Private Equity puzzles, solved step by step

About 9.27x: total price of 1,390 over total EBITDA of 150. The platform costs 1,100. The bolt-ons cost 60, 105 and 125, which is 290 for 50 of EBITDA, a 5.8x average of their own. Add prices, add EBITDA, divide. The simple average of the four multiples, 7.25x, is wrong because it weights a 10 of EBITDA deal the same as a 100.

Why can you not average the multiples?

Buy 10 kg of rice at Rs 60 a kg and 1 kg of saffron rice at Rs 600 a kg. The average price per kg is not Rs 330; it is Rs 1,200 over 11 kg, about Rs 109. A multiple is a price per unit of EBITDA, so blending multiples means total price over total EBITDA, which weights each deal by its size. The platform holds two thirds of the group's EBITDA, so it pulls the blend towards 11x.

What does the blend tell you about the buy-and-build story?

The sponsor has bought 150 of EBITDA for 9.27x. If the larger group is worth the platform's 11x at exit, the same EBITDA is worth 1,650, against 1,390 paid. That gap of 260 is the multiple arbitrage that buy-and-build plans are built on, and it exists only if the market values the combined group at the platform's multiple.

Say the limitation. Bolt-on EBITDA is often bought on the seller's figures, before integration costs, lost customers and the management time each deal eats. Small businesses sell for lower multiples partly because they are riskier. The blend is a fair measure of what was paid; whether the group deserves 11x at exit is a separate judgement.

About 22.6% more exit equity: from 2.49x the cheque to 3.05x. 1.2 to the fifth is 2.49 and 1.25 to the fifth is 3.05. Their ratio is (1.25 / 1.2) to the fifth, about 1.226. Five points of IRR compound every year, so over five years they add up to almost a quarter more money at exit, and more again over longer holds.

Why is the answer not five per cent?

Two runners leave together, one about 4% faster. After one lap the gap is small. After five laps it is five small gaps stacked, each measured on a longer distance. An extra five points of IRR is an extra growth factor of 1.25 / 1.20, about 1.042, applied every year, so over five years the exit has to be 1.042 to the fifth, about 1.226 times larger. That is a 22.6% increase in exit equity.

How much more enterprise value does that take?

Less than 22.6%, if the deal carries debt at exit. Say exit EV is 1,000 with 400 of debt, so equity of 600 earns 20%. Equity needs to rise to about 736, and with debt unchanged that needs EV of about 1136. Leverage works in your favour here: a 13.6% rise in enterprise value delivers the 22.6% rise in equity. That is also why sponsors stress-test exit multiples so carefully, since small EV moves swing equity hard either way.

Say the limitation: the sums assume a single cheque in and a single exit. With interim dividends or staged investments, the required change in exit value differs, and the IRR needs a full cash flow schedule.

About 7.5 before discounting, about 5.7 after. The rate falls 1.5 points on 300, saving 4.5 a year, or 13.5 over three years. The call premium is 2% of 300, which is 6, paid up front. Net, 13.5 less 6 is 7.5. Discount the savings at 7.5% and they are worth 11.7 today, so the net gain is about 5.7.

How should you think about a call premium?

Switching to a cheaper phone plan midway through a contract usually costs an exit fee. You switch only if the monthly saving, times the months left, beats the fee. A call premium is the lender's exit fee: it compensates the lender for losing a loan that pays above today's rate, so the borrower refinances only when the saving over the remaining term beats it. Here the fee is 6 and each year saves 4.5, so the fee is earned back in 1.33 years.

When would refinancing not be worth it?

Work out the breakeven rate gap. Spread the premium over the years left: 6 over 3 years is 2 a year, which is 0.67% of 300. Refinancing pays only if the rate falls by more than the premium divided by the years remaining, about 67 basis points here. With one year left the breakeven gap would be the full 2 points, and refinancing at 7.5% would lose money.

Say what the sums leave out. A new loan carries its own arrangement fee and legal costs, often a point or more, which would eat most of the 7.5. Lenders also build call protection to step down over time, so waiting a year can cut the premium. And if the business improves, a lender may simply reprice the existing loan, which avoids the premium altogether.

EBITDA rises from 96 to 117.1, a 22% increase, and the margin from 12% to 14.6%. Costs are 800 less 96, which is 704. A 3% cut saves 21.1, and with revenue fixed every rupee of it reaches EBITDA. Because costs are about 7.3 times EBITDA, a small cost change is a large profit change: 3% times 7.33 is 22%.

