Statistical Characteristics of Asset Returns

A list of past returns only becomes useful once it is summarized: where the returns are centred, how widely they spread, whether the distribution leans to one side or has fat tails, and how two assets move together. This module covers the measures analysts use every day to describe return data, how to calculate them from a sample, and what each one can and cannot tell you about risk.

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What you will learn

1. Calculate and interpret measures of central tendency and location, including the mean, median, mode, quantiles and the interquartile range.

Mean uses every observation and is sensitive to outliers; the median is robust.
A mean well above the median indicates a long right tail; well below indicates a long left tail.
Percentile position in sorted data: L_y = (n + 1) y / 100, with linear interpolation between neighbours.
Interquartile range = Q3 - Q1, the spread of the middle 50% of the data.
Box plots display the median, the IQR, the whiskers and outliers.

2. Calculate and interpret measures of dispersion, including the range, mean absolute deviation, sample variance and sample standard deviation.

Range = maximum - minimum; it depends only on the two extremes.
MAD = average absolute deviation from the mean.
Sample variance divides the sum of squared deviations by n - 1 (degrees of freedom).
Standard deviation is in the same units as the data; variance is in squared units.
With independent returns, annualize a standard deviation by multiplying by the square root of the number of periods per year.

3. Describe and interpret the skewness and kurtosis of return distributions and explain their implications for measuring risk.

Positive skew: long right tail, mean > median > mode; negative skew: long left tail, mean < median < mode.
Excess kurtosis = kurtosis - 3; the normal distribution has excess kurtosis of zero.
Leptokurtic distributions have fat tails: extreme outcomes are more likely than under a normal distribution.
Financial returns are typically leptokurtic, and equity returns are often negatively skewed.
Normal-based risk measures understate tail risk when returns are negatively skewed and fat-tailed.

4. Calculate and interpret the sample covariance and correlation between two return series, and recognize the limitations of correlation analysis.

Sample covariance = sum of products of paired deviations / (n - 1); its sign shows direction.
Covariance depends on units; correlation r = s_XY / (s_X s_Y) is unit-free and between -1 and +1.
Correlation measures linear association only.
Outliers, chance and third variables can produce spurious correlation; correlation is not causation.
Correlations change over time and often rise in market stress.

5. Calculate and interpret semideviation, target downside deviation and the coefficient of variation.

Semideviation uses only returns below the mean; target downside deviation uses only returns at or below a chosen target.
Both divide the sum of squared shortfalls by n - 1, using the full sample size.
A higher target raises measured downside risk.
Coefficient of variation = standard deviation / mean: risk per unit of return; lower is better.
CV is unreliable when the mean is near zero or negative.

Module at a glance

Mean uses every observation; the median is robust to outliers; their gap reveals skewness.
Locate percentiles at (n + 1) y / 100 and interpolate; IQR = Q3 - Q1.
Sample variance divides by n - 1; standard deviation is its square root, in the units of the data.
Negative skew means occasional large losses; positive excess kurtosis means fat tails.
Correlation = covariance / (s_X s_Y), between -1 and +1, and measures linear association only.
Semideviation and target downside deviation measure downside risk, dividing by n - 1 for the whole sample.
Coefficient of variation = s / mean: risk per unit of return, unreliable when the mean is near zero.

This free module has 5 lessons, 13 practice questions with full explanations.

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