Private Equity puzzles, solved step by step

About 10.5%, against a 9% coupon. The coupon pays 9 a year. The discount of 4 is collected when the loan repays at 100, which spread over three years adds about 1.33 a year, so the rough yield is 10.33%. Dividing by the average price of 98 refines it to 10.54%, and the exact figure is 10.63%. A shorter life spreads the same discount over fewer years.

Where does the extra yield come from if the coupon is fixed at 9%?

Picture buying a gift voucher worth 100 for 96, redeemable in three years. On top of whatever the voucher pays along the way, you pocket 4 when you redeem it. A loan bought below par pays you twice: the coupon every year, and the gap between the price and par when the borrower repays. The coupon here is the base rate plus the margin, 4% plus 5%, so 9 a year on each 100 of face value. The 4 points of discount are the second source, and the whole question is how to turn that one-off 4 into a per-year rate.

The shortcut is to spread it evenly: 4 divided by 3 years is 1.33 a year, so the rough yield is 9 plus 1.33, which is 10.33%. That figure is measured against 100, but you paid less than 100 for most of the life. Divide the 10.33 of annual income by the average of the purchase price and par, 98, and you get 10.54%. The exact yield, solving for the rate that discounts the coupons and the 100 back to 96, is 10.63%.

Why does the same discount matter more on a short loan?

Because the discount is a fixed sum and the life is the divisor. A discount adds roughly its size divided by the years until repayment, so halving the life doubles what the discount is worth per year. Bought at 96, a one-year loan yields about 13.5%, the three-year loan 10.6% and a six-year loan 9.9%. That is why credit investors care so much about when a loan actually repays. Leveraged loans can usually be prepaid at par, and many desks quote yields to an assumed life shorter than the legal maturity for exactly this reason: a borrower who refinances early hands the discount back sooner, which raises the yield.

What assumption sits underneath the number?

The coupon floats, so the 9% is only today's coupon. Every yield on a floating-rate loan assumes the base rate stays where it is, so the honest answer is a spread over the base rate, not a fixed percentage. Here the loan earns the 5% margin plus roughly 1.5 points from the discount, about 6.5 points over the base. Say that framing, then give 10.5% as the figure at a 4% base rate, and the interviewer hears that you know which part of the return the loan locks in and which part moves.

About 20%, and 20.1% exactly. Bracket it with round rates you can compound in your head. 1.2 to the fifth is 2.49 and 1.25 to the fifth is 3.05, so 2.5x sits right at the bottom of the bracket, a hair above 20%. Straight division, 150% over five years, gives 30% and overstates the return badly because it ignores compounding.

How do you compound 1.2 five times in your head?

Do it by squaring, the way you would fold a sheet of paper. 1.2 squared is 1.44. Square again to get the fourth power: 1.44 times 1.44 is about 2.07. One more 1.2 gives about 2.49. Chaining small multiplications you trust is safer than reaching for a formula you half remember, and it is exactly what the interviewer wants to watch. The same chain at 1.25 runs 1.56, then 2.44, then 3.05.

Why is the answer not 30%?

Think of a savings account that pays interest on interest. If it pays 20% a year, the gain in year five is far larger than the gain in year one, because it is earned on a bigger balance. An IRR is a compound rate, so the five years of gains are not equal slices of the 150% total; the later years carry more of it. Dividing 150 by five assumes every year earns 30 on the original 100, which is simple interest. The compound rate needed is lower: 20.1%.

What table is worth memorising before a buyout interview?

A small grid of multiples by holding period. Once you know that 2x in five years is about 15%, 2.5x is about 20% and 3x is about 25%, most return questions become a lookup with a small adjustment. If you need more precision, interpolate between rungs: 2.5 is 0.01 of the way along a 0.56 gap from 2.49 to 3.05, which adds about a tenth of a point to 20%, giving 20.1%. State that this assumes a single cash flow in and a single cash flow out; interim dividends would change it.

360, or 90% of the sponsor's cost. Debt at 2x EBITDA of 120 is 240; at 5x it is 600. The 360 of new borrowing is paid out as a dividend, and 360 over the original 400 is 90%. The sponsor gets most of its money back in year three while keeping full ownership, and the business now carries two and a half times the debt it did.

What exactly is a dividend recap doing?

Think of a family that has paid down most of its home loan, then takes a fresh, larger loan against the same house and spends the cash. Nothing about the house changed; the family simply took equity out and left a bigger loan behind. A dividend recapA recapitalisation in which a company borrows new debt and pays the proceeds to its shareholders as a dividend. borrows against the company's current earnings and hands the cash to the owner, so the only new money is the increase in debt. Here that increase is 600 less 240, which is 360.

Has the sponsor made money, or just moved it?

