In finance, the beta (β) of a stock or portfolio is a number describing the relation of its returns with that of the financial market as a whole.^{[1]}
An asset with a beta of 0 means that its price is not at all correlated with the market. A positive beta means that the asset generally follows the market. A negative beta shows that the asset inversely follows the market; the asset generally decreases in value if the market goes up and vice versa.^{[2]}
Correlations are evident between companies within the same industry, or even within the same asset class (such as equities), as was demonstrated in the Wall Street Crash of 1929. These correlations, measured by Beta, creates almost all of the risk in a diversified portfolio.^{[citation needed]}
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 the diversification provided by the portfolio of many risky assets, because it is correlated with the return of the other assets that are in the portfolio. Beta can be estimated for individual companies using regression analysis against a stock market index.
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The formula for the beta of an asset within a portfolio is
where r_{a} measures the rate of return of the asset, r_{p} measures the rate of return of the portfolio, and Cov(r_{a},r_{p}) is the covariance between the rates of return. The portfolio of interest in the CAPM formulation is the market portfolio that contains all risky assets, and so the r_{p} terms in the formula are replaced by r_{m}, the rate of return of the market.
Beta is also referred to as financial elasticity or correlated relative volatility, and can be referred to as a measure of the sensitivity of the asset's returns to market returns, its nondiversifiable risk, its systematic risk, or market risk. On an individual asset level, measuring beta can give clues to volatility and liquidity in the marketplace. In fund management, measuring beta is thought to separate a manager's skill from his or her willingness to take risk.
The beta coefficient was born out of linear regression analysis. It is linked to a regression analysis of the returns of a portfolio (such as a stock index) (xaxis) in a specific period versus the returns of an individual asset (yaxis) in a specific year. The regression line is then called the Security characteristic Line (SCL).
α_{a} is called the asset's alpha and β_{a} is called the asset's beta coefficient. Both coefficients have an important role in Modern portfolio theory.
For an 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 investors can expect the second manager to duplicate that performance in future periods is of course a different question.

The SML essentially graphs the results from the capital asset pricing model (CAPM) formula. The xaxis represents the risk (beta), and the yaxis represents the expected return. The market risk premium is determined from the slope of the SML.
The relationship between β and required return is plotted on the securities market line (SML) which shows expected return as a function of β. The intercept is the nominal riskfree rate available for the market, while the slope is E(R_{m} − R_{f}). The securities market line can be regarded as representing a singlefactor model of the asset price, where Beta is exposure to changes in value of the Market. The equation of the SML is thus:
It is a useful tool in determining if an asset being considered for a portfolio offers a reasonable expected return for risk. Individual securities are plotted on the SML graph. If the security's risk versus expected return is plotted above the SML, it is undervalued since the investor can expect a greater return for the inherent risk. And a security plotted below the SML is overvalued since the investor would be accepting less return for the amount of risk assumed.
There is a simple formula between beta and volatility (sigma):
That is, beta is a combination of volatility and correlation. For example, if one stock has low volatility and high correlation, and the other stock has low correlation and high volatility, beta can decide which is more "risky".
This also leads to an inequality (since r is not greater than one):
In other words, 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 one (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).
Published betas typically use a stock market index such as S&P 500 as a benchmark. The benchmark should be chosen to be similar to the other assets chosen by the investor. Other choices may be an international index such as the MSCI EAFE. The choice of the index need not reflect the portfolio under question; e.g., beta for gold bars compared to the S&P 500 may be low or negative carrying the information that gold does not track stocks and may provide a mechanism for reducing risk. The restriction to stocks as a benchmark is somewhat arbitrary. Sometimes the market is defined as "all investable assets" (see Roll's critique); unfortunately, this includes lots of things for which returns may be hard to measure.
By definition, the market itself has an underlying beta of 1.0, and individual stocks are ranked according to how much they deviate from the macro market (for simplicity purposes, the S&P 500 is usually used as a proxy for the market as a whole). A stock that swings more than the market (i.e. more volatile) over time has a beta whose absolute value is greater than 1.0. If a stock moves less than the market, the absolute value of the stock's beta is less than 1.0.
