Private Equity puzzles, solved step by step

Ten lockers are open: 1, 4, 9, 16, 25, 36, 49, 64, 81 and 100, the perfect squares. Locker n is toggled once by each person whose number divides n, so its final state depends on how many divisors n has. Divisors come in pairs that multiply to n, which makes the count even, except for a perfect square, where one divisor pairs with itself. An odd count leaves the locker open.

What decides whether one locker ends open?

Think of a light switch flicked by everyone who walks past: if an odd number of people flick it, the light ends on. Locker n is flicked by person d exactly when d divides n, so the number of toggles is the number of divisors of n, and the locker ends open only if that number is odd. The whole puzzle reduces to one question: which numbers have an odd number of divisors? Do not simulate a hundred people; reason about one locker.

Why do only perfect squares have an odd number of divisors?

Pair each divisor with its partner: for 12, 1 goes with 12, 2 with 6, 3 with 4. Every divisor has a different partner, so the count is even and the locker closes. For a perfect square, one divisor is its own partner, 6 x 6 for 36, so it is counted once and the total turns odd. There are ten perfect squares from 1 to 100, so ten lockers stay open. The answer for any number of lockers N is the whole part of the square root of N.

DPI is 0.75x, RVPI is 0.875x and TVPI is 1.625x; only DPI is cash. All three divide by the 800 paid in. DPI counts the 600 distributed, RVPI the 700 still held at estimated value, and TVPI adds them, 1,300 over 800. On paper the fund is well ahead, but in cash the investor has 600 back on 800 and is still short of break-even.

What does each ratio measure?

Imagine lending a friend Rs 800 to start a business. She has paid you back Rs 600 and says your share of the shop is worth Rs 700 more. Rs 600 is in your wallet; Rs 700 is her estimate. DPI counts the cash back, RVPI counts the estimated value still held, and TVPI adds the two, all measured against the money paid in. In fund language: distributions to paid-in, residual value to paid-in, total value to paid-in.

Why does the difference between DPI and TVPI matter so much?

Because NAV is a valuation made by the manager, and it only becomes cash when the holdings are sold. A fund with a high TVPI and a low DPI is telling you it is worth a lot on paper, and investors have learnt to ask how much of that has actually come home. Here more than half the reported value, 700 of 1,300, is still an estimate. A buyer of this fund stake on the secondary market would price that 700 at whatever discount it thinks the estimate deserves, which is exactly why the split matters.

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.

About 16.3%. The lender pays 100, receives 12 a year for five years and gets the 100 back at the end. On its own that is exactly a 12% return. The warrants are 2% of equity worth 1,500, which is 30, received with the final payment, taking year 5 to 142. That one extra payment lifts the IRR by about 4.3 points.

What does the lender earn before the warrants?

Think of a fixed deposit at 12% that pays its interest every year and returns the principal at maturity: it earns exactly 12%. A loan bought at par that pays its full coupon in cash and is repaid at par earns exactly its coupon rate, so the first 12 points of the IRR need no calculation at all. The question is only about what the warrants add on top.

How do you estimate the warrant's effect without a calculator?

The 30 arrives in five years. At around 16%, five years of discounting roughly halves money, so the warrants are worth about 14 to 15 in today's terms. Spread that extra 14 or so of value across five years of a 100 loan and it adds roughly 4 points a year to the coupon, which puts the IRR near 16%. Then confirm by trial: at 16%, the flows of 12 a year and 142 at the end discount to slightly above 100, so the IRR is a touch above 16%, 16.3%.

Say the risk too. The warrant is worth 30 only if the equity really is worth 1,500 at exit. If equity were worth 500, the warrants would bring 10 and the IRR would fall to about 13.5%. That is the point of mezzanine: a contractual coupon plus a slice of the upside, priced for sitting behind the senior lenders.

About 11.65%. In rupees the investment grows by a factor of 1.15 a year. Converting back, each dollar now costs 1.03 times as many rupees, so the dollar value grows by 1.15 divided by 1.03, which is 1.1165. Over five years that compounds to 2.01x in rupees but only 1.74x in dollars. Subtracting 3 from 15 gives 12%, a close but slightly low shortcut.

Why divide by the currency move rather than subtract it?

Picture a relative abroad who sends you dollars to invest in a rupee deposit. The deposit grows, but when you convert the money back, each dollar costs more rupees than before. The dollar investor's growth is the rupee growth divided by the rise in the rupee price of a dollar, because both are growth factors applied to the same money. It is the same arithmetic as turning a nominal return into a real one: a ratio, with subtraction as the rough version.

Does it matter how the depreciation is quoted?

Yes, and saying so earns credit. If instead the rupee loses 3% of its dollar value each year, the factor is 0.97 rather than 1 over 1.03, and the dollar IRR is 1.15 x 0.97 minus 1, 11.55%. The two conventions differ by only a tenth of a point here, but state which one you are using before you compute. The bigger point for a foreign fund is the size of the drag: 3 points a year off a 15% rupee return is a fifth of the return, every year, before any fees.

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.

Down 1%. Start from 100. A 20% rise takes it to 120, a 25% fall takes away 30 to leave 90, and a 10% rise adds 9 to reach 99. In one line, 1.2 x 0.75 x 1.1 = 0.99. Percentage changes multiply; adding them gives plus 5%, which never happened.

Why can you not just add the percentages?

A shop marks a Rs 100 kurta up 20% to Rs 120, then runs a 25% sale, then nudges the price up 10%. The sale takes 25% of 120, which is 30, not 25. Every percentage is measured against the level it starts from, so each change has a different base and the changes multiply rather than add. Adding them treats every change as if it were applied to the original 100.

How do you do 1.2 x 0.75 x 1.1 fast?

Pick the pair that cancels nicely. 1.2 times 0.75 is exactly 0.9, because three quarters of 1.2 is 0.9, and 0.9 times 1.1 is 0.99. Saying the intermediate 0.9 out loud lets the interviewer follow you and catch a slip early. The order does not matter: multiplication gives the same answer whichever change comes first.

In private equity this is the same arithmetic as a multiple that expands 20% at one point and contracts later, or a business that grows, shrinks and recovers. Revenue up 20% and back down 20% does not return to the start: it ends 4% lower.

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.

The ball costs Rs 5 and the bat Rs 105. Call the ball x. The bat is x plus 100, so together they are 2x plus 100, which must equal 110. That makes 2x equal to 10 and x equal to 5. The fast answer of Rs 10 fails the check: a Rs 100 bat is only Rs 90 more than a Rs 10 ball.

Why does Rs 10 jump out, and why is it wrong?

The numbers are built so that 110 minus 100 hands you 10 without thinking. That subtraction answers a different question: what is left if the bat costs exactly 100. The condition is a difference, the bat is 100 more than the ball, not a price, so the bat must contain a whole ball's worth plus 100. Check Rs 10 against the condition and it fails at once: 100 minus 10 is 90.

What is the interviewer actually testing?

Not algebra. They are testing whether you check a fast answer against the conditions before you say it. A deal model full of quick numbers has the same risk: a cell that looks right because it is round, never tested against the constraint it was meant to meet. Writing one line, x plus x plus 100 equals 110, takes three seconds and removes the risk.

Say the answer, then the check in the same breath: ball 5, bat 105, difference 100, total 110. Interviewers often use this as a warm-up and judge you more on the check than on the number.