Private Equity puzzles, solved step by step

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

Showing 31–40 of 78 · filtered from 100Clear filters

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?
041Estimate the annual power bill of a data centre with 20 MW of IT capacity, running at 70% utilisation, with a power usage effectiveness (PUE) of 1.4 and a tariff of Rs 7 per kWh.Market sizing and estimationHardInfrastructure fund
Try it first
Which is closest to the yearly bill?
Show the worked solution
About Rs 120 crore a year. 20 MW at 70% use is 14 MW of IT load. A PUE of 1.4 means the site draws 1.4 times that for cooling and losses, 19.6 MW. Running all year, 19.6 MW times 8,760 hours is about 171.7 million kWh. At Rs 7 a kWh that is about Rs 1,202 million, or Rs 120 crore. The tariff is an assumption to check.
What does PUE add to the bill?
A home air conditioner does not just cool the room; it also uses power to run its own compressor and fan. Power usage effectivenessTotal power drawn by the whole facility divided by the power used by the IT equipment alone. 1.0 would mean no overhead at all. is total site power divided by IT power, so a PUE of 1.4 means every 1 MW of servers needs another 0.4 MW for cooling, power conversion and lighting. Here that is 5.6 MW on top of 14 MW of IT load, and the bill pays for all 19.6.
20 MW of IT capacity at 70% use draws 19.6 MW once a PUE of 1.4 adds cooling and losses, which over 8,760 hours is 171.7 million kWh and about Rs 120 crore a year at an assumed Rs 7 per kWh.
How do you keep the units straight out loud?
Convert once, early, and say it. One megawatt running for one hour is 1,000 kWh, so 19.6 MW for a year is 19.6 x 8,760 x 1,000, about 171.7 million kWh. Then price it: Rs 7 times 171.7 million is about Rs 1,202 million. A crore is ten million, so divide by ten: Rs 120 crore. Most wrong answers to this question are right in method and a factor of ten off in units.
The relationship
$\text{Bill} = C \times u \times PUE \times 8760 \times 1000 \times p = 20 \times 0.7 \times 1.4 \times 8760 \times 1000 \times 7$
C IT capacity, 20 MW
u utilisation, 70%
PUE total power over IT power, 1.4
p tariff, Rs 7 per kWh, an assumption
What it says in wordsMultiply the power actually drawn by the hours in a year, convert to kWh, and price it.
Now the investor's view. Power is usually the largest running cost of a data centre, so PUE is an economic number, not a technical one: improving it from 1.4 to 1.3 saves about Rs 8.6 crore a year here. Many colocation contracts pass power through to tenants, which shifts who bears the tariff risk. Ask how the contracts treat power before you treat the bill as the owner's cost.
Where candidates lose it
The common loss is a factor of ten in the units: candidates convert MW to kWh correctly and then slip between million and crore, landing at Rs 12 crore or Rs 1,200 crore. Say each conversion out loud.
The second loss is pricing only the IT load and forgetting the PUE, which understates the bill by 40% and misses the point of the question.
What the interviewer asks next
What tariff would make the bill Rs 150 crore?
If utilisation rises to 90%, what happens to the bill and to the PUE?
In a colocation contract, who usually pays for power, and why does that matter to the investor?
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?
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?
045A seller wants 100 upfront plus an earnout of 50 if year-2 EBITDA hits a target. You think the chance of hitting it is 40%. At a 12% discount rate, what is the earnout worth to you today, and what is the total price in value terms?Probability and expected value in dealsHardMid-market buyout fund
Try it first
What is the earnout worth to you today?
Show the worked solution
The earnout is worth about 15.9 to you, so the price in value terms is about 115.9. A 40% chance of 50 is 20 on average, payable in two years, and 20 / 1.12^2 is 15.9. The headline price is 150, but you are paying about 116 in value. A seller who thinks the chance is 80% values the same earnout at 31.9, which is why earnouts close price gaps.
How do you value a payment that may not happen?
A shopkeeper who offers a Rs 500 voucher to anyone who spends Rs 5,000 next month books its cost by how many customers she expects to claim it, not by the face value. An earnout is a contingent payment, so you value it as chance times amount, discounted for the wait: 0.4 x 50 / 1.12^2 is about 15.9. That number, not 50, is what the earnout adds to your price.
The relationship
$V = \frac{p \times EO}{(1+r)^t} = \frac{0.4 \times 50}{1.12^2} \approx 15.9$
p your estimate of the chance the target is hit, 40%
EO the earnout payment, 50
r the discount rate, 12%
t years until payment, 2
What it says in wordsAn earnout is worth the chance of paying it, times the amount, discounted to today.
