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Beta, Correlation and Covariance

Beta Definition
Beta is a measure of the systematic, non-diversifiable risk of an investment

A misconception about beta is that it measures the volatility of a security relative to the volatility of the market. If this were true, then a security with a beta of 1 would have the same volatility of returns as the volatility of market returns.  In fact, this is not the case, because beta also incorporates the correlation of returns between the security and the market.

    Beta = Correlation of Asset to Market * (Std Dev of Asset / Std Dev of Market)       

For example, if one stock has low volatility and high correlation, and the other stock has low correlation and high volatility, beta cannot decide which is more risky.
Beta sets a floor on volatility.  For example, if market volatility is 10%, any stock (or fund) with a beta of 1 must have volatility of at least 10%.

Another way of distinguishing between beta and correlation is to think about direction and magnitude. If the market is always up 10% and a stock is always up 20%, the correlation is 1 (correlation measures direction, not magnitude).  However, beta takes into account both direction and magnitude, so in the same example the beta would be 2 (the stock is up twice as much as the market).

Correlation Definition
Correlation – measures the degree to which two variables relate to each other.  It’s a standardized measure (unlike Covariance) of the strength of the linear relationship between two variables.  When you say that two items are correlated, you are saying that the change in one item effects a change in another item.

    Correlation = Covariance of Asset to Market / (Std Dev of Asset * Std Dev of Market)

Covariance Definition
Covariance – measures the way two variables relate relative to each other (move together).  It illustrates the linear relationship between two random variables.  If you say that two items tend to move together then you are talking about the covariance between the two items which can be positive or negative covariance.

    Covariance = (1/n) * Sum [(Asset Price - Asset Price Avg) * (Market Price - Market Price Avg)]

The number that represents covariance depends on the units of the data, so it is difficult to compare covariances among data sets that have different scales.  The correlation coefficient addresses this issue by normalizing the covariance to the product of the standard deviations of the variables, allowing comparison across different data sets.

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