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.

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.