Private Equity puzzles, solved step by step

Yes: the study is worth Rs 7.5 crore in expectation and costs Rs 2 crore, a net gain of Rs 5.5 crore. Without it, the fund carries a 25% chance of losing Rs 30 crore, an expected loss of Rs 7.5 crore. With it, the fund walks away in the bad case and pays only the fee. The study stays worth buying as long as the chance of a problem is above 2 over 30, about 6.7%.

What exactly is the fund buying?

Think of paying a mechanic to inspect a second-hand car before you buy it. The inspection is worth nothing if the car is sound and a great deal if the gearbox is about to fail. A diligence study is worth the loss it can prevent, multiplied by the chance that the loss is really there. It is not worth the loss itself, and it is not worth zero because the problem is unlikely. Draw both choices as branches and the answer is visible.

What assumptions is that answer resting on?

Two, and you should say both. First, the study is perfect: it always finds the problem when it is there and never raises a false alarm. If the study catches the problem only 60% of the time, its value falls to 0.25 x 0.6 x 30, Rs 4.5 crore, still above the fee. Second, walking away costs nothing beyond the fee. In a live auction, pausing for a study can lose the deal to a faster bidder, and that cost belongs on the yes branch.

The implied enterprise value rises 32%, from 500 to 660. A 20% premium on equity of 800 is 160, taking the offer to 960. The 300 of cash is worth 300 to everyone, so the whole 160 is a premium on the operating business. With net cash, the premium on EV is always larger than the headline premium on equity; with net debt, it is smaller.

Why can the premium not land on the cash?

Picture buying a shop that has Rs 3 lakh sitting in its till. You may pay extra for the shop's location and customers, but nobody pays Rs 3.6 lakh for Rs 3 lakh of cash: the cash is worth its face value to every buyer. The equity price is cash plus the business, and since the cash is fixed at 300, the full 160 of premium has to sit on the business. That turns a 20% premium into a 32% premium on the part the buyer is really valuing.

Why would a buyout investor care which premium you quote?

Because the sponsor underwrites the business, not the cash. If the business earned EBITDA of 50, the market price would be 10x and the offer is 13.2x. A 20% headline premium sounds modest, but on a cash-rich target it can mean paying a third more for the operations. The limit runs the other way too: on a company with large net debt, a 20% equity premium is a much smaller premium on EV, so headline premiums are never comparable across capital structures without this step.

The company must repay 320, which is the whole equity cheque. The business costs 800, funded by 480 of debt and 320 of equity. With flat EBITDA and the same multiple it is still worth 800 at exit, so equity of 640 needs debt to fall to 160. That is 80 a year for four years, equal to all of EBITDA, so in practice this deal cannot reach 2x without growth.

Why does the repayment equal the equity cheque?

Think of a house bought for Rs 80 lakh with a Rs 48 lakh loan and Rs 32 lakh of your own money. If the house is still worth Rs 80 lakh years later, the only way your share doubles to Rs 64 lakh is for the loan to shrink to Rs 16 lakh. When the enterprise value does not move, equity gains exactly what the debt loses, so doubling the equity means repaying an amount equal to the original equity. That holds at any leverage, which makes it a fast check.

Is that repayment realistic?

Now test the answer against the business. Repaying 80 a year means every rupee of EBITDA goes to the lenders as principal. Interest on 480 of debt at 8% is about 38 in year one alone, before any tax or capex, so the cash to repay 80 a year simply does not exist. The honest conclusion is the useful one: a flat business bought at 10x with 6x debt cannot reach 2x from paydown. It needs EBITDA growth, a higher exit multiple, or a lower entry price.

1,000 / 7 is 142.857 and 22 / 0.35 is about 62.86. Seven goes into 1,000 142 times with 6 left, and six sevenths is 0.857142, repeating. For the second, multiply both numbers by 100 to get 2,200 / 35, divide both by 5 to get 440 / 7, and that is 62 with 6 left: 62 and six sevenths, about 62.86. Both answers end in the same sevenths.

Why is one seventh worth memorising?

Think of a clock face with six numbers on it instead of twelve. Every seventh, whatever it is, lands on the same circle of six digits, 1, 4, 2, 8, 5, 7, and only the starting point changes. Once you know 1/7 = 0.142857, every other seventh is a rotation of the same six digits: 2/7 starts at the 2, 3/7 at the 4, and 6/7 at the 8. That turns both of these divisions into one fact you already hold.

How do you show the long division out loud?

