{"id":54484,"date":"2019-07-31T08:42:18","date_gmt":"2019-07-31T07:42:18","guid":{"rendered":"https:\/\/valuesque.com\/?p=54484"},"modified":"2019-07-31T08:50:24","modified_gmt":"2019-07-31T07:50:24","slug":"judging-steinhoff-by-benfords-law-and-its-big-limits-the-digits-frequency-financial-analysis-problem","status":"publish","type":"post","link":"https:\/\/valuesque.com\/en\/blog\/2019\/07\/31\/judging-steinhoff-by-benfords-law-and-its-big-limits-the-digits-frequency-financial-analysis-problem\/","title":{"rendered":"Judging Steinhoff by Benford\u2019s Law (and its big limits!!) \u2013 The Digits-Frequency Financial Analysis Problem"},"content":{"rendered":"\n

In my recent articles on financial analysis (see https:\/\/valuesque.com\/aktuelles\/<\/a>) I mainly focussed on some fundamental analyses that should show whether companies stretched the discretion that they have in financial reporting too much from an economic point of view, or whether they possibly even stepped over the threshold of what is allowed in financial reporting. Today I want to approach this topic from a purely non-fundamental, a quantitative point of view. I look at an interesting approach which clearly targets at finding accounting fraud by just looking at the stated numbers: Benford\u2019s law!<\/p>\n\n\n\n

Benford\u2019s law \u2013 first clearly described in 1938 and named after the physicist Frank Benford (although detected far earlier, see Newcomb, 1881, Note on the Frequency of Use of Different Digits in Natural Numbers, American Journal of Mathematics) \u2013 describes the patterns and frequency of the first, second, third, etc. digits in numbers that are not bound per se and follow e.g. a natural exponential growth process such as economic or biological data. According to Benford\u2019s law the distribution of the first digits of a large set of such data look as follows.<\/p>\n\n\n\n

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

This means that at the first digit the \u201c1\u201d shows up with a probability of ca. 30%, the \u201c2\u201d with a probability of ca. 18% and so on until \u201c9\u201d shows up at a probability of 4.6%. But why is this the case? Stated in very simple terms: If you start by zero und count up, you get an even distribution of first digits until 9. Then you get the next ten numbers with a first digit \u201c1\u201d (10-19) \u2013 so until 19 a clear domination of the \u201c1\u201d as the first digit. This will then slowly even out with other digits being in the first place \u2013 first with \u201c2\u201d catching up in frequency (20-29), then \u201c3\u201d (30-39) and so on \u2013 until you reach 99. Then again you have a long phase of \u201c1\u201d in the first place (100-199), which again slowly evens out until 999, and so on. So it is always \u201c1\u201d which builds up frequency first, then \u201c2\u201d and so on. Hence, if you pick a random number out of an adequate data set following natural growth patterns, it is by nature much more probable that it starts with a \u201c1\u201d then with e.g. a \u201c2\u201d and so on. And to be clear in mathematical terms, the probability P<\/em> of a certain digit sequence x<\/em> being in the first place of a bigger number should follow the equation:<\/p>\n\n\n\n

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

We can observe Benford\u2019s law in many natural\nphenomena such as rainfall patterns or population distributions. And we can\nalso observe it in financial reporting data. In fact, the numbers in accounting\nfit very well into the Benford framework: no matter what the size of the company\nor the industry they are operating.<\/p>\n\n\n\n

This, however, is only true as long as the\nnumbers follow actual economic processes and are not somehow artificially\nclassified or altered. This again means that we can use Benford\u2019s law as an\nindication of whether accounting numbers are based on actual transactions or whether\nthey are somehow (perhaps fraudulently) made-up. The term \u201cindication\u201d is important\nhere. In fact, due to econometric issues it is not possible to state that the\nconformity with Benford\u2019s law means that the numbers are non-manipulated \u2013 nor\ncan we say that the numbers are manipulated if they are non-conform with\nBenford\u2019s law. But more on this below.<\/p>\n\n\n\n

We apply this approach to Steinhoff, the dual-listed (Germany and South-Africa) furniture company that \u2013 as a consequence of a PriceWaterhouseCoopers investigation in March 2019 \u2013 was convicted of accounting fraud amounting to roughly 7.4 bn USD, sending down the stock price by more than 90%. More specifically, we apply Benford\u2019s law to the disclosed numbers in the 2015-2017 balance sheets of Steinhoff and tried to find out whether these numbers already gave an indication of what later became publicly known. For the analysis below we just took the numbers from Thomson Reuters Eikon and ran our model (a very simple proceeding). These are the results:<\/p>\n\n\n\n

