Beta (finance)

In finance, the beta (β or market beta or beta coefficient) is a measure of the systematic risk of an investment arising from exposure to general market movements (as opposed to idiosyncratic risk). Beta is also sometimes called 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.

For other uses, see Beta (disambiguation).

"Beta coefficient" redirects here. For the general statistical concept, see Standardized coefficient.

By definition, a value-weighted portfolio of all investable assets has a beta of 1. A beta above (below) 1 indicates an investment whose prices move more (less) than 1-to-1 with the market on average.[1] Few stocks have negative betas (tending to go up when the market goes down). Most stocks have betas between 0 and 3. Treasury bills and commodities tend to have low or zero betas, call options tend to have high betas, put options tend to have negative betas.

Importance

Beta is the hedge ratio of an investment with respect to the stock market. For example, to hedge away the market-risk of a stock with a market beta of 2.0, an investor could short $2,000 in the stock market for every $1,000 invested in the stock. Thus insured, movements of the overall stock market no longer influence the combined position on average.

Beta thus measures the contribution of an individual investment to the risk of the market portfolio that was not reduced by diversification. It does not measure the risk when an investment is held on a stand-alone basis. In the idealized capital asset pricing model (CAPM), beta risk is the only kind of risk for which investors should receive an expected return higher than the risk-free rate of interest.[2]

Definition

The market beta of an asset i is defined by a linear regression of the rate of return of asset i on the rate of return on the (typically value-weighted) stock-market index:

${\displaystyle r_{i,t}=\alpha _{i}+\beta _{i}\cdot r_{m,t}+\varepsilon _{t},}$

where ε_t is an unbiased error term to be minimized. The solution is

${\displaystyle \beta _{i}={\frac {\mathrm {Cov} (r_{i},r_{m})}{\mathrm {Var} (r_{m})}},}$

where Cov and Var are the covariance and variance operators. The y-intercept is often referred to as the alpha.

By using the relationships between standard deviation, variance and correlation: ${\displaystyle \sigma _{a}={\sqrt {\mathrm {Var} (r_{a})}}}$, ${\displaystyle \sigma _{b}={\sqrt {\mathrm {Var} (r_{b})}}}$, ${\displaystyle \rho _{a,b}=\mathrm {Cov} (r_{a},r_{b})/{\sqrt {\mathrm {Var} (r_{a})\mathrm {Var} (r_{b})}}}$, this expression can also be written as

${\displaystyle \beta _{i}=\rho _{i,m}{\frac {\sigma _{i}}{\sigma _{m}}}}$,

where ρ_i,m is the correlation of the two returns, and σ_i and σ_m are the respective volatilities.[3] The equation shows that the idiosyncratic risk (σ_i) is related to but often very different market beta. Attempts have been made to estimate the three ingredient components separately, but this has not led to better estimates of market-betas.

Adding to a portfolio

Suppose an investor has all his money in an asset class M and wishes to move a small amount to an asset class Y. For example, X could be U.S. stocks, while Y could be stocks of a different country, or bonds. Then the new portfolio, Z, can be expressed symbolically

${\displaystyle Z=(1-\delta )M+\delta I.}$

The variance can be computed as

${\displaystyle \operatorname {Var} (Z)=(1-\delta )^{2}\operatorname {Var} (M)+2\delta (1-\delta )\operatorname {Cov} (M,I)+\delta ^{2}\operatorname {Var} (I)}$

which can be simplified by ignoring δ² terms:

${\displaystyle \operatorname {Var} (Z)\approx (1-2\delta )\operatorname {Var} (M)+2\delta \operatorname {Cov} (M,I).}$

The first formula is exact, while the second one is only valid for small δ. Using the formula for β of Y relative to X,

${\displaystyle \beta =\operatorname {Cov} (M,I)/\operatorname {Var} (I),}$

we can compute

${\displaystyle \operatorname {Var} (Z)/\operatorname {Var} (M)\approx 1+2\delta (\beta -1).}$

This suggests that an asset with β greater than 1 will increase variance, while an asset with β less than 1 will decrease variance, if added in a small amount.

Empirical Estimation

Only the historical data and thus a historical beta can be directly observed. (This is most often obtained with standard OLS linear regression.) However, there are two problems: The historical beta is based on one particular random realization of history, and market-betas are known to move over time. Thus, even the historical true market-beta is never known. Moreover, investors are more interested in the true prevailing (and thus future beta), which relates to the risk contribution going forward.

Despite the problems, the historical beta remains an obvious benchmark predictor. It is obtained as the slope of the fitted line from the linear least-squares estimator. The evidence suggests that the OLS regression should be estimated on about 1-3 years worth of daily stock (not weekly or monthly) stock returns.

Even better performing estimators reflect the tendency of betas (like rates of return) for regression toward the mean, induced by underlying changes in the true beta and/or historical randomness. (Intuitively, one would not suggest a company with high return [e.g., a drug discovery] last year also to have as high a return next year.) Better estimators include the Blume/Bloomberg beta[4] (used prominently on many financial websites), the Vasicek beta[5], the Scholes-Williams beta[6], the Dimson beta[7], and the Welch beta[8].

