Abstract
A major concern in business valuation is how to derive a beta value that adequately represents the assessment of long-term risk for a company. Against this background Morningstar (Ibbotson SBBI valuation yearbook 2012: market results for stocks, bonds, bills, and inflation 1926–2011. Ibbotson Associates, 2013), Bloomberg and Thomson Reuters recommend adjusting betas estimated for company valuation purposes (using \(\beta _{i}^{adj.}=.371+.635\beta _{i}^{raw}\) commonly named as the “\(\frac{1}{3}+\frac{2}{3}\)-adjustment”) to take into account research findings from Blume (J Finance 26(1):1–10, 1971) demonstrating that betas revert towards the mean value of one over time. Using theoretical analysis as well as a simulated data set reflecting real market patterns, we analyse the eligibility of this beta adjustment formula for company valuation practice. We show that derived adjustment formula coefficients are influenced by the variation of market returns, the length of the analysis period chosen, the measurement error for beta, as well as the distribution of true betas, quantifying the impact of all four elements, and confirm the regression to the mean fallacy interpretation as discussed by Friedman (J Econ Lit 30(4):2129–2132, 1992), Quah (Scand J Econ 95(4):427–443, 1993), Stigler (Stat Sci 11(3):244–252, 1996, Stat Methods Med Res 6(2):103–114, 1997), and Barnett et al. (Int J Epidemiol 34(1):215–220, 2004). We further demonstrate the biasing effect on company values when using the \(\frac{1}{3}+\frac{2}{3}\)-adjustment which is particularly intensified for small betas measured. Based on our analysis we conclude that the recommended \(\frac{1}{3}+\frac{2}{3}\)-adjustment as a justification for converging risk profiles lacks fundamental substance and, accordingly, its potential use in business valuation should be subject to critical consideration.
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Notes
According to Fama (1977) a constant equity cost of capital over multiple time periods is in line with the academic framework of the CAPM and is, therefore, an appealing practical approach due to its simplicity. For a literature review on the CAPM and asset pricing see Ross (1978), Dimson and Mussavian (1999), Subrahmanyam (2010) and Goyal (2012). When using the expression “Capital Asset Pricing Model” or the acronym CAPM we refer in our paper to the asset pricing model of Sharpe (1964), Lintner (1965), Mossin (1966), and Black (1972).
For modified betas in general see also Pratt and Grabowski (2010, p. 167).
For an up-to-date application see Ernstberger et al. (2011).
Consequently, the portfolio betas are estimated with fresh data and, therefore, without a statistically caused measurement error. But this approach reduces the data available for estimating beta since one period is lost. For the measurement error and the aligned errors-in-variables-problem see Shanken (1992).
See Morningstar (2013, p. 78), Blume (1971, p. 8), Table 4. Similar to the adjustment of Blume (1971) is the Merill Lynch, Pierce, Fenner, & Smith Inc. (MLPFS). Different approaches are the adjustments of Vasicek (1973) and the James–Stein Approach (Stein 1956, 1961) which are not discussed here. For further discussion on adjustments (see Klemkosky and Martin 1975, p. 1126; Schultz and Zimmermann 1989, pp. 200–201; Bauer 1992, pp. 101–102; Zimmermann 1997, p. 246, p. 251; Sharpe 1999, pp. 479–480).
As an alternative to analytical analysis, simulations help to understand the behaviour of frequentist statistical results and, therefore, contribute to assess inferences derived from empirical data. For a general discussion on Monte Carlo simulations see Mooney (1997).
Where \(\lim _{T\rightarrow \infty }\sigma _{\hat{\beta }_{i}}^{2}=\lim _{T\rightarrow \infty }\sigma _{\varepsilon _{i}}^{2}(( T-2) \sigma _{R_{M}}^{2}) ^{-1}=0\), \(\hat{\beta }_{i}\) denotes the beta estimator, \(\beta _{i}\) the true but unknown beta.
In addition, the assumption that the true but unobservable beta, \(\beta _{i}\), is constant over time may not be given.
See Ernstberger et al. (2011) for the advantages of sorting in general.
