Private Equity puzzles, solved step by step

2.5x and an IRR of about 20%. Entry EV is 50 x 8 = 400, half debt, so equity is 200. Exit EV is 75 x 8 = 600, and debt has fallen from 200 to 100, so exit equity is 500. 500 over 200 is 2.5x. Over five years that is about 20% a year, since 1.2 to the fifth is about 2.49.

What is the fastest way to do this in your head?

Buy a flat for Rs 40 lakh with a Rs 20 lakh loan. Five years later it is worth Rs 60 lakh and you have paid the loan down to Rs 10 lakh. Your stake went from Rs 20 lakh to Rs 50 lakh. Every paper LBO is two balance sheets, entry and exit, and the answer is exit equity over entry equity, where equity is EV less debt each time. Write the four numbers, subtract twice, divide once.

How do you get the IRR without a calculator?

Carry three anchors for a five-year hold. 2x in five years is about 15% a year, 2.5x is about 20%, and 3x is about 25%. 1.2 to the fifth is 1.2 x 1.2 = 1.44, then 1.73, 2.07 and 2.49, so 2.5x is a touch over 20%: 20.1%. Saying the anchor and then the check sounds far more assured than reaching for a formula.

Then split the gain, because the interviewer will ask where the return came from. EBITDA rose 25 at a constant 8x, adding 200 of value; cash flow repaid 100 of debt; the multiple did not move, so it added nothing. The limitation: the sums ignore fees at entry and any cash left on the balance sheet at exit, both of which a real model would carry.

Choose levered free cash flow, and Company A is the cheaper one. Both cost 9.0x EBITDA, but A keeps 90 after capex and B only 50, so the same 900 buys cash at 10.0x in A and 18.0x in B. After 25% tax the gap holds at 13.3x against 24.0x. Levered free cash flow is the one number that has already paid for capex, working capital, interest and tax: it is what an owner can actually take out.

What does EBITDA leave out that matters here?

Picture two taxis, each bringing in Rs 1 lakh a month after fuel and the driver. One is new and needs little upkeep; the other is old and eats half its takings in repairs. Before repairs they look identical, and nobody would pay the same for both. EBITDA is earnings before the business pays for the assets it uses up, so two companies with equal EBITDA can leave their owners very different amounts of cash. Company B spends half its EBITDA just to keep running.

Why levered free cash flow rather than EBIT?

EBIT does subtract depreciation, which here happens to equal capex, so EBIT would rank the two correctly. But depreciation is an accounting estimate of past spending, not the cash going out this year, and it ignores working capital, interest and tax. Levered free cash flow is the only one of the three that is cash in the owner's hand after every claim ahead of the owner has been paid. Assume no debt and depreciation equal to capex: A's free cash flow is 100 less tax of 22.5 less capex of 10, which is 67.5; B's is 37.5.

What is the limit of your answer?

Say two things. First, one year of free cash flow can be lumpy: a big capex year or a working capital swing can make a good business look expensive. With a single period of data, levered free cash flow is the most honest number, but it is also the noisiest. Second, levered cash flow belongs to the equity, so you compare it with equity value, not enterprise value. In this example there is no debt, so the two are the same; with debt you would divide by equity value instead.

391, about 1.26, 87.5% and 162. 17 x 23 is (20 - 3)(20 + 3), so 400 - 9. 1.08 cubed is about 1 + 3 x 0.08 + 3 x 0.0064, which is 1.259, close to the exact 1.2597. Seven eighths is one less one eighth, 100% - 12.5%. And 45% of 360 is half of 360 less a twentieth of it, 180 - 18.

Why name the trick rather than just give the number?

A shopkeeper who adds a bill in his head is not calculating faster than you; he has a handful of shortcuts he has used ten thousand times. Speed in a quick-fire round comes from recognising which shortcut a question is built for, so say the shortcut as you give the answer. It shows the interviewer the answer is reliable, and it protects you when a number comes out slightly off: the method is audible even if the last digit slips.

How do the two harder ones work?

For 1.08 cubed, expand (1 + x) cubed as 1 + 3x + 3x squared + x cubed with x = 0.08. When x is small, the first two terms carry almost everything and the third is a small correction: 1 + 0.24 + 0.0192 is 1.2592, within 0.001 of the exact 1.2597. That is three years of 8% growth, about 26%, which is why the trick earns its place on a returns desk. For 17 x 23, check the trick fits: it only works when the two numbers sit the same distance either side of a round number.

Fractions and percentages are best memorised in eighths: 12.5%, 25%, 37.5% and so on up to 87.5%. Percentages of awkward numbers split into easy pieces: 10%, 5%, 50%. With those four habits, most of what turns up in a quick-fire round is one step.

