Basic Equity Return Statistics

June 1, 2012 • Posted in Big Ideas

What do the basic statistics of stock market returns tell us about risk and predictability? Basic statistics are the measures of the moments of the return distribution: mean (average), standard deviation, skewness and kurtosis. Are these characteristics of stock market returns (and the risk-reward environment they imply), stable over time? To make the investigation manageable, we calculate these four statistics month-by-month based on daily returns for each month. Using daily closes of the Dow Jones Industrial Average (DJIA) for January 1930 through April 2012 (988 months) and the S&P 500 index for January 1950 through April 2012 (748 months), we find that:

The following chart summarizes the average of the means of daily returns by month for the U.S. stock market over the entire sample periods and decade-by-decade (with the 2010s partial only). It shows that the investing environment varies based on this equity market performance metric, with the 1950s, 1980s and 1990s standing out as “easy” decades and the 1930s, 1970s and especially the 2000s standing out as “hard” decades. The 2010s so far look easy. The reward from passively investing in equities varies considerably over long periods.

The next chart summarizes the average of the standard deviations of daily returns by month for the U.S. stock market over the entire sample periods and decade-by-decade (with the 2010s again partial only). It shows that the investing environment varies also based on volatility of equity market performance, with the 1950s and the 1960s being “calm” decades and the 1930s and the 2000s standing out as “jumpy” decades. The 2010s so far look more jumpy than calm. Risk measured by return volatility varies over long periods.

The next chart summarizes the average of the skewnesses of daily returns by month for the U.S. stock market over the entire sample periods and decade-by-decade (with the 2010s again partial only). It shows that the investing environment also varies considerably based on this measure of distribution tail asymmetry, with the 1940s and 1950s notable for negative skewness (long left tail) and the 1970s notable for positive skewness (long right tail). The 2010s so far look unremarkable. Skewness risk varies considerably over long periods.

The next chart summarizes the average of the kurtosis of daily returns by month for the U.S. stock market over the entire sample periods and decade-by-decade (with the 2010s again partial only). It shows that the investing environment varies also based on this measure of return distribution fat-tailedness, with the 1940s, 1950s and 1980s standing out as “fat-tailed” and the 1960s and 1970s standing out as “more normal” decades. The 2010s so far look quite fat-tailed. Kurtosis risk varies over long periods.

Do the statistics of monthly stock market return distributions predict next month’s returns?

The next chart summarizes the correlations between DJIA and S&P 500 Index monthly returns and the above four statistics for daily returns for the preceding month since the 1950s. Correlations are small, suggesting that daily return statistics for a given month have little or no power to predict next-month stock market behavior. The strongest indication is a negative relationship between return volatility (standard deviation) last month and return next month.

Are these relationships stable over time?

The final two charts summarize the correlations between DJIA monthly returns (upper chart) and S&P 500 Index monthly returns (lower chart) and the above four statistics for daily returns for the preceding month by decade since the 1930s and 1950s, respectively (with the 2010s again partial only). Relationships between last month’s daily return statistics and this month’s return are generally not stable over time.

In summary, evidence from simple statistics suggests that investors should not count on the long-run stability of the equity investing environment or on continuity of short-term relationships between past and future returns.

By extension, investors may want to demand a fairly high level of proof that more specialized stock return indicators offer reliable value.

Cautions regarding findings include:

  • The sample for the 2010s is small and could change substantially as new data becomes available.
  • Other ways of measuring the four moments of equity index returns (for example, measuring by year rather than by month) may produce different results.
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