Private Equity puzzles, solved step by step

391, about 1.26, 87.5% and 162. 17 x 23 is (20 - 3)(20 + 3), so 400 - 9. 1.08 cubed is about 1 + 3 x 0.08 + 3 x 0.0064, which is 1.259, close to the exact 1.2597. Seven eighths is one less one eighth, 100% - 12.5%. And 45% of 360 is half of 360 less a twentieth of it, 180 - 18.

Why name the trick rather than just give the number?

A shopkeeper who adds a bill in his head is not calculating faster than you; he has a handful of shortcuts he has used ten thousand times. Speed in a quick-fire round comes from recognising which shortcut a question is built for, so say the shortcut as you give the answer. It shows the interviewer the answer is reliable, and it protects you when a number comes out slightly off: the method is audible even if the last digit slips.

How do the two harder ones work?

For 1.08 cubed, expand (1 + x) cubed as 1 + 3x + 3x squared + x cubed with x = 0.08. When x is small, the first two terms carry almost everything and the third is a small correction: 1 + 0.24 + 0.0192 is 1.2592, within 0.001 of the exact 1.2597. That is three years of 8% growth, about 26%, which is why the trick earns its place on a returns desk. For 17 x 23, check the trick fits: it only works when the two numbers sit the same distance either side of a round number.

Fractions and percentages are best memorised in eighths: 12.5%, 25%, 37.5% and so on up to 87.5%. Percentages of awkward numbers split into easy pieces: 10%, 5%, 50%. With those four habits, most of what turns up in a quick-fire round is one step.

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.

DPI is 0.75x, RVPI is 0.875x and TVPI is 1.625x; only DPI is cash. All three divide by the 800 paid in. DPI counts the 600 distributed, RVPI the 700 still held at estimated value, and TVPI adds them, 1,300 over 800. On paper the fund is well ahead, but in cash the investor has 600 back on 800 and is still short of break-even.

What does each ratio measure?

Imagine lending a friend Rs 800 to start a business. She has paid you back Rs 600 and says your share of the shop is worth Rs 700 more. Rs 600 is in your wallet; Rs 700 is her estimate. DPI counts the cash back, RVPI counts the estimated value still held, and TVPI adds the two, all measured against the money paid in. In fund language: distributions to paid-in, residual value to paid-in, total value to paid-in.

Why does the difference between DPI and TVPI matter so much?

Because NAV is a valuation made by the manager, and it only becomes cash when the holdings are sold. A fund with a high TVPI and a low DPI is telling you it is worth a lot on paper, and investors have learnt to ask how much of that has actually come home. Here more than half the reported value, 700 of 1,300, is still an estimate. A buyer of this fund stake on the secondary market would price that 700 at whatever discount it thinks the estimate deserves, which is exactly why the split matters.

The ball costs Rs 5 and the bat Rs 105. Call the ball x. The bat is x plus 100, so together they are 2x plus 100, which must equal 110. That makes 2x equal to 10 and x equal to 5. The fast answer of Rs 10 fails the check: a Rs 100 bat is only Rs 90 more than a Rs 10 ball.

Why does Rs 10 jump out, and why is it wrong?

The numbers are built so that 110 minus 100 hands you 10 without thinking. That subtraction answers a different question: what is left if the bat costs exactly 100. The condition is a difference, the bat is 100 more than the ball, not a price, so the bat must contain a whole ball's worth plus 100. Check Rs 10 against the condition and it fails at once: 100 minus 10 is 90.

What is the interviewer actually testing?

Not algebra. They are testing whether you check a fast answer against the conditions before you say it. A deal model full of quick numbers has the same risk: a cell that looks right because it is round, never tested against the constraint it was meant to meet. Writing one line, x plus x plus 100 equals 110, takes three seconds and removes the risk.

Say the answer, then the check in the same breath: ball 5, bat 105, difference 100, total 110. Interviewers often use this as a warm-up and judge you more on the check than on the number.

About 8 years by the rule of 72, and 8.04 years exactly. Divide 72 by the rate in per cent: 72 over 9 is 8. The exact answer is the log of 2 over the log of 1.09. The rule is near exact around 8% a year and drifts at very low or very high rates, where 69 or 70 works better.

Why does dividing 72 by the rate work?

