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Cost Of Capital Technical Workshop Draft Position Paper, 1998 Beta Cost of Capital Technical Workshop Position Paper - Beta Page 2 DRAFT Table of contents 1. General considerations.............................................................................................................3 1.1 A need for general guidelines - objectives of the paper .....................................................3 1.2 Analysis of the stock market Beta .......................................................................................3 1.3 Country anomalies ..............................................................................................................4 2. CAPM and Beta ........................................................................................................................5 2.1 The Capital Asset Pricing Model .........................................................................................5 2.2 Estimating Beta ...................................................................................................................6 2.2.1 The market model .........................................................................................................6 2.2.2 Input data - security returns ..........................................................................................7 2.2.3 Measuring estimation error .........................................................................................10 2.3 Adjustments for estimation errors .....................................................................................11 2.4 External providers of Betas ...............................................................................................13 3. Sector and portfolio Betas ......................................................................................................15 3.1 Method of comparables - portfolio Beta ............................................................................15 3.2 Sector Beta........................................................................................................................16 4. Effects of capital structure ......................................................................................................17 C:\Documents and Settings\Harris\My Documents\Harris\CVC Papers\Beta.doc JSG-08/05/98 Cost of Capital Technical Workshop Position Paper - Beta Page 3 DRAFT 1. General considerations 1.1 A need for general guidelines - objectives of the paper It is quite obvious that we should avoid delivering different quality of work in different PWC offices involved. This is especially true for one of the major value drivers of the value of a company - the cost of capital. Using the Capital Asset Pricing Model (CAPM) to calculate cost of capital, Beta is one of the most important statistics. Beta incorporates the operating risk as well as the financial leverage of a company. There is a need for a general guideline which provides a standardised framework for calculating Beta across all PWC offices and provides basic information on models used for calculation. This paper is not a complete review of literature, and does not include a discussion of all different methodologies which one can use for daily corporate finance and consulting work. Only a few, commonly used methods are described in this paper. 1.2 Analysis of the stock market Beta According to the Capital Asset Pricing Model, the greater a security s systematic risk, the greater its required return. Systematic risk is measured by Beta and expresses to what extent a company s stock returns move in relation to the market portfolio. In mathematical terms, Beta is the slope of a linear regression where the stock return is regressed against the return on the market portfolio. The mathematical model behind the linear regression (OLS regression model) forces us to obey the underlying assumptions of this particular model. These are assumptions regarding the error term: normality, constant variance, and no residual correlation. These assumptions are important because they must be satisfied in order to make valid statistical inferences regarding the "true" underlying Beta. The main point is that the CAPM gives us no guidance in choosing a proxy for the market index, and it does not tell us what data frequency to use. Moreover, in empirical applications C:\Documents and Settings\Harris\My Documents\Harris\CVC Papers\Beta.doc JSG-08/05/98 Cost of Capital Technical Workshop Position Paper - Beta Page 4 DRAFT we also typically assume that Beta is constant over some time period. In this context it is also important to consider changes in operations of companies which are reflected in stock returns. Another practical point: Valuation and corporate finance specialists at PWC do not use raw data (eg. stock prices, stock returns) to estimate Beta based on their programmed regression algorithms. In general professional information systems and databases, eg. Barra, Datastream, are used as sources for Betas. Before we can decide whether we use a specific provider as a standard provider for PWC, at first it has to be decided if the calculation method is appropriate and second, it has to be analysed which method is used by common providers of Betas. 