Why does a 3% cost cut grow EBITDA by 22%?

A family earning Rs 1 lakh a month and spending Rs 88,000 saves Rs 12,000. Trim spending by 3%, about Rs 2,640, and savings jump to about Rs 14,640, up 22%. Profit is the thin slice left after costs, so any change to the large cost base is magnified when measured against the small profit. The magnifier is the ratio of costs to EBITDA, here 704 over 96, about 7.3.

How does an operating partner use this, and where does it break?

Multiply by the exit multiple to see what it is worth. At 8x EBITDA, 21.1 of lasting savings adds about 169 of enterprise value, which is why cost programmes sit at the centre of so many buyout plans. The thinner the margin, the bigger the magnifier: at a 5% margin costs are 19 times EBITDA and a 3% cut lifts EBITDA by 57%.

Say the limitations. Not all of 704 can be cut: raw materials may be fixed by contract and wages by law, so the 3% often has to come from a much smaller part of the base. Cuts that hurt service can cost revenue later. And the same magnifier works in reverse: a 3% cost overrun would take 22% off EBITDA.

The square root of 2.5 is about 1.58, so 2.5x over two years is about 58% a year. 1.5 squared is 2.25 and 1.6 squared is 2.56, so the root lies between them and much nearer 1.6: 2.5 is 25 of the 31 hundredths along, giving about 1.581. One Newton step from 1.6 gives 1.5813, against a true 1.5811.

How do you find a square root you have not memorised?

A train leaves one station at 6:00 and reaches the next at 9:00. A signal four fifths of the way along is passed at about 8:24: you slide between the two times you know. Find the two squares you know that bracket the number, then slide between them in proportion: the root sits about as far along as the number sits between the squares. 1.5 squared is 2.25 and 1.6 squared is 2.56. The target 2.5 is 0.25 above the lower square out of a gap of 0.31, about 0.81 of the way, so the root is about 1.5 plus 0.081, or 1.581.

How do you sharpen the estimate and check it?

Use one Newton step: take a guess, divide the number by it, and average the two. From 1.6, 2.5 divided by 1.6 is 1.5625, and the average of 1.6 and 1.5625 is 1.58125. One averaging step from a good guess gets you to three decimals, close enough for any interview. Check by squaring: 1.58 squared is 2.4964, just under 2.5, so the root is a hair above 1.58.

Now the investing use. A growth deal returning 2.5x in two years earns the square root of 2.5, less one, each year: about 58%. The fast answer, 150% over two years so 75% a year, ignores compounding: 1.75 squared is 3.06, not 2.5. Say the limitation: short holds produce eye-catching IRRs on modest money, so investors read the multiple alongside it.

About 9.4x EV/EBITDA. Paying 300 for 30% values all the equity at 300 / 0.3 = 1,000. Enterprise value is equity plus net debt, 1,000 + 500 = 1,500. Divided by EBITDA of 160, that is 9.38x. Two steps turn a minority cheque into a whole-company multiple: gross up to 100%, then add back the debt.

Why can you not divide the cheque by EBITDA?

Buying a 30% share of a flat for Rs 30 lakh tells you the owners think the whole flat's equity is worth Rs 1 crore. If the flat also carries a Rs 50 lakh home loan, the flat itself is worth Rs 1.5 crore. EBITDA is earned by the whole company for all its funders, so it has to be compared with the whole company's value, which is all the equity plus the net debt. The cheque is a fraction of only one of those two pieces.

Does it matter whether the money goes to the sellers or into the company?

Yes, and growth investors are asked this often. If the 300 is new money paid into the company, the company's cash rises by 300, net debt falls to 200, and the same price buys a business valued at only 1,200, or 7.5x. Buying existing shares sends the cash to the sellers, so the company's debt is unchanged and the multiple is 9.38x. Ask which it is before you answer.

Say the limitation. A minority price can carry a discount for lack of control, or a premium if the investor gets preference shares with downside protection. So the implied multiple is a starting point, not a clean comparison with what a buyer of 100% would pay.

Fund A ends ahead: 5.23x against 4.91x. 1.18 to the tenth is about 5.23. For B, 1.25 to the fifth is about 3.05 and 1.1 to the fifth about 1.61, which multiply to 4.91. B leads at year five, 3.05x against 2.29x, but its slow second half lets A catch up around year 9. B's ten-year compound rate is 17.3%, below A's 18%.

Why does the fast starter lose?

