# Beta (finance)

In finance, the Beta (β) of a stock or portfolio is a number describing the correlated volatility of an asset in relation to the volatility of the benchmark that said asset is being compared to. This benchmark is generally the overall financial market and is often estimated via the use of representative indices, such as the S&P 500.1 Some interpretations of beta are explained in the following table:2

Value of Beta Interpretation Example
β < 0 Asset generally moves in the opposite direction as compared to the index Gold, which often moves opposite to the movements of the stock market
β = 0 Movement of the asset is uncorrelated with the movement of the benchmark Fixed-yield asset, whose growth is unrelated to the movement of the stock market
0 < β < 1 Movement of the asset is generally in the same direction as, but less than the movement of the benchmark Stable, "staple" stock such as a company that makes soap. Moves in the same direction as the market at large, but less susceptible to day-to-day fluctuation.
β = 1 Movement of the asset is generally in the same direction as, and about the same amount as the movement of the benchmark A representative stock, or a stock that is a strong contributor to the index itself.
β > 1 Movement of the asset is generally in the same direction as, but more than the movement of the benchmark Volatile stock, such as a tech stock, or stocks which are very strongly influenced by day-to-day market news.

It measures the part of the asset's statistical variance that cannot be removed by the diversification provided by the portfolio of many risky assets, because of the correlation of its returns with the returns of the other assets that are in the portfolio. Beta can be estimated for individual companies using regression analysis against a stock market index.

## Definition

The formula for the beta of an asset within a portfolio is

$\beta_a = \frac {\mathrm{Cov}(r_a,r_b)}{\mathrm{Var}(r_b)},~$

where ra measures the rate of return of the asset, rb measures the rate of return of the portfolio benchmark, and cov(ra,rb) 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 rb terms in the formula are replaced by rm, 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 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. 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) (x-axis) in a specific period versus the returns of an individual asset (y-axis) in a specific year. The regression line is then called the Security characteristic Line (SCL).

$\mathrm{SCL}: r_{a,t} = \alpha_a + \beta_a r_{m,t} + \epsilon_{a,t}$

$\alpha_a$ is called the asset's alpha and $\beta_a$ is called the asset'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. Because 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.

### Security market line

 The Security Market Line

The SML graphs the results from the capital asset pricing model (CAPM) formula. The x-axis represents the risk (beta), and the y-axis 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 security market line (SML) which shows expected return as a function of β. The intercept is the nominal risk-free rate available for the market, while the slope is E(Rm)− Rf. The security market line can be regarded as representing a single-factor model of the asset price, where Beta is exposure to changes in value of the Market. The equation of the SML is thus:

$\mathrm{SML}: E(R_i) - R_f = \beta_i (E(R_M) - R_f).~$

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 because the investor can expect a greater return for the inherent risk. A security plotted below the SML is overvalued because the investor would be accepting a lower return for the amount of risk assumed.

## Choice of benchmark

In the US, published betas typically use a stock market index such as S&P 500 as a benchmark. Other choices may be an international index such as the MSCI EAFE. The benchmark should be chosen to be similar to the other assets chosen by the investor. The ideal index would match the portfolio; for example, for a person who owns S&P 500 index funds and gold bars, the index would combine the S&P 500 and the price of gold. In practice a standard index is used. 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. A model portfolio may be stocks plus bonds. 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.

## Investing

By definition, the market itself has a 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 sometimes used as a proxy for the market as a whole). A stock whose returns vary more than the market's returns over time can have a beta whose absolute value is greater than 1.0 (whether it is, in fact, greater than 0 will depend on the correlation of the stock's returns and the market's returns). A stock whose returns vary less than the market's returns has a beta with an absolute value less than 1.0.

A stock with a beta of 2 has returns that change, on average, by twice the magnitude of the overall market's returns; when the market's return falls or rises by 3%, the stock's return will fall or rise (respectively) by 6% on average. (However, because beta also depends on the correlation of returns, there can be considerable variance about that average; the higher the correlation, the less variance; the lower the correlation, the higher the variance.) Beta can also be negative, meaning the stock's returns tend to move in the opposite direction of the market's returns. A stock with a beta of −3 would see its return decline 9% (on average) when the market's return goes up 3%, and would see its return climb 9% (on average) if the market's return falls by 3%.

Higher-beta stocks tend to be more volatile and therefore riskier, but provide the potential for higher returns. Lower-beta stocks pose less risk but generally offer lower returns. Some have challenged this idea, claiming that the data show little relation between beta and potential reward, or even that lower-beta stocks are both less risky and more profitable (contradicting CAPM).3 In the same way a stock's beta shows its relation to market shifts, it is also an indicator for required returns on investment (ROI). Given a risk-free rate of 2%, for example, if the market (with a beta of 1) has an expected return of 8%, a stock with a beta of 1.5 should return 11% (= 2% + 1.5(8% − 2%)) in accordance with the financial CAPM model.

Academic theory claims that higher-risk investments should have higher returns over the long-term. Wall Street has a saying 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). Further, highly rational investors should consider correlated volatility (beta) instead of simple volatility (sigma). Theoretically, a negative beta equity is possible; for example, an inverse ETF should have negative beta to the relevant index. Also, a short position should have opposite beta.

