Calculate covariance and correlation and avoid the main errors in interpreting them.
The average product of two variables' deviations from their means. A positive value means they tend to move together. Its size depends on the units, so it is hard to read on its own.
Covariance divided by the product of the two standard deviations. It always lies between minus 1 and +1 and has no units.
Five paired observations.
| X | Y | X deviation | Y deviation | Product |
|---|---|---|---|---|
| 1 | 2 | -2 | -1 | 2 |
| 2 | 1 | -1 | -2 | 2 |
| 3 | 4 | 0 | +1 | 0 |
| 4 | 3 | +1 | 0 | 0 |
| 5 | 5 | +2 | +2 | 4 |
Both means are 3. The sum of products is 8, so the sample covariance is 8 / 4 = 2. Both sample variances are 2.5. Correlation = 2 / (square root of 2.5 x square root of 2.5) = 2 / 2.5 = 0.80.
Only the strength of a straight-line relationship. Two variables can be strongly related in a curved way and show zero correlation. A single outlier can create or destroy a correlation. Always look at a scatter plot.
In a two-variable relationship, the share of one variable's variance explained by the other is the correlation squared. A correlation of 0.6 explains 36%; a correlation of 0.3 explains only 9%.
t = r x square root of (n minus 2) / square root of (1 minus r^2). A correlation of 0.30 from 38 observations gives t = 0.30 x 6 / 0.954 = 1.89, just short of significance at the 5% level. Small-sample correlations are unreliable.
Three other explanations must be ruled out:
Two series that both trend upward over time will be highly correlated in levels even if unrelated. Correlate changes or returns, not levels (Module 9).
They are estimates over a window and drift with the regime. Compute rolling correlations (for example over 36 months) to see the variation. Correlations between risky assets tend to rise sharply in crises, which is when diversification is most needed. The stock-bond correlation has been negative for long stretches and positive in others, including 2022.
Spearman's correlation uses the ranks of the data instead of the values. It is less sensitive to outliers and captures any consistently rising or falling relationship.
A table of the correlations between every pair of assets in a portfolio. Together with the volatilities, it gives everything needed to compute portfolio risk (RS 302, Module 3).
X is 1, 2, 3, 4, 5 and Y is 1, 4, 9, 16, 100. Every increase in X brings an increase in Y, so the ranks match perfectly and Spearman's correlation is 1.0. Pearson's correlation is lower, about 0.8, because the relationship is far from a straight line and the last point dominates. When data are skewed or have outliers, compute both.
Beta = correlation x (stock volatility / market volatility). A stock with a correlation of 0.8 with the market, volatility of 30% and market volatility of 20% has a beta of 0.8 x 1.5 = 1.2. Two stocks with the same beta can have very different correlations: a volatile stock with a low correlation and a quiet stock with a high one.
The t-statistic for a correlation grows with the square root of the sample. A correlation of 0.30 from 38 observations gives t = 1.89, not significant; the same correlation from 102 observations gives t = 0.30 x 10 / 0.954 = 3.14, highly significant. With large samples, tiny correlations can be significant and still useless (Module 6 on economic significance).
An asset with a correlation of -0.2 to the rest of a portfolio reduces risk even if its own volatility is high, because it tends to rise when the portfolio falls. This is why investors pay a premium, through lower expected returns, for assets that perform well in crises, such as long government bonds in a deflationary recession.
Over a calm decade, two credit funds show a monthly correlation of 0.2. In the three months of a credit crisis, both fall sharply and the correlation over those months is 0.9. A full-period figure of 0.3 hides this. Compute correlations separately for down markets, or look at how often both assets lost money in the same month.
Use returns, not prices; check the scatter plot; compute rolling and down-market correlations; compare Pearson with Spearman; and ask whether there is an economic reason for the relationship.
Compute the correlation of monthly returns between your stock and the index over the full ten years, then as a rolling 36-month series, and chart it. Then compute the correlation between the two price levels and explain why it is misleadingly high.
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