In
finance
Finance refers to monetary resources and to the study and Academic discipline, discipline of money, currency, assets and Liability (financial accounting), liabilities. As a subject of study, is a field of Business administration, Business Admin ...
, the beta ( or market beta or beta coefficient) is a
statistic
A statistic (singular) or sample statistic is any quantity computed from values in a sample which is considered for a statistical purpose. Statistical purposes include estimating a population parameter, describing a sample, or evaluating a hypot ...
that measures the expected increase or decrease of an individual
stock price in proportion to movements of the
stock market
A stock market, equity market, or share market is the aggregation of buyers and sellers of stocks (also called shares), which represent ownership claims on businesses; these may include ''securities'' listed on a public stock exchange a ...
as a whole. Beta can be used to indicate the contribution of an individual
asset
In financial accounting, an asset is any resource owned or controlled by a business or an economic entity. It is anything (tangible or intangible) that can be used to produce positive economic value. Assets represent value of ownership that can b ...
to the
market risk
Market risk is the risk of losses in positions arising from movements in market variables like prices and volatility.
There is no unique classification as each classification may refer to different aspects of market risk. Nevertheless, the m ...
of a
portfolio when it is added in small quantity. It refers to an asset's non-diversifiable
risk
In simple terms, risk is the possibility of something bad happening. Risk involves uncertainty about the effects/implications of an activity with respect to something that humans value (such as health, well-being, wealth, property or the environ ...
,
systematic risk, or market risk. Beta is not a measure of
idiosyncratic risk.
Beta is the hedge ratio of an investment with respect to the stock market. For example, to hedge out the market-risk of a stock with a market beta of 2.0, an investor would
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 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.
The beta of an asset is compared to the market as a whole, usually the
S&P 500
The Standard and Poor's 500, or simply the S&P 500, is a stock market index tracking the stock performance of 500 leading companies listed on stock exchanges in the United States. It is one of the most commonly followed equity indices and in ...
. By definition, the value-weighted average of all market-betas of all investable assets with respect to the
value-weighted market index is 1. If an asset has a beta above 1, it indicates that its return moves more than 1-to-1 with the return of the market-portfolio, on average; that is, it is more volatile than the market. In practice, few stocks have negative betas (tending to go up when the market goes down). Most stocks have betas between 0 and 3.
Most fixed income instruments and
commodities
In economics, a commodity is an economic good, usually a resource, that specifically has full or substantial fungibility: that is, the market treats instances of the good as equivalent or nearly so with no regard to who produced them.
Th ...
tend to have low or zero betas;
call option
In finance, a call option, often simply labeled a "call", is a contract between the buyer and the seller of the call Option (finance), option to exchange a Security (finance), security at a set price. The buyer of the call option has the righ ...
s tend to have high betas; and
put option
In finance, a put or put option is a derivative instrument in financial markets that gives the holder (i.e. the purchaser of the put option) the right to sell an asset (the ''underlying''), at a specified price (the ''strike''), by (or on) a ...
s and
short positions and some
inverse ETFs tend to have negative betas.
Technical aspects
Mathematical definition
The market beta
of an asset
, observed on
occasions, is defined by (and best obtained via) a
linear regression
In statistics, linear regression is a statistical model, model that estimates the relationship between a Scalar (mathematics), scalar response (dependent variable) and one or more explanatory variables (regressor or independent variable). A mode ...
of the rate of return
of asset
on the rate of return
of the (typically value-weighted) stock-market index
:
:
where
is an unbiased error term whose squared error should be minimized. The coefficient
is often referred to as the
alpha
Alpha (uppercase , lowercase ) is the first letter of the Greek alphabet. In the system of Greek numerals, it has a value of one. Alpha is derived from the Phoenician letter ''aleph'' , whose name comes from the West Semitic word for ' ...
.
The
ordinary least squares
In statistics, ordinary least squares (OLS) is a type of linear least squares method for choosing the unknown parameters in a linear regression
In statistics, linear regression is a statistical model, model that estimates the relationship ...
solution is:
:
where
and
are the
covariance
In probability theory and statistics, covariance is a measure of the joint variability of two random variables.
The sign of the covariance, therefore, shows the tendency in the linear relationship between the variables. If greater values of one ...
and
variance
In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...
operators. Betas with respect to different market indexes are not comparable.
Relationship between own risk and beta risk
By using the relationship between
standard deviation
In statistics, the standard deviation is a measure of the amount of variation of the values of a variable about its Expected value, mean. A low standard Deviation (statistics), deviation indicates that the values tend to be close to the mean ( ...
and
variance
In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...
,
and the definition of
correlation
In statistics, correlation or dependence is any statistical relationship, whether causal or not, between two random variables or bivariate data. Although in the broadest sense, "correlation" may indicate any type of association, in statistics ...
