5.5 Stylized facts for daily and monthly asset returns

A stylized fact is something that is generally true but not always. From the data analysis of the daily and monthly returns on Microsoft and the S&P 500 index we observe a number of stylized facts that tend to be true for other individual assets and portfolios of assets. For monthly data we observe the following stylized facts:

  1. M1. Prices appear to be random walk non-stationary and returns appear to be mostly covariance stationary. There is evidence that return volatility changes over time.
  2. M2. Returns appear to be approximately normally distributed. There is some negative skewness and excess kurtosis.
  3. M3. Assets that have high average returns tend to have high standard deviations (volatilities) and vice-versa. This is the no free lunch principle.
  4. M4. Returns on individual assets (stocks) have higher standard deviations than returns on diversified portfolios of assets (stocks).
  5. M5. Returns on different assets tend to be positively correlated. It is unusual for returns on different assets to be negatively correlated.
  6. M6. Returns are approximately uncorrelated over time. That is, there is little evidence of linear time dependence in asset returns. 7. M7. Returns do not exhibit dynamic feedback. That is, there are no strong lead-lag effects between pairs of returns.
  7. M8 There is some informal evidence of non-constant volatilities and correlations, particularly during financial crisis periods.

Daily returns have some features in common with monthly returns and some not. The common features are M1 and M3-M7 above. The stylized facts that are specific to daily returns are:

  1. D2. Returns are not normally distributed. Empirical distributions have much fatter tails than the normal distribution (excess kurtosis).
  2. D7. Returns are not independent over time. Absolute and squared returns are positively auto correlated and the correlation dies out very slowly. Volatility appears to be auto correlated and, hence, predictable.

These stylized facts of daily and monthly asset returns are the main features of returns that models of assets returns should capture. A good model is one that can explain many stylized facts of the data. A bad model does not capture important stylized facts.