Here's how an optimized version of Berkshire Hathaway's (BRK.A,BRK.B) portfolio, consisting of 15 stocks with the longest history, grew over the years: *(Click to enlarge)*

Let's first build the backdrop to this from an optimization perspective. In my last article, we looked at Berkshire's latest release on its holdings. The stocks listed were held as of December 30, 2011:

**The holdings that did not change from the previous quarter were as follows:**

American Express Co. (AXP); Bank of New York Mellon Corp (BK); Comdisco Holdings (OTCQB:CDCO.PK); Coca Cola Co. (KO); Costco Wholesale (COST); ConocoPhillips (COP); Gannett Co. (GCI); Dollar General Corporation (DG); General Electric Corp (GE); GlaxoSmithKline (GSK); Ingersoll-Rand (IR); M&T Bank Corporation (MTB); Mastercard (MA); Moody's Corp (MCO); Procter & Gamble (PG); Sanofi-Aventis (SNY); Torchmark Corp (TMK); US Bancorp (USB); USG Corp (USG); United Parcel Service (UPS); Wal-Mart Stores (WMT); Washington Post (WPO).

**The holdings with larger positions were as follows:**

CVS Caremark Corp (CVS); DirecTV (DTV); General Dynamics Corp (GD); Intel Corp (INTC); Visa (V); Wells Fargo & Co. (WFC).

**The new holdings were as follows:**

Da Vita (DVA); International Business Machines Corporation (IBM); Liberty Media Corp (LMCA).

**The holdings with lower positions were as follows:**

Johnson & Johnson (JNJ); Kraft Foods (KFT); Verisk Analytics (VRSK).

**The holdings that were eliminated from the portfolio were:**

Exxon Mobil Corp (XOM).

**The Science of Optimization**

To calculate an efficient portfolio from this group of stocks, you minimize the portfolio variance which can be represented as a product of matrices. w_{t}.V.w where w is the column vector of portfolio weights, V is the variance-covariance matrix of investment returns, and w_{t} is the transform of w. The minimization is done using calculus subject to the constraints that w_{t}r=r_{T}

where r is the column vector of component stock returns and r_{T} is the portfolio return. Also, the weights should sum to one and, if we do not allow short selling, be non-negative. To calculate a whole frontier of efficient portfolios, quadratic programming methods can be used. Enough said.

**Mean Reversion**

What exactly does mean reversion mean? An interesting website with a chart you can play with is found at this site owned by Franklin Templeton which is aptly titled "History favors a return to the mean".

On the website, annual returns of the S&P 500 were positive approximately 70% of the time. Measuring returns over 10-year time periods, the results worked in investors' favor 95% of the time. Following a negative 10-year return, the subsequent historical 10-year return was positive.

**Does the Berkshire portfolio revert to the mean?**

Before we answer that question, let's look at some of the stocks that comprised the Berkshire portfolio. One of these stocks was American Express Co. The chart below shows the return differences for this stock over time. A return difference is defined as the actual annualized return minus the mean return.

The stock reverts to the mean whenever the return difference crosses the horizontal axis from positive to negative or vice versa. There were 4883 above-average returns and 5687 below-average returns.

*(Click to enlarge)*

Now let's look at another stock that was in the Berkshire portfolio --- Wal-Mart Stores. This stock had 3922 above-average returns and 6663 below-average returns.

*(Click to enlarge)*

Looking at the two graphs above, it is easy to tell that each stock eventually reverts to the mean, but difficult to decipher the fashion in which they do so.

When we chart the frequency histograms of the two stocks above, we get a better feel of how they revert to the mean. The histograms give us an idea not just of the number of above-average and below-average returns but also the extent to which these are above-average or below-average.

*(Click to enlarge)*

In the histogram for American Express Co. (the blue bars) above, we can see straightaway how below-average returns predominate, while in Wal-Mart Stores below, the above-average returns roughly balance out the below-average returns, despite the greater number of below-average returns. This may partially be due to the extent of some above-average returns.

*(Click to enlarge)*

It is therefore not by coincidence that a normality test done on American Express Co. rejected normality, while one done on Wal-Mart Stores tested normal. The red bars in the two histograms above are what the distributions would be like if they were normally distributed.

Note that normality is defined by 68% of data being within one standard deviation from the mean, 95% of data within two standard deviations and 99% within three standard deviations.