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 (βE) which, in turn, is a function of both leverage and asset risk (βA):

$K_{E} = R_{F} + \beta_E (R_{M} - R_{F})$

where:

• KE = firm's cost of equity
• RF = risk-free rate (the rate of return on a "risk free investment"; e.g., U.S. Treasury Bonds)
• RM = return on the market portfolio
• $\beta_E = \beta =\left[ \beta_A - \beta_D \left(\frac {D}{V}\right) \right] \frac {V}{E}$

because:

$\beta_A = \beta_D \left(\frac {D}{V}\right) + \beta_E \left(\frac {E}{V}\right)$

and

Firm Value (V) + Cash and Risk-Free Securities = Debt Value (D) + Equity Value (E)

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.4

## 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.

Multiple-factor models contradict CAPM by claiming that some other factors can influence return, therefore one may find two stocks (or funds) with equal beta, but one may be a better investment.

## Estimation of beta

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 least-squares calculation is the estimated Beta. The y-intercept is the alpha.

Myron Scholes and Joseph Williams (1977) provided a model for estimating betas from nonsynchronous data.5

Beta specifically gives the volatility ratio multiplied by the correlation of the plotted data. To take an extreme example, something may have a beta of zero even though it is highly volatile, provided it is uncorrelated with the market. Tofallis (2008) provides a discussion of this,6 together with a real example involving AT&T Inc. 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 stand-alone asset (because the correlation coefficient, R, has been removed from Beta), rather than part of a well-diversified portfolio. Because appraisers frequently value closely held companies as stand-alone 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) = risk-free rate + Total Beta*Equity Risk Premium. Once appraisers have a number of TCOE benchmarks, they can compare/contrast the risk factors present in these publicly traded benchmarks and the risks in their closely held company to better defend/support their valuations.

## Extreme and interesting cases

• Beta has no upper or lower bound, and betas as large as 3 or 4 will occur with highly volatile stocks.
• Beta can be zero. Some zero-beta assets are risk-free, such as treasury bonds and cash. However, simply because a beta is zero does not mean that it is risk-free. A beta can be zero simply because the correlation between that item's returns and the market's returns is zero. An example would be betting on horse racing. The correlation with the market will be zero, but it is certainly not a risk-free endeavor.
• A negative beta simply means that the stock is inversely correlated with the market.
• A negative beta might occur even when both the benchmark index and the stock under consideration have positive returns. It is possible that lower positive returns of the index coincide with higher positive returns of the stock, or vice versa. The slope of the regression line in such a case will be negative.
• Using beta as a measure of relative risk has its own limitations. Most analyses consider only the magnitude of beta. Beta is a statistical variable and should be considered with its statistical significance (R square value of the regression line). Higher R square value implies higher correlation and a stronger relationship between returns of the asset and benchmark index.
• If beta is a result of regression of one stock against the market where it is quoted, betas from different countries are not comparable.
• Staple stocks are thought to be less affected by cycles and usually have lower beta. Procter & Gamble, which makes soap, is a classic example. Other similar ones are Philip Morris (tobacco) and Johnson & Johnson (Health & Consumer Goods). Utility stocks are thought to be less cyclical and have lower beta as well, for similar reasons.
• 'Tech' stocks typically have higher beta. An example is the dot-com bubble. Although tech did very well in the late 1990s, it also fell sharply in the early 2000s, much worse than the decline of the overall market.
• Foreign stocks may provide some diversification. World benchmarks such as S&P Global 100 have slightly lower betas than comparable US-only benchmarks such as S&P 100. However, this effect is not as good as it used to be; the various markets are now fairly correlated, especially the US and Western Europe.citation needed
• Derivatives and other non-linear assets. Beta relies on a linear model. An out of the money option may have a distinctly non-linear payoff. The change in price of an option relative to the change in the price of the underlying asset (for example a stock) is not constant. For example, if one purchased a put option on the S&P 500, the beta would vary as the price of the underlying index (and indeed as volatility, time to expiration and other factors) changed. (see options pricing, and Black Scholes).

## Criticism

Seth Klarman of the Baupost group wrote in 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 lower-risk 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."7

## Notes

1. ^ Levinson, Mark (2006). Guide to Financial Markets. London: The Economist (Profile Books). pp. 155–6. ISBN 1-86197-956-8.
2. ^ Definition of Beta Definition via Wikinvest
3. ^ McAlpine, Chad (2010). "Low-risk TSX stocks have outearned riskiest peers over 30-year period", The Financial Post Trading Desk, June 22, 2010
4. ^
5. ^ Scholes, Myron; Williams, Joseph (1977). "Estimating betas from nonsynchronous data". Journal of Financial Economics 5 (3): 309–327. doi:10.1016/0304-405X(77)90041-1.
6. ^ Tofallis, Chris (2008). "Investment Volatility: A Critique of Standard Beta Estimation and a Simple Way Forward". European Journal of Operational Research 187 (3): 1358–1367. doi:10.1016/j.ejor.2006.09.018.
7. ^ Klarman, Seth; Williams, Joseph (1991). "Beta". Journal of Financial Economics 5 (3): 117. doi:10.1016/0304-405X(77)90041-1.