Read the shape of a return sample

Learning goals

• Choose location and dispersion measures that fit the data.
• Calculate covariance, correlation, downside dispersion and relative variability.

Draft · approximately 60 minutes, excluding further revision.

Averages need a distribution around them

The mean uses every observation; the median is the central ordered observation and is less sensitive to extremes. Quantiles locate observations within an ordered distribution. Sample variance divides the sum of squared deviations from the sample mean by n−1 when estimating population variance from independent observations. Population variance divides by N when describing the complete population. Keep that choice consistent when computing covariance.

Standard deviation restores the return unit after squaring. Interquartile range measures the spread of the middle half and is less driven by extremes. The coefficient of variation is standard deviation divided by mean, but becomes unstable near a zero mean and difficult to interpret with a negative mean. A small ratio is not proof of low downside risk.

Co-movement and asymmetric tails

Sample covariance sums products of matching deviations and divides by n−1. Correlation normalizes covariance by both standard deviations and lies between −1 and +1 when defined. Zero linear correlation does not imply independence. Correlations can change in stressed markets, so a historical estimate is not a permanent diversification contract.

Negative skewness indicates a longer left tail, not necessarily a negative mean. Kurtosis concerns tail weight relative to a reference distribution; excess kurtosis subtracts 3 from the fourth standardized moment. State whether kurtosis is raw or excess. For downside deviation relative to target τ, one useful convention is √[Σ min(0,Rᵢ−τ)²/n], including all observations in the denominator; other conventions must be labeled rather than mixed.

Calculate dispersion before interpreting a small sample

For three hypothetical returns −2%, 1%, 4%, the arithmetic mean is 1%. Deviations are −3,0,3 percentage points, so sample variance is (9+0+9)/(3−1)=9 squared percentage points and sample standard deviation is 3 percentage points. Dividing by n would instead describe the population variance of these three values. State which object you estimate and keep decimal versus percentage-point units consistent.

The coefficient of variation, standard deviation divided by mean, is unstable when the mean is near zero and may be hard to interpret with negative means. A symmetric three-observation sample cannot establish normality, stable skewness or a reliable extreme-loss estimate. Inspect the time sequence, outliers, missing observations and data definitions before treating a summary statistic as a risk model.

Formula and assumptions

s² = Σ(Rᵢ−R̄)²/(n−1); ρ = Cov(A,B)/(σAσB); downside deviation = √[Σ min(0,Rᵢ−τ)²/n]

Three return observations

1. Returns are −2%, 2%, 6%. Mean is 2%; deviations are −4, 0 and +4 percentage points.
2. Sample variance = (16+0+16)/(3−1) = 16 squared percentage points; sample standard deviation = 4%. In decimal-return units the variance is 0.0016.
3. With a 0% target and all-observations denominator, downside deviation = √(4/3) = 1.1547%. A second asset with deviations exactly twice these has correlation +1, although its variance is four times as large.

Calculation lab

Errors to catch

• Mixing decimal returns with percentage-point variances.
• Treating zero correlation as independence or ignoring downside-measure denominators.

Original knowledge check

Answer every question before checking. Review explanations after submitting; retry starts a new attempt.

Sources and coverage status

These original lessons are drafts. Public outlines are used to check scope; the official learning-outcome text and proprietary question banks are not reproduced here. Follow the source for exam-year requirements and updates.

• Reference · www.cfainstitute.org