Calculating the beta of a portfolio

The punch line: The beta of a portfolio is the sum of its weighted betas.

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It is not difficult to see why.

Beta, as originally defined by I believe Sharpe sometime around 1962, is the slope of a linear regression line when the market's excess return is plotted on the x-axis and the stock's excess return is plotted on the y-axis.*

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Accordingly, per the formula for the slope of a linear regression line:

\[ \beta=\frac{Cov\left(r,R\right)}{Var(R)} \]

Here, Cov stands for covariance and Var for variance. The lower-case r is for the stock and the upper-case R is for the market.

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For a portfolio of two stocks, A and B, because the return of a portfolio is the sum of its weighted returns, the formula becomes:

\[ \beta=\frac{Cov\left(w_{a}r_{a}+w_{b}r_{b},R\right)}{Var(R)} \]

Here the w's represent the weights of each stock in the portfolio. For example, if a portfolio is 70% IBM and 30% JNJ, the weights are 0.7 for IBM and 0.3 for JNJ.

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Conveniently, because the covariance function is linear in its arguments, in particular its first argument:

\[ \beta=\frac{w_{a}Cov\left(r_{a},R\right)+w_{b}Cov\left(r_{b},R\right)}{Var(R)} \]

Or:

\[ \beta=\frac{w_{a}Cov\left(r_{a},R\right)}{Var(R)}+\frac{w_{b}Cov\left(r_{b},R\right)}{Var(R)} \]


Or:

\[
\beta=w_{a}\beta_{a}+w_{b}\beta_{b}

\]

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In other words, the beta of a portfolio of two stocks is the sum of its weighted betas. The result extends to three or more stocks and, indeed, to not just stocks but to any assets -- as long as the definition of beta makes sense.

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To see how this works in real life, suppose you own a portfolio, 70% of which is in IBM, with a beta of 0.7, and 30% of which is in JNJ, with a beta of 0.5. The weighted beta for IBM is 0.7 * 0.7 or 0.49. The weighted beta for JNJ is 0.3 * 0.5 or 0.15. The beta for the portfolio is the sum of the weighted betas, that is, 0.49 + 0.15, or 0.64. Piece of cake, right?

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* Excess return is return over and above the return of a risk-free asset. In other words, excess return of an asset is the return of the asset minus the return of a risk-free asset. If a stock's return is 5% and a risk-free asset's return is 1%, the stock's excess return is 4%.