By Robert Ferguson and David Schofield

It is a fact that a simple, equal-weighted portfolio will beat a capitalisation-weighted benchmark over time, if the distribution of capital across the stocks in the benchmark is reasonable. This can be proved mathematically using stochastic portfolio theory.

Let’s take the S&P 500 Index and suppose its largest stock weighting is 5 per cent and its smallest stock weighting is 0.0001 per cent. If you arrange these 500 stocks from largest to smallest along the horizontal axis of a chart and plot the stocks’ S&P 500 weights on the vertical axis, the resulting descending curve is called the capital distribution curve.

For more than 80 years, the capital distribution curve of actual broad capitalisation-weighted equity benchmarks has conformed to a typical shape that has a reasonable balance of large, medium and small stocks. It should not be surprising, therefore, that from January 1 1966 to June 30 2010, a simulated equal-weighted portfolio of the S&P 500 stocks beat the capitalisation-weighted S&P 500 by an average of about 3 per cent annually.

Why is that? It has long been thought that the outperformance of equal-weighted portfolios versus cap-weighted portfolios is due to the “small-stock effect” – small-cap stocks have greater returns than large-cap stocks.

However, our alternative interpretation is that equal-weighted portfolios outperform the market over time by capturing relative volatility.

This is a consequence of the trading rule required to maintain equal weights, which is to buy after a negative relative return and to sell after a positive. To the extent that stocks move up and down relative to the cap-weighted benchmark, the trading rule has a “buy low/sell high” character, which generates outperformance.

Remarkably, it can be shown that the underlying alpha source is always positive, which is another indication that the excess return has something to do with volatility. Things such as stock returns move up and down; however, stock volatilities are always positive. Why then do equal-weighted portfolios have great tracking error over time relative to the cap-weighted index? The answer involves diversity.

Diversity is a measure of the concentration of capital among large and small stocks and depends on the capital distribution curve. Diversity is at a maximum if all stock capitalisations are equal; it is at a minimum when one stock’s capitalisation accounts for virtually the entire market. If large stocks outperform small, the capital distribution curve gets steeper and diversity declines.

Changes in diversity influence relative return of the equal-weighted portfolio. Since the capital distribution curve has a typical shape, diversity must have a typical value, and no long-term trend. In this case, the influence due to changes in diversity will be close to zero over time, and the outperformance of the equal-weighted portfolio will be close to its volatility-based source, which is necessarily positive.

To summarise, stochastic portfolio theory shows mathematically that the relative return of an equal-weighted portfolio with respect to its capitalisation-weighted benchmark is the sum of two components.

The first is the relative-volatility capture rate, which is the true, underlying source of alpha for the equal-weighted portfolio, and is always positive. The second is the influence of a true size factor, called diversity. Changes in diversity are the source of tracking error. The power of this approach is that it provides a precise and useful decomposition of relative return, which is not explained by the “small-stock effect”.

The establishment of an always positive alpha due to volatility capture is meaningful. Any time one observes a true positive alpha source, it raises the following question: can it be harnessed to generate long-term returns? An example is an equal-weighted portfolio, but this is volatile. There are more efficient ways to capture relative volatility.

Such approaches require only a forecast of the volatilities and correlations among stocks. The result is that more efficient relative volatility capture strategies have higher information ratios, higher expected alphas and experience fewer and less severe underperformance periods, attempting to gain benefit of the alpha source while “smoothing out the ride”.

It is a fact that a simple, equal-weighted portfolio will beat a capitalisation-weighted benchmark over time, if the distribution of capital across the stocks in the benchmark is reasonable. This can be proved mathematically using stochastic portfolio theory.

Let’s take the S&P 500 Index and suppose its largest stock weighting is 5 per cent and its smallest stock weighting is 0.0001 per cent. If you arrange these 500 stocks from largest to smallest along the horizontal axis of a chart and plot the stocks’ S&P 500 weights on the vertical axis, the resulting descending curve is called the capital distribution curve.

For more than 80 years, the capital distribution curve of actual broad capitalisation-weighted equity benchmarks has conformed to a typical shape that has a reasonable balance of large, medium and small stocks. It should not be surprising, therefore, that from January 1 1966 to June 30 2010, a simulated equal-weighted portfolio of the S&P 500 stocks beat the capitalisation-weighted S&P 500 by an average of about 3 per cent annually.

Why is that? It has long been thought that the outperformance of equal-weighted portfolios versus cap-weighted portfolios is due to the “small-stock effect” – small-cap stocks have greater returns than large-cap stocks.

However, our alternative interpretation is that equal-weighted portfolios outperform the market over time by capturing relative volatility.

This is a consequence of the trading rule required to maintain equal weights, which is to buy after a negative relative return and to sell after a positive. To the extent that stocks move up and down relative to the cap-weighted benchmark, the trading rule has a “buy low/sell high” character, which generates outperformance.

Remarkably, it can be shown that the underlying alpha source is always positive, which is another indication that the excess return has something to do with volatility. Things such as stock returns move up and down; however, stock volatilities are always positive. Why then do equal-weighted portfolios have great tracking error over time relative to the cap-weighted index? The answer involves diversity.

Diversity is a measure of the concentration of capital among large and small stocks and depends on the capital distribution curve. Diversity is at a maximum if all stock capitalisations are equal; it is at a minimum when one stock’s capitalisation accounts for virtually the entire market. If large stocks outperform small, the capital distribution curve gets steeper and diversity declines.

Changes in diversity influence relative return of the equal-weighted portfolio. Since the capital distribution curve has a typical shape, diversity must have a typical value, and no long-term trend. In this case, the influence due to changes in diversity will be close to zero over time, and the outperformance of the equal-weighted portfolio will be close to its volatility-based source, which is necessarily positive.

To summarise, stochastic portfolio theory shows mathematically that the relative return of an equal-weighted portfolio with respect to its capitalisation-weighted benchmark is the sum of two components.

The first is the relative-volatility capture rate, which is the true, underlying source of alpha for the equal-weighted portfolio, and is always positive. The second is the influence of a true size factor, called diversity. Changes in diversity are the source of tracking error. The power of this approach is that it provides a precise and useful decomposition of relative return, which is not explained by the “small-stock effect”.

The establishment of an always positive alpha due to volatility capture is meaningful. Any time one observes a true positive alpha source, it raises the following question: can it be harnessed to generate long-term returns? An example is an equal-weighted portfolio, but this is volatile. There are more efficient ways to capture relative volatility.

Such approaches require only a forecast of the volatilities and correlations among stocks. The result is that more efficient relative volatility capture strategies have higher information ratios, higher expected alphas and experience fewer and less severe underperformance periods, attempting to gain benefit of the alpha source while “smoothing out the ride”.

## 4 comments:

Of course equal waiting portfolio will perform better, it is a time series analysis.

Its funny, the number one investment over the last 40 years has been MUNI BONDS! Tax adjusted of course. Also over the last 10 years long term gov bonds have been the highest performers. I wonder how m

So great job focusing on this issue.

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This is absolutely correct. Equal weighted captures the value premium in the market.

I liked the idead

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