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

In this analysis we can see that that there is\nan obvious mismatch, in particular between the positioning of \u201c1\u201d and \u201c2\u201d. But\nis this enough to state non-conformity with Benford\u2019s law? We applied a test of\n\u2018goodness of fit\u2019 of the two lines \u2013 the actual and the predicted one \u2013 in\norder to find out more on this. The typical test for this is the\nKolmogorov-Smirnov-Test which has the attractiveness that it can be applied to\nsmall numbers of input data (the econometric-afine readers might look over my\nhands-on approach here). Our analysis shows that the patterns are just NOT significant\nat the 10% level (meaning \u2013 roughly speaking – that we still have to state\nconformity at a confidence level of 90% but not for slightly lower levels).\nAdditionally, we ran some more analyses which also just missed the\n10%-significance level \u2013 but were not far away from it \u2013 in every single case.\nSo admittedly, the presentation of the results is not very academic \u2013 but also\nnot far away from reality. We were always just slightly missing the 10%-level.<\/p>\n\n\n\n

So what do we do with the results? Obviously\nBenford\u2019s law here shows no statistical indication but perhaps a very weak economic\nindication of manually dressed-up numbers at Steinhoff. With hindsight, however\n\u2013 as Steinhoff has already committed that it dressed-up the numbers \u2013 this\ndoesn\u2019t tell us a lot. But if you get similar results in analyses of companies\nwhich are not convicted of financial fraud, but are still doing business as\nusual, this does tell you a lot. In business valuation or investing your job is\nnot to follow any results from statistical analyses. But such econometric\nanalyses should definitely serve as a screen to trigger further analyses. We\nusually go deeper into fundamental analyses if we spot weaker\nsignificance-levels than the roughly 10% of Steinhoff. This is exactly how an\nanalyst should use Benford\u2019s law for analysis \u2013 as a trigger for further\nanalyses!<\/p>\n\n\n\n

So Benford-analysis is a good tool to spot indications\nof irregularities? Not necessarily because what I wrote above IS NOT THE END of\nthe Benford-story, not at all! It would be incomplete if we do not set it into\nthe framework of real world developments of these days. And this is certainly an\ninteresting point for fundamental financial statement analysts.<\/p>\n\n\n\n

Let\u2019s look at this first from a more general\nperspective first. Sometimes in finance, effects disappear once they become\ncommon knowledge \u2013 the buzzword is here: \u201carbitrage\u201d. E.g. Rolf Banz discovered\nthe effect that small cap stocks significantly outperform large cap stocks \u2013\nall other things being equal \u2013 in 1981 (Banz, 1981, The relationship between\nreturn and market value of common stocks, Journal of Financial Economics). But once\nhe found out this effect seemed to disappear in the years thereafter and it is\nstill today at a much lower level (and partially better explained by other\nfactors) than in the years before 1981. This development was the consequence of\ninvestors just starting to arbitrage out this anomaly once it became public. So\non the good side, Banz has helped to make markets more efficient. On the bad\nside, he limited the ability to generate outperformance of those investors who\nknew about this effect before but didn\u2019t talk publicly about it. And this is quite\nsimilar to Benford\u2019s law. Now, that it became public knowledge over the years\n(no, sorry, I don\u2019t tell you any secrets here in this article), also dishonest\nCFOs are aware of this rule. And if you were an evil CFO, knowing about\nBenford\u2019s law, how would you make up your numbers? Exactly! You would check the\ndressed-up numbers for any non-conformity. Because otherwise you would get too quickly\ncaught by Benford-afine, semi-quantitative investors!<\/p>\n\n\n\n

Hence from my experience I can tell you that it\nwas easier to spot indications for accounting fraud using Benford\u2019s law in\nearlier days than it is today. E.g. for Enron (the US energy giant that\nspectacularly went into bankruptcy in 2001) the analysis usually shows a high\nsignificance of non-conformity of the numbers with Benford\u2019s law. And Benford also\ndid not stop at corporate accounts in the good old day. There are research\npapers which found non-conformity in public data such as the Greek ones at the\ntime they applied for the Euro-zone (e.g. Rauch et al. (2011), Fact and Fiction\nin EU\u2010Governmental Economic Data, German Economic Review). And in fact: That Greece\nhelped itself manually with the presentation of the numbers at that time is now\ncommon knowledge today. Good job of Benford! But for today\u2019s investment\nsituations? What does Benford tell us? Difficult!<\/p>\n\n\n\n

