Private Equity puzzles, solved step by step

The MOIC is 2.5x and the IRR is about 25.7%. MOIC is money out over money in: 750 divided by 300. The IRR is the yearly rate that compounds 300 into 750 over four years, the fourth root of 2.5. Take the square root twice: 2.5 becomes 1.581, then 1.257. Check: 1.25 to the fourth is 2.44, just short of 2.5, so the rate sits a little above 25%.

Why work out the multiple before the rate?

Picture a plant that grew from 30 centimetres to 75 in four years. Asking how many times bigger it got is one division; asking how fast it grew each year needs a root. Getting the multiple first strips the problem down to one number, 2.5, so the only hard step left is a fourth root. In a paper LBO the multiple is also what the interviewer will ask for first, because it is the number a sponsor quotes when it talks about a deal.

How do you take a fourth root in your head?

A fourth root is a square root of a square root. The square root of 2.5 is about 1.58, because 1.58 squared is about 2.5, and the square root of 1.58 is about 1.257, because 1.26 squared is 1.59. Then check from the other side with a round rate you can compound: 1.25 squared is 1.5625, and squared again is 2.44. That is a little short of 2.5, so the IRR is a little above 25%, which matches 25.7%.

What would change the IRR without changing the multiple?

Time and interim cash. The same 2.5x earned over five years is about 20%, and over three years about 36%, so a sponsor always quotes both numbers together. Say also that this calculation assumes one cash flow in and one out. A dividend paid in year two, or a second equity cheque in year one, means the IRR has to be solved over several cash flows, and the shortcut gives only an approximation.

11.5% compounded monthly pays more: an effective 12.13% against 12.00%. Monthly compounding pays 0.958% each month, and each payment then earns interest for the rest of the year. Over twelve months that interest on interest adds about 0.63 points to the 11.5% headline. Compare effective annual rates, never headline rates with different compounding.

Why can a lower headline rate pay more?

Imagine two jobs paying the same annual salary, one paid monthly and one in a single lump in December. The monthly earner can put January's pay in a deposit and earn on it for eleven months. The more often interest is paid, the sooner it starts earning interest of its own, so the effective rate rises above the headline rate. 11.5% paid monthly is 0.958% a month, and each month's interest joins the balance that earns the next month's.

How do you estimate the extra without a calculator?

Use the second term of the expansion. Compounding r over n periods adds roughly r squared times (n minus 1) over 2n on top of r, which for 11.5% monthly is about 0.0132 x 11/24, or 0.61 points. So 11.5% plus about 0.6 is roughly 12.1%, enough to beat 12%. The exact figure is 12.13%. A quicker sanity check: continuous compounding is the ceiling, e to the 0.115, which is 12.19%, and monthly sits just below it.

Where does this show up on a private equity desk?

In debt terms and in returns reporting. Loan margins, PIK interest and preferred returns are quoted with different compounding conventions, and comparing them on headline rates is the same mistake as answering 12% here. A PIK note that compounds quarterly costs more than its headline suggests. Fund returns quoted as an IRR are already annual effective rates, which is why they can be compared across funds with different cash flow timing.

2,496, 9,991 and 249,999. Each pair sits evenly either side of a round number, so the product is that number squared less the gap squared. 48 x 52 is 50 squared less 2 squared, 2,500 less 4. 97 x 103 is 10,000 less 9. 499 x 501 is 250,000 less 1. Say the rule first, then the three answers come in seconds.

What do the three pairs have in common?

Look at where each pair sits. 48 and 52 are both 2 away from 50; 97 and 103 are both 3 away from 100; 499 and 501 are both 1 away from 500. Two numbers placed symmetrically around a round number multiply to that number squared, less the gap squared. In symbols, (a minus b) times (a plus b) is a squared minus b squared. The round number is easy to square, the small gap is easy to square, and one subtraction finishes the job.

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

Think of a square garden 50 metres on each side. If you make it 2 metres longer and 2 metres narrower, the strip you gain along one side is 2 by 48, but the strip you lose is 2 by 50. You lose slightly more than you gain, and the shortfall is exactly the small 2 by 2 corner, which is why the answer is always the square less the gap squared. That is the picture worth having in your head when the interviewer asks why the trick works.

