Statistical Moments of REAL Stock Return Distributions

Jeffery Owen Katz, Ph.D. and Donna L. McCormick (Advanced Option Pricing Models) conducted a statistical moment analysis of stock returns for 2,241 stocks, segmented over something close a decade (since Jan 2, 96). They then compared the results to simulated, Monte Carlo stochastic distributions popularly assumed in many financial models, some interesting results came forth.

Sample Mean Comparisons (5-bars, 20 bars)
Values measured in terms of Standard Deviations, or Z-score.

Growth
Real: (0, 0.001)
Simulated: (0, 0)

Volatility
Real: (2.189, 4.217)
Simulated: (2.225, 4.378)

So far so good.

Skew
Real: (-0.035, -0.051)
Simulated- (-.001, 0)

Kurtosis
Real: (1.093, -0.273)
Simulated- (0.009, -0.395)
-------------------------------------------------------------------------------------------------

Conclusions
Real expected returns over in the long run seems to concur with existing beliefs of randomness, where real (before inflation) returns remain at zero, and volatility roughly multiples of square-root of time elapsed.

The skew and kurtosis values do not agree with conventional assumptions of stock price movements following a Monte Carlo stochastic style. More importantly, the negative skew value reveals a key many public investors will never understand, that the stock markets, like a casino, gives a slight edge to those who bet on the downside (short sellers).

Kurtosis on the other hand tells a much more realistic picture of the markets. Price rarely move in a nice and smooth fashion. Instead, they either crawl at a blistering pace, or jump/dive spectacularly, and the high short-term kurtosis tells it like it is.

The skew of a statistical distribution measures relative asymmetry, and in this case a negative value reveals a fatter tail to the left hand side. It suggests that yes, stocks tend to go up a bit more often than down, but when prices drop they move more violently. This explains why in bearish periods, average investors tend to lose their shirts, short-exclusive funds make fortunes, and in bullish moments crowds make relatively "small" returns, but are happy, while the downside betters sit quietly.

SPY Distribution Skew VS. Monte Carlo Simulation (1bar, 10bar,20bar)
Real: (-0.287, -0.513, -0.377)
Simulated: (-0.005, 0.0370, 0.253)

The stock indice return distribution further supports the notion. Stocks come down hard, serious traders should embrace this truth.

Once understood, it becomes clear why professional traders favor shorts instead of long positions. Of course it is not very politically correct, then again what has political correctness done for you lately?