, market beta can also be written as
:
,
where
is the correlation of the two returns, and
,
are the respective
volatilities. This equation shows that the idiosyncratic risk (
) is related to but often very different to market beta. If the idiosyncratic risk is 0 (i.e., the stock returns do not move), so is the market-beta. The reverse is not the case: A coin toss bet has a zero beta but not zero risk.
Attempts have been made to estimate the three ingredient components separately, but this has not led to better estimates of market-betas.
Adding an asset to the market portfolio
Suppose an investor has all his money in the market
and wishes to move a small amount into asset class
. The new portfolio is defined by
:
The variance can be computed as
:
For small values of
, the terms in
can be ignored,
:
Using the definition of
this is
:
This suggests that an asset with
greater than 1 increases the portfolio variance, while an asset with
less than 1 decreases it ''if'' added in a small amount.
Beta as a linear operator
Market-beta can be weighted, averaged, added, etc. That is, if a portfolio consists of 80% asset A and 20% asset B, then the beta of the portfolio is 80% times the beta of asset A and 20% times the beta of asset B.
:
Financial analysis
In practice, the choice of index makes relatively little difference in the market betas of individual assets, because broad value-weighted market indexes tend to move closely together. Academics tend to prefer to work with a value-weighted market portfolio due to its attractive aggregation properties and its close link with the
capital asset pricing model
In finance, the capital asset pricing model (CAPM) is a model used to determine a theoretically appropriate required rate of return of an asset, to make decisions about adding assets to a Diversification (finance), well-diversified Portfolio (f ...
(CAPM). Practitioners tend to prefer to work with the
S&P 500
The Standard and Poor's 500, or simply the S&P 500, is a stock market index tracking the stock performance of 500 leading companies listed on stock exchanges in the United States. It is one of the most commonly followed equity indices and in ...
due to its easy in-time availability and availability to hedge with stock index futures.
In the idealized 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. When used within the context of the CAPM, beta becomes a measure of the appropriate expected rate of return. Due to the fact that the overall rate of return on the firm is weighted rate of return on its debt and its equity, the market-beta of the overall
unlevered firm is the weighted average of the firm's debt beta (often close to 0) and its levered equity beta.
In fund management, adjusting for exposure to the market separates out the component that fund managers should have received given that they had their specific exposure to the market. For example, if the stock market went up by 20% in a given year, and a manager had a portfolio with a market-beta of 2.0, this portfolio should have returned 40% in the absence of specific stock picking skills. This is measured by the
alpha
Alpha (uppercase , lowercase ) is the first letter of the Greek alphabet. In the system of Greek numerals, it has a value of one. Alpha is derived from the Phoenician letter ''aleph'' , whose name comes from the West Semitic word for ' ...
in the market-model, holding beta constant.
Occasionally, other betas than market-betas are used. The
arbitrage pricing theory
In finance, arbitrage pricing theory (APT) is a multi-factor model for asset pricing which relates various macro-economic (systematic) risk variables to the pricing of financial assets. Proposed by economist Stephen Ross (economist), Stephen Ross i ...
(APT) has multiple factors in its model and thus requires multiple betas. (The
CAPM has only one
risk factor
In epidemiology, a risk factor or determinant is a variable associated with an increased risk of disease or infection.
Due to a lack of harmonization across disciplines, determinant, in its more widely accepted scientific meaning, is often use ...
, namely the overall market, and thus works only with the plain beta.) For example, a beta with respect to
oil price
The price of oil, or the oil price, generally refers to the spot price of a Oil barrel, barrel () of benchmark crude oil—a reference price for buyers and sellers of crude oil such as West Texas Intermediate (WTI), Brent Crude, Dubai Crud ...
changes would sometimes be called an "oil-beta" rather than "market-beta" to clarify the difference.
Betas commonly quoted in
mutual fund
A mutual fund is an investment fund that pools money from many investors to purchase Security (finance), securities. The term is typically used in the United States, Canada, and India, while similar structures across the globe include the SICAV in ...
analyses often measure the exposure to a specific fund benchmark, rather than to the overall stock market. Such a beta would measure the risk from adding a specific fund to a holder of the mutual fund benchmark portfolio, rather than the risk of adding the fund to a portfolio of the market.
Special cases
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.
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 are examples of
non-linear
In mathematics and science, a nonlinear system (or a non-linear system) is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathe ...
assets. Whereas Beta relies on a linear model, an
out of the money option will have a distinctly non-linear payoff. In these cases, then, the change in
price of an option relative to the change in the price of its
underlying asset is not constant. (True also - but here, far less pronounced - for
volatility,
time to expiration,
and other factors.) Thus "beta" here, calculated traditionally, would vary constantly as the price of the underlying changed.
Accommodating this,
mathematical finance
Mathematical finance, also known as quantitative finance and financial mathematics, is a field of applied mathematics, concerned with mathematical modeling in the financial field.
In general, there exist two separate branches of finance that req ...
defines a specific volatility beta.