1.3 Country anomalies In empirical studies it has been observed, that there are stock market anomalies in every national market. Because these anomalies are not the same across all stock markets is has to be investigated whether and what kind adjustments to Beta have to be made in some countries. As far as the mentioned studies show it is not possible to transfer findings from on country to another. On example is in the size effect . In the US strong empirical evidence indicates a size effect where small caps realise higher returns than large caps. But this size effect is not reflected in Beta and therefore adjustments have to be made to the cost of capital. There are some countries where there has been identified a stock market anomaly which is widely accepted under scientists and practitioners. If this is the case, a guideline can be designed, which provides adjustments for this specific country anomaly. The VSG Manual of former PW US practice gives a good example how to adjust cost of capital for the Small Stock Premium taking into account that risk is not fully captured by Beta. A position paper on adjustments for country anomalies will be left to position papers of the next generation. Further surveys focusing on specific country anomalies should be carried out by PWC offices in each country with a developed stock market. C:\Documents and Settings\Harris\My Documents\Harris\CVC Papers\Beta.doc JSG-08/05/98 Cost of Capital Technical Workshop Position Paper - Beta Page 5 DRAFT 2. CAPM and Beta 2.1 The Capital Asset Pricing Model In the theory of the CAPM the expected return of security I equals the risk free rate plus an individual equity risk premium. The individual equity risk premium equals Beta of security I times the market premium. [ E(R i ) = Rr + i E ( R M ) Rr ] E(R i ) Expected stock return of security i Rr Risk free rate i Beta of security i [ i E ( R M ) Rr [ E (R ) R ] M r ] Individual equity risk premium of security i Market risk premium, or excess return of the market portfolio Beta is an estimate of the historical alignment of the stock return with the market return. Although in practice historical stock returns are used to estimate Beta, Beta is used for a prediction of expected security return as stated in the CAPM. But the theory gives no guidance how to estimate Beta and how to measure the return of the market portfolio. This issues will be discussed in the next sections. C:\Documents and Settings\Harris\My Documents\Harris\CVC Papers\Beta.doc JSG-08/05/98 Cost of Capital Technical Workshop Position Paper - Beta Page 6 DRAFT 2.2 Estimating Beta 2.2.1 The market model Statistically, Beta is measured as the slope coefficient in a linear regression with the market portfolio (proxy: market index) as the independent variable and the individual security return as the dependent variable. The regression equation is derived from Sharpe s Market Model, in which is stated that the expected equity return is a function of the market return. rit = i + i RMt + et i Regression coefficient i Intercept of the regression line RMt Independent variable, market return et Residuals, deviations of the regression line According to the Model above the driving parameters behind Beta can be analysed within the formula for Beta: i = cov(R r , RM ) (RM ) 2 = (R i , RM ) (R I ) (R M ) cov(R r , RM ) Covariance between security i and the market index 2 ( RM ) Variance of the market index (R i , RM ) Correlation coefficient between security i and the market index (R I ) Standard deviation of returns of security i (R M ) Standard deviation of market returns C:\Documents and Settings\Harris\My Documents\Harris\CVC Papers\Beta.doc JSG-08/05/98 Cost of Capital Technical Workshop Position Paper - Beta Page 7 DRAFT The formula for Beta as described above is equivalent to the Ordinary Least Squares Method (OLS) when estimating the slope of the regression line. There are some assumptions of the OLS model which have to be obeyed to make valid statistical inferences regarding the "true" underlying Beta. These are assumptions regarding the error term: normality, constant variance, and no residual correlation. Although these assumptions are important for the model we have to consider that in practice we are not able to test them for each Beta we estimate. 