QM 5 Statistical Characteristics of Asset Returns

Summarizing a series of returns begins with a single number that captures where the data cluster. Two ideas frame the whole discussion. The population is the entire set of observations that could be recorded, such as the complete history of the US dollar to euro exchange rate. A sample is a subset drawn from it, such as the rates observed during the COVID-19 pandemic. A measure of central tendency is one number meant to represent a typical value in the dataset.

Mean, median, and mode

The mean, or average, is the sum of all values divided by their count. It carries the same unit as the data, so if every return is a percentage the mean is a percentage, and if every value is doubled the mean doubles too.

The mean is not robust to outliers: a single extreme value can drag it by an arbitrary amount. Take the dataset [2, 3, 4, 5, 100]. Its mean is 114 divided by 5, which is 22.8, a figure pulled far above four of the five observations by the lone value of 100. In statistics, robust describes methods that stay reliable even when the usual assumptions fail, when outliers appear, or when the data come from an unexpected distribution. An outlier is an unusual point that differs sharply from the rest, often through measurement error, data-entry error, sampling error, or genuine natural variability.

The median is the middle value once the data are sorted. With an odd count it is the single central observation; with an even count it is the average of the two central observations. It is extremely robust: nearly half the points can be distorted without moving it. For the same dataset [2, 3, 4, 5, 100] the median is 4, against a mean of 22.8, which shows its resistance to the outlier. The cost is that the median ignores the magnitude of the observations and uses only their rank, and it requires sorting.

The mode is whichever value turns up with the highest frequency. A dataset may be unimodal, multimodal, or have no mode at all. It is useful when the point of interest is where values clump. For [1, 2, 2, 3, 4, 5, 5] the modes are 2 and 5, since each occurs twice, and modes can also be defined locally: [1, 2, 2, 2, 3, 4, 5, 6, 6, 6] has one mode at 2 and another at 6. Unlike the mean and median, the mode allows for more than one center.

Dealing with outliers

Extreme values can be legitimate unusual events, data-handling errors, or contamination from a different population. After confirming the data are accurate, three strategies remain for valid extremes.

1. Leave them unchanged. Appropriate when the values are legitimate and the whole distribution matters. Using every point avoids the subjective judgment of what counts as an outlier and preserves potentially valuable information.
2. Remove them. A trimmed mean drops a set fraction of the smallest and largest observations, then averages whatever is left. A 1 percent trimmed mean drops the lowest 0.5 percent and the highest 0.5 percent, leaving 99 percent of the data. This replaces subjective judgment with an objective rule, but it does discard information.
3. Substitute them. A winsorized mean pins a set fraction of the smallest observations to a chosen floor, for instance the value at the 2.5th percentile, and the same fraction of the largest observations to a chosen cap, for instance the value at the 97.5th percentile, before averaging. It keeps the information that outliers were present while removing their raw magnitude, striking a balance between the mean and the median.

The effect depends on the shape of the original data. Across 100 back-to-back monthly returns on the front-month Brent crude oil contract, the raw mean sat at 0.035 percent. Trimming 1 percent removed the two extremes, −79.82 percent and 33.51 percent, and lifted the average to 0.83 percent; trimming 10 percent removed ten observations and raised it to 0.99 percent. Winsorizing 1 percent and 10 percent both produced 0.80 percent. Trimming thins the tails and pulls the mean toward the center more aggressively than winsorizing, which keeps every observation but caps the extremes.

Quantiles and location

Location measures pick out the value beneath which a given proportion of the data sit. Quantile is the general term. The common cases are the quartile (four parts), the quintile (five parts), the decile (ten parts), and the percentile (one hundred parts). For quartiles, the first quartile Q1, second quartile Q2 (the median), and third quartile Q3 split the sorted data into four equal groups. The interquartile range (IQR) is Q3 minus Q1, and it measures the spread of the middle 50 percent of the data around the median.

Labeling conventions differ by field, so context matters. In Excel the 0th percentile is the smallest value and the 100th is the largest, yet a manager in the first quartile of a performance ranking usually sits among the highest returns while the fourth quartile holds the lowest. Splitting data into more segments raises the resolution: ranking companies by market capitalization into deciles, with the first decile the smallest firms and the tenth the largest, lets an analyst compare performance across sizes.

Dispersion captures how widely observations scatter about their center. An average return tells us how much value changed; a dispersion figure tells us how uncertain that change was, which is exactly why dispersion serves as a proxy for risk. The range, the mean absolute deviation, the variance, and the standard deviation are absolute measures, gauging spread on their own terms rather than against any benchmark.

Range and mean absolute deviation

The range is the maximum value minus the minimum value. It is intuitive and simple, but because it depends only on the two most extreme points it says nothing about how the rest of the data are distributed.

Averaging the raw deviations from the mean is useless because positive and negative deviations always cancel to zero. The mean absolute deviation (MAD) fixes this by averaging the absolute deviations, so it uses every observation. It is considered better than the range, but it is awkward to manipulate mathematically and is seldom used for building distributional models.

A related robust measure is the median absolute deviation, obtained by taking the median of how far, in absolute terms, each point sits from the data median; it resists outliers even more strongly.

Variance and standard deviation

These two rest on squared deviations from the mean, which sidesteps the cancellation problem. The population variance is the average squared deviation around the population mean, and the population standard deviation is its positive square root, which returns the measure to the original units.

