Equity Research puzzles, solved step by step

The revenue mix shifted towards the lower-margin segment. Say segment A earns 30% on 80 of revenue and B earns 10% on 20: the group earns 26%. Next year A earns 32% on 40 and B earns 12% on 60. Both improved by 2 points, but the group earns 20 on 100, 20%, because most revenue now sits in the 12% business.

How can an average fall when every part of it rises?

Think of a class where both the science and arts sections improve their average marks, yet the school's overall average falls, because far more students joined the lower-scoring section this year. An overall average depends on the weights as well as the parts, so a big enough shift in weights can reverse the direction of the average. Statisticians call this Simpson's paradoxThe pattern where a trend that holds in every group reverses when the groups are combined, because the group sizes changed..

How do you show the numbers work?

Check the group line each year. Year 1: 80 x 30% plus 20 x 10% is 24 plus 2, or 26 on 100. Year 2: 40 x 32% plus 60 x 12% is 12.8 plus 7.2, or 20 on 100. The whole fall comes from moving 40 of revenue out of a 30% business and into a 10% one, which costs 8 points of group margin; the segment improvements claw back only 2.

In a research note this is the difference between a margin miss that signals trouble and one that simply reflects mix. Split the change into a mix effect and a rate effect before judging management. The limitation: the split depends on which year's weights you use, so state the convention.

About 9.6% a year, not 14%. The 14% average only covers funds that survived, and funds usually close because they did badly. Weight both groups by their count: 60 funds at 14% and 40 at 3% gives 0.6 x 14% plus 0.4 x 3%, or 9.6%. A database that drops closed funds overstates the average investor's experience by 4.4 points a year.

Why does the survivor average mislead?

Ask the toppers of a coaching class how hard the entrance exam was, and they will say it was manageable. The students who failed have gone home and nobody asked them. Survivorship bias is judging a group by the members still standing, when the ones that dropped out left because their results were poor. A fund database that lists only live funds is the coaching class that only asks its toppers.

How big is the gap once it compounds?

Over ten years, 14% a year turns Rs 100 into Rs 371, while 9.6% turns it into Rs 250. A gap of 4.4 points a year looks modest, but it is the difference between multiplying money 3.7 times and 2.5 times. That is why performance studies that ignore closed and merged funds tend to make active management look better than it was.

Say the limitation too. Averaging annual returns across funds of different lives is a simplification; a careful study would weight by assets and by years in existence. The direction of the bias does not change, though: leaving out the losers always flatters the average.

About 19%. Start from the sector's 10%, the base rate, and keep only the part of the outperformance that tends to persist. The excess was 40% - 10% = 30 points, and a coefficient of 0.3 keeps 9 of them, so the forecast is 10% + 0.3 x 30 = 19%. Extreme results are partly luck, and luck does not repeat, so the forecast sits much nearer the average than last year's number.

Why not forecast 40% again?

A student who scores 95 in one mock exam, when the class averages 60, is probably both able and lucky that day. On the next mock she is likely to beat the class again, but by less, because the luck part does not come back. An extreme result is part skill and part luck, and only the skill carries forward, so the best forecast pulls the result back towards the average. The persistence coefficient says how much pulling to do: 0.3 means 30% of the gap survives and 70% fades.

Where does the 0.3 come from, and what if you are not given it?

It is an empirical fact about a set of companies, not a law. You would estimate it by comparing one year's growth with the next across many companies in your universe over several years. Without data, the question to ask is how much of last year's jump came from something that recurs, such as a new plant or share won from rivals, and how much from something that does not, such as a one-off order, a weak base year or a tax credit. A jump built on recurring drivers deserves a higher coefficient.

How does this show up on a research desk?

Forecasts for last year's winners are often extrapolations with a haircut too small. A stock priced as if 40% growth continues is exposed when growth drifts back towards the sector, even if the company stays better than average. Say that link after the number: it shows you understand why the interviewer framed the question around a forecast rather than a fact.

27 reports. Find the unit rate first: 3 analysts writing 3 reports in 3 days means one analyst writes one report in 3 days, a third of a report per analyst-day. Nine analysts working nine days put in 81 analyst-days, and 81 x 1/3 = 27. Tripling the team triples the output and tripling the time triples it again: 3 x 3 x 3 = 27.

Why does the fast answer of 9 feel right?