2.2.2 Input data - security returns Providers of financial data usually use the same methodology or apply the same transformation to input data for the regression model when estimating Beta. This is quite easy to achieve, but to choose the appropriate input data or transformation method is the more difficult task. In the following paragraphs the major issues concerning input data for the Beta formula are discussed and adjustments are evaluated. Shareholder return including capital changes and dividends Nearly all company stock prices available from databases is adjusted for capital changes, splits, etc.. But there is a choice between ordinary stock returns, excluding dividend payments and the Total Shareholder Return (TSR), in which dividend payments are included. In the case of the Total Shareholder Return, dividend payments are theoretically reinvested into the stock the adjustment results in a higher stock return. The theory demands returns including the dividends therefor TSR should be used whenever possible, although in praxis it doesn t result in large difference in Beta. C:\Documents and Settings\Harris\My Documents\Harris\CVC Papers\Beta.doc JSG-08/05/98 Cost of Capital Technical Workshop Position Paper - Beta Page 8 DRAFT Frequency of security returns To choose the frequency (daily, weekly, monthly or quarterly returns) of returns is one of the major practical issues when estimating Beta. The CAPM is based on maximising expected utility, therefore the security returns have to be normally distributed and the distribution is fully described by standard deviation and the expected return. Fama 1 showed that the distribution of monthly continuos returns comes close to normal distribution. This is not the case especially for daily or weekly stock returns. But the normality assumption it is expected to have little effect on Beta estimation. In the best case one would like to use daily data if possible because this allows a maximum number of data points for the regression. In addition, we would not have to assume beta is constant for extremely long time periods when using other frequencies. Graph 2: Hoechst Beta changes over time - 60 time periods Betas, monthly returns Hoechst 60 time period beta, monthly returns 1.2 1.1 1 0.9 0.8 0.7 30/05/97 31/10/96 29/03/96 31/08/95 31/01/95 30/06/94 30/11/93 30/04/93 30/09/92 28/02/92 31/07/91 31/12/90 31/05/90 31/10/89 31/03/89 31/08/88 29/01/88 30/06/87 28/11/86 30/04/86 30/09/85 28/02/85 31/07/84 30/12/83 31/05/83 29/10/82 31/03/82 31/08/81 30/01/81 30/06/80 30/11/79 30/04/79 29/09/78 0.6 Date of calculation Changes in Beta can also be observed within the 5-year period. Economically these changes over time could be caused by operational changes of the business or changes of the 1 Fama, E.F., 1976, Foundations of Finance, Basic Books C:\Documents and Settings\Harris\My Documents\Harris\CVC Papers\Beta.doc JSG-08/05/98 Cost of Capital Technical Workshop Position Paper - Beta Page 9 DRAFT company s leverage (See discussion in section 4). That explains why the assumption that Beta remains constant over many years is too restrictive for most practical applications. To get a reasonable number of observations for the regression with quarterly return data many years of data must be employed and Beta would more likely change over a longer than a shorter period. Graph 1: Standard error of Betas across 60 German industrials as a function of the number of observations available for regression Additionally infrequent trading is a severe problem with daily and weekly data, but much less so with monthly or quarterly data. To summarise it: The real issue here is a trade-off between the number of observations and infrequent trading. Associated issue is the assumption that Beta remains constant over time. Combining the relevant arguments stated in empirical research what frequency one should use and what to do with infrequently traded stocks a simple policy for estimating Betas can be derived: Monthly returns to take into account the trade-off between number of observations and infrequent trading 36-60 months of stock price observations (35 to 59 returns). Infrequent trading can still be a problem for small firms even at monthly frequencies. Therefore it is reasonable to exclude stocks which are not traded or the turnover in shares is close to zero. Proxy for the market Portfolio C:\Documents and Settings\Harris\My Documents\Harris\CVC Papers\Beta.doc JSG-08/05/98 Cost of Capital Technical Workshop Position Paper - Beta Page 10 DRAFT In theory the market portfolio consists of all assets in the economy weighted by their market value. In praxis a stock market index is used as an estimate for the market portfolio. According to the CAPM the stock market index should include all assets and their returns should be weighted by market value of the member assets. The question is now whether the relevant market index is always the home country index or the world index. For example, suppose a German firm has access to world-wide capital markets. This firm can raise capital from investors throughout the world. These investors would not necessarily care about a Beta computed using a German stock market index because they can invest in a global portfolio. This argument seems to suggest using a world index. The underlying assumption for this suggestion is that capital markets are integrated and that there is a world-wide Capital Asset Pricing Model and it s other coefficients, world market risk premium and world-wide risk free rate. So far empirical evidence supports that some markets are integrated and some are not, but there are tendencies for conversion to an integrated market. This issue is discussed in another position paper. Until this issue is solved, the number of members of the index should be as high as possible. If there is no large index available one should use which represents a large portion of the total market capitalisation of the national stock market. In countries where there is no suitable index available the world index should be used. The concept of the market index has to be consistent to the calculation method of the corresponding stock return. 