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Covariance tells you whether two return series tend to be above or below their own averages together. Correlation divides that covariance by the series’ standard deviations, leaving a unitless measure of linear association. A positive value signals movement in the same direction; neither measure says that one return causes the other.
Why the two measures differ
For paired observations, first find each series’ mean. Multiply the deviations from those means pair by pair, add the products, and divide by the sample denominator. The result retains the squared units of the inputs, so its sign is easier to interpret than its size.
Standardising removes that scale problem. Correlation uses the same covariance but expresses it relative to the variability of each series. Its possible values run from negative one to positive one; zero means no linear association, which need not mean no relationship of any kind.
An illustrative calculation from paired returns
The following invented observations are percentages, paired by period. They are teaching numbers, not market returns.
| A | 2% | 2% | −2 points | −2 points | 4 points squared |
| B | 4% | 6% | 0 points | 2 points | 0 |
| C | 6% | 4% | 2 points | 0 points | 0 |
Both illustrative means are 4%. The cross-products total 4 percentage points squared. With three pairs, the sample denominator is two, so sample covariance is 2 percentage points squared. Each series has squared deviations totalling 8; dividing by two gives a sample variance of 4 and a standard deviation of 2 percentage points. Correlation is therefore 2 divided by the product of 2 and 2, or 0.5.
The positive covariance says the deviations tend to have matching signs. The correlation of 0.5 communicates the strength of that linear pattern without carrying percentage-point units. If the returns were entered as decimals, the covariance number would change while correlation would stay 0.5.
What the exam is testing
CFA Institute’s portfolio mathematics learning outcomes call for calculating and interpreting both measures. The distinction matters because portfolio risk depends on how holdings vary together, not only on how much each varies alone. Its reading on tests of independence also treats correlation as a linear relationship that may need statistical testing.
Keep the paired periods aligned, use the same sample convention for covariance and standard deviations, and check the scatter plot before making a broad claim. A curved pattern or an unusual observation can make one correlation coefficient a poor summary. Most importantly, association alone cannot establish a causal mechanism.
For the wider topic sequence, see the CFA program page. The study methodology provides a way to organise revision after working the calculation yourself.
Sources
- Portfolio Mathematics | CFA Institute · cfainstitute.org · Retrieved
- Parametric and Non-Parametric Tests of Independence | CFA Institute · cfainstitute.org · Retrieved
