Private Equity puzzles, solved step by step

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.