Private Equity puzzles, solved step by step

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 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.

Contribution rises about 6.9%. On a price of 100 with variable cost of 60, each unit contributes 40. A 5% price rise adds 5 straight to that margin, making 45, a 12.5% rise. Selling 95 units at 45 gives 4,275 against 4,000 before. Volume would have to fall 11.1% before the price rise stopped paying.

Why does a 5% price rise beat a 5% volume loss?

A tea stall sells a cup for Rs 20 that costs Rs 12 to make, keeping Rs 8. Raise the price by Rs 1 and the stall keeps Rs 9: a 5% price rise is a 12.5% rise in what it earns per cup. A price increase falls entirely on the margin because variable cost does not change, so the lower the margin, the bigger the percentage boost. Here the margin is 40% of price, so a 5% price rise lifts contribution per unit by 5 divided by 40, 12.5%.

What would an operating partner check before raising prices?

The arithmetic says volume could fall 11.1% before this price rise lost money, so the real question is how customers respond. A business whose customers cannot easily switch can usually hold more than 89% of volume after a 5% rise; one in a crowded market with lookalike products may lose far more. Operating teams test price on one region or product line first. Revenue here actually falls 0.25%, which is why a revenue-only dashboard would wrongly call the move a failure.

Expected loss is about 2% a year, 8 on 400, so the loan earns about 8% after losses. The chance of default is 5% and a default costs the 40% not recovered: 5% times 40% is 2%. That takes a fifth of the 10% coupon. Assuming a 5% base rate, the 5 point spread shrinks to about 3 points once expected losses are paid for.

Why multiply the default chance by the loss, not use it alone?

If you lend a friend Rs 1,000 and think there is a one in twenty chance they cannot repay, but you know you would get Rs 600 back from them anyway, your expected loss is not Rs 50 but one twentieth of Rs 400, Rs 20. Expected loss is the probability of default times the loss given defaultThe share of the loan a lender loses when a borrower defaults, after recoveries. A 60% recovery means a 40% loss given default., so recovery matters as much as the chance of default. Here: 5% times 40%, which is 2% a year.

What changes over the full three years?

Default can happen in any year, so the chance of at least one default over three years is 1 minus 0.95 cubed, about 14.3%. Expected loss in rupees falls slightly each year because a loan that has already defaulted cannot default again: 8, then 7.6, then 7.2, about 22.8 over the life. The simple 2% a year is a good approximation. It also ignores the coupon lost in the year of default, which a fuller model would include.

A credit interviewer then asks whether 8% is enough. Compare it with the fund's cost of capital and with what safer loans pay. If senior loans to stronger borrowers yield 9% with expected losses of 0.5%, this loan is the worse risk-adjusted deal despite its higher coupon.

The team needs 150 of new revenue at a 40% margin, and about 171 at 35%. A fixed cost is paid for out of contribution, and each rupee of revenue brings only 40 paise of it: 60 divided by 0.40 is 150. If the team wins sales by discounting and the margin falls to 35%, the hurdle rises to 60 divided by 0.35, about 171. Breakeven revenue moves from 750 to 900.

Why is the answer not simply 60?

A shop hires a helper for Rs 6,000 a month. If every Rs 100 of sales costs Rs 60 in stock, the helper must bring in Rs 15,000 of extra sales, not Rs 6,000, because only Rs 40 of each Rs 100 is left to pay wages. New fixed costs are paid out of contribution, so the revenue needed is the cost divided by the contribution margin. At 40%, 60 of cost needs 150 of revenue.

What would an operating partner ask before approving the hire?

Breakeven is the floor, not the case for the hire: the team should bring in well above 150 within a reasonable ramp, at a margin close to the existing 40%. Sales teams often win volume with discounts, which is exactly what drops the margin to 35% and lifts the hurdle to 171. The partner would also ask how long the ramp takes, because a team that needs a year to sell anything costs 60 before it earns a rupee.

About 393 of equity, and an entry enterprise value of about 786. Exit equity is 1,500 less 300 of net debt, 1,200. A 25% IRR for five years means multiplying the money by 1.25 to the fifth, about 3.05x, so the most the sponsor can invest is 1,200 divided by 3.05. If equity is half the price, the business is worth at most twice that.

Why work backwards from the exit?

