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.

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.

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.

About 1,133, so the winner overpays by about 133 on average. Each estimate is right on average, but the auction picks the highest of five. Five draws spread evenly between 800 and 1,200 sit on average at the sixths of the range, and the top one averages 800 plus five sixths of 400, about 1,133. To avoid overpaying, each bidder should bid below its own estimate, and by more as the number of bidders grows.

If every estimate is unbiased, how can the winner be wrong?

Picture five friends guessing the number of sweets in a jar. On average they are right, but the one who guesses highest is almost certainly too high. Each estimate is unbiased on its own, but the auction does not pick a random estimate; it picks the highest, and the highest of several noisy guesses is biased upwards. That is the {term('winner’s curse', 'The tendency for the winner of an auction, chosen because its estimate was highest, to have overestimated the value of what it bought.')}: winning is itself evidence that you overestimated.

Why does the top estimate average five sixths of the range?

Five points dropped at random on a line split it, on average, into six equal gaps. So the expected sorted estimates sit at one sixth, two sixths and so on up to five sixths of the way from 800 to 1,200: 867, 933, 1,000, 1,067 and 1,133. The top one is 400 x 5/6, or 333, above 800, which is 1,133. With n bidders the top estimate averages n over n plus 1 of the range, so ten bidders push it to about 1,164.

What does a disciplined bidder do with this?

Shade the bid. A bidder that wants to break even when it wins must bid as though its estimate is the highest of five, which in this setup means bidding about 133 below its estimate, and more in a crowded auction. That is the case for walking away from processes with many bidders and for building value on what the buyer can change rather than on a higher estimate of the same cash flows. Say the limits: real bidders have different information and different synergies, so not all of the gap is error, and a bidder with genuine private information is less exposed.

About 1.52x. The buyer pays 85 for the NAV today and 20 more when the unfunded commitment is drawn, 105 in all. It receives 130 when the NAV realises at 1.3x and 30 when the drawn 20 returns 1.5x, 160 in all. 160 over 105 is 1.52x, roughly 15% a year if the money went out at the start and came back at year three. At full NAV the same deal would be 1.33x.

Why does the unfunded commitment belong in the sum at all?

Think of taking over a friend's gym membership halfway through. You pay him something for the months already paid up, but you also inherit the remaining monthly instalments, and in return you get every month of the membership that is left. Buying an LP interest transfers both the assets already in the fund and the obligation to fund the rest of the commitment, so the cheque you write today is only part of what the position costs. Here the buyer pays 85 now and 20 later, and the unfunded commitmentThe part of an LP commitment that the fund has not yet called. The buyer of the interest takes over the obligation to pay it when called. is money the buyer must find whether or not it wants to.

What are the numbers, line by line?

Money out: 85% of 100 is 85, plus 20 when it is drawn, so 105. Money in: the existing NAV of 100 grows 1.3x to 130, and the 20 of new capital earns 1.5x, which is 30, so 160. 160 over 105 is 1.52x, and the discount is what lifts it, because at full NAV the cost would be 120 and the multiple only 1.33x. Note that the 15-point discount is 15% of NAV but only 12.5% of the 120 of total exposure. Quoting the discount on NAV alone makes a secondary look cheaper than it is when a lot of the commitment is still to be drawn.

What does the simple version leave out?

Timing, mostly. The 20 is drawn after the purchase and the 160 arrives in pieces, so the 15% figure is only a placeholder for an IRR that depends on when each call and distribution lands. A later draw raises the IRR because less money is out for less time, which is one reason secondary buyers like interests with a large unfunded tail: the headline discount applies to a small NAV and the rest of the exposure is priced at cost. Say that out loud, then say what else is missing: the GP's management fee on the commitment, transfer costs and the fact that 1.3x and 1.5x are assumptions, not knowledge.

Reinvest 25% and the P/E is 15x; at a 10% return on capital it falls to 10x. To grow 5% a year on a 20% return, the business reinvests 5 over 20, a quarter of profit, and pays out 75 of every 100. 75 over 10% less 5% is 1,500, fifteen times profit. At a 10% return it must reinvest half, pays out 50, and 50 over 5% is 1,000, ten times profit, the same value it would have with no growth at all.

Why does growth cost anything in the first place?

Picture a tiffin service that wants 5% more customers next year. It needs more tiffin boxes and a bigger kitchen, and that comes out of this year's profit before the owner can take anything home. Growth has to be funded, and the amount of profit that must be ploughed back is the growth rate divided by the return the business earns on new capital. A business earning 20% on capital needs to reinvest 5 over 20, a quarter of profit, to add 5% a year. One earning 10% needs 5 over 10, half its profit, for the same growth. The owner of the second business gives up twice as much to get the same thing.

How does the reinvestment rate turn into a multiple?

Value is the cash the owner actually receives, growing at 5%, discounted at 10%. With 20% returns the owner receives 75 of every 100, so value is 75 over 5%, which is 1,500 and a P/E of 15x; with 10% returns the owner receives only 50, and 50 over 5% is 1,000, a P/E of 10x. Compare that with no growth at all: the owner receives the whole 100, value is 100 over 10%, which is also 1,000 and 10x. So growth at a 10% return on capital has moved the multiple from 10x to exactly 10x. The 5% growth consumed exactly as much as it created, because each rupee reinvested earned precisely the rate the owner demanded.

What is the lesson for a buyer, and where does the formula break?