Here, analogous to the above, this beta represents the covariance between the derivative's return and changes in the value of the underlying asset, with, additionally, a correction for instantaneous underlying changes.
See
volatility (finance)
In finance, volatility (usually denoted by "sigma, σ") is the Variability (statistics), degree of variation of a trading price series over time, usually measured by the standard deviation of logarithmic returns.
Historic volatility measures a t ...
,
volatility risk
Volatility risk is the risk of an adverse change of price, due to changes in the volatility of a factor affecting that price. It usually applies to derivative instruments, and their portfolios, where the volatility of the underlying asset is a ...
, .
Empirical estimation
A true beta (which defines the true expected relationship between the rate of return on assets and the market) differs from a realized beta that is based on historical rates of returns and represents just one specific history out of the set of possible stock return realizations. The true market-beta is essentially the average outcome if infinitely many draws could be observed. On average, the best forecast of the realized market-beta is also the best forecast of the true market-beta.
Estimator
In statistics, an estimator is a rule for calculating an estimate of a given quantity based on Sample (statistics), observed data: thus the rule (the estimator), the quantity of interest (the estimand) and its result (the estimate) are distinguish ...
s of market-beta have to wrestle with two important problems. First, the underlying market betas are known to move over time. Second, investors are interested in the best forecast of the true prevailing beta most indicative of the most likely ''future beta'' realization and not in the ''historical market-beta''.
Despite these problems, a historical beta estimator remains an obvious benchmark predictor. It is obtained as the slope of the fitted line from the
linear least-squares estimator. The OLS regression can be estimated on 1–5 years worth of daily, weekly or monthly stock returns. The choice depends on the trade off between accuracy of beta measurement (longer periodic measurement times and more years give more accurate results) and historic firm beta changes over time (for example, due to changing sales products or clients).
Improved estimators
Other beta estimators reflect the tendency of betas (like rates of return) for
regression toward the mean
In statistics, regression toward the mean (also called regression to the mean, reversion to the mean, and reversion to mediocrity) is the phenomenon where if one sample of a random variable is extreme, the next sampling of the same random var ...
, induced not only by measurement error but also by underlying changes in the true beta and/or historical randomness. (Intuitively, one would not suggest a company with high return
.g., a drug discoverylast year also to have as high a return next year.) Such estimators include the Blume/Bloomberg beta (used prominently on many financial websites), the Vasicek beta, the Scholes–Williams beta, the Dimson beta, and the Welch beta.
* The ''Blume beta''
shrinks the estimated OLS beta towards a mean of 1, calculating the weighted average of 2/3 times the historical OLS beta plus 1/3. A version based on monthly rates of return is widely distributed by Capital IQ and quoted on all financial websites. It predicts future market-beta poorly.
* 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 under the (violated) assumption that the underlying market-beta does not move. It is modestly difficult to implement. It performs modestly better than the OLS beta.
* The ''Scholes–Williams and Dimson betas'' are estimators that 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) or in markets with rare trading activity.
* The ''Welch beta'' is a
slope-winsorized beta estimator that bounds daily stock returns within the range of −2 and 4 times the contemporaneous daily market return. The slope-winsorized daily return of a stock follows
, effectively restricts beta estimates to be between −2 and 4. The beta is estimated with the weighted least squares (WLS) estimation on slope-winsorized daily stock returns and the market returns. It outperforms OLS beta, Blume beta, Vasicek beta, and Dimson betas in forecasting the future realizations of market betas and hedging.
These estimators attempt to uncover the instant prevailing market-beta. When long-term market-betas are required, further regression toward the mean over long horizons should be considered.
See also
*
Alpha (finance)
Alpha is a measure of the active return on an investment, the performance of that investment compared with a suitable market index. An alpha of 1% means the investment's return on investment over a selected period of time was 1% better than the ...
*
Betavexity
*
CSS Theory - Beta
*
Cost of capital
In economics and accounting, the cost of capital is the cost of a company's funds (both debt and equity), or from an investor's point of view is "the required rate of return on a portfolio company's existing securities". It is used to evaluate ne ...
*
Financial risk
Financial risk is any of various types of risk associated with financing, including financial transactions that include company loans in risk of default. Often it is understood to include only downside risk, meaning the potential for financi ...
*
Hamada's equation
*
List of financial performance measures
*
Macro risk
*
Pure play method
*
Risk factor (finance)
*
Treynor ratio
*
WACC
References
Further reading
*
External links
ETFs & Diversification: A Study of CorrelationsCalculate Beta in a SpreadsheetFree Beta Calculator for any Asset-Index pairCalculate Sharpe Ratio in ExcelCalculate Beta in ExcelOnline Portfolio Beta Calculator
{{DEFAULTSORT:Beta (Finance)
Mathematical finance
Fundamental analysis
Financial ratios
Statistical ratios
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