The other stocks in the Berkshire portfolio that tested non-normal (or more precisely, where normality was rejected at the 1% significance level) were Costco WholeSale Corp, General Dynamics Corp, Gannett Co, General Electric Co., International Business Machines and United Parcel Service:

*(Click to enlarge)*

**The Berkshire Portfolio is Normal**

It may come as a surprise that despite the fact that the above component stocks did not exhibit normal behavior, the Berkshire portfolio did. Here is a graph that charts the portfolio return differences (but over a shorter period). The portfolio component weights were as stated in Berkshire's December 2011 release. Out of the data points charted, 484 were above the mean while 588 were below.

*(Click to enlarge)*

Note that the ratio of above-average to below-average returns was not much different from that for non-normal American Express Co. You need to view the histogram below, where the Berkshire portfolio return differences are the blue bars and a normal distribution with the same mean and standard deviation are the red bars, to get the fuller picture.

*(Click to enlarge)*

**Mean Reversion vs Normality**

While the illustration at the Franklin Templeton website sounds encouraging to the investor, volatility plus the effect of compounding long-term annualized returns will have its impact on total portfolio value. Even though time diversifies return volatility, the effects of compounding will increase your portfolio value-at-risk.

For example, if you had a 5-year investment that promises a 10% annual return and you lose 30% of your capital at the end of the first year, you will need about 24% return every year for the next 4 years to achieve what you had originally hoped to achieve in that 5 year time frame.

A non-normal distribution, while still reverting to the mean in an irregular way, exacerbates the need to recover at high returns in order to achieve what you had originally set out to do with your portfolio.

Non-normality can be the result of a non-stationary series. A non-stationary series simply implies that the mean and variance are not constant over time. Non-normality can also be due to a greater number of extreme values in the tail, resulting in what is called a "fat tail". Such distributions may be stable but have infinite variance (which is effectively variance undefined).

For portfolio optimization to occur with minimal impact on value-at-risk, mean reversion under a normal distribution (and not mean reversion alone) is a prerequisite. In the Berkshire portfolio, the portfolio distribution reverts to the mean under a normal distribution.

**Optimization vs Forecasting**

There is considerable confusion in the market place about these two concepts.

Mean-value optimization calculations do not entail the use of time-dependent equations. Rather, optimization estimates the probability distribution of the portfolio and assumes that this probability distribution governs the behavior of the returns. Unlike a forecast, it does not know and it does not seek to know the time a specific return will occur in the future.

So if you are like the Oracle of Omaha with investment time frames of 5 to 10 years or more, mean reversion under a normal distribution can be a very good proposition. With a normal distribution, mean reversion occurs in a regular way and optimization calculations are more stable. This in turn lends itself to a buy-and-hold strategy.

**Buy and Hold?**

Let's consider the following 15 stocks in Berkshire's portfolio (we take the ones with the longest history) assuming a buy-and-hold strategy:

American Express Co; Bank of New York Mellon Corp; Coca Cola Co; ConocoPhillips; Gannett Co; General Dynamics Corp; General Electric Corp; GlaxoSmithKline; Ingersoll-Rand; Intel Corporation; International Business Machines Corporation; Johnson & Johnson; Procter & Gamble; Wal-Mart Stores; Wells Fargo & Co.

The results show how $100 grew over the 15-year period beginning February 1998. The blue bars depict an optimized portfolio which tested normal and was calculated as at February 1998, while the red bars depict an equal weighted portfolio.

*(Click to enlarge)*

As the results showed at the start of this article, the Berkshire portfolio compounded at a rate of about 7% per annum over the 15 years. During years of steady increases from 2003 to 2008, the compound rate reached 22% per annum. Not bad.

**Look for Noise**

In practice notes 4,5 and 6 in my article on Complexity vs. Simplicity in portfolio optimization, I talked about keeping it simple. While exotic and academically exciting research is being done to overcome the problems of non-normality in optimization, the advantages are too often outweighed by over-fitted data.

Here's the simple art of optimization: Look not just for noise but white noise.

Mean reversion will occur when the underlying distribution is normally distributed (the Berkshire portfolio). But it will also occur when the distribution is non-normal (American Express Co).

In the context of optimization, mean reversion under a normal distribution is white noise. That is what you should be looking for.

**Conclusion**

In the Berkshire portfolio as of December 30, 2011, it is tempting to suggest that some form of the Central Limit Theorem contributed to the normality of the portfolio returns despite certain component stocks being non-normal.

I don't think this is case. Rather, it is simply that except for American Express Co. and International Business Machines, Berkshire put very little weight into the other five non-normal stocks. My guess is that the two stocks (American Express and IBM) will also see lesser weight put on them in the future.

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