spare. Interest cover ignores capex, so the real cushion is free cash flow after interest, 30 here, not the gap between EBITDA and interest. Cover would suggest EBITDA can halve; the cash says it can fall 30%.
EBITDA of 100 pays capex of 20 and interest of 50 and leaves 30 of free cash; at EBITDA of 70, a 30% fall, free cash is zero while interest cover still reads 1.4x.
The relationship
$\text{cover} = \frac{100}{50} = 2.0\times \qquad \text{breakeven EBITDA} = 20 + 50 = 70$
100 EBITDA
50 interest, 10% on debt of 500
20 capex the business must spend
What it says in wordsCover compares EBITDA with interest; the breakeven adds the capex that must also be paid.
What does a private credit lender do with this?
A lender tests the downside in cash, not in ratios: how far EBITDA can fall before the company must borrow more, cut capex, or miss a payment. Here the answer is 30%, and a lender would compare it with how far EBITDA fell in the sector's last downturn. If capex could be cut to 10 in a crisis, the cushion widens to a 40% fall, which is why lenders ask how much of capex is maintenance and how much is growth.
One limitation: with tax ignored the picture is generous. Tax would take a slice of the 30, and working capital swings in a downturn usually absorb cash too.
Where candidates lose it
Candidates say EBITDA can halve because cover is 2.0x. That forgets capex, which the business must pay whether or not EBITDA falls.
The second miss is giving the breakeven as an EBITDA level only. Say the percentage fall too, 30%, because that is the number a credit committee compares with history.
What the interviewer asks next
Add tax at 25% on EBITDA less D&A of 20 less interest. Where is the breakeven now?
If the rate rises to 12%, how much cushion is left?
Which covenant would you set for this loan, and at what level?
096A buyout is financed with senior debt of 4x EBITDA at 8% and mezzanine of 1.5x at 13%. What is the blended cost of debt? If the lender's floor is 2.0x interest cover, how much all-senior debt at 8% could EBITDA of 100 carry?Leverage and capital structureHardPrivate credit
Try it first
What is the blended cost of the 5.5x of debt?
Show the worked solution
The blended cost is about 9.36%, and all-senior debt at 8% could reach 625, 6.25x, at 2.0x cover. Interest is 32 on the senior and 19.5 on the mezzanine, 51.5 on 550 of debt. A 2.0x floor on EBITDA of 100 allows 50 of interest, which buys 625 at 8%. The planned 5.5x mix costs 51.5, cover of 1.94x, just under the floor.
Why is the blended rate not the average of 8% and 13%?
If you borrow Rs 4 lakh from a bank at 8% and Rs 1.5 lakh from a relative at 13%, your average cost is pulled towards 8% because most of the money is cheap. A blended cost of debt is a size-weighted average: total interest divided by total debt. Here that is 51.5 divided by 550, about 9.36%, well below the simple average of 10.5%.
Drawn with width equal to size and height equal to rate, the 400 of senior debt at 8% and 150 of mezzanine at 13% cost 51.5 of interest, a blended 9.36%; at a 2.0x cover floor EBITDA of 100 carries 625 of all-senior debt but only about 534 at the blended rate, so the 5.5x mix sits just over the limit.
The relationship
$r_{blend} = \frac{400 \times 8\% + 150 \times 13\%}{550} = \frac{51.5}{550} = 9.36\% \qquad D_{max} = \frac{100 / 2.0}{8\%} = 625$
400, 150 senior and mezzanine debt
51.5 total interest
100 / 2.0 the most interest a 2.0x cover floor allows
What it says in wordsBlend the rates by size; then divide the interest the cover floor allows by the rate to find how much debt it supports.
Why does cover limit the debt before the leverage multiple does?
A lender's cover test caps interest, not turns of debt. The cheaper the debt, the more of it fits under the same interest ceiling, so 8% senior debt supports 6.25x while debt at the blended 9.36% supports only about 5.34x. That is the surprise in this question: replacing the mezzanine with more senior debt would let the sponsor borrow more, not less, if a senior lender would go that far. In practice senior lenders also cap leverage directly, which is why mezzanine exists at all.
Close with the practical point. The planned structure fails a 2.0x floor by a whisker, 1.94x. A credit interviewer wants you to notice that and suggest a fix: trim the mezzanine to about 1.4x, or negotiate the cover test on a measure that adds back something, and say which you would try first.
Where candidates lose it
The common slip is averaging the two rates, 10.5%, which overweights the small expensive tranche. Weight by size every time.
The second is assuming more debt always means more interest cover pressure in the same proportion. Cover depends on the rate as well as the amount, so a cheaper tranche changes the capacity even at the same leverage.
What the interviewer asks next
How much mezzanine can stay in if senior is fixed at 4x and cover must be 2.0x?
Rates rise by 2 points on the senior debt only. What is cover now?
Why would a sponsor accept 13% mezzanine at all if senior debt is cheaper?

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

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

r_A	the unlevered asset return, 9%
r_D	the cost of debt, 12%
D/E	debt over equity, 70/30 at 70% debt

Debt share	Equity return, debt at 12%	Equity return, debt at 6%
0%	9%	9%
40%	7%	11%
70%	2%	16%

Try it first

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?

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

How does leverage change the equity return?

Why does a buyout investor worry about this?

Where candidates lose it

What the interviewer asks next

Try it first

Why does leverage fall faster than the debt?

How much of the drop came from growth?

Where candidates lose it

What the interviewer asks next

Try it first

Why does the equity take the whole fall?

What does a buyout investor take from this?

Where candidates lose it

What the interviewer asks next

Try it first

Why is cover of 2.0x not the cushion it looks like?

What does a private credit lender do with this?

Where candidates lose it

What the interviewer asks next

Try it first

Why is the blended rate not the average of 8% and 13%?

Why does cover limit the debt before the leverage multiple does?

Where candidates lose it

What the interviewer asks next