Mostly moved it. Suppose the business is worth 10x EBITDA, which is 1,200. Before the recap the sponsor's equity is worth 1,200 less 240 of debt, which is 960. After it, the equity is worth 1,200 less 600, which is 600, plus 360 already in the bank: the same 960. A recap does not create value; it changes when the sponsor is paid and how much risk sits on the remaining equity. The returns arithmetic still improves, because cash returned in year three lifts the IRR and takes 90% of the cost off the table, which is why sponsors use the tool when credit markets are open.

Who carries the risk afterwards?

The company and its lenders. At an assumed 8% interest rate, interest rises from 19.2 to 48 a year, so EBITDA covers interest 2.5 times instead of 6.25. A 20% fall in EBITDA, to 96, now takes cover down to exactly 2x, a level at which many lenders start to worry, while the sponsor's money is already mostly home. Say the limitation too: lenders price this risk, covenants may cap the payout, and a board must be satisfied the company stays solvent after paying the dividend.

Half the cost base is fixed. Take revenue of 100, so EBITDA is 20 and costs are 80. Revenue grows to 110 and EBITDA to 26, so costs rose by only 4. A 4 rise on 10 of extra revenue means variable costs are 40% of revenue, which is 40. The remaining 40 of the 80 does not move, so fixed costs are 50% of costs.

Why does profit grow faster than revenue at all?

A tea stall pays the same rent whether it sells 100 cups or 110. The milk and sugar go up with each cup; the rent does not. When some costs are fixed, every extra unit of revenue only has to cover its own variable cost, and the rest drops straight to profit. The bigger the fixed slice, the more of each new rupee of revenue falls through, and the faster profit grows relative to sales. That gap between the two growth rates is the clue the question hands you.

So pick round numbers and let the gap tell you the split. Revenue of 100 at a 20% margin means EBITDA of 20 and costs of 80. Ten percent growth takes revenue to 110; thirty percent growth takes EBITDA to 26. Costs are therefore 110 less 26, which is 84, up 4. Those 4 are the variable costs riding on 10 of new revenue, so variable cost is 40% of revenue: 40 of the original 80.

Is there a formula you can say out loud to check it?

Yes. The ratio of profit growth to revenue growth is the degree of operating leverageHow many per cent profit changes for each one per cent change in revenue, equal to contribution divided by profit., here 30 over 10, which is 3. Operating leverage equals contribution divided by EBITDA, so contribution must be three times EBITDA: 60 on revenue of 100. Revenue of 100 less contribution of 60 leaves variable costs of 40, and the cost base of 80 less 40 leaves fixed costs of 40. Two routes landing on the same 50% is the check the interviewer wants to hear.

Why would an operating partner care about this number?

Because it cuts both ways. The same 3x leverage that turned 10% growth into 30% profit growth turns a 10% revenue fall into a 30% profit fall. Say the limitation: real costs are rarely cleanly fixed or variable. Staff can be cut with a lag, rent steps up when a site is added, and a one-year jump can include price rises that carry no variable cost at all, which would make the fixed share look larger than it is.

It is worth 1,050, and growth of about 4.76% makes it worth exactly 20x. Next year's cash flow is 50 x 1.05, or 52.5, and dividing by the 10% discount rate less 5% growth gives 1,050. For 20x, value must be 1,000: solving 50(1 + g) / (0.10 - g) = 1,000 gives g = 50/1,050, about 4.76%. A quarter of a point of growth moves value by 50.

Why does the formula use next year's cash flow?

If you value a fruit tree today, this season's crop has already been picked and sold; what you are buying is next season's crop and every one after it. A perpetuity values the stream of future cash flows, and the first one in that stream arrives a year from now, already grown by 5%. So the numerator is 50 x 1.05, which is 52.5. The denominator is the discount rate less growth, 10% less 5%, which is 5%. 52.5 divided by 0.05 is 1,050, or 21x this year's cash flow.

How do you solve for the growth rate behind a 20x multiple?

Set the value to 1,000, which is 20 x 50, and solve. 50(1 + g) = 1,000 x (0.10 - g), so 50 + 50g = 100 - 1,000g, which gives 1,050g = 50 and g = 4.76%. Dropping growth by just 0.24 of a point, from 5% to 4.76%, takes value down by 50, from 1,050 to 1,000. That is the real lesson of the question, and it is worth saying as a sentence rather than leaving inside the algebra.

Why is a perpetuity so sensitive near the discount rate?

Because the growth rate sits in the denominator as a subtraction. Each point of growth shrinks the gap between the discount rate and growth, and value is one divided by that gap, so the closer growth gets to 10%, the faster value explodes. At 6% the business is worth 1,325; at 8%, 2,700. This is why a terminal value in a buyout model is usually cross-checked against an exit multiple: a small, unexaminable change in a perpetual growth rate can swing the answer by more than any operational assumption. Say also that growth above the long-run growth of the economy cannot last forever, so the formula breaks down when g is set high.