We do not know whether Steinhoff incorporated\nthe idea of masking Benford as part of their fraud. But Steinhoff did not pop\nup massively in our screens over the years. In fact, if we run an analysis of\nlarge indices and want to detect fraud by Benford\u2019s law today we usually get at\na 10% significance level roughly 10% of the companies which fall below the\nconformity level \u2013 absolutely in line with what our statistical confidence-level\ntells us (at a 5% level we get roughly 5%, also absolutely in line with our\nmodel). Very, very rarely, we find sufficient evidence of fraud simply by\nBenford\u2019s law today (but we find a couple of cases where we go deeper into\nthings and indeed find some problems). So Benford\u2019s law has lost relevance over\nthe years. Too bad! Perhaps it is also just a nice gimmick which is no longer\napplicable in our modern days because too many people know about it.<\/p>\n\n\n\n

But stop! This is also NOT THE END of the\nstory of Benford\u2019s law! There is an interesting twist to it (which will perhaps\nalso disappear soon) which is worth one more important anecdote. Once, when we\nwere in the middle of our analyses and scratched our heads about why we\ncouldn\u2019t detect fraudulent companies as easily as in earlier days by applying\nBenford\u2019s law, we thought about a super-correct, Benford-knowing, but dishonest\nCFO. How would he mask the dressed-up numbers? Certainly not at a probability\nof 10% of non-conformity! He would rather mask away everything \u2013 so were our\nthoughts. As a consequence we simply ran our screen again, now checking for\nsuper-conformity with Benfor\u2019s law, e.g. where all the numbers match the\nprediction at a high degree of confidence. And of course here we also spotted\nsome companies. We continued with fundamental analysis for these candidates and\n(not surprisingly) we found most of the companies being simply caught in our\nscreen by chance. But for one company we exactly found what we were looking\nfor. It turned out that a specific company exactly dressed-up the numbers in a\nway that does not allow a Benford-analyst to spot it. In the end we found\nseveral accounting irregularities in this company with \u2013 at our opinion \u2013 a\nhigh degree of criminal energy of the CFO. As this case is still ongoing, we\ncannot disclose it here. But it is a very nice example of over-fulfilment.<\/p>\n\n\n\n

By the way, criminal investigators have been\nusing this over-positive evidence strategy already for a long time to spot\nsuspicious people. E.g. the German police was looking in the aftermath of the\n9\/11 attacks on the World Trade Center actively \u2013 amongst other criteria \u2013 for\nstudents who actively paid their fee to the German broadcasting TV authority\n(the GEZ). This is a mandatory fee which however can hardly be collected by the\nauthority if one does not pay it deliberately. Their thinking was that usually\nstudents don\u2019t pay it, but if they do so they have something to hide. They had\nquite a success with this strategy and \u2013 helped by this screen \u2013 managed to\nuncover the so called \u2018Hamburg terror cell\u2019.<\/p>\n\n\n\n

So what is the message for investors here? Four\npoints are worth mentioning in a summary.<\/p>\n\n\n\n

  • Always check for Benford\u2019s law if you are suspicious of accounting fraud!\nIt is not too tricky to apply it. Use it as a starting point for further\ninvestigations. We are quite sure that you can still spot some indication of\nirregularities with this tool (and we do it as well).<\/li>
  • Don\u2019t put too much evidence into the findings. They are only a starting\npoint of further analyses. Times have changed over the last years. CFOs also\nknow about this law and if they are dishonest and know it, they will be able to\nmask it.<\/li>
  • Also look for super-conformers. This is not rarely a similar indication\nof something going wrong as the non-conformity.<\/li>
  • And finally, the story of Benford\u2019s law is even NOT NOW AT THE END, neither.\nThere are more patterns we look for when trying to detect fraud or aggressive\naccounting. What is it? I cannot tell you here. Otherwise the story would\nunfold similar as the simple Benford-story described in this article.<\/li><\/ul>\n","protected":false},"excerpt":{"rendered":"

    In my recent articles on financial analysis (see https:\/\/valuesque.com\/aktuelles\/) I mainly focussed on some fundamental analyses that should show whether […]<\/p>\n","protected":false},"author":4,"featured_media":54480,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-54484","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-allgemein"],"yoast_head":"\nJudging Steinhoff by Benford\u2019s Law (and its big limits!!) \u2013 The Digits-Frequency Financial Analysis Problem - VALUESQUE EN<\/title>\n<meta name=\"robots\" content=\"noindex, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Judging Steinhoff by Benford\u2019s Law (and its big limits!!) \u2013 The Digits-Frequency Financial Analysis Problem - 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