# Beta

Beta can be used to refer to a security’s or portfolio’s exposure to the market or to other risk factors.

In its most common usage beta is intended to indicate how closely related a security’s returns are to a broad market index (the S&P 500 in the United States, the Nikkei 223 in Japan, the Euro Stoxx in Europe, etc.). If a stock has a beta of 0.35, and the market has a return of 2.00%, we would expected the stock to move 0.70% more than it would otherwise, 0.70% = 0.35 × 2.00%.

For a portfolio, we can measure beta as a percentage of AUM or in dollar terms. If a portfolio has a beta of \$500 million and the market falls 4.00%, we would expect the portfolio to lose \$20 million more than it would if the market had been flat.

While beta is most frequently used to refer to a security’s or portfolio’s sensitivity to broad market indexes, we can measure a beta to any risk factor. We can refer to a portfolio’s beta to sector indexes, excess return factors, commodity indexes, and just about anything else. A portfolio with a beta of \$50 million to oil, for example, would be expected to gain \$5 million more if oil prices increased by 10%, relative to if oil prices remained constant.

To determine beta — whether for a security or portfolio, as a percentage or in dollar terms — we perform a regression analysis. If Rt is the return at time t of the security or portfolio of interest, and Rm,t is the return at time t of the market or risk factor, then we have:

Rt = α + βRm,t + εt

where, α and β are constants, and εt is a mean zero error term. The calculation of beta is often done using excess returns, that is we would subtract the risk free rate from both Rt and Rm,t before calculating the regression equation. Defined this way, beta is closely related to the Capital Asset Pricing Model (CAPM).

While the preceding regression equation could be calculated using the manager’s realized returns, the regression calculation assumes that β is constant. For active managers with changing exposures a more accurate procedure is to measure the beta of the portfolio each day based on the current holdings using backcast returns.

Regression analysis assumes that the relationship between portfolio returns and factor returns are linear. Portfolios with significant non-linear exposures may need to use other methods, including stress testing and scenario analysis, in addition to regression analysis.

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