The same 50 earnout is worth 15.9 today to a buyer who puts the chance at 40% and 31.9 to a seller who puts it at 80%, so the buyer feels it is paying 115.9 while the seller feels it is receiving 131.9.
Why does an earnout make a deal possible?
Because both sides can be right in their own terms. The seller, confident in the plan, values the package at about 132; the buyer, more cautious, values it at about 116; and the contract lets each side bet on its own view. If the target is hit, the buyer pays more but has a business that performed. If not, the buyer has paid 100 for a business that did not.
Say the risks. Earnouts are a common source of disputes: how EBITDA is defined, which costs the buyer loads onto the business after closing, and whether the seller's management stays to run it. A buyer controls the business that decides the payout, so the contract needs a tight EBITDA definition and agreed accounting policies.
Where candidates lose it
The common loss is counting the earnout at its face value of 50, which makes the price look like 150 and either overpays or kills the deal. The other is applying the probability but forgetting two years of waiting.
The second loss is treating the 40% as a fact. It is your estimate; the seller's is higher. Say that the gap between the two estimates is exactly what the earnout is for.
What the interviewer asks next
What chance of hitting the target makes the earnout worth 25 to you?
Would you rather the earnout be all-or-nothing or paid in proportion to EBITDA achieved, and why?
How would you protect the seller from the buyer managing EBITDA down after closing?
047Fund A compounds at 18% a year for 10 years. Fund B earns 25% a year for the first 5 years, then 10% a year for the next 5. Which ends with more money?Compounding and time valueCoreSecondaries and fund of funds
Try it first
Which fund ends ahead after ten years?
Show the worked solution
Fund A ends ahead: 5.23x against 4.91x. 1.18 to the tenth is about 5.23. For B, 1.25 to the fifth is about 3.05 and 1.1 to the fifth about 1.61, which multiply to 4.91. B leads at year five, 3.05x against 2.29x, but its slow second half lets A catch up around year 9. B's ten-year compound rate is 17.3%, below A's 18%.
Why does the fast starter lose?
A batsman who scores fast for 20 overs and then crawls can end below a partner who kept a steady rate all innings. Over a long period what matters is the compound rate across the whole span, and two halves at 25% and 10% compound to 17.3% a year, below a steady 18%. The early lead is real, but the second half compounds on a big base at a low rate.
Fund B's 25% start puts it at 3.05x after five years against A's 2.29x, but at 10% thereafter it is overtaken around year 9.1 and ends at 4.91x against A's 5.23x.
The relationship
$1.18^{10} \approx 5.23 \qquad 1.25^5 \times 1.10^5 \approx 3.05 \times 1.61 \approx 4.91$
1.18^10 Fund A's ten-year multiple
1.25^5 x 1.10^5 Fund B's two halves multiplied together
What it says in wordsMultiply the growth factors of each period; never average the rates.
Why is the average rate misleading here?
The simple average of 25% and 10% is 17.5%, but the compound rate is the geometric mean of the factors: the square root of 1.25 x 1.10, less one, which is 17.3%. The geometric mean is always below the simple average when rates differ, so a fund with uneven returns compounds more slowly than its average suggests. That gap widens the more the rates vary.
This matters to a secondaries or fund-of-funds buyer reading a track record. A fund that shows a strong early IRR may owe it to a few quick exits; what the LP takes home depends on the whole life. Say the limitation: real funds call and return capital over time, so their IRRs are not simple compound rates on one cheque.
Where candidates lose it
The common loss is backing B because it starts faster and its rates average almost the same. Averaging rates is the error: compounding multiplies factors, and the geometric mean sits below the arithmetic one.
The second loss is computing 1.25 to the fifth correctly and then adding the second half's growth instead of multiplying. Say each half as a factor, then multiply.
What the interviewer asks next
What rate in B's second half would make the two funds end level?
Why is a fund's early IRR often higher than its final IRR?
What is the geometric average of +50% and -50%, and why does that matter for volatile returns?
048Without a calculator, what is 1.1 to the power 7, to two decimal places?Mental mathsCoreLarge-cap buyout fund
Try it first
Which is right to two decimals?
Show the worked solution
About 1.95. Multiply up one step at a time: 1.21, 1.331, 1.4641, 1.6105, 1.7716, then 1.7716 plus a tenth of itself is 1.9487. Each step is the last number plus a tenth of it, which is easy to do in your head. Check with the rule of 72: at 10%, money doubles in about 7.2 years, so after 7 years it should be just under 2.
