# Beta factor

In finance, and in particular in capital market theory , the beta factor ( factor) represents a key figure based on the Capital Asset Pricing Model (CAPM) for the systematic risk (also called market risk ) assumed with an investment or financing . ${\ displaystyle \ beta}$

## description

Put simply, the beta is a gauge of how much the stock fluctuates compared to the market. At a value of 1.0, the stock fluctuates as much as the average. If the value is below 1, this indicates a smaller fluctuation. If the value is above 1.0, the stock fluctuates more than the average. A negative beta means that the return on the asset is contrary to the overall market.

The beta factor of a security compared to an efficient market portfolio results from mathematical derivation as the quotient of the covariance between the return on the security and the return on the market and the variance in the return on the market: ${\ displaystyle i}$ ${\ displaystyle M}$${\ displaystyle r_ {i}}$${\ displaystyle r_ {M}}$

${\ displaystyle \ operatorname {Beta factor} = {\ frac {\ operatorname {Cov} (r_ {i}, r_ {M})} {\ operatorname {Var} (r_ {M})}} = \ operatorname {Corr} (i, M) {\ frac {\ operatorname {SD} (r_ {i})} {\ operatorname {SD} (r_ {M})}}}$

or.

${\ displaystyle \ beta = {\ frac {\ sigma _ {iM}} {\ sigma _ {M} ^ {2}}} = \ rho _ {iM} {\ frac {\ sigma _ {i}} {\ sigma _ {M}}}}$

d. H. as the quotient of the covariance of the expected return of the security with the expected return of the market portfolio to the variance of the market portfolio , ${\ displaystyle i}$${\ displaystyle M}$${\ displaystyle M}$

or equivalent as the product of the correlation coefficient of the expected return of the security to that of the market portfolio with the ratio of the standard deviation of the expected return of the security to the standard deviation of the expected return of the market portfolio . ${\ displaystyle i}$${\ displaystyle M}$${\ displaystyle i}$${\ displaystyle M}$

The beta of a market portfolio is, by definition, 1.

That says what change the expected return of an individual security or security portfolio undergoes if the return of the market portfolio changes by one percentage point. It shows a linear relationship between the expected return on a risky investment and the expected return on the market portfolio. ${\ displaystyle \ beta}$

Three groups of securities can be formed with the beta factor:

1. ${\ displaystyle | \ beta |> 1}$ means: The security moves with greater fluctuations than the overall market.
2. ${\ displaystyle | \ beta | = 1}$ means: The security moves in the same way as the overall market.
3. ${\ displaystyle | \ beta | <1}$ means: The security moves less than the market as a whole.

## Application in practice

### Company valuation

The beta factor of the CAPM is used in particular to determine risk-adjusted discount rates ( cost of capital ) in company valuation . The CAPM also forms the basis of the assessment in accordance with the assessment standards of the Institute of German Auditors (IDW S1). The return demand of the equity capital provider arises as follows:

${\displaystyle r_{EK}=r_{F}+\beta \times \left(r_{M}-r_{F}\right)}$ .

The return on a risky investment results from the return on a risk-free investment r F plus a risk premium. The risk premium is obtained by multiplying the risk premium of the market ( r M -r F ) by the measure for the company-specific risk β . The determination of the individual sizes can be found in the article about the CAPM .

### Development of investment strategies

In investment practice, the use of the beta factor leads to the risk weighting of securities and thus has a significant influence on the level of individual assessments and the construction of investment strategies, in particular market-driven .

Decisive for the inclusion of a security in the portfolio is its contribution to the risk of the overall portfolio. If the beta and the standard deviations of the security (WP) and the portfolio (PF) are known, the correlation coefficient can be calculated.${\ displaystyle \ beta _ {WP} = {\ frac {\ rho _ {EP} \ sigma _ {WP}} {\ sigma _ {PF}}} \ quad \ Leftrightarrow \ quad \ rho _ {EP} = { \ frac {\ beta _ {WP} \ sigma _ {PF}} {\ sigma _ {WP}}}}$

This is required for calculating the variance of the extended portfolio (EP).