About 9 crore tyres a year, on the assumptions stated. Take about 20 crore two-wheelers in use, each riding about 7,000 km a year on 2 tyres: that is 2.8 lakh crore tyre-kilometres. If a tyre lasts about 30,000 km, roughly 9.3 crore tyres wear out each year. Tyre life is the sensitive input: 20,000 km gives 14 crore and 40,000 km gives 7 crore.

Why start from the bikes on the road rather than new sales?

Think of how many toothbrushes a household buys. It depends on how many people live there and how often a brush wears out, not on how many babies were born this year. A replacement market is sized off the installed base, the stock in use, multiplied by how fast each unit wears out. New two-wheelers arrive with tyres already fitted at the factory, which is a separate market sold to manufacturers, so counting new sales here would mix two different customers.

How do you defend each assumption out loud?

Say where each number comes from and how you would check it. Registered vehicle counts overstate the bikes in use, because scrapped vehicles stay on the register, so cut the registered figure down and say you are doing so. Seven thousand km is about 20 km a day, a commute and errands. Then run the cross-check a rider can judge: at 7,000 km a year a 30,000 km tyre lasts about four years, which sounds right for a commuter bike. All three inputs are illustrations to be confirmed against current industry data, not facts.

Close by naming the input you would research first. Tyre life halves or doubles the answer across a plausible range, and rear tyres wear faster than front ones, so a sharper version splits the two. A sponsor looking at a tyre retailer would care about exactly that split, because the rear tyre is the more frequent purchase.

Levered free cash flow is 92.5. EBITDA is 250. Take off D&A of 40 and interest of 60 to get pre-tax profit of 150, tax of 37.5 and net income of 112.5. Add back the 40 of D&A because it is not cash, then subtract capex of 50 and the 10 of working capital the growth absorbs. What remains, 92.5, is cash the owners could take out.

Why does depreciation come off and then go back on?

Think of a delivery van bought last year. Each year the accounts charge a slice of its cost as depreciation, but no money leaves the business that year: the van was paid for already. Depreciation reduces taxable profit, which saves real tax, but it is not a cash payment, so it is subtracted to work out tax and then added back to reach cash. The cash cost of assets enters through capex instead, here 50, which is more than the 40 of depreciation because the business is growing.

What makes it levered, and why does a buyout investor want that version?

Levered means after interest: the lenders have been paid. Levered free cash flow is the cash left for the equity holders after the lenders, the taxman and the business's own reinvestment, which is exactly the cash that repays debt in a buyout. Check it a second way from EBITDA: 250 less 60 less 37.5 less 50 less 10 is 92.5. The limit worth saying: working capital is modelled here as 10% of growth, and a real business can swing by far more in a single year.

The rule of 72 gives 36%; the exact IRR is about 41.4%. Eight times is three doublings, so the money doubles every two years, and 72 divided by 2 is 36. The exact rate is 8 to the power one sixth, which is the square root of 2, so the IRR is 1.414 minus 1, 41.4%. The rule runs low because it is tuned for rates near 8%.

How does the rule of 72 apply to a multiple bigger than 2?

Think of folding a sheet of paper: each fold doubles the thickness, so eight layers is three folds. Any money multiple that is a power of 2 is a count of doublings, and once you know the years per doubling, the rule of 72 gives the rate. Eight is 2 x 2 x 2, so six years hold three doublings of two years each. The rule says a doubling every two years needs 72 over 2, which is 36% a year.

How do you get the exact answer with a basic calculator?

You need the sixth root of 8. Split it: the sixth root is the square root of the cube root, and the cube root of 8 is 2, so the answer is the square root of 2. One press of the square root key gives 1.414, an IRR of 41.4%. Without a square root key, use trial and error: 1.4 squared is 1.96, and 1.96 cubed is about 7.53, a little short of 8, while 1.42 to the sixth is about 8.20, a little over. The answer sits between 40% and 42%.

Then say why the rule ran low. The rule of 72 is built around rates near 8%; the higher the rate, the bigger the number you should divide into. For a two-year doubling the exact rate implies a numerator of about 83, not 72. Checking the shortfall out loud, 36% for six years gives only 6.33x, shows the interviewer you know the rule's limit.

Break-even is Rs 5,000 a passenger; a 10% margin on revenue needs about Rs 5,556. A 2-hour flight costs 2 x Rs 3.6 lakh, Rs 7.2 lakh. An 80% load factor fills 144 of 180 seats, so each passenger must cover Rs 7.2 lakh / 144, Rs 5,000. For cost to be 90% of revenue, divide by 0.9, not multiply by 1.1, which gives about Rs 5,556.

Why divide by passengers and not by seats?