What is the cleanest way to do it in your head?
Adding 10% is the friendliest step there is: move the decimal one place and add. A bill of 1,771 plus 10% is 1,771 plus 177.1, which is 1,948.1. Multiplying by 1.1 is the same as adding a tenth of the number to itself, so seven careful additions beat any half-remembered formula. Keep four decimals until the last step, then round.
Seven 10% steps take 1 to 1.949, with each step adding a tenth of the bar before it, which lands just under the 2.0 that the rule of 72 predicts for seven years at 10%.
How do you check it a second way?
Two checks. First, the rule of 72: 72 divided by 10 is 7.2 years to double, so seven years should give a little under 2. Second, expand the bracket. (1 + 0.1) to the 7th is 1 + 0.7 + 0.21 + 0.035 + 0.0035 + smaller terms, which sums to about 1.9487. The terms shrink fast, so the first five get you there.
The relationship
$(1+0.1)^7 = 1 + 7(0.1) + 21(0.1)^2 + 35(0.1)^3 + 35(0.1)^4 + \dots \approx 1.9487$
7, 21, 35, 35 the binomial coefficients for the seventh power
0.1 the 10% growth rate
What it says in wordsExpanding the bracket term by term gives 1.9487, with each term much smaller than the one before.
Why a buyout interviewer asks this: a business growing 10% a year for seven years almost doubles. That is the sort of number you need to sanity-check a model's exit EBITDA in two seconds. The limitation of the rule of 72 is that it drifts at high rates; at 10% it is close, at 50% it is not.
Where candidates lose it
The common loss is adding seven lots of 10% to get 1.70, the simple-interest answer. The interviewer is checking whether compounding is automatic for you.
The second loss is rounding hard at every step, 1.2, 1.3, 1.5 and so on, which drifts well below the true value. Carry the decimals and round only once.
What the interviewer asks next
What is 1.1 to the 10th, roughly?
Use the same method for 1.08 to the 5th.
EBITDA of 100 grows 10% a year for 7 years. What is the exit EBITDA, and what if growth is 12%?
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?
050You have two ropes and a lighter. Each rope takes exactly 60 minutes to burn, but they burn unevenly, so half a rope's length does not take 30 minutes. How do you measure exactly 45 minutes?Logic and brainteasersCoreLarge-cap buyout fund
Try it first
What is the key move?
Show the worked solution
Light rope A at both ends and rope B at one end, at the same moment. Rope A burns out after exactly 30 minutes, because two flames together consume 60 minutes of burning in half the time. At that moment rope B has 30 minutes of burning left, so light its other end. It burns out 15 minutes later, at 45 minutes in total.
Why does lighting both ends give exactly 30 minutes?
Two people painting a fence from opposite ends finish in half the time, even if some planks take longer than others, because they always work together on whatever is left. A rope holds 60 minutes of burning in total; two flames each burn their own share, and between them they use it up in 30 minutes, wherever they happen to meet. The uneven burn only changes where the flames meet, not when.
Rope A, lit at both ends, burns out at 30 minutes whatever its burn rate; rope B, lit at one end, then has exactly 30 minutes of burning left, and lighting its other end at that moment makes it last 15 more, so it goes out at 45.
Why does rope B have exactly 30 minutes left?
Because time, not length, is what the rope stores. Rope B has burned for 30 minutes at the moment rope A goes out, so whatever length remains holds exactly 30 minutes of burning, and two flames finish that in 15. You never need to know where on the rope the flame is. That is the general lesson: when a quantity is uneven in one dimension, measure it in the dimension that is reliable.
Say it in order, with the two events that trigger each action: at the start, light three ends; when A goes out, light B's last end; when B goes out, 45 minutes have passed. Interviewers mark you on the clean sequence as much as on the idea. The limitation of the puzzle is its assumption that both ends can be lit at exactly the same moment.
Where candidates lose it
The common loss is trying to use length: cutting a rope in half or marking three quarters of it. The question says the burn is uneven precisely to rule that out, and candidates who miss the word lose the room.
The second loss is lighting rope B's second end at the start as well, which gives 30 minutes, not 45. B must burn from one end for the first 30 minutes.
What the interviewer asks next
With the same two ropes, how would you measure 15 minutes?
What times between 0 and 120 minutes can you measure with two such ropes?
How many ropes would you need to measure 7.5 minutes?
051A sponsor buys a business at 12x EBITDA, funded 50% with debt, and repays none of the debt. It expects to sell at 9x. By how much must EBITDA grow before the sponsor simply gets its money back?Mental paper LBOsHardLarge-cap buyout fund