• The Blume beta estimates the future beta as 2/3 times the historical OLS beta plus 1/3 times the number 1.
• The Vasicek beta varies the weight between the historical OLS beta and the number 1 (or the average market beta if the portfolio is not value-weighted) by the volatility of the stock and the heterogeneity of betas in the overall market. It can be viewed either as an optimal Bayesian estimator or a random-effects estimator under the (violated) assumption that the underlying market-beta does not move. It is modestly difficult to implement.
• The Scholes-Williams and Dimson betas are procedures to account for infrequent trading causing non-synchronously quoted prices. They are rarely useful when stock prices are quoted at day's end and easily available to analysts (as they are in the US), because they incur an efficiency loss when trades are reasonably synchronous. However, they can be very useful in cases in which frequent trades are not observed (e.g., as in private equity).
• The Welch beta is a recent innovation that winsorizes (truncates) investment returns according to the contemporaneous market rate of return before running the OLS regression. It furthermore applies a simple exponential decay. This estimator outperforms all other known methods in the context of U.S. stocks and is relatively easy to implement.

These estimators attempt to uncover the prevailing current market-beta. When long-term market-betas are required, further regression toward the mean over long horizons should be considered.

Use in Performance Measurement

In fund management, measuring beta is thought to separate a manager's skill from his or her willingness to take risk.

The portfolio of interest in the CAPM formulation is the market portfolio that contains all risky assets, and so the r_b terms in the formula are replaced by r_m, the rate of return of the market. The regression line is then called the security characteristic line (SCL).

${\displaystyle \mathrm {SCL} :r_{a,t}=\alpha _{a}+\beta _{a}r_{m,t}+\varepsilon _{a,t}}$

${\displaystyle \alpha _{a}}$ is called the asset's alpha and ${\displaystyle \beta _{a}}$ 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

Main article: 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 R_f available for the market, while the slope is E(R_m)− R_f (for market return R_m). 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, giving the expected value of the return on asset i, is thus:

${\displaystyle \mathrm {SML} :E(R_{i})-R_{f}=\beta _{i}(E(R_{M})-R_{f}).~}$

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 U.S., published betas typically use a stock market index such as the S&P 500 as a benchmark. The S&P 500 is a popular index of U.S. large-cap stocks. Other choices may be an international index such as the MSCI EAFE. The benchmark is often chosen to be similar to the assets chosen by the investor. 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 many things for which returns may be hard to measure.

Investing

By definition, the market itself has a beta of 1, and individual stocks are ranked according to how much they deviate from the macro market (for simplicity purposes, the S&P 500 Index 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 1.0 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; 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).[9] 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

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

${\displaystyle K_{E}=R_{F}+\beta _{E}(R_{M}-R_{F})}$

where:

• K_E = firm's cost of equity
• R_F = risk-free rate (the rate of return on a "risk free investment"; e.g., U.S. Treasury Bonds)
• R_M = return on the market portfolio
• ${\displaystyle \beta _{E}=\beta =\left[\beta _{A}-\beta _{D}\left({\frac {D}{V}}\right)\right]{\frac {V}{E}}}$

because:

${\displaystyle \beta _{A}=\beta _{D}\left({\frac {D}{V}}\right)+\beta _{E}\left({\frac {E}{V}}\right)}$

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.[10]

Multiple beta model

Occasionally, other betas than market-betas are used. For example, a beta with respect to oil-price changes would sometimes be called an "oil-beta" rather than "market-beta" to clarify the difference. The beta commonly quoted in mutual fund analyses often measures the exposure to a specific fund benchmark, rather than to the overall stock market. Such a beta would measure the risk this fund would add to a holder of the benchmark, rather than to a portfolio diversified among all stocks or fund types.[11]

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.

Interpretations

Some interpretations of beta are explained in the following table:[12]

Value of Beta	Interpretation	Example
β < -1	Asset moves in the opposite direction, and in a greater amount than the negative of the benchmark	An inverse fund, such as the Direxion 3X Bear ETF, which seeks 300% the inverse of the S&P 500 Index.
-1 < β < 0	Asset movement is in the opposite direction of the benchmark	An inverse exchange-traded fund or a short position
β = 0	Asset movement is uncorrelated to the benchmark	Fixed-yield asset, whose growth is unrelated to the movement of the stock market
0 < β < 1	Asset moves in the same direction, but in a lesser amount than 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	Asset moves in the same direction and in the same amount as the benchmark	A representative stock, or a stock that is a strong contributor to the index itself.
β > 1	Asset moves in the same direction, but in a greater amount than the benchmark	Stocks which are very strongly influenced by day-to-day market news, or by the general health of the economy.

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. An alternative to standard beta is downside beta.

Beta is always measured in respect to some benchmark. Therefore, an asset may have different betas depending on which benchmark is used. Just a number is useless if the benchmark is not known.