The LPs earn 1.56x net. The 850 invested at 2.0x returns 1,700. LP profit is 1,700 less the 1,000 they paid in, fees included, which is 700. Carry at 20% of 700 is 140, so LPs receive 1,560 on 1,000. Fees cost 0.30x and carry 0.14x, so a 2.0x gross fund delivers 1.56x.

Why do fees and carry need separate treatment?

Think of a farmer who hands a contractor 100 kilos of seed. The contractor keeps 15 kilos as a fee and sows 85, then takes a fifth of whatever extra grain the harvest brings. Fees shrink the base that gets invested, while carry takes a share of the profit after the LPs have their money back. They hit at different points, so they cannot simply be added as percentages, and trying to do it in one step is where most answers go wrong.

So run it in order. Fees of 150 mean only 850 reaches companies. At 2.0x, 850 becomes 1,700. LPs paid in 1,000 in total, fees included, so the profit is 700. Carry at 20% of 700 is 140. The LPs receive 1,700 less 140, which is 1,560, a net multiple of 1.56x.

How big is the gap between gross and net, really?

Measured as a multiple, fees and carry take 0.44x off 2.0x, just over a fifth. Measured as profit, the gap is far larger: the LPs keep 560 of what would have been 1,000 of gross profit had every rupee been invested at 2.0x, so fees and carry together take 44% of the gain. That framing is why LPs negotiate hard over the fee base and the fee step-down after the investment period. A fee that looks small as an annual percentage of committed capital takes a large share of the profit over a ten-year life.

What has the simple version left out?

Two things worth naming. Most funds pay carry only after LPs earn a hurdleA minimum return, often expressed as an annual rate, that LPs must receive before the GP earns carried interest. and then let the GP catch up; at a 2.0x outcome a full catch-up still leaves the GP with 20% of all profit, so the answer here holds. Timing is the bigger omission: net IRR falls further than net multiple, because fees are paid early and proceeds arrive late. Recycling of proceeds and fee offsets from portfolio companies would also change the figure.

About 360 halls, and the wedding calendar sets that figure, not annual demand. Fifty lakh people marrying once in about 70 years gives roughly 36,000 weddings a year. If half use a hall for 1.5 events each, that is 27,000 hall bookings. Spread evenly that needs only 74 halls, but if 40% fall on 30 peak dates, those dates alone need 360.

How do you get from a population to a number of weddings?

Start with something every listener can check: most people marry once, and a city's population turns over roughly once a lifetime. Take a lifetime of about 70 years. Then 50 lakh people divided by 70 gives about 71,000 people marrying each year, and two people make one wedding, so about 36,000 weddings. A rate built from one human fact, how often a person marries, is easier to defend than a recalled statistic, because the interviewer can follow every step. You can refine it later for a young city or for weddings held in hometowns elsewhere.

Not every wedding uses a banquet hall; some are at home, at temples, on open lawns or in hotels. Assume half use a hall, so 18,000. Many families book a hall for more than one function, a reception or a sangeet as well as the main day, so assume 1.5 hall events each: 27,000 bookings a year. Every one of these is an illustrative assumption to be tested, not a sourced figure.

Why is 27,000 divided by 365 the wrong answer?

Think of an umbrella shop. Averaged over the year, it sells a few a day, but it must hold stock for the first heavy rain, when everyone wants one at once. A banquet hall cannot store an empty Tuesday and sell it on an auspicious Saturday, so capacity has to match the peak, not the average. Weddings bunch into a season, and within the season onto a small number of favoured dates. If 40% of bookings land on 30 dates, those dates carry about 360 events each, and with one main event a hall per evening, the city needs about 360 halls.

What does this tell an investor about the business?

Utilisation is the economics. A city sized for its peak runs most halls far below capacity most of the year, so a hall's earnings depend on how it fills off-peak dates with corporate events, smaller parties and exhibitions. The ratio here, 360 halls against 74 if demand were even, says average utilisation is around 20% on wedding business alone. Say the weak assumptions out loud: the peak share and the hall share move the answer most, so they are what you would test with local operators.

28%. The easy route is through the opposite event. The chance there is no restatement is 80% and the chance the customer stays is 90%; because the risks are independent, the chance of neither is 0.8 x 0.9, which is 72%. At least one problem is everything else, 100% less 72%, or 28%. Adding 20% and 10% gives 30% and double counts the 2% chance of both.

Why not just add the two chances?

