Correlations Across Windows: An Invented Six-Observation Panel
Every return in this hypothetical note is invented for arithmetic demonstration. The numbered observations are not calendar dates, and the unnamed series do not represent actual assets. Their extreme correlations are deliberately constructed, not discovered market relationships.
The first three hypothetical pairs have correlation +1, the last three have correlation -1, and the complete six-pair panel has correlation 0. Re-centering each sample explains the results. The example demonstrates sensitivity to the selected window; it does not estimate a historical regime, population correlation, or dependable hedge.
Define the hypothetical panel and its limited purpose
Use six matched, equally spaced hypothetical return pairs, expressed in percentage points. There are no prices to reconstruct and no claim that these observations occurred. Both series use the same interval, currency convention, and missing-data treatment by assumption. The exercise isolates window selection; it does not simulate realistic market tails, estimate an economic process, or provide evidence about any named investment.
CFA Institute's Portfolio Risk and Return: Part I discusses covariance and correlation as inputs to portfolio risk. Here those definitions are applied to a deliberately small constructed panel. The extreme outcomes make the mechanics visible, but six chosen observations cannot establish the behavior of a real population. The sample's design is disclosed before any coefficient is calculated, preventing a classroom construction from masquerading as an empirical discovery.
| Invented observation | Series A return | Series B return |
|---|---|---|
| 1 | -2% | -2% |
| 2 | -1% | -1% |
| 3 | 0% | 0% |
| 4 | 0% | 0% |
| 5 | +1% | -1% |
| 6 | +2% | -2% |
Calculate Pearson correlation with window-specific means
For each chosen window, subtract that window's A mean and B mean from their respective returns. Sum the products of centered pairs, then divide by the square root of the product of the two centered sums of squares. Sample-covariance normalization cancels from the ratio when applied consistently. Correlation is undefined if either series has zero variation within the selected window.
In observations 1 through 3, both means are negative 1%. Each centered sequence is negative 1, zero, positive 1 in percentage-point units. The centered cross-product sum is 2 and each squared sum is 2, so correlation is 2 divided by the square root of 4, or +1. This is exact proportionality in the invented values, not a statement about a stable economic relationship.
Recompute the shorter and longer windows independently
For observations 4 through 6, A has mean +1% and B has mean -1%. Centered A values are negative 1, zero, positive 1, while centered B values are positive 1, zero, negative 1. Their product sum is negative 2, giving correlation -1. The change follows from the supplied numbers; it is not evidence that an actual market switched regimes.
Across all six observations, A's mean is zero and B's mean is negative 1%. The centered cross-product sum is zero, while centered squared sums are 10 and 4. Full-panel correlation is consequently zero. Pooling requires recalculating means and sums rather than averaging previously calculated correlations. The arithmetic average of the two coefficients also happens to be zero here, but that coincidence is not a general pooling formula.
| Hypothetical window | Pairs | Mean A | Mean B | Pearson correlation |
|---|---|---|---|---|
| Observations 1-3 | 3 | -1% | -1% | +1.00 |
| Observations 4-6 | 3 | +1% | -1% | -1.00 |
| Observations 1-6 | 6 | 0% | -1% | 0.00 |
Distinguish a changing estimate from a changing process
A rolling coefficient can change because the underlying relationship changes, because the sample composition changes, or because estimation is noisy. Short windows react quickly but use fewer observations. Longer windows mix more information and may combine genuinely different conditions. CFA Institute's Estimation and Inference distinguishes sample statistics from population parameters, which is essential before interpreting a displayed coefficient as an enduring property.
The fictional panel cannot distinguish these explanations because its values were selected by construction. Nor should confidence intervals designed for independent random samples be attached automatically to three purpose-built pairs. In actual work, serial dependence, non-normality, and structural change require additional judgment about uncertainty. A coefficient calculated exactly can still be an uncertain or inappropriate estimate of the relationship the researcher cares about.
Predeclare windows and preserve measurement consistency
For an empirical extension, specify window lengths and sampling frequency before reviewing which result supports the desired conclusion. Changing the start date repeatedly and reporting only the most striking coefficient introduces selection. Adjacent rolling windows overlap, so their estimates are not independent confirmations. A long smooth correlation chart can contain much less independent information than its many plotted points appear to suggest.
CFA Institute's Backtesting & Simulation discusses structural breaks and biases relevant to historical research. Consistent total-return definitions, currency conversion, and market-close alignment are prerequisites before comparing windows. None of those real-data issues is solved by the fictional table. They are requirements for a separate investigation, not grounds for treating these constructed coefficients as evidence that a particular window length is optimal.
Report the result without implying investment evidence
Publish the input pairs and window definitions with the formula, so a reader can reproduce every coefficient without an external price feed. Identify the observation count and any zero-variance cases. Distinguish correlation from portfolio loss: a full-sample coefficient of zero does not preclude simultaneous declines on particular dates, and a negative coefficient does not ensure that the sizes of offsetting moves meet a liability.
The date on this note is its educational publication date. No historical sample period, earlier analyst record, or live market conclusion is implied. The retained methodological finding is that sample selection and re-centering matter. A practical claim about changing diversification would require genuine source data, an explicit uncertainty analysis, and a separately stated research design rather than more elaborate interpretation of these six invented pairs.
What to retain
- All six return pairs are hypothetical, with no invented historical prices or dates.
- Window-specific means produce correlations of +1, -1, and 0 in the stated samples.
- A changing sample coefficient is not proof of a changing economic relationship.
Source notes
Related research
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Version 1, 15 October 2026. First publication. Illustrative calculations are identified in context; source observations retain their stated dates.