How Beta Depends on Correlation and Relative Volatility

Summary

The note explains that beta and correlation measure different aspects of how an asset moves with an index, so a high correlation can coexist with a low beta. It gives the relationship between them: beta equals correlation multiplied by the ratio of the asset’s standard deviation to the index’s standard deviation, assuming beta is calculated against the index.

The example describes a low-volatility series that closely follows another series with a small lag. Their returns can move together closely, producing high correlation, while the lower relative volatility keeps beta small. The note also mentions oscillating series as a way to illustrate the distinction. It offers a conceptual explanation rather than empirical market evidence, and the example may be difficult to identify in real assets. The result depends on using the same observation period and consistent return definitions for both statistics.

Key ideas

• High correlation describes how closely two return series move together, while beta also reflects their relative volatility.
• Beta equals correlation multiplied by the ratio of the asset’s standard deviation to the index’s standard deviation.
• A low-volatility asset can have high correlation with an index and still have low beta.
• The relationship assumes beta and correlation use the same period and comparable return series.

Tags

# Low beta and high correlation


# Low beta and high correlation












Assuming that time period used to calculate the beta and correlation between an index and an asset is the same, is it possible to observe low beta while having high correlation?

If yes, how would you explain it?

## Answer by rrg (score 0, accepted)

https://quant.stackexchange.com/a/30763

Yes, it's possible if one instrument has higher volatility that the other. Consider a low volatility series that is slightly lagged. For example, take a large sine wave and duplicate with tiny lag.

In practice it may be hard to find a well-correlated asset - perhaps alternatively consider a very high freq sine of a lower freq sine function (oscillating oscillations) versus the lower freq sine.

From these formula: Corr(1,2) = Cov(1,2)/(Var(1).Var(2))^1/2 and Beta = Cov(1,2)/Var(1)

Therefore Beta = Corr(1,2) * 1/Var(1) * [Var(1) * Var(2)]^0.5 = Corr(1,2) * [Var(2) / Var(1)]^0.5

A good reference may be: Do two stocks with the same beta have a correlation of 1?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.