Unmasking the Hidden Patterns: Cross-Sectional Skewness and Kurtosis in Stocks and Portfolios

The Mystery at Hand

Why does the behavior of skewness and kurtosis in a universe of stocks not always align with expectations when they are aggregated into portfolios? Unraveling this mystery can significantly impact investment decisions, potentially enhancing returns and minimizing risk.

The Relevance of Cross-Sectional Skewness and Kurtosis

Skewness and kurtosis are crucial measures of a dataset's distribution. Skewness indicates the asymmetry of data around the mean, while kurtosis reveals the 'tailedness' or the presence of outliers in the data. In finance, these measures help investors understand the risk profile of their investments better. However, when moving from individual stocks to portfolios, the expected reduction in skewness and kurtosis does not always occur as anticipated.

A Glimpse into the Past

The peculiar phenomenon was first observed by a keen observer who noticed that the cross-sectional distribution of simple returns of stocks did not conform to normal expectations. This led to an investigation into whether the aggregation of stocks into portfolios would result in a distribution closer to normality.

Unveiling the Truth: Skewness and Kurtosis in Stocks vs. Portfolios

The Setup

Daily prices of nearly all S&P 500 stocks from early 2006 to late February 2012 were used. Two sets of random portfolios were created, each consisting of 10,000 portfolios, with varying constraints on the number of names and maximum/minimum weights. These portfolios were not rebalanced, allowing for a more comprehensive analysis.

Skewness: A Rolling Tale

The 200-day rolling skewness for the stock universe and the two sets of portfolios is displayed in Figure 2. The 200-name portfolios exhibit relatively small skewness, while the expectation of lower skewness in 20-name portfolios compared to stocks is not always met.

During mid-2010 (Figure 3), an anomaly in skewness was observed. In this period, the 30-day returns suggest that the 20-name portfolios had significantly higher skewness than the stock universe, contradicting conventional expectations.

Kurtosis: A Twist in the Tale

The 200-day rolling kurtosis for stocks and portfolios is shown in Figure 4. Here, expectations align better with reality, but an anomaly was observed in early 2011. The 30-day returns in Figure 5 further emphasize this abnormality, primarily affecting the 20-name portfolios.

Surprisingly, the kurtosis of the 200-name portfolios (Figure 6) remains remarkably well-behaved, indicating a relatively normal distribution.

The Big Question: Why Do Skewness and Kurtosis Diverge?

The discrepancy between skewness and kurtosis of portfolios and the underlying universe's measures can be attributed to various factors. These deviations could be due to:

1. Changes in stock composition within portfolios 2. The impact of extreme events on individual stocks affecting overall portfolio metrics 3. Differences in liquidity among stocks, leading to inconsistent weighting in the portfolio construction process