A batsman who scores fast for 20 overs and then crawls can end below a partner who kept a steady rate all innings. Over a long period what matters is the compound rate across the whole span, and two halves at 25% and 10% compound to 17.3% a year, below a steady 18%. The early lead is real, but the second half compounds on a big base at a low rate.

Why is the average rate misleading here?

The simple average of 25% and 10% is 17.5%, but the compound rate is the geometric mean of the factors: the square root of 1.25 x 1.10, less one, which is 17.3%. The geometric mean is always below the simple average when rates differ, so a fund with uneven returns compounds more slowly than its average suggests. That gap widens the more the rates vary.

This matters to a secondaries or fund-of-funds buyer reading a track record. A fund that shows a strong early IRR may owe it to a few quick exits; what the LP takes home depends on the whole life. Say the limitation: real funds call and return capital over time, so their IRRs are not simple compound rates on one cheque.

About 1.95. Multiply up one step at a time: 1.21, 1.331, 1.4641, 1.6105, 1.7716, then 1.7716 plus a tenth of itself is 1.9487. Each step is the last number plus a tenth of it, which is easy to do in your head. Check with the rule of 72: at 10%, money doubles in about 7.2 years, so after 7 years it should be just under 2.

What is the cleanest way to do it in your head?

Adding 10% is the friendliest step there is: move the decimal one place and add. A bill of 1,771 plus 10% is 1,771 plus 177.1, which is 1,948.1. Multiplying by 1.1 is the same as adding a tenth of the number to itself, so seven careful additions beat any half-remembered formula. Keep four decimals until the last step, then round.

How do you check it a second way?

Two checks. First, the rule of 72: 72 divided by 10 is 7.2 years to double, so seven years should give a little under 2. Second, expand the bracket. (1 + 0.1) to the 7th is 1 + 0.7 + 0.21 + 0.035 + 0.0035 + smaller terms, which sums to about 1.9487. The terms shrink fast, so the first five get you there.

Why a buyout interviewer asks this: a business growing 10% a year for seven years almost doubles. That is the sort of number you need to sanity-check a model's exit EBITDA in two seconds. The limitation of the rule of 72 is that it drifts at high rates; at 10% it is close, at 50% it is not.

Light rope A at both ends and rope B at one end, at the same moment. Rope A burns out after exactly 30 minutes, because two flames together consume 60 minutes of burning in half the time. At that moment rope B has 30 minutes of burning left, so light its other end. It burns out 15 minutes later, at 45 minutes in total.

Why does lighting both ends give exactly 30 minutes?

Two people painting a fence from opposite ends finish in half the time, even if some planks take longer than others, because they always work together on whatever is left. A rope holds 60 minutes of burning in total; two flames each burn their own share, and between them they use it up in 30 minutes, wherever they happen to meet. The uneven burn only changes where the flames meet, not when.

Why does rope B have exactly 30 minutes left?

Because time, not length, is what the rope stores. Rope B has burned for 30 minutes at the moment rope A goes out, so whatever length remains holds exactly 30 minutes of burning, and two flames finish that in 15. You never need to know where on the rope the flame is. That is the general lesson: when a quantity is uneven in one dimension, measure it in the dimension that is reliable.

Say it in order, with the two events that trigger each action: at the start, light three ends; when A goes out, light B's last end; when B goes out, 45 minutes have passed. Interviewers mark you on the clean sequence as much as on the idea. The limitation of the puzzle is its assumption that both ends can be lit at exactly the same moment.

About 2.25x. Three repayments take debt from 600 to 300, and three years of 10% growth take EBITDA from 100 to 133.1. Leverage is 300 over 133.1, or 2.25x, down from 6.0x. Halving the debt alone would have left 3.0x; growth removes most of another turn, and that matters to a lender reading the covenant.

Why does leverage fall faster than the debt?

Think of a family loan measured against the family's income. Paying down the loan helps, and so does a pay rise, because the bank looks at how many years of income the loan represents. Leverage is debt divided by EBITDA, so it falls when the top shrinks and when the bottom grows, and here both are happening at once. Track both lines year by year rather than jumping to the end.

How much of the drop came from growth?

Split the move multiplicatively, which is exact for a ratio. Halving the debt divides leverage by 2.0; growing EBITDA by 33.1% divides it by 1.331. Six divided by 2.0 and then by 1.331 is 2.25x. Measured that way, repayment does about 71% of the work and growth about 29%, so growth is worth roughly three quarters of a turn here. Say the limit too: the 100 a year of repayment is an assumption, and in a real deal the cash for it comes from the same EBITDA that is growing.