Picture two friends who each might be late for dinner. If you add their chances of being late, the evening where both are late gets counted once for each of them. Adding probabilities works only when the events cannot happen together; when they can, the overlap is counted twice. Here both problems happen together 0.2 x 0.1, or 2%, of the time, so 20 plus 10 overshoots by exactly that: 30 less 2 is 28.

Why is one minus none the safest route?

Because the opposite of at least one is a single clean case: nothing goes wrong. At least one problem is everything except the case where neither happens, so you multiply the two chances of no problem and subtract from one. It scales without effort. With five independent risks of 10% each, the chance none happens is 0.9 to the fifth, about 59%, so at least one is about 41%, a number that adding would put at 50%.

Is independence a fair assumption in diligence?

Usually not, and saying so earns credit. A company that needs a revenue restatement may have weak controls or stretched customer relationships, so the two risks tend to move together, which raises the chance of both and lowers the chance of at least one below 28%. If the risks were perfectly linked, so the customer leaves only when revenue is also restated, the answer would fall to 20%. The 28% is the answer the question asks for; the correlation point is the judgement the interviewer is listening for.

Carry of 50 is paid, and a 10% LP that sells receives 45. The transfer is a sale at 500 against a cost of 250, so the old fund books a profit of 250 and the GP takes 20% of it, 50. The remaining 450 belongs to the old fund's LPs; a 10% holder gets 45 in cash. The GP earns that carry at a price it helped set, which is the conflict LPs examine.

What is a continuation vehicle, in plain terms?

Picture a shopkeeper who manages a shop for a group of owners and is paid a share of the profit when the shop is sold. Instead of selling to an outsider, he sets up a new group, with some new owners and some old ones, and sells the shop to that group. A continuation vehicleA new fund, managed by the same GP, that buys one or more assets from the GP’s older fund so they can be held for longer. is the same GP selling an asset from its old fund to a new fund it also manages, so existing LPs can take cash or roll into the new vehicle. To the old fund the transfer is a sale, so its carry is calculated as though the asset had been sold.

Why does the transfer price matter so much?

Because the GP is on both sides of it. A higher price raises the carry the GP collects today; a lower price makes the new vehicle, which the GP also manages and earns carry from, a cheaper purchase. At 450, carry falls to 40 and old LPs get 410; at 550, carry is 60 and they get 490. Each 10 of price moves carry by 2 and the selling LPs by 8. That is why these deals usually rely on a third-party lead buyer setting the price and a fairness opinion, so the number is tested by someone without the conflict.

What has the simple version left out?

Carry in most funds is calculated across the whole fund, not deal by deal. If the old fund has losses elsewhere, or has not yet returned LP capital and the hurdle, the crystallised carry could be lower or held back in escrow. Transaction costs, a discount to NAV and the GP rolling part of its carry into the new vehicle all change the cash figures. Name those three, then give 50 and 45 as the answer on the question's terms.

He proposes 98 for himself, 0, 1, 0 and 1. Solve from the end. With two left, partner 4 takes everything. With three, partner 3 buys partner 5 for one unit. With four, partner 2 buys partner 4 for one. With five, partner 1 needs two votes besides his own and buys the two partners who would get nothing in the next round, 3 and 5, for one unit each.

Why start from the end rather than the beginning?

Think of planning a train journey with a fixed arrival time: you work back from when you must arrive to when you must leave. Each partner's vote depends on what they would get if the current proposer were removed, so you can only value a vote once you know the next round, and the only round you can solve directly is the last one. With two partners left, partner 4 proposes 100 for himself and 0 for partner 5. His own vote is one of two, which is half, and half is not fewer than half, so it passes.

With three left, partner 3 needs two yes votes. Partner 5 gets nothing if partner 3 is removed, so one unit buys him: 99, 0, 1. With four left, partner 2 needs two votes. Partner 4 gets nothing in the three-partner round, so one unit buys him: 99, 0, 1, 0.

Which votes does the most senior partner buy, and why so cheaply?

With five partners he needs three yes votes: his own and two more. Look at the four-partner row. Partners 3 and 5 get nothing there, while partner 4 gets one unit and partner 2 gets 99. The cheapest votes always belong to whoever would be worst off in the next round, and here that is partners 3 and 5, so one unit each beats their alternative of zero. He keeps 100 less 2, which is 98. The answer feels unfair, and that is the point: in this game power comes from position in the sequence, not from any idea of a fair share.

Which assumptions does the answer rest on?

Three, and naming them is part of the answer. The tie rule, the tie-breaking preference and pure self-interest each change the split if you alter them. If a plan needed a strict majority, the proposer would need more votes. If an indifferent partner voted yes, the bought votes would cost zero instead of one, and the senior partner would keep 100. And real partners care about fairness and future rounds of carry, which is why actual carry allocations are set by negotiation and track record rather than by this logic.