The sentence has a rhythm, three, three, three, then nine and nine, and the ear wants to finish the pattern with another nine. It is the same pull as a kitchen puzzle where three cooks bake three cakes in three hours. Matching the pattern of the words is the trap; the fix is to reduce the problem to one worker and one day before scaling anything. Once you have the rate per analyst-day, the answer is multiplication.

How do you do it in ten seconds?

Scale each dimension on its own. Three times the analysts gives three times the reports, and three times the days gives three times again, so the answer is 3 reports x 3 x 3 = 27. The general rule is output equals the rate per worker per day, times the workers, times the days, and it handles any version of the item, including the ones where the numbers do not match so neatly.

What is a timed test like this really checking?

Wonderlic-style tests put many short items against a tight clock, so the difficulty is the time, not any single item. The format, number of items and time allowed vary by version, so confirm what the firm uses. The habit that scores on the test is the same one that keeps a model right: write the unit rate, then scale it. On a timed test, skipping an item you cannot set up in a few seconds and coming back to it is usually worth more than grinding.

About 1.54 standard deviations too high. Each forecast is right on average, but choosing the highest selects for the biggest positive error. The expected maximum of ten independent draws from a normal distribution is 1.54 standard deviations above the mean. If each analyst's error has a spread of Rs 2 around a true EPS of Rs 50, the top forecast averages Rs 53.08.

Why does choosing the highest create bias when nobody is biased?

Weigh yourself on ten bathroom scales, each accurate on average but each a little off, and write down only the heaviest reading. You will always look heavier than you are. The bias is not in any one estimate; it comes from the choosing, because the maximum picks whichever estimate had the largest positive error. Auction theory calls the same effect the winner's curseThe tendency for the winning bid in an auction to overestimate the value of the prize, because the highest estimate of many noisy ones is usually too high.: the bidder with the highest estimate wins, and usually overpaid.

How do you get to 1.5 without tables?

Ask where the maximum has an even chance of landing. All ten estimates must fall below a level for the maximum to fall below it, so you need the level where the chance for one estimate, raised to the tenth power, is one half. That single-estimate chance is 0.5 to the power one tenth, about 0.933, which a normal table puts at 1.50 standard deviations. The mean of the maximum is a touch higher, 1.54, because the distribution of the maximum has a longer right tail. Either number is a fine answer in the room if you show the route.

Where does this bite an analyst?

Anywhere the top of a list is chosen after the fact. The most bullish forecast in the consensus, the best of ten back-tested strategies and the top-ranked fund of the year all carry a selection premium that will not repeat. The effect grows with the list: the highest of five is 1.16 standard deviations high, of twenty about 1.87. The fix is to shrink the chosen number back towards the average in proportion to how noisy the estimates are. The limit of this answer is the independence assumption: analysts who talk to the same management team share errors, and correlated errors shrink the gap.

The average of the three P/Es is 43.3x, but the sector trades at 18.75x. The P/Es are 10x, 100x and 20x, and their simple average is 43.3x. The sector is worth 300 in total and earns 16 in total, so its P/E is 300 over 16, 18.75x. The average is pulled up by the company earning only 1, which says more about that company's depressed profit than about the sector's valuation.

Why does one company dominate the average?

Think of three friends who each spend Rs 100 on lunch, one buying ten samosas, one buying one fancy sandwich and one buying five. Average the price per item and the sandwich makes lunch look absurdly expensive; divide total spend by total items and you get the real average price. A ratio with a small denominator explodes, and a simple average of ratios gives that explosion full weight, however little of the sector's earnings it represents. Company B earns 1 of the sector's 16 but contributes 100 of the 130 P/E points summed.

What is the right way to get a sector multiple?

Sum first, then divide. The sector P/E is total market value over total earnings, 300 over 16, 18.75x, which is what you would pay for a slice of the whole sector's profit. An equivalent route is to average the earnings yields, 10%, 1% and 5%, which gives 5.33%, and turn that upside down: 18.75x. Earnings yields do not explode as profit falls, so averaging them is safe when the companies are the same size.

When is the median or the average still useful?

When you want a typical company rather than the sector as a whole. The median, 20x, ignores the outlier and is a fair description of a normal company in the group. But when you compare a stock with its sector, or value a basket, you need the aggregate multiple, because that is the price of the sector's earnings. In practice, the fix for a comps table is to drop or flag companies with depressed or negative earnings, whose P/Es mean little, rather than let them skew the answer.