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Monday, May 4, 2009

Relation to beta

Besides an investment manager simply making more money than a passive strategy, there is another issue: Although the strategy of investing in every stock appeared to perform better than 75 percent of investment managers, the price of the stock market as a whole fluctuates up and down, and could be on a downward decline for many years before returning to its previous price.

The passive strategy appeared to generate the market-beating return over periods of 10 years or more. This strategy may be risky for those who feel they might need to withdraw their money before a 10-year holding period, for example. Thus investment managers who employ a strategy which is less likely to lose money in a particular year are often chosen by those investors who feel that they might need to withdraw their money sooner.

The measure of the correlated volatility of an investment (or an investment manager's track record) relative to the entire market is called beta. Note the "correlated" modifier: an investment can be twice as volatile as the total market, but if its correlation with the market is only 0.5, its beta to the market will be 1.

Investors can use both alpha and beta to judge a manager's performance. If the manager has had a high alpha, but also a high beta, investors might not find that acceptable, because of the chance they might have to withdraw their money when the investment is doing poorly.

These concepts not only apply to investment managers, but to any kind of investment.

Origin of the concept

The concept and focus on Alpha comes from an observation increasingly made during the middle of the twentieth century, that around 75 percent of stock investment managers did not make as much money picking investments as someone who simply invested in every stock in proportion to the weight it occupied in the overall market in terms of market capitalization, or indexing. Many academics felt that this was due to the stock market being "efficient" which means that since so many people were paying attention to the stock market all the time, the prices of stocks rapidly moved to the correct price at any one moment, and that only random variation beyond the control of the manager made it possible for one manager to achieve better results than another, before fees or taxes were considered. A belief in efficient markets spawned the creation of market capitalization weighted index funds that seek to replicate the performance of investing in an entire market in the weights that each of the equity securities comprises in the overall market. The best examples are the S&P 500 and the Wilshire 5000 which approximately represent the 500 largest equities and the largest 5000 securities respectively, accounting for approximately 80%+ and 99%+ of the total market capitalization of the US market as a whole.

In fact, to many investors, this phenomenon created a new standard of performance that must be matched: an investment manager should not only avoid losing money for the client and should make a certain amount of money, but in fact should make more money than the passive strategy of investing in everything equally (since this strategy appeared to be statistically more likely to be successful than the strategy of any one investment manager). The name for the additional return above the expected return of the beta adjusted return of the market is called "Alpha".

Alpha (investment)

Alpha is a risk-adjusted measure of the so-called active return on an investment. It is the return in excess of the compensation for the risk borne, and thus commonly used to assess active managers' performances. Often, the return of a benchmark is subtracted in order to consider relative performance, which yields Jensen's alpha.

The alpha coefficient (αi) is a parameter in the capital asset pricing model (CAPM). It is the intercept of the Security Characteristic Line (SCL). Alternatively, it is also the coefficient of the constant in a market model regression.

It can be shown that in an efficient market, the expected value of the alpha coefficient equals the return of the risk free asset: Ei) = rf.

Therefore the alpha coefficient indicates how an investment has performed after accounting for the risk it involved:

  • αi < rf: the investment has earned too little for its risk (or, was too risky for the return)
  • αi = rf: the investment has earned a return adequate for the risk taken
  • αi > rf: the investment has a return in excess of the reward for the assumed risk

For instance, although a return of 20% may appear good, the investment can still have a negative alpha if it's involved in an excessively risky position.

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 securities 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 and the market 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. Many precious metals and precious-metal-related stocks are beta-negative as their value tends to increase when the general market is down and vice versa.
  • 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, i.e. the beta, in such a case will be negative.
  • Using beta as a measure of relative risk has its own limitations. Most analysis 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.
  • Since beta is a result of regression of one stock against the market where it is quoted, betas from different countries are not comparable.
  • Staples stocks are thought to be less affected by cycles and usually have lower beta. Procter & Gamble is a classic example, who makes soap. Other similar ones are Philip Morris (tobacco) and Anheuser-Busch (alcohol). Utility stocks are thought to be less cyclical and have lower beta as well, for similar reasons.
  • 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.

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. 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 the estimated Beta. The y-intercept is the alpha.

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

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, 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 stand-alone asset (since the correlation coefficient, R, has been removed from Beta), rather than part of a well-diversified portfolio. Since 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.

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.


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 above 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%.

Higher-beta stocks mean greater volatility and are therefore considered to be riskier, but are in turn supposed to provide a potential for higher returns; low-beta 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%.

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} \frac{}{})


  • 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}


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


Firm Value (V) = 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. th Geared Beta.

Choice of benchmark

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: beta for e.g. 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.

Beta volatility and correlation

β = (σ / σm)r

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".

\sigma \ge |\beta| \sigma_m

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).


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

\beta_a = \frac {\mathrm{Cov}(r_a,r_p)}{\mathrm{Var}(r_p)} ,

where ra measures the rate of return of the asset, rp measures the rate of return of the portfolio, and Cov(ra,rp) is the covariance between the rates of return. In the CAPM formulation, the portfolio is the market portfolio that contains all risky assets, and so the rp 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. On a portfolio level, 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).

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

αa is called the asset's alpha coefficient and βa 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.

Beta (finance)

The beta coefficient, in terms of finance and investing, describes how the expected return of a stock or portfolio is correlated to the return of the financial market as a whole.

An asset with a beta of 0 means that its price is not at all correlated with the market; that asset is independent. 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 (as is common with precious metals).

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. This correlated risk, measured by Beta, creates almost all of the risk in a diversified portfolio.

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.

Earnings yield

Earnings yield is the quotient of earnings per share divided by the share price. It is the reciprocal of the P/E ratio.

The earnings yield is quoted as a percentage, allowing an easy comparison to going bond rates.


The earnings yield can be used to compare the earnings of a stock, sector or the whole market against bond yields. Generally, the earnings yields of equities are higher than the yield of risk-free treasury bonds reflecting the additional risk involved in equity investments. The average P/E ratio for U.S. stocks from 1900 to 2005 is 14,[citation needed] which equates to an earnings yield of over 7%.

The earnings yield is also the cost to a publicly traded company of raising expansion capital through the issuance of stock.

Other uses

Earnings yield is one of the factors discussed in Joel Greenblatt's The Little Book That Beats the Market. However, Greenblatt uses an adjusted earnings yield formula to account for the fact that different companies have different debt levels and tax rates.