${\ displaystyle \ sigma _ {EP} ^ {2} = \ sigma _ {1} ^ {2} x_ {1} ^ {2} + \ sigma _ {2} ^ {2} x_ {2} ^ {2 } + 2x_ {1} x_ {2} \ sigma _ {1} \ sigma _ {2} \ rho _ {12}}$

## Determination of beta factors

Estimation of beta factors

### Determination based on linear regressions

The beta factor of a listed company i results from the ratio of the covariance between the company's rate of return and the market rate of return to the variance of the rate of return of the market rate . The betas can be calculated using time series data with a simple linear regression in the form ${\ displaystyle r_ {i}}$${\ displaystyle r_ {M}}$${\ displaystyle \ sigma _ {M} ^ {2}}$

${\ displaystyle r_ {i, t} = \ alpha + \ beta r_ {M, t} + \ varepsilon _ {t} \ quad {\ text {with}} \ quad \ Delta \ epsilon _ {t} = u_ { t} {\ sqrt {\ Delta t}} \ quad {\ text {where}} \ quad u_ {t} \; \ sim \; {\ mathcal {N}} (0,1) \ quad, t = 1 , 2, \ ldots, T}$

to be appreciated. Here referred to the return of the subject company and the market return. This model is also known as the market model. Note that this approach does not require a time series for a risk-free return. The estimated parameter α provides an estimate for the risk-free return. Alternatively, can the β also based on excess returns ( english excess returns ) to determine: ${\ displaystyle r_ {i}}$${\ displaystyle r_ {M}}$

${\ displaystyle r_ {i, t} -r_ {f, t} = \ alpha + \ beta (r_ {M, t} -r_ {f, t}) + \ varepsilon _ {t} \;}$

Using the example of Apple shares, the illustration on the right shows that the estimation results can be very different. The considerable differences can be explained by the different degrees of freedom in the estimation:

• Regression model : Using the market model gives different estimates than using excess returns. The betas can differ by a few percentage points. The choice of the estimation method ( method of least squares (KQ estimation), maximum likelihood method (ML estimation)) has no influence on the results, since these provide identical results for linear models.
• Estimation period (phase) : The estimated betas are not stable over time. The currently estimated betas from Apple are about 30% higher than the betas that were estimated about 10 years ago.
• Estimation period (length of the estimate) : For one-year estimates, the range of estimates is 0.5-1.2! The range narrows to 0.25 for estimates over a period of 5 years. With an estimate of 10 years, the range is hardly available.

It can be seen that estimating suitable beta factors is not trivial. Although the CAPM is a one-period model, the betas must be estimated over an observation period. During this observation period, the individual business risks and financial risks can change considerably and lead to an inexclusive estimate of the systematic risk. When estimating betas with the help of a peer group , experience shows that some of these distortions cancel each other out.

### Determination of beta factors with the help of a peer group

Estimating beta factors with the help of comparable companies has the advantage that the estimated parameters are more stable than with an individual estimate of the market model. In addition, a company must be listed if the CAPM is to be used to determine risk premiums. Otherwise, the returns of the company to be valued cannot be observed on the capital market. In the case of companies that are not listed on the stock exchange, the beta factor can therefore only be approximated using beta factors from comparable companies (“branch betas” or betas of a “peer group”).

The following table shows the estimated beta factors for a Microsoft peer group. The mean of the estimated betas (levered betas) is 1.11. This beta represents the typical systematic risks of indebted companies in the software industry quite well. It must be noted, however, that the systematic risks include market risks , operational risks and financial risks . If these risks of peer group companies and the company to be valued are comparable, the determination of betas using a peer group is appropriate.

 Peer group Levered beta Debt Val. Mkt. Val. Equity Debt / Equity Debt / Total cap. Marginal Tax rate Unlevered beta VMware, Inc. 1.24 4,981 62,529 8.0% 7.4% 27.0% 1.17 Oracle Corporation 0.86 56,776 177,617 32.0% 24.2% 27.0% 0.69 Alphabet Inc. 0.91 14,226 861.202 1.7% 1.6% 27.0% 0.90 salesforce.com, inc. 1.55 6,432 124,588 5.2% 4.9% 27.0% 1.49 Workday, Inc. 1.37 1,543 34,889 4.4% 4.2% 27.0% 1.33 ServiceNow, Inc. 1.51 1.107 42,805 2.6% 2.5% 27.0% 1.48 Adobe Inc. 1.17 4.137 126,388 3.3% 3.2% 27.0% 1.14 Splunk Inc. 1.83 1,906 17,048 11.2% 10.1% 27.0% 1.69 Citrix Systems, Inc. 0.64 989 13,627 7.3% 6.8% 27.0% 0.61 Average 1.11 9,476 146.069 7.5% 6.5% 27.0% 1.05 Microsoft Corporation 86,455 1,049,415 8.2% 7.6% 27.0% 1.03 Relevered Beta (Hamada) 1,114 Relevered Beta (Miles / Ezzel) 1.137 Relevered Beta (Harris / Pringle) 1.138 Calculations: Hamada : ø Unlevered Beta x (1 + Debt / Equity x (1 - Tax Rate)) = 1.05 x (1 + 8.2% x (1 - 27.0%)) = 1.114 Miles / Ezzel : ø Unlevered Beta x (1 + Debt / Equity x (1 - s * r f / 1 + r f )) = 1.05 x (1 + 8.2% x (1 - 0.6%) ) = 1.137 Harris / Pringle : ø Unlevered Beta x (1 + Debt / Equity) = 1.05 x (1 + 8.2%) = 1.138