Try it first
Before you work it: how much EBITDA growth returns the equity, and no more?
Show the worked solution
EBITDA must grow by 33.3% just to hand the sponsor its money back. Take EBITDA of 100: the business costs 1,200, funded by 600 of debt and 600 of equity. With no paydown the debt is still 600 at exit, so equity is 600 only if the business is again worth 1,200. At 9x that needs EBITDA of 133.3, which is 12 over 9 minus 1.
Why is the answer not 25%?
Picture a flat bought for 12 years of rent and sold later when buyers will pay only 9 years of rent. To get your price back, the rent has to rise until 9 years of it equals the old 12. That is a rise of 12 over 9, a third, not a quarter. A 25% cut in the multiple needs a 33.3% rise in earnings to undo it, because the recovery is measured on the smaller multiple. It is the same base effect that makes a 10% fall need an 11.1% rise.
Entry and exit both show an enterprise value of 1,200: twelve turns of EBITDA 100 going in, nine turns of EBITDA 133.3 coming out, so EBITDA has to grow 33.3% before the 600 of equity is returned.
Why does the 50% debt not change the answer?
Write the equity at exit as exit value less debt. The debt was 600 going in and, with no repayment, is 600 coming out. Equity is returned in full only when the exit value equals the entry value, so the break-even growth is the same at 30% debt, 50% debt or no debt at all. Leverage changes how fast equity grows once you pass break-even, and how fast it is wiped out below it. It does not move the break-even itself when nothing is repaid.
The relationship
$g = \frac{M_{\text{entry}}}{M_{\text{exit}}} - 1 = \frac{12}{9} - 1 = 33.3\%$
g EBITDA growth needed to get the equity back
M_entry the purchase multiple, 12x
M_exit the exit multiple, 9x
What it says in wordsWith no debt repaid, the break-even EBITDA growth is the entry multiple divided by the exit multiple, less one.
What does this do to a real growth plan?
Say the plan grows EBITDA 50% to 150. Sold at 12x, the business would be worth 1,800 and the equity 1,200, double the cheque. Sold at 9x it is worth 1,350 and the equity 750, only 1.25x. The first third of the growth plan is spent refilling the hole left by the lower multiple. This is why sponsors who pay high multiples underwrite their exit at the entry multiple or lower, and why the interviewer asks the question.
Where candidates lose it
The fast wrong answer is 25%, read straight off the drop from 12x to 9x. It measures the cut on the old multiple, when the earnings have to climb back on the new one: 12 over 9 is 1.333, not 1.25.
The second trap is letting the 50% debt pull you into a leverage calculation. With no paydown the debt cancels out of the break-even, and saying so in one sentence shows the interviewer you saw it.
What the interviewer asks next
The company repays 200 of debt over the hold. What EBITDA growth now returns the equity?
EBITDA grows 50% and the exit is at 9x after five years. What is the IRR? (equity 750 on 600)
Why might a buyer at 9x in five years pay less than you did, even for a better business?

D	floating debt, 500
b	the base rate, now 7%
m	the lender's margin, 4%

C	IT capacity, 20 MW
u	utilisation, 70%
PUE	total power over IT power, 1.4
p	tariff, Rs 7 per kWh, an assumption

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

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

Try it first

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

What does a sponsor do about it?

Where candidates lose it

What the interviewer asks next

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What does PUE add to the bill?

How do you keep the units straight out loud?

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Why does the higher IRR make less money?

So which would an LP prefer?

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How does leverage change the equity return?

Why does a buyout investor worry about this?

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How do you value a payment that may not happen?

Why does an earnout make a deal possible?

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Why does the fast starter lose?

Why is the average rate misleading here?

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

How do you check it a second way?

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Why is 25% wrong?

How do you find a sixth root in your head?

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Why does lighting both ends give exactly 30 minutes?

Why does rope B have exactly 30 minutes left?

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Why is the answer not 25%?

Why does the 50% debt not change the answer?

What does this do to a real growth plan?

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