More specifically, a stock that has a beta of 2 follows the market in an overall decline or growth, but does so by a factor of 2; meaning when the market has an overall decline of 3% a stock with a beta of 2 will fall 6%. Betas can also be negative, meaning the stock moves in the opposite direction of the market: a stock with a beta of 3 would decline 9% when the market goes up 3% and conversely would climb 9% if the market fell by 3%.
Higherbeta stocks mean greater volatility and are therefore considered to be riskier, but are in turn supposed to provide a potential for higher returns; lowbeta stocks pose less risk but also lower returns. In the same way a stock's beta shows its relation to market shifts, it also is used as an indicator for required returns on investment (ROI). If the market with a beta of 1 has an expected return increase of 8%, a stock with a beta of 1.5 should increase return by 12%.
Academic theory claims that higher risk investments should have higher return longterm. Wall Street saying is that "higher return requires higher risk", not that a risky investment will automatically do better. Some things may just be poor investments (e.g., playing roulette or lighting money on fire). Further, highly rational investors should use correlated volatility (beta) instead of simple volatility (sigma).
This expected return on equity, or equivalently, a firm's cost of equity, can be estimated using the Capital Asset Pricing Model (CAPM). According to the model, the expected return on equity is a function of a firm's equity beta (β_{E}) which, in turn, is a function of both leverage and asset risk (β_{A}):
where:
because:
and
An indication of the systematic riskiness attaching to the returns on ordinary shares. It equates to the asset Beta for an ungeared firm, or is adjusted upwards to reflect the extra riskiness of shares in a geared firm., i.e. the Geared Beta.^{[3]}
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.
To estimate beta, one needs a list of returns for the asset and returns for the index; these returns can be daily, weekly or any period. Then one uses standard formulas from linear regression. The slope of the fitted line from the linear leastsquares calculation is the estimated Beta. The yintercept is the alpha.
Myron Scholes and Joseph Williams (1977) provided a model for estimating betas from nonsynchronous data.^{[4]}
There is an inconsistency between how beta is interpreted and how it is calculated. The usual explanation is that it gives the asset volatility relative to the market volatility. If that were the case it should simply be the ratio of these volatilities. In fact, the standard estimation uses the slope of the least squares regression line—this gives a slope which is less than the volatility ratio. Specifically it gives the volatility ratio multiplied by the correlation of the plotted data. Tofallis (2008) provides a discussion of this,^{[5]} together with a real example involving AT&T. The graph showing monthly returns from AT&T is visibly more volatile than the index and yet the standard estimate of beta for this is less than one.
The relative volatility ratio described above is actually known as Total Beta (at least by appraisers who practice business valuation). Total Beta is equal to the identity: Beta/R or the standard deviation of the stock/standard deviation of the market (note: the relative volatility). Total Beta captures the security's risk as a standalone asset (since the correlation coefficient, R, has been removed from Beta), rather than part of a welldiversified portfolio. Since appraisers frequently value closelyheld companies as standalone assets, Total Beta is gaining acceptance in the business valuation industry. Appraisers can now use Total Beta in the following equation: Total Cost of Equity (TCOE) = riskfree rate + Total Beta*Equity Risk Premium. Once appraisers have a number of TCOE benchmarks, they can compare/contrast the risk factors present in these publiclytraded benchmarks and the risks in their closelyheld company to better defend/support their valuations.
Beta is not without its own criticisms. Seth Klarman of the Baupost group wrote in his timely classic Margin of Safety: "I find it preposterous that a single number reflecting past price fluctuations could be thought to completely describe the risk in a security. Beta views risk solely from the perspective of market prices, failing to take into consideration specific business fundamentals or economic developments. The price level is also ignored, as if IBM selling at 50 dollars per share would not be a lowerrisk investment than the same IBM at 100 dollars per share. Beta fails to allow for the influence that investors themselves can exert on the riskiness of their holdings through such efforts as proxy contests, shareholder resolutions, communications with management, or the ultimate purchase of sufficient stock to gain corporate control and with it direct access to underlying value. Beta also assumes that the upside potential and downside risk of any investment are essentially equal, being simply a function of that investment's volatility compared with that of the market as a whole. This too is inconsistent with the world as we know it. The reality is that past security price volatility does not reliably predict future investment performance (or even future volatility) and therefore is a poor measure of risk."^{[6]} Beta is also used to analyze the underlying implication of capital structure.