When compiling the peer group, it is therefore important to ensure that the comparable companies present risks similar to those of the company to be assessed. As a rule, companies in an industry are exposed to comparable business risks. However, you will often find that these companies can differ significantly in terms of their financial risks, despite their industry affiliation. In these cases it is suggested to adjust the estimated betas for the individual financial risk.

For this purpose, betas are first calculated for the peer group companies, which would result if these companies were not in debt. The betas of non-indebted companies (“unlevered betas”) can be determined from the estimated betas of indebted companies (“levered betas”) using the so-called Hamada formula (see appendix below):

${\ displaystyle \ beta _ {u, i} = {\ frac {\ beta _ {l, i}} {1+ (1-s) {\ frac {FK_ {i}} {EK_ {i}}}} }}$

With

• β u, i: beta factor of the non-indebted company i,
• β l, i: beta factor of the partially leveraged company i,
• s: constant corporate income tax rate,
• FK: market value of the debt of company i,
• EK: market value of the company's equity i.

The arithmetic mean or the median of the betas of the peer group ( β l, PG ) is determined from the unlevered betas . This beta represents the systematic business risks of a non-indebted company in the industry or peer group. So that it also reflects the financial risks of a company, this beta must be »relevered« with the capital structure of the company to be valued:

${\ displaystyle \ beta _ {l, i} = \ beta _ {l, PG} \ cdot \; \ left (1+ (1-s) {\ frac {FK_ {i}} {EK_ {i}}} \ right)}$ .

You can see from the table above that Microsoft's estimated beta in accordance with the Hamada formula is 1.114. There is no significant difference to the average of the betas owed. In addition to the Hamada formula, the Miles / Ezzel and Harris / Pringel approaches were also calculated (see Appendix below). There are also no significant differences.

However, it is noticeable that in the peer group the betas of the non-indebted companies still have a similar spread as the betas of the indebted companies. This indicates that the “unlevering” did not level the risks. The betas of the non-indebted companies should actually all be quite similar within an industry and only depict the - quite comparable - business risks.

Strong fluctuations in the unlevered betas indicate that the production and cost structures of the companies within the peer group or branch are different. When analyzing the operating leverage , it becomes clear that the proportion of fixed costs can have a considerable influence on the risks for the capital provider. When calculating betas with the help of a peer group, it therefore makes sense to also factor out the leverage effect of the “Degree of Operating Leverage” (DOL). The relationship between DOL and beta is mostly described in the literature as follows:

${\ displaystyle \ beta _ {\ text {unlevered}} = \ beta _ {\ text {Operating Business}} \ cdot \ left (1 + {\ frac {\ text {Fixed costs}} {\ text {Variable costs} }} \ right)}$

The time-consuming “unlevering” and “relevant” of empirical betas is very vulnerable. The financial risks are determined exclusively by the level of indebtedness; other major causes of corporate financial problems are ignored. This ignores the reality. Payment problems occur e.g. B. far more frequently than problems of over-indebtedness. A similar criticism applies to the elimination of operating leverage. Fluctuations in operating results are also based solely on the proportion of fixed costs, but the reality is far more complex. Fluctuations in operating results also depend on the extent to which a company is diversified in terms of products, customers, suppliers and sales markets. The sophisticated ignorance and relevance of different types of risk (financial risks, operational risks) appears against the background of the numerous estimation inaccuracies (selection of the market index, calculation of the market return, selection of estimation periods, selection of the estimation method, selection of the peer group) rather than academic sophistication. A sham accuracy is suggested that the underlying data does not usually provide.

