Descriptive statistics that matter

Before any model, you describe the data. The four moments of a return series tell you most of what you need.

The four moments

• Mean — the average return. Over long horizons this is your edge (or lack of one).
• Variance / standard deviation — the spread. In finance the standard deviation of returns is called volatility, and it is the dominant measure of risk.
• Skewness — the asymmetry. Negative skew (common in equities) means the big surprises tend to be downward — crashes are sharper than rallies.
• Kurtosis — the fatness of the tails. High kurtosis warns of frequent extremes.

Why volatility is not constant

A beginner treats volatility as a fixed property of an asset. It is not. Volatility clusters — calm periods follow calm periods, turbulent days follow turbulent days. A statistic measured over a quiet month badly misleads you about a crisis month. We return to this with GARCH later in the chapter.

Risk-adjusted return

A raw return number is meaningless without its risk. The standard summary is the Sharpe ratio:

Sharpe = (mean return − risk-free rate) / standard deviation of return

It asks: how much return did you earn per unit of risk taken? A strategy returning 20 percent with wild swings can have a worse Sharpe than one returning 8 percent smoothly. Quants chase risk-adjusted return, not headline return — and we will see in the validation chapter how easily a Sharpe ratio can be inflated by accident.