Doubling needs the growth factor to reach 2, and the log of 2 is about 0.693. For small rates, the log of 1 plus r is close to r, so doubling time is about 69.3 divided by the rate in per cent. 72 is used instead of 69 because it divides cleanly by 2, 3, 4, 6, 8, 9 and 12, and because it corrects for the approximation at the rates people use most. It is a calculator in your head, like knowing that a dozen eggs at Rs 6 each is Rs 72.

Where does a buyout interviewer use this?

Everywhere returns are quoted. A deal that doubles the money in about four years has an IRR near 18%, and one that doubles in about three years is near 24%, because 72 over 4 is 18 and 72 over 3 is 24. That lets you check a quoted IRR against a quoted money multiple in seconds. At 24% the rule says 3 years and the truth is about 3.2, so for high-return deals you shade the rule slightly.

EBITDA is 100. Net income of 40 at 15x gives equity value of 600. Add net debt of 200 to reach enterprise value of 800. The EV/EBITDA multiple of 8x then gives EBITDA of 800 divided by 8, which is 100. Two multiples and the bridge between equity and enterprise value pin down the missing line.

Why can you not go straight from net income to EBITDA?

Net income sits below interest, tax, depreciation and amortisation, and the question gives none of them. The two multiples work on different values, P/E on equity and EV/EBITDA on the whole business, so the route runs through value, not through the income statement. It is like knowing a flat's price per square foot and the loan on it: you get to the total value first and then back out what you need.

What follow-up can you get ahead of?

Interviewers often ask what the gap between EBITDA of 100 and net income of 40 is made of. Sixty of EBITDA goes on depreciation, interest and tax, and with net debt of 200 interest is only a modest slice, so depreciation or tax must be large. At an assumed 8% rate, interest would be 16, leaving 44 for depreciation and tax. Saying that shows you read the result, not only compute it.

State one assumption as you go: net debt here is all the claims that sit between equity and enterprise value. If there were minority interests or preference shares, they would be added too and EBITDA would come out higher.

994. Treat 12.5% as one eighth and halve 1,368 three times: 684, 342, 171. For a third of 2,469, split it into 2,400 and 69: a third of each is 800 and 23, so 823. Then 171 plus 823 is 994. Common percentages are fractions in disguise, and fractions are faster in your head.

Why turn 12.5% into a fraction?

Multiplying by 0.125 in your head means three digits of decimals to keep track of. Halving is something you have done since school, like splitting a restaurant bill between two, then four, then eight friends. 12.5% is exactly one eighth, so three halvings give the answer with no decimals at all. 1,368 halves to 684, then 342, then 171.

How do you divide by three cleanly?

Split the number into a part that divides easily and a small remainder. 2,469 is 2,400 plus 69, and a third of each is 800 and 23, so a third of the whole is 823 with no long division. Check it: 823 times 3 is 2,469. The same trick works for any divisor: pick the nearest round multiple, then handle the leftover.

Buyout interviews use questions like this as a warm-up, often before a paper LBO, to see whether you will cope with arithmetic out loud. Saying each step lets the interviewer follow you and makes a slip easy to catch.

About 352.5 is owed, about 32.5 more than a cash-pay note's 320. PIK interest is added to the balance each year: 224, 250.9, 281.0, 314.7, then 352.5. A cash-pay note returns 24 a year plus 200 at the end, 320 in all. The extra 32.5 is interest on interest, which is why a PIK claim grows faster than its coupon suggests.

What does paid in kind actually mean for the balance?

A PIKPayment in kind: interest that is not paid in cash but added to the amount owed, so the loan balance grows instead. note pays its interest with more debt. It is like a credit card where you pay nothing and the interest is added to the bill each month, so next month's interest is charged on the bigger bill. Because each year's interest joins the balance, the 12% is earned on a growing base, and the claim compounds. After one year the balance is 224, after two 250.9.

Why does a private credit lender or a sponsor care?

The borrower keeps its cash today, which helps a company that is growing or stretched. The price is a claim that grows every year, so at exit the sponsor's equity sits behind a bigger debt than the original 200. For the lender, PIK means more risk: nothing is received for five years, and if the company fails the larger balance may not be recovered. Lenders usually charge a higher rate for PIK than for the same loan paid in cash.