2.2.3 Measuring estimation error As described above the regression equation is derived from the Market Model, in which expected equity return is expressed as a function of the market return. The assessment regarding the accuracy of the Beta estimate can be done by the standard error. Standard error of Beta: ( i RMt ) = 2 (e i ) (T 1) 2 (RM ) C:\Documents and Settings\Harris\My Documents\Harris\CVC Papers\Beta.doc JSG-08/05/98 Cost of Capital Technical Workshop Position Paper - Beta Page 11 DRAFT Holding everything constant, the standard error of beta should be as low as possible. Another measure provided by commercial databases is R-Square. This is a ratio stating the systematic risk of security I in relation to the total risk of security i. R-Square: 2 (R i , R M ) = 1 2 (e i ) 2 (R i ) The r-square is relatively low for individual securities because of unsystematic risk and relatively high for portfolios since unsystematic risk is alleviated. This result is consistent with the CAPM. An appropriate way of checking a Beta estimate is to combine the more intuitive economic measures with the standard error. But standard error should not be used as the one and only kick out criteria. In the previous section frequent trading and the assumption of a constant Beta over time is mentioned. The assessment has to be made by the individual who is valuing the company. 2.3 Adjustments for estimation errors In many cases analysts have no faith in firm-specific Beta, in absence of other guiding estimates. In that case a Bayesian adjustment is used to compensate for estimation error. The idea is that there are two balancing considerations for assessing the relative risk of any business: one is the stock market Beta observed, the other is the recognition that this business is one among many and therefore likely to be somewhat comparable to the average of all businesses. The average of all assets/businesses is represented in the market Beta, 1.0 (Beta of the market portfolio). adjusted = raw p + 1 (1 p ) C:\Documents and Settings\Harris\My Documents\Harris\CVC Papers\Beta.doc JSG-08/05/98 Cost of Capital Technical Workshop Position Paper - Beta Page 12 DRAFT Each estimate is weighted according to its accuracy as a predictor. Absent all other information an analyst s best estimate of Beta is 1. The less accurate the Beta estimate (the greater the estimation error) the more weight should be given to the market Beta of one. But the greater the diversity of all firms around the average, the less precise is the average as a predictor for any one firm, and therefore the less weight that should be given to the value 1.0, the Beta of the market portfolio. Professional providers of financial information on companies quite often use the Bayesian technique to adjust their Betas. E.g. Bloomberg and Merrill Lynch use weights of 0.66 on stock market Beta and 0.33 on the Beta of the market portfolio. Bayesian adjustment reduces the spread of Beta across companies and therefore reduces possibilities of differentiation between company Betas, therefore this adjustment should only be applied when the quality of data is poor or there are not enough confidence in the data. One idea to overcome the disadvantage of information reduction is to adjust the Beta towards the sector Beta instead of Beta of the market portfolio. adjusted = raw p + sec tor (1 p ) Other popular adjustments are applied to input data. E.g. Datastream adjusts the returns used for regression against the index for very high and very low observation. This adjustments results in lower Betas compared to Betas calculated on the unadjusted returns. Because the economic explanation behind the adjustment for extreme observations in returns are not clear and that the result is a general reduction of most of the Betas, the adjustment should be avoided. In case of the Bayesian adjustment the use of the sector Beta seems to be more appropriate than the use of the Beta of the market portfolio. In there is also some evidence that these adjustments often little help in predicting future returns2. C:\Documents and Settings\Harris\My Documents\Harris\CVC Papers\Beta.doc JSG-08/05/98 Cost of Capital Technical Workshop Position Paper - Beta Page 13 DRAFT 2.4 External providers of Betas There are many databases which provide Betas estimates for the purpose of cost of capital calculation on a professional basis. Some of the well known are: Bloomberg Datastream Barra London Business School, LBS Most of the services of these providers are available to many PWC offices or PWC employees have access through the PWC Information Centre in London. Although