Seven into 1,000 is 142, because 7 x 142 is 994, leaving 6. Bring down a zero: 60, seven goes 8 times, 4 left. 40 gives 5, with 5 left. 50 gives 7, with 1 left. The moment a remainder you have seen before comes back, the digits start repeating, so you can stop dividing and write the cycle. Three decimals is 142.857; the next digit is a 1, so no rounding is needed.

Interviewers ask these at desks that live in spreadsheets because quick division is how you sanity check a multiple or a margin in a meeting. A 22 crore profit on a 0.35 crore unit is about 63 units. Saying the rescaling step out loud matters more than speed: it shows a method you would trust on any number.

The split stack is cheaper, at a blended 8.33% against 8.5%. Two thirds of the debt costs 7% and one third costs 11%, so the blend is 7% plus a third of the 4-point gap. On EBITDA of 100, that is 37.5 of interest a year against 38.25, a gap of only 0.75. Sponsors often pay that for one lender, one set of documents and a surer, faster close.

How do you blend two rates quickly?

If you borrow Rs 2 lakh from a bank at 7% and Rs 1 lakh from a relative at 11%, you are not paying 9% on the whole: most of the money is the cheap kind. A blended rate is a weighted average, so start from the cheaper rate and add the gap times the share of debt that is expensive. Here the second lien is a third of the stack, so the blend is 7% plus a third of 4 points, 8.33%.

Why would a sponsor pay more for one lender?

Because the 0.75 a year buys simplicity that has real value. With two tranches there are two lender groups and an intercreditor agreementThe contract between two groups of lenders to the same company that sets who is paid first, who controls enforcement and what each may do without the other. between them, and every waiver or amendment means two negotiations. A unitranche puts one lender across the table, which usually means a faster close, fewer parties who can block a change, and more certainty in a competitive auction. The limit: compare all-in costs, because upfront fees and call protection can differ between the two and move the answer by more than 0.17 points.

The real return is about 1.9% a year, and after 10 years the money buys about 1.21x what it did. Real growth is the money growth divided by the price growth: 1.07 / 1.05 = 1.0190. Over ten years money grows 1.97x while prices grow 1.63x, so purchasing power grows 1.21x. Subtracting 5% from 7% gives 2% and 1.22x, close but slightly high.

Why divide instead of subtract?

Think in plates of biryani. If a plate costs Rs 100 and you have Rs 100, you can buy one. A year later you have Rs 107 and a plate costs Rs 105, so you can buy 107 / 105 plates, about 1.019. Real return measures how many more things your money buys, which is a ratio of two growth factors, not a difference of two rates. The subtraction shortcut ignores that inflation also eats into the 7% you earned, which is why it comes out slightly high.

When does the shortcut stop being good enough?

At low rates the gap is small: here 1.90% against 2.00%, and 1.21x against 1.22x after a decade. The error grows with the size of the rates, so with 20% returns and 15% inflation the shortcut says 5% when the truth is about 4.3%. A fund comparing returns across countries with different inflation, or across decades, should divide. In an interview, give the shortcut first, then the exact figure, and say which you would use when.

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.

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.

The manager receives 160 of carry. Investors first get 1,000 of capital and 469.3 of preferred return, 8% compounded for five years. The manager then takes 100% of the next 117.3 as catch-up. The last 213.3 splits 80/20. Manager's total: 117.3 plus 42.7, which is 160, exactly 20% of the 800 profit.

What order does the money flow in?

A distribution waterfallThe agreed order in which a fund pays out cash: capital back, then the preferred return, then catch-up, then the profit split. pays tier by tier, like a row of glasses being filled in turn: the next glass gets nothing until the one before is full. Capital comes back first, then the hurdle, then the catch-up, and only then the 80/20 split. Compounding 1,000 at 8% for five years gives 1,469.3, so the preferred return is 469.3.

How big is the catch-up, and why does it end at 20% of all profit?

The catch-up runs until the manager holds 20% of everything distributed above capital. Investors already hold 469.3 of profit, so the manager needs c where c equals 20% of 469.3 plus c. That solves to a quarter of the preferred return, 117.3, after which every rupee splits 80/20 and the manager's share of total profit stays at exactly 20%. There was enough money to finish the catch-up, so the answer is simply 20% of 800.

Say the shortcut, then show the tiers as proof. If the fund had returned only 1,500, the catch-up would not complete: after the hurdle only 30.7 would be left, all of it to the manager, and carry would fall well short of 20% of the 500 profit.

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%.

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.