## Appendix: Leveraging / delevaraging of beta factors

The classic Modigliani / Miller (MM) formula for calculating the return on equity with simple income taxes and risk-free debt is:

${\ displaystyle r_ {EK} = r_ {GK} + (r_ {GK} -r_ {FK}) \ cdot \ (1-s) \ cdot {\ frac {FK} {EK}}}$

By integrating the costs of indebted and non-indebted equity

${\ displaystyle r_ {EK} = r_ {FK} + \ beta _ {lev} \ cdot MRP}$
${\ displaystyle r_ {GK} = r_ {FK} + \ beta _ {unlev} \ cdot MRP}$

In the MM formula one obtains a connection between the indebted and not indebted betas:

${\ displaystyle r_ {FK} + \ beta _ {lev} \ cdot MRP = r_ {FK} + \ beta _ {unlev} \ cdot MRP + \ beta _ {unlev} \ cdot MRP \ cdot (1-s) \ cdot {\ frac {FK} {EK}}}$

from which the so-called Hamada formula results:

${\ displaystyle \ beta _ {lev} = \ beta _ {unlev} \ cdot \ left (1+ (1-s) \ cdot {\ frac {FK} {EK}} \ right)}$

This formula is most often found in textbooks and it is also used regularly in practice. The Hamada formula is also known as the MM fitting formula or the textbook formula.

This formula is based on the assumption that the debt can be raised at the risk-free interest rate. This simplification can only be justified for companies with a very good rating (investment grade). In addition, it is assumed that the amount of borrowed capital remains constant in absolute terms. This would mean that over time, as company values ​​rise, the level of debt falls. This assumption also implies that the interest rate on borrowed capital is the same in every period and that the tax shield can be assumed to be secure.

Due to the unrealistic assumptions, the MM formula was modified. Miles / Ezzel (1980) assume that borrowed capital remains constant on a market value basis. The market value of equity also indirectly determines the amount of debt in a certain period. As a result, only the tax shield for the first few periods is secure. In contrast, the future tax shields are to be regarded as uncertain and are subject to the same risk as equity. Accordingly, the tax shields are discounted with the equity costs of the non-indebted company. Miles / Ezzel derive the following formula based on these assumptions:

${\ displaystyle \ beta _ {lev} = \ beta _ {\ text {unlev}} \ cdot (1+ \ left ((1 - {\ frac {sr_ {F}} {1 + r_ {F}}}) \ cdot {\ frac {FK} {EK}} \ right))}$

These formulas can be further simplified if the creation of tax shields is assumed to be risky from the start. The so-called Harris / Pringle formula is derived on the basis of this assumption:

${\ displaystyle \ beta _ {lev} = \ beta _ {\ text {unlev}} \ cdot \ left (1 + {\ frac {FK} {EK}} \ right)}$

This simple formula is also often used in practice.

## literature

The calculation of the beta factor is discussed in detail in:

• Franziska Ziemer: T he beta factor: Theoretical and empirical findings after half a century of CAPM. Springer-Gabler, Wiesbaden 2017, ISBN 978-3-658-20244-6 .

The representation of the influence of the capital structure and the CAPM can also be found in every good book on finance, e.g. B. in:

• Richard Brealey, Steward C. Myers, Franklin Allen: Principles of Corporate Finance. 12th edition, McGraw-Hill 2016, ISBN 978-1-259-25333-1 .
• David Hillier, Stephen A. Ross, Randolph W. Westerfield: Corporate Finance , 2nd Edition. McGraw-Hill 2013, ISBN 978-0-07-713914-8 .

The most influential essays on leveraging / deleveraging betas are:

• Robert S. Harris, John J. Pringle: Risk-adjusted discount rates extension from the averagerisk case. In: Journal of Financial Research . Volume 8, No. 3, 1985, pp. 237-244.
• James Miles, John R. Ezzell: The Weighted Average Cost of Capital, Perfect Capital Markets, and Project Life: A Clarification . In: Journal of Financial and Quantitative Analysis . Volume 15, No. 3, 1980, pp. 719-730.
• Franco Modigliani, Merton H. Miller: The Cost of Capital, Corporation Finance and the Theory of Investment. In: The American Economic Review. Volume 48, No. 3, 1958, pp. 261-297.

## Individual evidence

1. Franziska Ziemer: The beta factor in science . In: The beta factor . Springer Fachmedien Wiesbaden, Wiesbaden 2017, ISBN 978-3-658-20244-6 , p. 139–333 ( dnb.de [accessed February 18, 2020]).
2. ^ Damodaran, Aswath .: The dark side of valuation: valuing old tech, new tech, and new economy companies . Financial Times Prentice Hall, 2002, ISBN 0-13-040652-X .