Why would a growth equity interviewer bother with this?

Because deal conversations are full of quick products: 48 months of a 52 lakh run rate, a multiple of 9.7x on 10.3 crore of EBITDA. The test is not whether you know the identity but whether you look for structure before you start grinding through digits. If the numbers are not symmetric, make them so: 47 x 55 is 51 squared less 4 squared, 2,601 less 16, which is 2,585. Say the method aloud so the interviewer hears the route, not just the result.

At 800, senior recovers 100%, the notes 75% and equity nothing. Value is paid strictly by priority. Senior takes its 500 first, leaving 300 for the 400 of notes, which is 75%. At 1,000, value covers all 900 of debt, so both classes recover in full and equity is left with 100. The 200 increase in value goes 100 to the notes and 100 to equity.

In what order does the value get paid out?

Think of a row of buckets under one tap, each bucket overflowing into the next. The first fills completely before a drop reaches the second. A capital structure works the same way: each class is paid in full before the class below it receives anything. With 800 of value, the senior bucket takes 500 and is full. The remaining 300 flows into the notes bucket, which holds 400, so it is 75% full. Nothing reaches equity.

Which security does the interviewer care about most?

The one the value line cuts through. That tranche is the fulcrum securityThe most senior class of claims that is not repaid in full, and so typically ends up owning the restructured company., here the subordinated notes at an EV of 800. Every change in enterprise value between 500 and 900 lands entirely on the notes, so they are the class whose price moves with the valuation argument while senior sits at par and equity sits at zero. In a restructuring the fulcrum holders usually swap their claim for the new equity, which is why distressed investors spend their time on it.

What changes if enterprise value is 1,000?

The line clears the debt. Senior is still paid 500, the notes are now paid their full 400, and the 100 left over belongs to equity. The 200 of extra value is split 100 to the notes and 100 to equity, and none of it reaches senior, which was already whole. That asymmetry is the lesson: a senior lender's upside is capped at par, while the junior classes capture the swing. Say the limitations too: real cases add administrative claims ahead of senior debt, and negotiated outcomes sometimes give junior classes a little value to secure their agreement.

The score is 25 today, and at 25% growth the margin must reach 15%. The Rule of 40 adds revenue growth and profit margin: 35 plus minus 10 is 25, well short of 40. If growth slows to 25%, the margin has to be 40 less 25, or 15%. That is a swing of 25 points of margin, from minus 10% to plus 15%, just to hold the bar.

What is the Rule of 40 actually measuring?

Think of a young restaurant that can either open new branches fast and lose money doing it, or open slowly and keep every branch profitable. Either can be healthy; losing money while also growing slowly is not. The Rule of 40 adds growth and margin so that a software company can trade one for the other, and treats a combined score of 40 or more as a sign of a healthy balance. Growth of 35% and a margin of minus 10% score 25, which says the company is spending more on growth than the growth is earning.

Why does slowing growth make the margin job so large?

Because the score falls point for point. Losing 10 points of growth drops the score from 25 to 15 if nothing else moves, so the margin has to cover both the existing 15-point shortfall and the 10 points just lost. That is 25 points of margin. In practice it means cutting sales and marketing spend that was buying the growth, which is often exactly what happens when a buyout fund takes over a software business: growth slows by choice and margin is rebuilt.

What should you say about the rule's limits?

That it is a heuristic, not a law. Different investors use different margin definitions, EBITDA, operating or free cash flow, and the same company can pass on one and fail on another. The rule also treats a point of growth and a point of margin as equal, which is untrue for a small company where growth compounds into a much larger future business. Use it to frame the trade-off, then look at what is driving each side: retention, sales efficiency and gross margin.

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.

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.

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.

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.

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.

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.

