{"id":104,"date":"2019-08-26T12:32:00","date_gmt":"2019-08-26T12:32:00","guid":{"rendered":"http:\/\/www.guythomas.org.uk\/blog\/?p=104"},"modified":"2025-07-09T09:10:42","modified_gmt":"2025-07-09T09:10:42","slug":"fractional-shares-and-dodgy-pie-charts","status":"publish","type":"post","link":"https:\/\/www.guythomas.org.uk\/blog\/fractional-shares-and-dodgy-pie-charts\/","title":{"rendered":"Fractional shares and dodgy pie charts"},"content":{"rendered":"\n

TLDR:<\/strong> When fractional shares (revenue by product, market shares by company, or similar quantities) are ranked and grouped in buckets, any ratio of successive bucket means<\/em> >0.5 is suspect.<\/p>\n\n\n\n

————–<\/p>\n\n\n\n

<\/p>\n\n\n\n

Listening to this interview<\/a> with John Hempton<\/a> reminded me of his forensic scepticism about the following pie chart, taken from an investor presentation about the pharma company Valeant.<\/p>\n\n\n\n

<\/p>\n\n\n

\n
\"\"<\/figure>\n<\/div>\n\n\n
\"\"<\/figure>\n\n\n\n

John wrote<\/a> (in 2014, when Valeant was still ostensibly a very successful company) that the chart looked implausible<\/p>\n\n\n

\u201cWhat we are saying is the top ten products average 1.8 percent of sales each. The next ten average 1.2 percent.<\/em><\/p>\n

Now lets just – for the sake of argument suggest that the top product is 3 percent of sales. And the next four average 2.3 percent. Well then the first five are 12.2 percent of sales – and the next five can only be 5.8 percent of sales (or they average 1.16 percent of sales).<\/em><\/p>\n

But oops, that is too much concentration – because we know the next ten average 1.2 percent of sales. In fact it is far\u00a0too much concentration as product number 11 needs to be more than 1.2 percent of sales.<\/em><\/p>\n

I have fiddled with these numbers and they imply a distribution of sales flatter than I have ever seen in any product or category. In order to make the numbers work the differences between product sales have to be trivial all through the first fifteen products.\u201d<\/em><\/p>\n\n\n

 This is an acute observation to make just skimming a presentation, and I\u2019m not sure I would have got it. And what if the buckets (sectors on the pie chart) each contained 3 or 5 items instead of 10 \u2013 what difference would that make? Can we generalise this?<\/p>\n\n\n\n

One way to think about it is to assume revenue shares by product form a geometric progression with a constant scaling factor, say r<\/em>. E.g. if first product has 5% share, the second has 5 x 0.8 = 4% share, then the third has 4% x 0.8 = 3.2% share, etc. This won\u2019t be exact in reality, but the sorting by size means it can\u2019t be far off.<\/p>\n\n\n\n

<\/p>\n\n\n\n

Then if the first term of the geometric progression \u2013 i.e. the share of the largest product \u2013 is x<\/em>1<\/sub>, the usual formula for the sum of the first 10 terms is<\/p>\n\n\n

\n
\"\"<\/figure>\n<\/div>\n\n\n

and the mean over the first bucket is just this divided by ten.<\/p>\n\n\n\n

We want to find {mean over bucket of first 10 \/ mean over bucket of next 10}.  Replacing 10 by n<\/em> for generality, some trivial algebra shows that this ratio of successive bucket means<\/em> is given by<\/p>\n\n\n

\n
\"\"<\/figure>\n<\/div>\n\n\n

We can display this quantity in a 2-way look-up table against the scaling factor, r,<\/em> and the number of items in each bucket, n:<\/em><\/p>\n\n\n

\n
\"\"<\/figure>\n<\/div>\n\n\n

For the original Valeant example, n<\/em> = 10, and the observed ratio of successive bucket means is 1.2% \/ 1.8% \u2248 0.66. So reading off the top number in the third column, this implies a scaling factor of around 0.96, which seems incredibly flat. Hence John\u2019s intuition that the chart looked dodgy.<\/p>\n\n\n\n

More generally, what\u2019s \u201cdodgy\u201d depends on what the true scaling factor is.\u00a0 E.g. If the largest holdings in an investor\u2019s portfolio are sorted by size, this might typically give a scaling factor of 0.8 or less; but not if the investor constantly rebalances the portfolio to equal weights (which some people say is a good idea, although the maths is\u00a0subtle<\/a>\u00a0and a lot of published work on \u201cvolatility pumping\u201d is flawed). \u00a0<\/p>\n\n\n\n

For most quantities, my intuition is that a scaling factor as high as 0.9 (the third row of the table) would be rare. It\u2019s also striking how quickly the values fall away in the lower part of the table, i.e. for lower (more typical?) scaling factors. <\/p>\n\n\n\n

Overall, the following seems a reasonable rule of thumb:<\/p>\n\n\n\n

When fractional shares (revenue by product, market shares by company, or similar quantities) are sorted by size and grouped into descending buckets, any ratio of successive bucket means<\/em> above 0.5 (or perhaps even 0.3) is suspect.<\/strong><\/p>\n","protected":false},"excerpt":{"rendered":"

TLDR: When fractional shares (revenue by product, market shares by company, or similar quantities) are ranked and grouped in buckets, any ratio of successive bucket means >0.5 is suspect. ————– Listening to this interview with John Hempton reminded me of his forensic scepticism about the following pie chart, taken from an investor presentation about the pharma company Valeant. John wrote (in 2014, when … Read more<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[5],"tags":[],"class_list":["post-104","post","type-post","status-publish","format-standard","hentry","category-investment"],"_links":{"self":[{"href":"https:\/\/www.guythomas.org.uk\/blog\/wp-json\/wp\/v2\/posts\/104","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.guythomas.org.uk\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.guythomas.org.uk\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.guythomas.org.uk\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.guythomas.org.uk\/blog\/wp-json\/wp\/v2\/comments?post=104"}],"version-history":[{"count":18,"href":"https:\/\/www.guythomas.org.uk\/blog\/wp-json\/wp\/v2\/posts\/104\/revisions"}],"predecessor-version":[{"id":646,"href":"https:\/\/www.guythomas.org.uk\/blog\/wp-json\/wp\/v2\/posts\/104\/revisions\/646"}],"wp:attachment":[{"href":"https:\/\/www.guythomas.org.uk\/blog\/wp-json\/wp\/v2\/media?parent=104"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.guythomas.org.uk\/blog\/wp-json\/wp\/v2\/categories?post=104"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.guythomas.org.uk\/blog\/wp-json\/wp\/v2\/tags?post=104"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}