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.
• On the other hand, if a stock has a moderately low but positive correlation with the market, but a high volatility, then its beta may still be high.
• 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). Closer to 1 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.
• Utility stocks commonly show up as examples of low beta. These have some similarity to bonds, in that they tend to pay consistent dividends, and their prospects are not strongly dependent on economic cycles. They are still stocks, so the market price will be affected by overall stock market trends, even if this does not make sense.
• 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).
• 'Tech' stocks are commonly equated with higher beta. This is based on experience of the dot-com bubble around year 2000. 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. More recently, this is not a good example.
• During the 2008 market fall, finance stocks did very poorly, much worse than the overall market. Then in the following years they gained the most, although not to make up for their losses.
• 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.
• 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 model).

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."[13]

At the industry level, beta tends to underestimate downside beta two-thirds of the time (resulting in value overestimation) and overestimate upside beta one-third of the time resulting in value underestimation.[14]

Another weakness of beta can be illustrated through an easy example by considering two hypothetical stocks, A and B. The returns on A, B and the market follow the probability distribution below:

Probability	Market	Stock A	Stock B
0.25	−30%	−15%	−60%
0.25	−15%	−7.5%	−30%
0.25	15%	30%	7.5%
0.25	30%	60%	15%

The table shows that stock A goes down half as much as the market when the market goes down and up twice as much as the market when the market goes up. Stock B, on the other hand, goes down twice as much as the market when the market goes down and up half as much as the market when the market goes up. Most investors would label stock B as more risky. In fact, stock A has better return in every possible case. However, according to the capital asset pricing model, stock A and B would have the same beta, meaning that theoretically, investors would require the same rate of return for both stocks. Of course it is entirely expected that this example could break the CAPM as the CAPM relies on certain assumptions one of the most central being the nonexistence of arbitrage, however, in this example buying stock A and selling stock B is an example of an arbitrage as stock A is worth more in every scenario. This is an illustration of how using standard beta might mislead investors. The dual-beta model, in contrast, takes into account this issue and differentiates downside beta from upside beta, or downside risk from upside risk, and thus allows investors to make better informed investing decisions.[14]

See also

• Alpha (finance)
• Beta Coefficient via Wikinvest
• Betavexity
• CSS Theory - Beta
• Cost of capital
• Financial risk
• Hamada's equation
• Macro risk
• Risk factor (finance)
• Treynor ratio
• WACC

References

1. https://www.investopedia.com/terms/b/beta.asp
2. Fama, Eugene (1976). Foundations of Finance: Portfolio Decisions and Securities Prices. Basic Books. ISBN 978-0465024995.
3. 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. arXiv:1109.4422. doi:10.1016/j.ejor.2006.09.018.
4. Blume, Marshall E. (1975). "Betas and Their Regression Tendencies". The Journal of Finance. 30 (3): 785–795. doi:10.1111/j.1540-6261.1975.tb01850.x. ISSN 1540-6261.
5. Vasicek, Oldrich A. (1973). "A Note on Using Cross-Sectional Information in Bayesian Estimation of Security Betas". The Journal of Finance. 28 (5): 1233–1239. doi:10.1111/j.1540-6261.1973.tb01452.x. ISSN 1540-6261.
6. Scholes, Myron; Williams, Joseph (1977-12-01). "Estimating betas from nonsynchronous data". Journal of Financial Economics. 5 (3): 309–327. doi:10.1016/0304-405X(77)90041-1. ISSN 0304-405X.
7. Dimson, Elroy (1979-06-01). "Risk measurement when shares are subject to infrequent trading". Journal of Financial Economics. 7 (2): 197–226. doi:10.1016/0304-405X(79)90013-8. ISSN 0304-405X.
8. Welch, Ivo (2020-05-18). "Simpler Better Market Betas". Rochester, NY. doi:10.2139/ssrn.3371240. Cite journal requires |journal= (help)
9. McAlpine, Chad (2010). "Low-risk TSX stocks have outearned riskiest peers over 30-year period", The Financial Post Trading Desk, June 22, 2010
10. "Click here definition of Equity Beta, what is Equity Beta, what does Equity Beta mean? Finance Glossary - Search our financial terms for a definition - London South East". Lse.co.uk. Retrieved 2012-12-03.
11. Ilmanen, Antti (2011). Expected Returns: An Investor's Guide to Harvesting Market Rewards. John Wiley & Sons. ISBN 978-1119990727.
12. Definition of Beta Definition via Wikinvest
13. Klarman, Seth; Williams, Joseph (1991). "Beta". Journal of Financial Economics. 5 (3): 117. doi:10.1016/0304-405X(77)90041-1.
14. James Chong; Yanbo Jin; Michael Phillips (April 29, 2013). "The Entrepreneur's Cost of Capital: Incorporating Downside Risk in the Buildup Method" (PDF). Retrieved 25 June 2013.

External links

• ETFs & Diversification: A Study of Correlations
• Leverage and diversification effects of public companies
• Calculate Beta in a Spreadsheet
• Free Beta Calculator for any Asset-Index pair
• Calculate Sharpe Ratio in Excel
• Calculate Beta in Excel
• Online Portfolio Beta Calculator