beta (finance) - wikipedia

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Beta (finance)From Wikipedia, the free encyclopedia

For other uses, see Beta (disambiguation).

In finance, the beta (β) of an investment is a measure of the risk arising from exposure to general market

movements as opposed to idiosyncratic factors. The market portfolio of all investable assets has a beta of

exactly 1. A beta below 1 can indicate either an investment with lower volatility than the market, or avolatile investment whose price movements are not highly correlated with the market. An example of the

first is a treasury bill: the price does not go up or down a lot, so it has a low beta. An example of the second

is gold. The price of gold does go up and down a lot, but not in the same direction or at the same time as the

market.[1]

A beta greater than one generally means that the asset both is volatile and tends to move up and down with

the market. An example is a stock in a big technology company. Negative betas are possible for investments

that tend to go down when the market goes up, and vice versa. There are few fundamental investments with

consistent and significant negative betas, but some derivatives like equity put options can have large

negative betas.

[2]

Beta is important because it measures the risk of an investment that cannot be reduced by diversification. It

does not measure the risk of an investment held on a stand-alone basis, but the amount of risk the investment

adds to an already-diversified portfolio. In the capital asset pricing model, beta risk is the only kind of risk

for which investors should receive an expected return higher than the risk-free rate of interest.[3]

The definition above covers only theoretical beta. The term is used in many related ways in finance. For

example, the betas commonly quoted in mutual fund analyses generally measure the risk of the fund arising

from exposure to a benchmark for the fund, rather than from exposure to the entire market portfolio. Thus

they measure the amount of risk the fund adds to a diversified portfolio of funds of the same type, rather

than to a portfolio diversified among all fund types.[4]

Beta decay refers to the tendency for a company with a high beta coefficient (β > 1) to have its beta

coefficient decline to the market beta. It is an example of regression toward the mean.

Contents

1 Statistical estimation

1.1 Security market line

2 Choice of benchmark

3 Investing

4 Adding to a portfolio

5 Academic theory

6 Multiple beta model

7 Estimation of beta

8 Interpretations of Beta

9 Extreme and interesting cases

10 Criticism

11 See also12 References

13 External links

a (finance) - Wikipedia, the free encyclopedia https://en.wikipedia.org/wiki/Beta_(finance)

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Statistical estimation

Beta is estimated by linear regression. Given an asset and a benchmark that we are interested in, we want to

find an approximate formula

where r a is the return of the asset and r b is return of the benchmark.

Since the data are usually in the form of time series, the statistical model is

,

where εt is an error term (the unexplained return). Click here for a definition of Alpha (α).

The best (in the sense of least squared error) estimates for α and β are those such that Σεt 2 is as small as

possible.

A common expression for beta is

,

where Cov and Var are the covariance and variance operators.

This can also be expressed as

where ρa,b is the correlation of the two returns, and σa and σb are the respective volatilities. Relationships

between standard deviation, variance and correlation:

Beta can be computed for prices in the past, where the data is known, which is historical beta. However,

what most people are interested in is future beta, which relates to risks going forward. Estimating future beta

is a difficult problem. One guess is that future beta equals historical beta.

From this, we find that beta can be explained as "correlated relative volatility". This has three components:

correlated

relative

volatility

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 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).


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The Security Market Line

is called the asset's alpha and 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 wouldexpect 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 available for the market, while the

slope is E( Rm)− R f . The security market line can be regarded as

representing a single-factor model of the asset price, where Beta isexposure 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 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 theS&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.


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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).[5] 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.

Adding to a portfolio

Suppose an investor has all his money in an asset class X 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

The variance can be computed as

which can be simplified by ignoring δ2 terms:

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,

we can compute

This suggests that an asset with β greater than one will increase variance, while an asset with β less than one

will decrease variance, if added in the right amount. This assumes that variance is an accurate measure of


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risk, which is usually good. However, the beta does need to be computed with respect to what the investor

currently owns.

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 investorsshould 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):

where:

K E = 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

because:

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.[6]

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,


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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.[7]

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,[8] 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. Onceappraisers 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.

Interpretations of Beta

Some interpretations of beta are explained in the following table:[9]

Value of Beta Interpretation Example

β < 0Asset generally moves in the opposite

direction as compared to the index

An inverse exchange-traded fund or a short

position

β = 0Movement 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


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


same direction as, but more than the

movement of 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 otherassets 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


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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). 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.

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. They are

still higher beta.

Foreign stocks may provide some diversification. World benchmarks such as S&P Global 100 haveslightly 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


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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 assumesthat 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."[10]

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.[11]

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 muchas 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. 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.[11]

See also

Alpha (finance)

Beta Coefficient via

Wikinvest

Beta decay (finance)

Capital asset pricing model

CSS Theory - Beta

Cost of capital

Downside beta

Downside risk

Dual-beta

Financial risk

Hamada's equation

Macro risk

Treynor ratio

Upside beta

Upside risk

WACC

References

Sharpe, William (1970). Portfolio Theory and Capital Markets. McGraw-Hill Trade. ISBN 978-0071353205.1.

Markowitz, Harry (1958). Portfolio Selection. John Wiley & Sons. ISBN 978-1557861085.2.

Fama, Eugene (1976). Foundations of Finance: Portfolio Decisions and Securities Prices. Basic Books.

ISBN 978-0465024995.

3.


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Ilmanen, Antti (2011). Expected Returns: An Investor's Guide to Harvesting Market Rewards. John Wiley &

Sons. ISBN 978-1119990727.

4.

McAlpine, Chad (2010). "Low-risk TSX stocks have outearned riskiest peers over 30-year period"

(http://business.financialpost.com/2010/06/22/low-risk-tsx-stocks-have-outearned-riskiest-peers-over-30-

year-period-analyst/), The Financial Post Trading Desk , June 22, 2010

5.

"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" (http://www.lse.co.uk

/financeglossary.asp?searchTerm=equity&iArticleID=1688&definition=equity_beta). Lse.co.uk. Retrieved

2012-12-03.

6.

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 (https://dx.doi.org

/10.1016%2F0304-405X%2877%2990041-1).

7.

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

(https://dx.doi.org/10.1016%2Fj.ejor.2006.09.018).

8.

Definition of Beta Definition via Wikinvest9.

Klarman, Seth; Williams, Joseph (1991). "Beta". Journal of Financial Economics 5 (3): 117.

doi:10.1016/0304-405X(77)90041-1 (https://dx.doi.org/10.1016%2F0304-405X%2877%2990041-1).

10.

James Chong; Yanbo Jin; Michael Phillips (April 29, 2013). "The Entrepreneur's Cost of Capital: Incorporating

Downside Risk in the Buildup Method" (http://www.macrorisk.com/wp-content/uploads/2013/04/MRA-WP-

2013-e.pdf) (PDF). Retrieved 25 June 2013.

11.

External links

ETFs & Diversification: A Study of Correlations (http://www.etf.com/sections/research/5911-etfs-a-

diversification.html?iu=1)

Leverage and diversification effects of public companies (http://rdcohen.50megs.com

/IDRHEqabstract.htm)

Calculate Beta in a Spreadsheet (http://investexcel.net/367/calculate-stock-beta-with-excel)

Retrieved from "https://en.wikipedia.org/w/index.php?title=Beta_(finance)&oldid=677781644"

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