Friday, August 30, 2013

Beta and Correlation

It was my 2nd class of risk management yesterday. During a calculation I saw that  beta's formula  is identical to correlation. Searched out some resourceful stuff form internet.
  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)
Beta and Correlation both measure the relationship between 2 variables. If you study the formulas you will see that correlation is just covariance expressed as a normalisation of covariance - that is it is scaled by the standard deviations of the 2 variables. Beta is however the Covariance of the 2 assets divided by the variance of the benchmark asset.

Intuitively what it really means is Beta is distinct from correlation in that correlation is more indicative of direction, while beta is also incorporating magnitude. If we say a market is up 15% always and the stock is always up 30%, then the correlation will be 1. However as beta gives us both the direction and the magnitude we would get a beta of 2.
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