these databases are wide spread it is not known to every user that the calculation method for Beta differs from one provider to the other and therefore there are differences in Betas from one provider to the other. Barra Time Period: Bloomberg 5 years Option to choose any time period, if < 100 points, a warning will occur daily, weekly, monthly, quarterly, yearly Maturity Definition: Regression: Historical Beta (HIST , local) historical Beta The measurement of a dependent variable s (e.g. stock returns) volatility relative to an independent variable (i.e. an index). Beta is the percent change in the price of the dependent variable given in a 1% change in the independent variable. This will reveal if the dependent variable moves in step with the independent variable; where a Beta of 1 indicates perfect alignment. Beta is used as a risk measurement; where a higher Beta equals a more volatile and therefore riskier security Beta (raw) is calculated by comparing, through linear 2 Fama, E.F. and K.R. French, 1997, "Industry Costs of Equity," Journal of Financial Economics (February), 153-193. C:\Documents and Settings\Harris\My Documents\Harris\CVC Papers\Beta.doc JSG-08/05/98 Cost of Capital Technical Workshop Position Paper - Beta Page 14 DRAFT Barra Index and share return used for calculation: Bloomberg of the stock relative to the local portion of the Morgan Stanley Capital International World Index (MSWLD); measures the past response of an asset (or portfolio) to market return Predicted Beta (Pred , local & world) predicted Beta of the stock relative to the MSWLD as well as the local component of the MSWLD; based on the BARRA risk model; predicts how an asset (or portfolio) will respond to future market returns using the past price behaviour of the stock and the local market historical Beta is calculated by running a regression (often over 60 months) on a stock s excess returns against the market s excess returns predicted Beta is calculated as the covariance of the portfolio return with the market return dividend by the variance of the market p=COV(rp,rm)/VARm regression, the chosen stock s week-to-week percent price change for a given time period equation for the line of regression: Y = (Beta) * X + Alpha Formula: Adj Beta= (0.66)*Raw Beta + (0.33)*1.0 Adjustments: Beta (raw) is adjusted using the assumption that all Betas gravitate toward 1.0 Beta is unlevered, i.e. just price regression with no adjustments for dividends, etc. the index against which Beta is calculated can be changed to any index which has a price history available Other: Datastream LBS Time Period: Maturity Definition: 5 years monthly Regression: The figure resulting out of the Bayesian adjustment is the geared (or levered) Beta coefficient an ungeared Beta can be estimated by extracting total debt figures: Ungeared Beta = Geared Beta * market value/(market value + total debt) total debt is all borrowings plus loan stock outstanding The share price is regressed against the respective Datastream total market index using log changes of the closing price on the first day of each month Adjustment for extreme observations in two directions Bayesian adjustment 5 years monthly The sensitivity of the share market moves. A share with a Beta of 1.0 tends to perform in line with the index; one with a Beta of 1.2 tends to change by 1.2 percent for each percent move in the index LBS offers industry and company tables industry tables summarise risk and returns associated with industry groupings company tables provide detailed information on some 2000 British quoted shares Index and share return used for calculation: Adjustments: Continuos compounded return Dividend is included Certain adjustments prediction/adjustment) are made (Bayesian Other: Bloomberg appears to be the most flexible information system and the system with the highest degree of transparency. It offers to adjust for the number of observations and the regression graph allows to spot extreme values and shows trading statistics for the individual security. Furthermore it is available in most of the PWC offices. C:\Documents and Settings\Harris\My Documents\Harris\CVC Papers\Beta.doc JSG-08/05/98 Cost of Capital Technical Workshop Position Paper - Beta Page 15 DRAFT 3. Sector and portfolio Betas 3.1 Portfolio and sector Beta The method of comparables often is applied when estimating Betas for companies, which are not quoted on any stock market. In this case, when one wants to deriving cost of capital from capital market, comparable quoted companies have to be identified. For those Betas can be calculated and grouped in a portfolio. Additionally for quoted companies the idea is intuitively appealing. The advantage of a portfolio Beta is that the standard error is generally lower than that of an individual stock. It is nearly impossible to identify an exact copy of the target company with the same operating risk structure but comparable companies which have similar operations which are likely to be exposed to the same general market risks (for financial risk see section 4.). The result is a portfolio Beta