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 MOIC is 2.5x and the IRR is about 25.7%. MOIC is money out over money in: 750 divided by 300. The IRR is the yearly rate that compounds 300 into 750 over four years, the fourth root of 2.5. Take the square root twice: 2.5 becomes 1.581, then 1.257. Check: 1.25 to the fourth is 2.44, just short of 2.5, so the rate sits a little above 25%.

Why work out the multiple before the rate?

Picture a plant that grew from 30 centimetres to 75 in four years. Asking how many times bigger it got is one division; asking how fast it grew each year needs a root. Getting the multiple first strips the problem down to one number, 2.5, so the only hard step left is a fourth root. In a paper LBO the multiple is also what the interviewer will ask for first, because it is the number a sponsor quotes when it talks about a deal.

How do you take a fourth root in your head?

A fourth root is a square root of a square root. The square root of 2.5 is about 1.58, because 1.58 squared is about 2.5, and the square root of 1.58 is about 1.257, because 1.26 squared is 1.59. Then check from the other side with a round rate you can compound: 1.25 squared is 1.5625, and squared again is 2.44. That is a little short of 2.5, so the IRR is a little above 25%, which matches 25.7%.

What would change the IRR without changing the multiple?

Time and interim cash. The same 2.5x earned over five years is about 20%, and over three years about 36%, so a sponsor always quotes both numbers together. Say also that this calculation assumes one cash flow in and one out. A dividend paid in year two, or a second equity cheque in year one, means the IRR has to be solved over several cash flows, and the shortcut gives only an approximation.

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.

11.5% compounded monthly pays more: an effective 12.13% against 12.00%. Monthly compounding pays 0.958% each month, and each payment then earns interest for the rest of the year. Over twelve months that interest on interest adds about 0.63 points to the 11.5% headline. Compare effective annual rates, never headline rates with different compounding.

Why can a lower headline rate pay more?

Imagine two jobs paying the same annual salary, one paid monthly and one in a single lump in December. The monthly earner can put January's pay in a deposit and earn on it for eleven months. The more often interest is paid, the sooner it starts earning interest of its own, so the effective rate rises above the headline rate. 11.5% paid monthly is 0.958% a month, and each month's interest joins the balance that earns the next month's.

How do you estimate the extra without a calculator?

Use the second term of the expansion. Compounding r over n periods adds roughly r squared times (n minus 1) over 2n on top of r, which for 11.5% monthly is about 0.0132 x 11/24, or 0.61 points. So 11.5% plus about 0.6 is roughly 12.1%, enough to beat 12%. The exact figure is 12.13%. A quicker sanity check: continuous compounding is the ceiling, e to the 0.115, which is 12.19%, and monthly sits just below it.

Where does this show up on a private equity desk?

In debt terms and in returns reporting. Loan margins, PIK interest and preferred returns are quoted with different compounding conventions, and comparing them on headline rates is the same mistake as answering 12% here. A PIK note that compounds quarterly costs more than its headline suggests. Fund returns quoted as an IRR are already annual effective rates, which is why they can be compared across funds with different cash flow timing.

2,496, 9,991 and 249,999. Each pair sits evenly either side of a round number, so the product is that number squared less the gap squared. 48 x 52 is 50 squared less 2 squared, 2,500 less 4. 97 x 103 is 10,000 less 9. 499 x 501 is 250,000 less 1. Say the rule first, then the three answers come in seconds.

What do the three pairs have in common?

Look at where each pair sits. 48 and 52 are both 2 away from 50; 97 and 103 are both 3 away from 100; 499 and 501 are both 1 away from 500. Two numbers placed symmetrically around a round number multiply to that number squared, less the gap squared. In symbols, (a minus b) times (a plus b) is a squared minus b squared. The round number is easy to square, the small gap is easy to square, and one subtraction finishes the job.

Why is the product always a little less than the square?

Think of a square garden 50 metres on each side. If you make it 2 metres longer and 2 metres narrower, the strip you gain along one side is 2 by 48, but the strip you lose is 2 by 50. You lose slightly more than you gain, and the shortfall is exactly the small 2 by 2 corner, which is why the answer is always the square less the gap squared. That is the picture worth having in your head when the interviewer asks why the trick works.

Why would a growth equity interviewer bother with this?

Because deal conversations are full of quick products: 48 months of a 52 lakh run rate, a multiple of 9.7x on 10.3 crore of EBITDA. The test is not whether you know the identity but whether you look for structure before you start grinding through digits. If the numbers are not symmetric, make them so: 47 x 55 is 51 squared less 4 squared, 2,601 less 16, which is 2,585. Say the method aloud so the interviewer hears the route, not just the result.