Salil Mehta
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Salil Mehta

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## Statistics Topics [View instapost]

Meanwhile I have put together some interesting peer-reviewed research (liquidity risk for the Society of Actuaries, and Federal Reserve forecasting statistics for a leading academic review- this latter one is a topic you seem to be into), and a Kids' Statistics internet-playground, both of which have taken up considerable time recently.

P.S.- BRK shares have now fallen back to below the S&P YTD, on market value basis. Where's that massive outperformance some wrong-headed critics/believers said is returning? ;)

http://cnb.cx/1lDHybP

http://bit.ly/1lDHybQ

## Statistics Topics [View instapost]

## Being Good, Until One's Not [View article]

## Forever Elusive Alpha [View article]

You might like also just some raw BRK data, which you can see on this link from the Statistical Ideas web portal:

http://bit.ly/1fcCvyl

There are no statistics nor econometric analysis on the link immediately above. Just raw data, which one can see and download. Also note that the data is illustrated against the S&P per the initial pages of the annual report. If one were instead to use other equity benchmark indices, then for some periods of time they would show marginally different results. The bottom line is that BRK's performance has certainly been slipping from the lofty levels (by a statistically large amount that can not be explained by luck), and not just over the past five years, but for much of this millennium.

It wouldn't be difficult to find that research as well.

## Being Good, Until One's Not [View article]

This weekend it was one of the most read and shared research in the news.

## Being Good, Until One's Not [View article]

## Being Good, Until One's Not [View article]

## Being Good, Until One's Not [View article]

Now to just see an extreme illustration of how the requirement to demonstrate skill rises as the number of trials is reduced, see this illustration at the bottom of this link (http://bit.ly/1kuXKcJ) concerning a fair coin flip (so now completely unhinged from empirical stock market analysis). We see the probability of flipping this 50/50 coin consistently with one outcome (e.g., heads) is inversely proportional to the number of chained attempts (and greatly reduces well under 1/2% if done over 50 times). Notice the small jaggedness in the red line, due to the discrete approximation error with the number of coin flips.

## Being Good, Until One's Not [View article]

It's corollary is: Buffett's underperformance was 10 of 48 years, or 21%. They force us to use these summary bullets, which is annoying for you as it is for me.

A 21% underperformance rate is good. What's stated is that it is expected that he will experience continued underperformance rate at this same (good) 21% rate in the future.

Your final comment piques my interest since I have studied Tufte's work and recommend it to my students as some, among many, best-practice examples. I find often that people either really find my charts great and clear, or they tend to have different stylistic tastes (which is generally more personal and not a reflection of the original quality). I do think that often people should still be able to learn the material, even if presented in a less preferable way for their own interests (while Tufte is great, no one person's style works best for everyone). I take feedback like this seriously through, so please let me know exactly what you liked or would see differently.

Send it as a public response, or shoot me a personal e-mail.

## Being Good, Until One's Not [View article]

The S&P is the benchmark, not momentum stocks. And rising markets (S&P 500) does not excuse underperformance (using 1995-1999, or 1984-1988 as examples). Any astute analyst should be able to connect the dots and not use Buffett's underperformance as a contra-indicator for relative performance. That's confounding and below the level expected of Seeking Alpha.

If someone has issues with the S&P 500 (or some other sector of market) being frothy, then just sell the market short. Buffett goes down if the market crashes. So does your portfolio. Else this aggressive idea that people are magically owed good performance since the S&P 500 has had it good is getting tiring.

## Being Good, Until One's Not [View article]

"Still, if statistical mean-reversion has anything to say about market patterns, then Buffett should experience continued underperformance at the same good rate in the future."

Even Michael Jordan misses some shots at the hoop, just at a good (lower) rate, and every once in a small while can have a small negative streak. Doubt that when a lengthy career shows sign of fatigue, (A) whether you can pick that up, (B) if so, know what that can portend about the future, and (C) understand that there is a rich battery of probability and statistics modeling that can be explored on skill versus luck risk decomposition, which goes beyond any one senior investing legend and his cult of followers.

## Forever Elusive Alpha [View article]

http://seekingalpha.co...

## Being Good, Until One's Not [View article]

So now the chance of seeing the 5 year "abberative" streak, of at least 4 losses in 5 years (either at the start or the conclusion of the 48-year pattern) would come to roughly 3%. This can be solved at least two general ways, using both a combinatorial mathematics estimate and another with a difference in means estimate (or a t-distribution test). The bottom line is that we can best explain the pattern with an unfair coin analogy, and even then it is rare to see such a dichotomous drop-off at one end to be simply a normal (non-significant) continuation of the other 43-trial period.

There are also multiple other ways to look at this, and some are not so meaningful. What would lose statistical credibility, for example, is if one were to go above and beyond these paragraphs, and suggesting there is one pattern completely "flanked" by a different pattern.

## Being Good, Until One's Not [View article]

Still, mixing now over 6 years, as you suggest, you get a monthly overperformance of 0.3% (continuous rate). Less than the 0.8% that show outperformance by skill given the stock dispersion opportunities that were offered over this time! A haphazard investor is more likely to achieve a 0.3% outperformance over 6 years, then would have occured by a skilled investor.

## Being Good, Until One's Not [View article]

I did mention in a couple of my blog notes that Buffett's overall outperformance is statistically significant, and he's of course in a small fraction of people in his decade of investment experience to get there since the odds stiffen over the long haul. To see this up close, see the levels at 47 years of the [“The Forever Elusive α” (Statistical Ideas)] chart on this link: http://cfa.is/QCORoW

For fun, and this deviates from the probabilistic pattern I show above that has deteriorated, here is the probability of getting at least 38 heads out of 48 flips:

Unfair coin weighted to land on heads 25% of the time: 0% (0.0 even in bps)

Fair coin weighted to land on heads 50% of the time: 0% (0.3 in bps to be precise)

Unfair coin weighted to land on heads 2/3 of the time: 4%

Unfair coin weighted to land on heads 75% of the time: 32%