which can be used as a benchmark for the target company. The company Betas within the portfolio, especially when they are in the same sector, should be similar - the variance within the portfolio of the comparables should be limited. Criteria for identifying comparable quoted companies can be drawn from various directions: Business description Turnover in different sectors Investments in different sectors Price Earnings, Book-to-Market, Other information on fundamentals But current and future operations of the target company are most valuable information to choose the right comparables. In some cases comparables of the target company are well known and information on those companies is updated regularly. But if comparables have to be identified initially, the amount of data analysed for each comparable company should be sufficient to explain possible differences in Beta across all comparables. This implies the assumption that there is a relation between company s fundamentals and Beta, which will be discussed in a following paragraph. C:\Documents and Settings\Harris\My Documents\Harris\CVC Papers\Beta.doc JSG-08/05/98 Cost of Capital Technical Workshop Position Paper - Beta Page 16 DRAFT 3.2 Sector Beta There are two reasons to calculate sector/portfolio Betas: The first is to calculate a sector Beta as an estimate for non quoted companies, the second is to take into account that there are significant and persistent differences between industry subgroups (eg. telecommunication industry with service providers and network providers) which can be used to improve estimation of business risk. In any case the sector Beta is extremely useful and should be calculated each time when estimating cost of capital of a target company. It can be used as a benchmark for the comparables Beta and it can be used for the Bayesian adjustment as stated in chapter 2.4. In the absence of useful data for comparables the sector Beta is the best proxy for the company Beta, because the individual Betas are less precise than sector Betas3. An extensive sector analysis should to be carried out first to identify comparable companies. If suitable comparables can not be identified a portfolio of different sector Betas can be generated which are weighted according to the targets operations. Within one portfolio the spread between Betas should be limited and if not, a plausible explanation for it has to be found by analysing each company separately. If there are no comparables available the sector Beta is the best proxy for the Beta of the target company. 3 Fama, E.F. and K.R. French, 1997, "Industry Costs of Equity," Journal of Financial Economics (February), 153-193. C:\Documents and Settings\Harris\My Documents\Harris\CVC Papers\Beta.doc JSG-08/05/98 Cost of Capital Technical Workshop Position Paper - Beta Page 17 DRAFT 4. Effects of capital structure The required rate of return depends on operating risk and financial risk. When a Beta is derived from stock market data it includes the operating risk as well as the financial risk, which reflects the increase in risk due to leverage or debt financing. It is the so called levered or unadjusted Beta. When applying the method of comparables it is important to take into account that comparable companies have different leverages from the target company and adjustments have to be made to Beta. To remove the effect of financial leverage from the comparable Betas, the stock market Beta has to be unlevered in a first step and then it has to be relevered with the appropriate capital structure of the target company. The formulas to unlever and relever a stock market Beta are: Unlever Beta of comparable companies: unlevered = levered 1 + (1 t ) DMV E MV Relever Beta of target company with appropriate target capital structure: levered = unlevered 1 + (1 t ) DMV E MV With t DMV Effective income tax rate E MV Market value of debt to market value of equity ratio C:\Documents and Settings\Harris\My Documents\Harris\CVC Papers\Beta.doc JSG-08/05/98 Cost of Capital Technical Workshop Position Paper - Beta Page 18 DRAFT The following graph shows relevered and unlevered Betas when applying different capital structures (assumption on effective tax rate: 0.36). Graph 4.: Hoechst Beta (levered and unlevered) - as a function of debt/equity ratio Error! Not a valid link. The amount data needed for unlevering Betas of comparable companies is large, but the impact of individual leverages on Beta can be very high, as demonstrated in the example. Therefore adjustments for differences in capital structures should be made, unless the assumption is made that the average capital structure of comparables applies to the target company. A reasonable estimate for the target market value of debt is the targeted book value of debt. In some cases Beta can change over time due to change of company s leverage. Therefore it is helpful to analyse the gearing of the company over time when changes in Beta occur. C:\Documents and Settings\Harris\My Documents\Harris\CVC Papers\Beta.doc JSG-08/05/98

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