Private Equity puzzles, solved step by step

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.

1.0x with no paydown and 1.75x with 300 repaid. The purchase price is 1,000, with 600 of debt and 400 of equity. With flat EBITDA and the same multiple, the exit value is again 1,000. If debt is still 600, equity is still 400: 1.0x. If 300 has been repaid, debt is 300 and equity is 700, which is 1.75x, an IRR of about 12% over five years.

Where does the return come from if nothing grows?

Think of buying a flat for 1 crore with a 60 lakh home loan, then renting it out and using the rent to pay down 30 lakh of the loan over five years. If the flat is still worth 1 crore, your share has gone from 40 lakh to 70 lakh, though the flat itself is unchanged. Debt paydown moves value from the lenders to the owner: the business is worth the same, but a larger slice of it belongs to the sponsor. The company's own cash flow is doing the repaying, so the sponsor's cheque never changes.

What are the numbers, step by step?

Entry: 10 x 100 is 1,000. Sixty per cent debt is 600, so equity is 400. Exit with nothing repaid: 1,000 less 600 is 400 of equity, 1.0x, a zero return over five years. Exit with 300 repaid: 1,000 less 300 is 700 of equity, and 700 over 400 is 1.75x. Over five years, 1.75x is about 11.8% a year: 1.12 to the fifth is 1.76, so a shade under 12%.

Is 300 of paydown realistic, and what is left out?

It means 60 a year of free cash flow after interest and tax on a business with EBITDA of 100, which is possible for a capital-light company and unlikely for a capital-hungry one. Paydown is the most reliable of the three return levers, alongside EBITDA growth and multiple change, because it depends on cash the business already generates rather than on the future. Say what the simple version ignores: transaction fees at entry and exit, any cash left on the balance sheet, and the risk that interest rates or a downturn absorb the cash meant for repayment.

The EBITDA margin is 20%. Both multiples share the same enterprise value on top. Divide EV/Sales by EV/EBITDA and EV cancels, leaving EBITDA divided by sales: 2 over 10, which is 20%. Check it with any EV you like: at 1,000, sales are 500 and EBITDA is 100, and 100 is 20% of 500.

Why does the enterprise value not matter?

A cricket bat costs as much as 10 balls, and as much as 2 sets of pads. How many balls is a set of pads worth? Five, and you never needed the price of the bat. When two ratios share the same top line, dividing one by the other cancels it and leaves the ratio of the two bottom lines. Here the shared top line is enterprise value, and the bottom lines are sales and EBITDA.

How do you check the direction of the division?

A margin must be smaller than 100%, and EBITDA is a slice of sales, so EBITDA has to be the smaller number. The higher multiple sits on the smaller number, so the margin is the low multiple over the high multiple, never the other way. Dividing 10 by 2 gives 5, which would be a 500% margin and is impossible. Dividing 2 by 10 gives 0.2.

A buyout investor uses this the other way round all the time. If comparable companies trade at 2x sales and the target earns a 10% margin, then 2x sales is 20x EBITDA for this target, which is expensive. Sales multiples hide margin differences; converting to EBITDA puts them back. The limitation: EBITDA multiples carry their own blind spots, such as heavy capital spending that EBITDA leaves out.

The blended margin falls from 40% to 36%, with no change in either product's economics. Before, 0.5 x 60% + 0.5 x 20% = 40%. After, 0.4 x 60% + 0.6 x 20% = 24% + 12% = 36%. Moving 10 points of the mix from a 60% product to a 20% product costs 10% of the 40-point gap between them, which is 4 points of blended margin.

How can margin fall when nothing got worse?

A cafe earns 70% on coffee and 25% on sandwiches. If more customers come at lunch and order sandwiches, the cafe's overall margin drops even though neither the coffee nor the sandwich got less profitable. A blended margin is a sales-weighted average, so shifting weight towards the lower-margin product pulls the average down by itself. Each 1 point of mix moved costs 1% of the margin gap, here 0.4 points.

Why does a buyout team care, and what should it check next?

When a target's margin falls, the first question is whether the products got worse or the mix moved. A mix-driven margin fall is a different problem from a pricing or cost problem, and it can come with rising profit. If the low-margin product is growing fast, total gross profit may still rise even as the percentage falls: a business at a 36% margin on revenue of 150 earns more than one at 40% on 100.

Say the limitation. Gross margin ignores the overheads each product uses. A low-margin product that needs little selling effort can be more attractive after overheads than it looks here, so ask for contribution by product before judging the mix.

Interest cover falls from 2.44x to 2.0x. The all-in rate is 5% plus 4%, so 9% on 500 is 45 of interest, and 110 / 45 is 2.44x. After a 200 basis point rise the rate is 11%, interest is 55, and 110 / 55 is 2.0x. A two-point rise in the base rate lifts the interest bill by 22% and cuts cash left after interest from 65 to 55.

Why does a two-point rate rise do so much damage?

A household with a floating-rate home loan feels every rate rise in the next EMI, while one with a fixed rate does not. On floating-rate debt the borrower carries the rate risk, so a rise in the base rate goes straight into the interest bill. Two points on 500 is 10 more a year, which is 22% more than the 45 being paid today, with no change in how the business is running.

What does a sponsor do about it?

Most leveraged loans are floating, so sponsors usually hedge part of the debt. An interest rate swap fixes the rate on a portion of the loan, and a cap limits how high it can go in return for an upfront premium. Lenders often require a minimum hedged share in the loan agreement. The trade-off is cost and lost upside: a swap set before rates fall locks the borrower into the higher rate.

Say what a credit analyst would look at next. EBITDA interest cover ignores capital spending and tax, so cash cover is tighter than 2.0x. A cover covenant set at, say, 2.0x would now be right at its limit, which turns a market move into a negotiation with lenders.

