Beta coefficient
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- This article discusses the 'beta coefficient' as used in economics. For a more basic statistical term often used in regression, see standardized coefficient.
The Beta coefficient, is a key parameter in the Capital asset pricing model (CAPM). It measures the part of the asset's statistical variance that cannot be mitigated by sufficient diversification of the portfolio, because it is correlated with the variability of the other assets that are available in that market.
For example, if every stock in the New York Stock Exchange was uncorrelated with every other stock, then every stock would have a Beta of zero, and it would be possible to create a portfolio that was nearly risk free, simply by diversifying it sufficiently so that the variations in the individual stocks' prices averaged out. This would be like owning a casino: essentially none of the business risk of owning a casino comes from the uncertain outcomes of the games of chance played by the customers, because those are uncorrelated, and average out over any significant period of time. In reality, investments tend to be highly correlated, perhaps within an industry, or, as demonstrated most spectacularly in the Wall Street crash of 1929, in a market as a whole. This correlated risk, measured by Beta, is what actually creates almost all of the risk in a diversified portfolio.
The formula for the Beta of an asset is <math>\beta_a = \frac {\mathrm{Cov}(r_a,r_m)}{\mathrm{Var}(r_m)}</math> ,
where <math>r_m</math> measures the rate of return of the market and <math>r_a</math> measures the rate of return of the asset.
Beta is also referred to as financial elasticity or correlated relative volatility, and can be referred to as a measure of the asset's sensitivity of the asset's returns to market returns, its non-diversifiable risk, its systematic risk or market risk. On an individual asset level, measuring beta can give clues to volatility and liquidity in the marketplace. On a portfolio level, measuring beta is thought to separate a manager's skill from his or her willingness to take risk.
The beta movement should be distinguished from the actual returns of the stocks. For example, a sector may be performing well and may have good prospects, but the fact that its movement does not correlate well with the broader market index may decrease its beta. However, it should not be taken as a reflection on the overall attractiveness or the loss of it for the sector, or stock as the case may be. Beta is a measure of risk and not to be confused with the attractiveness of the investment.
The beta coefficient was actually born out of linear regression analysis. It is linked to a regression analysis of the returns of the stock index (x-axis) in a specific period versus the returns an individual security (y-axis) in a specific year. The regression line is then called the Security Characteristic Line (SCL).
- <math>SCL : r_{it} = \alpha_i + \beta_i r_{mt} + \epsilon_{it} \frac{}{}</math>
<math>\alpha_i</math> is called the stock's alpha coefficient and <math>\beta_i</math> is called the stock's beta coefficient. Both coefficients have an important role in Modern portfolio theory.
For example, in a year where the broad market or benchmark index returns 25% above the risk free rate, suppose two managers gain 50% above the risk free rate. Since this higher return is theoretically possible merely by taking a leveraged position in the broad market to double the beta so it is exactly 2.0, we would expect a skilled portfolio manager to have built the outperforming portfolio with a beta somewhat less than 2, such that the excess return not explained by the beta is positive. If one of the managers' portfolios has an average beta of 3.0, and the other's has a beta of only 1.5, then the CAPM simply states that the extra return of the first manager is not sufficient to compensate us for that manager's risk, whereas the second manager has done more than expected given the risk. Whether we can expect the second manager to duplicate that peformance in future periods is of course a different question.
[edit] Multiple Beta Model
The Arbitrage Pricing Theory (APT) has multiple betas in its model. In contrast to the CAPM that has only one risk factor, namely the overall market, APT has multiple risk factors. Each risk factor has a corresponding beta indicating the responsiveness of the asset being priced to that risk factor.
[edit] Calculation of Beta
To calculate Beta, one needs a list of returns for the asset and returns for the index; these returns can be daily, weekly or any period. Next, a plot should be made, with the index returns on the x-axis and the asset returns on the y-axis, in order to check that there are no serious violations of the linear regression model assumptions. The slope of the fitted line from the linear least-squares calculation is Beta. The y-intercept is the alpha.
