In my article Looking for Noise in Berkshire's Portfolio, I mentioned that it was not enough for returns to revert to the mean, they had to revert under a normal distribution in order for optimization to do what it is supposed to do.

In this two part article, I shall talk about rebalancing a portfolio. There are of course many methods that can be used. Some advisers rebalance regularly on a calendar basis, for example, once a year. Others rebalance when the portfolio moves away from its intended mix by a certain percent. And some don't even rebalance -- read this article by the CEO of SeekingAlpha.

In Part 1 of this article, I shall talk about rebalancing a portfolio that has been tested normal. Such a portfolio reverts to the mean in a regular fashion.

**Berkshire's** (BRK.A,BRK.B) **holdings as of Dec 30, 2011:**

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 Corp (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); CVS Caremark Corp (CVS); DirecTV (DTV); General Dynamics Corp (GD); Intel Corp (INTC); Visa (V); Wells Fargo & Co (WFC); Da Vita (DVA); International Business Machines Corp (IBM); Liberty Media Corp (LMCA); Johnson & Johnson (JNJ); Kraft Foods (KFT); Verisk Analytics (VRSK).

_{As usual we will leave it to readers to include the risk free rate in their own analysis. And at no time do we suggest that this is how Berkshire does its rebalancing.}

In "Looking for Noise in Berkshire's Portfolio," I showed that while there were some stocks that tested non-normal in Berkshire's portfolio, the portfolio itself tested normal. This was possibly due to the fact that Berkshire's weightings in the non-normal stocks were not heavy enough to skew the entire portfolio out of normality.

To discuss about rebalancing, let's choose 15 stocks from the holdings above that have a longer history and see how different rebalancing techniques affect an optimized portfolio, as well as an equal-weights portfolio, created in 1997. The optimized portfolio was computed using a mean variance algorithm while you could think of the equal-weights portfolio as simple diversification.

The 15 stocks tested are as follows:

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 stocks are displayed below on a volatility-return chart where the green stars denote individual stocks that have tested normal while the blue stars are the stocks that tested non-normal (or more precisely, where normality was rejected at the 1% significance level).

*Click to enlarge.*

In mean-variance optimization, volatility is defined as the standard deviation of annualized stock returns.

Standard deviation goes hand-in-hand with the concept of normality. And normality is the cornerstone of all of Modern Portfolio Theory.

Outside of mean-variance optimization, there are various ways to define volatility. Volatility can also be illustrated by taking a moving average of absolute log returns. The volatility of the 5 stocks above where normality was rejected can be seen in the following graphs:

Compare this with the volatility exhibited by a stock that tested normal. For example, the Coca-Cola Company below:

As mentioned in my previous article, mean reversion under a normal distribution can be a good proposition for a buy-and-hold strategy. Under a normal distribution, mean reversion occurs in a regular way and optimization calculations are more stable.

Now, let's go back in time, and see how our optimized portfolio consisting of the 15 stocks above performed over the years. The results below show how $150 grew over the 15-year period beginning February 1997. The reason for choosing $150 as a starting investment amount will become clear in the next paragraph. It is simply to facilitate ease of interpretation. Also, note that we do not take expenses into consideration.

In the graph below, the blue bars depict an optimized mix which tested normal and the red bars depict an equal-weights mix which in this case also tested normal. The mean return and volatility for the equal-weights mix were 18.3% and 16.9% respectively as at 22 Feb 1997 while the optimized mix had a volatility of 12% given the same mean return.

## Separating the Normals from the Non-Normals

What if we separated the normals from the non-normals and viewed their performances separately?

The graph below shows the growth in a portfolio with a starting value of $100 consisting of the 10 stocks that tested normal out of the 15 stocks above using a buy-and-hold strategy.

It can be observed that over the longer term, the optimized portfolio outdoes the equal-weights portfolio. For the equal-weights mix, the mean return and volatility were 18.4% and 15.4% respectively as at 22 Feb 1997. The mean return and volatility for the optimized mix were 18.4% and 13.2% respectively.

The next graph is a portfolio consisting of the 5 non-normal stocks using a buy-and-hold strategy. Unlike the portfolio above, the optimized mix lags behind the equal-weights mix in this instance. For the equal-weights mix, the mean return and volatility were 18.1% and 26.3% respectively as at 22 Feb 1997 while that for the optimized mix were 18.1% and 21.9% respectively.

## Observations

1. Where normality reigns, the optimized mix will always beat the equal-weights mix in the long term. After all, probability and probability distributions are long term concepts.

2. The portfolio value for the equal-weights mix at any year can be obtained by summing the portfolio value of the normals with that for the non-normals. For example, the equal-weights mix has the portfolio value of $492 which is the sum of 177 (non-normals) and 315 (normals) after 15 years.

3. This is not the case for the optimized mix, however, as $552 from the combined portfolio is distinctly higher than $475 (=154+321) from the sum of the normal and non-normal portfolios. This is because optimization capitalizes on the correlations (in the negative direction) between the normal and non-normal stocks in the combined portfolio.

4. The histogram of return differences for the combined portfolio is shown below where the blue bars denote the actual frequencies while the read bars are generated from a normal distribution with the same mean and standard deviation. Data was taken from Jul 1987 to Feb 1997.

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.

## Betting on Rebalancing

So what is the best way to rebalance the combined portfolio over the years? While the above graphs assume a buy-and-hold strategy, the graph below shows a yearly rebalance to the original mix. $150 invested at the start results in $612 after 15 years (9.8% compounded annual return). This is a little more than the $552 (9.1% compounded annual return) we ended with using the buy-and-hold strategy.

The article I referenced in the opening paragraphs mentions a study by Vanguard that a portfolio consisting 48% S&P 500, 16% small cap, 16% international, and 20% bond index, had "over the past 20 years, earned a 9.49% annual return without rebalancing and a 9.71% return if rebalanced annually. That's worth describing as "noise," and suggests that formulaic rebalancing with precision is not necessary."

Compare this with the portfolio we just discussed where a buy-and-hold strategy yielded a compounded annual return of 9.1% and a yearly rebalancing strategy yielded 9.8%.

These results should not be surprising as buy-and-hold and yearly rebalancing to the original mix are both based on the same premise which is reversion to the mean under a normal distribution.

In the case of buy-and-hold you hold on to the same quantity for each stock in the belief prices will settle at levels that translate to average returns that are near the originally calculated mean returns.

In the case of rebalancing to the original mix, you adjust the quantities as the prices move so that the original weights (where weight is the product of price and quantity) are forcibly restored. You do this because you believe future average returns will be near the originally calculated mean returns.

But what if the premise of normality does not hold? Will "formulaic rebalancing" then be necessary? In Part 2 of this article I will discuss other considerations in the art of the rebalance.

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