Warren Buffet and the “Fundamental Law”

Posted on January 15, 2011 by



“Wide diversification is only required when investors do not understand what they are doing.”  — Warren Buffet

Bufffet’s quote seems to contradict quantitative investment approaches: Quant funds often diversify across hundreds or thousands of positions.  Do quant funds know what they are doing?

BRK-A versus SPY since 1990

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Asset Allocation

You already know Warren Buffet is an effective fund manager, perhaps the best ever (see figure).  What might we learn from his investing style?  We can learn a lot of course, but I want to focus here on his allocation strategy: How does he apportion Berkshire Hathaway’s (BRK-A & BRK-B) assets across the various equities they hold?

As of September 2010, 54% of Buffet’s holdings were in just 3 stocks.

Buffet invests strongly in a small number of companies: As of September 2010, 54% of BRK-A’s holdings were in just 3 stocks: Coca Cola (KO), American Express (AXP), and Wells Fargo (WFC).  90% of their holdings were in just 12 stocks.  You can find more detail here.

Let’s compare Buffet’s allocation with Renaissance Technologies’ Medallion Fund.  Renaissance Technologies (RenTec) is perhaps the most successful quantitative hedge fund management firm, with consistent annual returns of 35% or more for their Medallion Fund each year.  In contrast to BRK, RenTec’s portfolio is distributed more or less uniformly across hundreds of positions.

So we have two very different but successful allocation strategies. They are at either end of a spectrum: One with just a few holdings (BRK-A) and another with hundreds (RenTec). You may be interested to learn about an investment theory that explains how they can both be successful.

The Fundamental Law of Active Investing

In the 1980s Richard Grinold introduced what he calls the Fundamental Law of Active Investing.  It is described nicely in his book with Ronald Kahn.  I paraphrase his law as:

performance = skill * √breadth

Skill is a measure of how well a manager transforms information about an equity into an accurate prediction of future return, and breadth represents the number of investment decisions (trades) the manager makes each year.

This law suggests that as a manager’s skill increases, returns increase. Of course!   That is not surprising.  What is interesting and perhaps surprising, is that to double performance at the same level of skill, a manager must find four times as many opportunities to trade.

The law also implies that Buffet could improve his performance significantly by expanding his portfolio.  Why doesn’t he do that?  We can only speculate, but it is likely because his skill does not scale.  Not only that, he is probably aware of the diminishing returns suggested by the Fundamental Law: If he worked twice as hard by looking at more companies, returns would only improve by 41%.  Buffet spends a great deal of time understanding a small number of companies deeply.  His attention and depth enables him to make accurate predictions for the companies he thinks about.  But he can’t apply this depth of attention to all of the 4,000 equities traded on the New York Stock Exchange.

On the other hand, RenTec’s  predictive power is not as strong as Buffet’s, but it is scalable.  They use computerized techniques to assess thousands of equities in a more shallow way.  But they do it rapidly, perhaps hundreds of times a second.  The accuracy of each of their predictions is not as high as Buffet’s, but because they are able to apply a modest level of skill to so many equities their breadth substantially outpaces his.

What does it mean for you?

The lesson is that if you are able to understand equities deeply, you can afford to hold fewer positions.  But if your skill or strategy is scalable you can benefit substantially with diversification.

Something I didn’t explicitly mention above is that at low skill levels decreased breadth becomes dangerous.  In other words, if you’re going to put 54% of your portfolio in only 3 stocks, you better know what you’re doing.  Let me suggest a corollary to Buffet’s quote:

If you’re less skilled than Buffet, be sure you are widely diversified.

I don’t think Mr. Buffet would disagree.

A little bit of math for those who are interested

Let’s consider how the Fundamental Law might relate the strategies of Buffet and RenTec.  For simplicity I’ll assume they both return 20% per year, and I’ll also assume Buffet makes about 120 investment decisions (trades) per year.  For Buffet we have

20% = skill * √120


skill = 20% / √120

skill = 1.825

Suppose RenTec’s skill were about 1/10th of Buffet’s.  How many trading opportunities would they need to find in order to equal the 20% return? Turning the equation around, we have:

breadth = (performance / skill)^2

breadth = (20%/.1825)^2

breadth = 12,009

In other words RenTec has to find about 100 times more trading opportunities if their skill on each trade is 1/10th Buffet’s.  To be honest, 1/10th is generous.  It is more likely that each of RenTec’s predictions is about 1/1000th as accurate as each of Buffet’s.  They probably make many more than 120,000,000 trades a year.