Think of a shared taxi to the airport that costs Rs 1,000 whether four people ride or two. If only two turn up, each pays Rs 500, not Rs 250. A flight costs the same to operate with empty seats, so the fare has to be set on the seats that are actually sold. That is why the load factor sits at the centre of airline economics: at 80%, every paying passenger carries a quarter of an empty seat.

What would you ask the interviewer for next?

In this format you can ask for more data, and asking well scores points. The two inputs that move the answer most are the load factor and what is inside the cost per hour, because fuel and aircraft leases dominate it. At a 70% load factor the break-even rises to about Rs 5,714. You might also ask about ancillary revenue, bags and seats, which lets the base fare sit lower. The interviewer's follow-up, why airlines have often earned poor returns, sits on the same logic: high fixed cost per flight and thin margins mean small swings in load factor or fuel swing profit hard.

Equity earns 17% when the asset earns 12%, and only 3% when it earns 5%. Per Rs 100 of asset, lenders take a fixed 3.5. At 12% equity keeps 8.5 on its 50; at 5% it keeps 1.5 on its 50. Leverage adds the spread between the asset return and the cost of debt, times debt over equity, in both directions.

Why does borrowing push the equity return above the asset return?

Think of buying a flat for Rs 100 lakh with Rs 50 lakh of your own money and a Rs 50 lakh loan at 7%. If the flat earns 12 lakh a year in rent and value, the bank takes 3.5 lakh and everything else is yours. Lenders get a fixed amount whatever the asset earns, so any return above the cost of debt on the borrowed half lands on the equity. The borrowed 50 earns 12% and costs 7%; that 5 point spread on 50 is 2.5, added to the 6 the equity's own half earns.

What happens when the asset earns less than the debt costs?

The same formula runs backwards. At 5%, the spread is minus 2 points, and with one rupee of debt per rupee of equity the equity return falls to 5 minus 2, which is 3%. Leverage is not a return booster; it is a multiplier on the spread, and the spread can be negative. Below a 3.5% asset return equity earns nothing at all, and below zero it loses money faster than the asset does.

In the room, give both numbers and then the rule in one sentence. A private credit interviewer is checking that you see the downside as clearly as the upside, because their job is to sit on the other side of that fixed 3.5.

About Rs 21,250 crore a year on these assumptions. Take roughly 25 crore school students, assume 85% attend schools that require uniforms, two sets a year and an average of Rs 500 a set: 25 x 0.85 x 2 x 500. Then say which input is weakest. Price is, because government and private sets differ several times over.

How do you structure the estimate before any number?

Say the chain out loud first: students, times the share who must wear a uniform, times sets a year, times price. It is how a shopkeeper would size the season: how many children in the area, how many of their schools insist on a uniform, how many sets each family buys, what they pay. The structure earns most of the marks, because the interviewer can disagree with a number and still follow your answer.

Then put a number on each branch and label it an assumption. Enrolment is the one input with a public source, the government's school statistics; use about 25 crore and say you would confirm the current figure. The uniform share, sets a year and price are judgement calls: 85%, two sets and Rs 500 a set are reasonable starting points, not facts.

Which branch do you name as the biggest uncertainty, and why?

Swing each input across a range you could defend, holding the others. The uniform share from 75% to 95% moves the pool by about Rs 5,000 crore. Sets a year from 1.5 to 2.5 moves it by about Rs 10,625 crore. Price from Rs 300 to Rs 800 moves it by Rs 21,250 crore, so price is where you would spend your first hour of real work. The reason is the mix: many state schools supply uniforms cheaply in bulk, while private schools sell branded sets at a premium.

A strong close splits the pool into those two segments and says which one a buyout fund would care about: the private school segment, where the price is set by the school and the supplier relationship is sticky.

About 2,000 shops on these assumptions. Of 2 crore people, assume 30% drink coffee, a cup a day each: 4.2 crore cups a week. If 10% are bought outside the home, that is 42 lakh a week, 6 lakh a day. At 300 cups a shop a day the metro supports about 2,000. The share bought outside is the branch to test.

What is the chain, and why build it before any number?

Shops exist to serve demand, so go from people to cups to shops. Picture a single street: how many of the people passing drink coffee, how often, and how many of those cups are bought rather than made at home. A sizing answer is a chain of stated assumptions, and the interviewer is testing the chain, not the final figure. Say the chain first, then fill it.

How do you sanity check the supply side?

Three hundred cups a day is the number to defend. A shop open twelve hours selling 25 cups an hour reaches it; a quiet neighbourhood outlet may sell half that and a busy office-district one double. Check the answer from the other end: about one shop for every 10,000 people, which you can compare with what you see on a busy street. If your answer implied a shop for every 500 people, a branch is wrong, and you would say which.

A growth equity interviewer often follows with how many of those shops a single chain could own, which turns the sizing into a market share question. Have the number of shops in your answer split into chains and independents if you can, even roughly.