The 20% for 5 years makes more money: 2.49x against 1.95x. 1.25 cubed is about 1.95 and 1.2 to the fifth is about 2.49. On a cheque of 100 that is a profit of 149 against 95. The 25% deal is faster, but the money is back after three years, and it only catches up if it can be reinvested at about 13% for the remaining two.

Why does the higher IRR make less money?

A car doing 100 km an hour for three hours covers 300 km; one doing 80 km an hour for five hours covers 400. Speed and distance are different questions. IRR measures how fast money grows, the multiple measures how much money you end with, and a longer hold at a lower speed can end further ahead. Here 1.25 cubed is 1.95 and 1.2 to the fifth is 2.49.

So which would an LP prefer?

It depends on what happens to the money after year three. If the LP can redeploy A's proceeds at more than about 12.9% a year, A ends ahead; below that, B wins. Reinvested at 20%, A reaches 2.81x by year five; parked at 8%, only 2.28x. That is why LPs read IRR and the money multiple together, and why GPs who sell early to protect a high IRR are sometimes accused of leaving money on the table.

Say the limitation. Both deals here are a single cheque in and out. A fund's IRR also depends on when capital is called and returned, and a high IRR on a small, quick deal can flatter a fund that made little money overall.

About 16.5% a year. CAGR is the ratio of end to start, 2.5, raised to one over the number of years, less one. 1.15 to the sixth is about 2.31 and 1.17 to the sixth about 2.57, so the rate sits about three quarters of the way between them, near 16.5%. Dividing the 150% gain by 6 gives 25%, which would compound to 381.

Why is 25% wrong?

A savings account that pays interest on interest grows faster each year, so it needs a lower rate than you might think to reach a target. CAGR is the single rate that, compounded every year, turns the start into the end, so it is the sixth root of 2.5, not 150% divided by six. 25% compounded for six years would reach 381, far past 250.

How do you find a sixth root in your head?

Bracket it with rates whose sixth powers you can build. 1.15 squared is 1.3225, cubed that is about 2.31. 1.17 squared is 1.3689, cubed that is about 2.57. 2.5 sits about 0.74 of the way from 2.31 to 2.57, so the rate is about 15% plus 0.74 of 2 points, near 16.5%. A cross-check with the rule of 72: at 16.5% money doubles in about 4.4 years, and 2.5x in 6 years is a little more than one doubling, which fits.

Say what the number hides. A CAGR smooths the path: a business could have been flat for four years and then jumped, and the CAGR would be the same. A buyout investor asks for the yearly figures before trusting the rate, and checks whether the 250 includes acquisitions.

Two weighings. Put three bars on each pan and three aside. If one pan rises, the light bar is among those three; if they balance, it is among the three aside. Then take that group of three and weigh one bar against one: the pan that rises holds it, and a balance means it is the bar left out. Each weighing has three outcomes, so two weighings tell nine bars apart.

Why split into three groups and not two?

Think of a balance scale as a question with three possible answers: left is lighter, right is lighter, or they balance. Asking a yes or no question wastes one of those answers. A weighing that splits the suspects into three equal groups uses every outcome, so each weighing cuts the suspects to a third rather than a half. Nine bars become three after one weighing and one after two.

How do you prove that one weighing is not enough?

Count the answers a single weighing can give: three. There are nine bars, so nine different possible answers to the question which bar is light. One weighing can distinguish at most three cases, two weighings at most nine, so with nine bars two is both enough and the minimum. The same count gives the general rule: n weighings can find one light bar among up to 3 to the power n bars, so three weighings handle 27.

About 60, released once. Revenue of 730 is 2 a day. At 90 days, customers are holding 180 of unpaid invoices; at 60 days, 120. Collecting 30 days faster brings in the 60 difference as cash, one time. After that, receivables simply stay at the lower level, so the cash flow benefit does not repeat each year.

What does a day of DSO actually hold?

Picture a tailor who lets regular customers pay at the end of the month. On any given day, a month's worth of stitched clothes is out there unpaid, and that money is not in the tailor's drawer. Days sales outstanding counts how many days of sales are sitting with customers, so each day of DSO is one day of revenue held as receivables instead of cash. Here one day is 730 over 365, which is 2.

Why does a buyout fund care that it happens only once?

Because a one-off release must not be valued like a recurring profit. The 60 can pay down debt or fund a dividend once, but it adds nothing to EBITDA and nothing to next year's cash flow. If a seller's numbers show a strong cash year driven by a receivables squeeze, a buyer strips it out before using that year to set the price. And as the company grows, receivables grow with it: 20% more revenue at 60 days lifts receivables to 144, absorbing 24 of cash.

Rs 160 crore, or 16% of commitments. In years one to five the fee is 2% of Rs 1,000 crore, Rs 20 crore a year, Rs 100 crore in all. In years six to ten it is 1.5% of the Rs 800 crore invested, Rs 12 crore a year, Rs 60 crore in all. The early fees are charged on the full commitment, whether or not the money has been invested yet.

Why does the base change halfway through?

Think of a gym that charges full membership from the day you sign, even before you start going, and then a lower fee once you only use part of the facilities. During the investment period, fees are charged on what investors have promised, because the manager is busy finding deals; afterwards they usually drop to what is actually invested, because the job shifts to managing what was bought. The rates and bases differ from fund to fund, so read the agreement rather than assuming.

Why does an investor in the fund care about this number?

Because fees come out of the investors' money before any profit is shared. Rs 160 crore of fees on Rs 1,000 crore of commitments means the investments must earn back 16% before the investors are even whole on what they put in. The early fees bite hardest: if only Rs 200 crore were invested in year one, an illustrative figure, the Rs 20 crore fee would be 10% of the money actually at work. That is why investors care about the fee base as much as the rate.