Roughly 0.8 to 0.9 billion tickets a year, on round assumptions. Demand: 1,400 million people, a quarter of whom go in a year, about 2.5 times each, gives about 875 million. Supply: about 9,000 screens, 4 shows a day, 200 seats, 30% full, gives about 788 million. Two routes agreeing within about 11% is the check; every input should be tested against an industry report.

Why build it twice?

If you want to know how many people eat at a canteen each day, you can count students and guess how often they eat there, or count tables, sittings and how full they look at lunch. Either guess on its own could be badly off; if both land near the same number you can trust it. A sizing built from demand and checked from supply catches the assumption that is wildly off, which a single chain never reveals. Interviewers care more about the check than about the final digit.

How do you defend each assumption out loud?

Segment before you guess. A quarter of the population going at least once a year is an average of very different groups: urban young adults may go many times, while much of rural India has no screen nearby. Say which segment drives the answer and why your share and frequency are reasonable for it, then give one number. On supply, the interviewer will push on occupancy: 30% is plausible because weekday afternoon shows run thin while weekend evenings fill. The screen count is an assumption to confirm, not a fact to recite.

A growth investor would go one step further: tickets times an average price gives box office revenue, and food and drink add a second stream on the same visits. Say the limitation: streaming changes how often people go, so a figure built on old habits may overstate tomorrow's market.

About Rs 49.6 crore off the price, but a cut is the weaker fix. Expected damages are 30% x 200 = Rs 60 crore, payable in two years, so 60 / 1.1^2 is Rs 49.6 crore today. A price cut is fair on average yet leaves you Rs 200 crore exposed if the case is lost. A specific indemnity from the seller backed by an escrow moves the bad branch to the seller instead.

How do you put a number on something that may never happen?

A shopkeeper who knows that 3 in 10 customers return a gadget within the warranty sets aside money for returns before counting profit. A contingent liability is priced as chance times cost, then discounted for time: here 0.3 x 200 = Rs 60 crore, and two years at 10% brings it to Rs 49.6 crore. That is the average cost across all the ways the case could end, in today's money.

Why is a price cut not the best answer?

Because a buyer does not live the average; it lives one branch. Cut the price by 49.6 and lose the case, and you have paid for a Rs 200 crore liability with a Rs 49.6 crore discount. A specific indemnityA promise by the seller to reimburse the buyer for one named liability, here the lawsuit, if it costs money after completion. puts the bad branch back with the seller, who knows the case best. Backing it with an escrow, part of the price held back until the case ends, makes sure the money is there to collect.

Say the trade-offs. Sellers dislike escrows because cash is locked up, so expect to give something back, perhaps a smaller cut or a cap on the indemnity. Warranty and indemnity insurance usually excludes known issues like this one, so check the policy before relying on it. And the 30% is itself an estimate: ask the legal team how they reached it, because the answer moves one for one with it.

About 8.2x. Exit EV is 140 x 9 = 1,260. Debt falls from 500 to 300, so exit equity is 960. A 3x return allows entry equity of 960 / 3 = 320. Add the 500 of entry debt and the most you can pay is an EV of 820, which is 8.2x entry EBITDA. Pay more and the 3x target is missed; fees and costs would lower the ceiling further.

Why work backwards instead of guessing an entry price?

If you want to arrive at a wedding by 7 and the drive takes two hours with half an hour of traffic, you leave at 4:30. You fix the end point and walk back. A target multiple fixes the exit equity you need relative to the cheque, so the maximum price is found by running the LBO in reverse: exit value, less exit debt, divided by the target, plus entry debt. Every number you need is in the question.

What does 8.2x tell the investment committee?

It is a ceiling, not an offer. Entering at 8.2x and exiting at 9x means the plan relies on a little multiple expansion as well as 40% EBITDA growth and 200 of debt paydown, so the committee will test each of those three. 3x over five years is an IRR of about 25%: 3 to the power one fifth is about 1.246.

Say the limitations. Transaction fees and financing costs come out of the equity at entry, so the true ceiling on the headline price is lower. A real model would also hold some cash at exit and might pay interest in kind. And if the lenders will not provide 5x at an 8.2x price, the equity cheque grows and the ceiling falls.