Usually the data are a sample, so the sample mean estimates the population mean and the divisor changes from n to n minus 1. This is Bessel correction, and it removes the downward bias that dividing by n would introduce in the variance. The value n minus 1 also counts the degrees of freedom, the independent pieces of information left for estimating the parameters, so a dataset of 100 observations carries 99 of them.

Frequency and the root of time

Statistical estimates depend on how often the data are sampled. Lower-frequency data, such as annual returns, allow more time between observations for prices to move, so their returns and their measured dispersion tend to be larger than for daily data. Volatility across horizons is often bridged by scaling: variance grows roughly in proportion to time, while standard deviation grows with the root of time. To move from daily to monthly figures, scale the daily variance up by about 22 trading days, or scale the daily standard deviation up by the root of 22.

Real return series obey this only approximately, because market frictions, shifting sentiment, sudden jumps, and fat tails distort the scaling. Gold from 1989 to 2023 shows the pattern clearly.

Scaling the annual standard deviation of 13.51 percent down by the square root of 252 gives about 0.85 percent as an estimated daily volatility, which is roughly 15 percent below the 0.98 percent measured directly from daily data. The longer horizon masks short-term movement, so observed dispersion in lower-frequency data can exceed what scaling would predict.

Mean and variance are the first two moments of a distribution, describing its center and its spread. The third and fourth moments, skewness and kurtosis, describe its shape: how symmetric it is and how heavy its tails are. Both are free of scale, meaning they do not change if every observation is multiplied by a constant, so they compare cleanly across datasets with different units.

Skewness

Skewness measures asymmetry. It rescales every gap between an observation and the mean using the standard deviation, cubes each rescaled value so the sign survives, and then averages.

A positive skew means a long right tail: high or positive returns are stretched out, so there are more large positive outliers, and a positively skewed return distribution offers a higher chance of large gains. A negative skew means a long left tail: the biggest moves are the losses, so the pattern is many small gains broken by a handful of severe losses. In a symmetric bell shape the mean, median, and mode coincide. When the distribution is positively skewed the mean exceeds the median, which exceeds the mode; when it is negatively skewed the order reverses, with the mean below the median below the mode. Investors generally prefer positive skew.

Kurtosis

Kurtosis measures the combined weight of the tails relative to the rest of the distribution. It raises the standardized deviations to the fourth power, so extreme observations dominate. Because a normal distribution has a kurtosis of 3, results are usually reported as excess kurtosis, the kurtosis minus 3, so that the normal case sits at zero.

A leptokurtic distribution has fatter tails than the normal and a positive excess kurtosis, meaning more frequent extreme deviations. A platykurtic distribution has thinner tails and a negative excess kurtosis. A mesokurtic distribution matches the normal, with zero excess kurtosis. Most financial asset returns are leptokurtic, so a model that assumes normality will underestimate the chance of extreme outcomes. Because some packages report kurtosis and others report excess kurtosis, it is important to know which is shown: a reported 4.49 is a kurtosis, while its excess counterpart would be 1.49.

The measures so far describe a single variable. Covariance and correlation describe how two variables move together. Covariance is a bivariate extension of variance: the average product of the paired deviations of two variables from their respective means. A positive covariance means the two tend to move the same way around their means; a negative covariance means they tend to move in opposite directions.

Covariance is hard to read on its own, because it is measured in the product of the two variables units and its size depends on their magnitudes. Correlation solves this by dividing the covariance by the product of the two standard deviations, producing a scale-free number between −1 and 1.

A correlation near +1 signals a strong positive linear relationship, and exactly +1 is a perfect one; near −1 signals a strong inverse relationship, and exactly −1 is a perfect inverse one; zero means no linear relationship. Notice the sign of the correlation is fixed entirely by the sign of the covariance, since the denominator, a product of standard deviations, is always positive. Two return series can each have a negative average and still be positively correlated.

Variance and standard deviation treat upside and downside variability alike, but investors usually worry only about the downside: returns below the average, or below a target they need to beat. Two specialized measures address this and the related question of risk per unit of return.

Semi-deviation and target downside deviation

Semi-variance is the variance computed using only the observations that fall below the mean, so it isolates unfavorable variability.

Often the concern is returns below a chosen threshold rather than the mean. The target downside deviation, also called the target semi-deviation, measures dispersion below a target B. Only observations at or below the target enter the numerator, but the sum is still divided by n minus 1, since the data are a sample of all returns. Because semi-variance uses fewer observations than the full variance, the two cannot be compared directly.

Coefficient of variation

The coefficient of variation (CV) is the standard deviation divided by the mean, a measure of relative dispersion. Because it is a ratio, it compares variability across datasets with different means or units, and for returns it expresses risk taken per unit of return, so a lower CV is more favorable. Its inverse can be read as return earned per unit of risk. Two cautions apply: if the mean return is negative the statistic becomes hard to interpret, and a low mean paired with high dispersion, common in daily or weekly data, can distort it, which annualizing the figures helps to remedy.

As a quick illustration, consider three sectors. Utilities average a 2.10 percent monthly return with a 1.23 percent standard deviation, materials average 1.25 percent with 1.35 percent, and industrials average 3.01 percent with 1.52 percent. Their coefficients of variation are 1.23 divided by 2.10, which is 0.59, for utilities; 1.35 divided by 1.25, which is 1.08, for materials; and 1.52 divided by 3.01, which is 0.51, for industrials. Materials carries the most risk per unit of return at 1.08, while industrials offers the most favorable balance at 0.51.