A house buyer who wants to double their money by the time they sell, and expects to sell for Rs 1 crore after clearing the loan, knows they cannot put in more than Rs 50 lakh today. A target return plus an expected exit turns into a ceiling on the price, which is how sponsors set their bids. The exit equity is fixed by the assumptions; the only free number is what you pay today.

How do you get 1.25 to the fifth without a calculator?

Square twice and multiply once. 1.25 squared is 1.5625, about 1.56; squared again is about 2.44; times 1.25 is about 3.05. A 25% IRR over five years is roughly a 3x money multiple, worth knowing by heart because buyout targets are often quoted that way. Then check consistency: entry debt of about 393 falling to 300 at exit means about 93 repaid over five years, which you would test against the business's cash flow.

About 6 years. 1.2 to the sixth power is 2.99, so six years of 20% a year turns 1 into very nearly 3. Build it from anchors: 1.2 cubed is 1.728, and squaring that gives 2.99. The exact answer, the log of 3 over the log of 1.2, is 6.03 years.

How do you find the hold without logarithms?

Climb the powers of 1.2 until you pass 3. It is like counting how many 20% pay rises it takes to triple a salary: each rise is on the new salary, so they compound. 1.2 cubed is 1.728, and 1.728 squared is about 2.99, so six steps of 20% take you to 3. Seven would be 3.58, well past.

Why is this worth memorising for buyout interviews?

Interviewers ask for any one of the three numbers, multiple, IRR or hold, given the other two. Knowing a few powers of 1.2 and 1.25 answers most of them instantly: 2x at 20% is about 4 years, 3x is about 6, and 3x at 25% is about 5. The rule of 72 gives a cross-check: doubling at 20% takes about 3.6 years, and tripling takes about 1.6 times as long as doubling, near 5.8.

One caveat worth saying: this assumes all the money went in at the start and came out at the end. If proceeds came back in stages, the same 3x and 20% would imply a longer total hold, because early returns lift the IRR.

The expected multiple is 2.2x, and the chance of at least one zero in three deals is 48.8%. Expected value is 20% x 0 plus 50% x 2 plus 30% x 4, which is 2.2. For the zero, take the complement: each deal avoids a zero 80% of the time, so all three do 0.8 cubed, 51.2% of the time. A good average hides a near coin-flip chance of a loss.

How do you get the expected multiple?

Weight each outcome by its chance and add. It is the same as asking what a hundred identical deals would return on average: twenty return nothing, fifty return 2x and thirty return 4x, which is 0 plus 100 plus 120, or 220 on 100 put in. The expected multiple is 2.2x, but no single deal ever returns 2.2x: each one returns 0, 2 or 4.

Why does at least one zero use the complement?

There are many ways to get at least one zero: the first deal, the second, the third, or any two, or all three. There is only one way to get no zero, so count that and subtract it from one. Adding 20% three times gives 60% because it counts the two-zero and three-zero cases more than once. The chance all three are zeros is 0.2 cubed, 0.8%.

The fund-level point is the one to close on: a manager with a 2.2x average still writes off a deal in about half of all three-deal stretches. That is why investors judge a manager over many deals, and why a single write-off early in a fund says little on its own.

About 1.8 deals close, and about 111 must be screened to expect 5. Forty screened becomes 10 at investment committee, 3 signed and 1.8 closed. The stage rates multiply to 4.5%, so expecting 5 closes takes 5 divided by 0.045, about 111 deals screened. Conversion rates multiply, which is why sourcing volume matters so much.

Why do the stage rates multiply?

Each rate applies only to what survived the stage before, like a cricket trial where a quarter of players make the camp, a third of those make the squad and most of the squad play. A funnel's overall conversion is the product of its stage rates, so three reasonable-looking rates combine into a small number: 25% x 30% x 60% is 4.5%.

What would you do with this as a deal team?

You can lift closes by screening more deals or by improving any one stage, and improving a stage is often cheaper. Raising the committee rate from 25% to 35%, by screening sharper, takes closes from 1.8 to 2.5 without a single extra deal. One honest caveat: 1.8 is an expected number. In a year with 40 screens the fund could close zero or four, so a target of 5 closes should be planned with a margin above 111.

Interviewers in Indian mid-market funds often ask this to see whether you understand that origination is a numbers game. A candidate who says a team needs to see more than a hundred opportunities to close five has understood why associates spend so much time on screening.