That a growth story is only worth paying for when the growth is earned at a return above the cost of capital. Below 10% in this example, growth destroys value: the business would reinvest more than it earns on the reinvestment, and the multiple would fall below the no-growth 10x. That is why a buyout investor asks what return the incremental capital earns before asking how fast revenue grows. The limits are real: the formula assumes the growth and the return last forever, that profit is a fair proxy for cash, and that the business can keep finding projects at the same return, which it usually cannot as it grows.

EBITDA can fall only 10%, from 100 to 90, before breach; after the drop to 85, about 110 of debt must be repaid. Breach comes at 450 divided by 5.0, which is 90. At 85 leverage is 5.29x, already in breach. For EBITDA to be able to fall 20% from 85, to 68, debt must be no more than 5.0 x 68 = 340. That is a repayment of 110.

Why measure headroom in EBITDA rather than in turns of leverage?

Think of a lift rated for 500 kg carrying 450 kg. The spare 50 kg sounds like a tenth of the rating, and it is: one more passenger and the alarm goes. A covenant works the same way, except that the load is fixed and the rating shrinks. Debt does not move between test dates, EBITDA does, so the only honest measure of headroom is how far EBITDA can fall before the ratio hits the limit. Here that point is 450 divided by 5.0, which is 90, a fall of 10% from today.

Half a turn of spare leverage, 4.5x against 5.0x, feels like comfort. It is not much. A business that misses budget by one bad quarter can lose 10% of its trailing EBITDA. The trailing twelve monthsThe covenant usually tests EBITDA over the last four quarters, so one weak quarter feeds straight into the ratio. test makes this worse, because a weak quarter enters the ratio in full and stays there for a year.

How do you size the repayment that restores a 20% cushion?

Define the cushion the same way you measured headroom: the share by which EBITDA could fall before breach. A 20% cushion on EBITDA of 85 means the covenant must still hold at 85 x 0.8 = 68, so debt can be at most 5.0 x 68 = 340. Debt is 450, so 110 has to be repaid. Leverage after the repayment is 340 / 85 = 4.0x.

Say where the money comes from, because the interviewer will ask. A business that just lost 15% of its EBITDA rarely has 110 of spare cash, so this usually means an equity cureA sponsor injection of new equity, allowed by many loan agreements, used to repay debt or to count towards EBITDA for a covenant test. The terms vary by document., an asset sale, or a negotiated reset with lenders. Which of these the loan agreement allows is a question for the document, not for arithmetic.

Management receives about 29.5, roughly 5.9x its money. The preference compounds to 400 x 1.12^5 = 704.9 and is paid first. That leaves 295.1 for the ordinary shares, and management's 10% is 29.5. The sponsor gets 970.5 on 445, about 2.18x. The structure gears management's small cheque on the ordinary hard.

Why does management do so much better than the sponsor on the same deal?

Imagine two friends buy a flat for Rs 50 lakh. One lends Rs 45 lakh at a fixed rate; both put a little cash in for the ownership. When the flat sells, the loan and its interest are repaid first and whatever is left belongs to the owners. If the price rises, the owners' small stake multiplies; the lender just gets the fixed rate. The preference share is the lender here: it takes a fixed 12% a year first, so all of the upside above that sits on the thin layer of ordinary equity where management's 10% lives. This is called sweet equityOrdinary shares sold to management at the same price as the sponsor, made valuable because most of the sponsor money sits in a senior preference instrument..

What happens to management if the deal goes less well?

Run it at different exits. Because the preference keeps compounding whether the business grows or not, management's payout swings from nothing to many times its money over a narrow range of exit values. The table shows management's cheque and multiple at four exit values.

Sponsors measure this with the envy ratioThe price per 1% of ordinary equity paid by the sponsor, counting all its money, divided by the price per 1% paid by management.. The sponsor pays 445 for 90%, about 4.94 per point; management pays 5 for 10%, 0.5 per point, an envy ratio of about 9.9x. The limitation: the sums assume no leaver clauses, ratchets or management loan notes, all of which change who gets what in a real deal.

The GP receives about 30.7, all of it from the catch-up, which is 6.1% of the profit. LPs first get their 1,000 back, then the 8% compounding preferred return: 1,000 x (1.08^5 - 1) = 469.3. That leaves 30.7. A full catch-up sends 100% of the next distributions to the GP until it holds 20% of profit, which would take 117.3, so the money runs out inside that tier.

What order does the money flow in?

Think of a restaurant partnership where the investor gets her money back first, then an 8% a year return on it, and only then does the chef share in the profit. A waterfall pays in tiers: capital back, then the preferred returnThe minimum return LPs receive before the GP shares in profit, often 8% a year, here compounding., then the GP's catch-up, then the 80/20 split, and each tier must fill before the next one starts. Here capital takes 1,000, the preferred return takes 469.3, and only 30.7 is left to flow further.

Why does the GP's share jump so fast just above the hurdle?

Because inside the catch-up tier every extra unit goes to the GP. Between 1,469 and 1,587 of proceeds, each additional 1 of exit value adds 1 to carry, so the GP's share climbs from zero to 20% across a band of only 117. Above 1,587 the split settles at 80/20 and stays there.

This is why secondaries buyers and LPs model carry near the hurdle so carefully. A fund sitting just above its hurdle has a GP with a strong incentive to push value up through that band, and a buyer of the LP interest must remember that much of the next gain goes to the GP. The limitation: real waterfalls run on dated cash flows, often deal by deal, with clawbacks; a single drawdown and exit is the clean case.