How to Calculate Beta From Volatility and Correlation

Investors understand intuitively that some stocks are riskier than others. The capital asset pricing model attempts to quantify the common perception of risk using a term called beta. By understanding how beta works, you can better understand how the individual stocks in your portfolio are likely to move based on how the overall market performs.

What beta isThe capital asset pricing model uses beta to describe how the returns of a given stock or portfolio stocks will compare to the returns of the overall market. The statistical definition of beta is that it equals the covariance between the returns of the stock and the returns of the overall market, divided by the variance in the returns of the market.

Most investors understand beta more easily by example. A stock that tracks the market perfectly has a beta of 1, in that its returns always match the overall market. A stock with a beta of 2 that's perfectly correlated with the market has returns that are twice as extreme as the market's returns. Therefore, if the market rises 1%, a perfectly correlated stock with a beta of 2 would rise 2%. If the market fell 1%, the stock would fall 2%.

Focusing on correlated relative volatilityHowever, the simple example-based description of beta often leaves out a key factor: correlation. Even if a stock tends to be twice as volatile as the market, its beta won't be 2 if its returns aren't perfectly correlated to the market's returns. In fact, if the stock's returns are completely uncorrelated with the market, then its beta will be 0. If they're perfectly correlated in a negative way, then its beta will be -2.

From the statistical definition above, an alternative definition of beta is that it equals the correlation between the stock's returns and the market's returns multiplied by the standard deviation of the stock's returns and divided by the standard deviation of the market's returns. In other words, beta equals correlation times the relative volatility of the stock versus the market.

What beta doesn't tell youWhat that means is that beta by itself doesn't tell you how volatile a stock is. For any given value of beta, the less correlated the stock's returns are to the overall market, the more volatile the stock is relative to the market's volatility. Only for highly correlated stocks or portfolios will the beta value come close to the relative volatility figure.

Many investors look at beta as a measure of risk. Beta can be useful in that way, but only if you recognize its shortcomings. Otherwise, it's all too easy to make erroneous conclusions from a stock's beta figure.

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