In the framework of empirical time series tests of the CAPM, a rotation of the security market line can be observed. Thus, empirically observed returns are too high for the low beta portfolios and too low for the high beta portfolios when compared to the predicted returns received from the standard CAPM based on beta estimates from the sorting period. See e.g. Stambaugh (1982), Black et al. (1972), Stambaugh (1982), Reichling (1995), Fama and French (2004, p. 33), Black et al. (1972) addressed this problem by separating the estimation period from the portfolio formation period which reduces the time series of utilisable data.
See Blume (1975, p. 794–795).
The index \(i\) describes a single company (where \(i=1,\ldots ,n\)), e.g. \(\hat{\beta }_{i}\), \(p\) a portfolio (where \(p=1,\ldots ,P\)), e.g. \(\hat{\beta }_{p}\), \(T\) the specific the cross section, e.g. \(\hat{\beta }_{T}\), and no index the pooled sample, e.g. \(\hat{\beta }\).
The risk free rate is assumed to be fixed and is, therefore, not considered throughout calculations.
Spurious measurement errors may occur due to the estimation process as will be illustrated in Sect. 3.
Although the S&P 500 captures the market only partially, the choice of the market proxy is not critical in our case since we use the data for demonstration purposes only. In addition, empirical analyses show that there are high correlations between the monthly returns of the S&P 500, the NYSE, NYSE/AMEX and NYSE/AMEX/NASDAQ according to Morningstar (2013, p. 73), indicating a minor impact by the selected market proxy used here. Furthermore, we select a long time period in order to capture various economic patterns.
Due to the assumptions of true beta \(\beta \sim N\left( 1;.16\right)\) the intercept is \(\gamma _{0}=1-\gamma _{1}\).
Hence, true betas are distributed around the market mean of one, \(\mu _{\beta }=1\), with a standard deviation of \(\sigma _{\beta }=.4\), \(\beta \sim N\left( 1;.16\right)\).
For simplicity the risk-free rate is set fix over each time period analysed and is, therefore, left out of the regression analysis.
With reference to real market data as reported by Morningstar (2013, pp. 76–77), there are more outliers which result in a peak at a \(R^{2}\) of nearly zero and consequently outliers with high standard deviations of beta. Due to the fixed standard deviation used in the simulation, this pattern cannot be completely reproduced. Nevertheless, the simulated values for the cross section are comparable to real market data as reported by Morningstar (2013, pp. 76–77), which is based on a 5 year window regression.
Portfolio betas are calculated as the mean of single betas estimated in that portfolio.
The correspondent 7 years variance of market returns for the first regression window, \(T=1\), is \(\sigma _{R_{M}}^{2}=.001248\).
Accordingly, the corresponding sum squared error, for the portfolio betas estimated in the first regression window (first dashed vertical line in Fig. 1), \(\hat{\beta }_{p,T=1}\), therefore, decreases from \(.009867\) to \(.002929\) for adjusted portfolio betas, \(\hat{\beta }_{p,T=1}^{adj.}\). Thereby, the sum squared error, \(SSE\), is calculated as the squared difference between the portfolio betas estimated and the true underlying portfolio betas scaled by the total number, \(P\), of portfolios, \(SSE=\sum _{p}^{P}( \hat{\beta }_{p}-\beta _{p}) ^{2}P^{-1}\) (here \(P=10\)).
Blume (1975) observes this pattern in the first as well as in the second subsequent period even after considering and adjusting the portfolio betas of the sorting period by a revised adjustment formula concluding that “companies of extreme risk [...] have less extreme risk characteristics over time” (Blume 1975, p. 794). He argues, therefore, that the mean reversion effect exists beyond measurement errors. Similarly, CAPM tests reveal a rotation of the security market line (e.g. Friend and Blume 1970; Black et al. 1972; Stambaugh 1982; Fama and French 2004). This has been observed with returns rather than with betas indicating that the linear form of the CAPM may not hold or that there may be additional pricing factors or that beta may vary over time. See Elgers et al. (1979) and Blume (1979) for a discussion about possible reasons for mean reversion effects. Generally, the reasons are seen in the order bias and in non-stationary betas, according to Jacobs and Z’graggen (1996, p. 94). For a practical application of adjustments for portfolios as well as for single stocks see Bauer (1992, p. 104), Blume (1971), Blume (1975), Klemkosky and Martin (1975), Elton et al. (1978), Eskew (1979), Eubank and Zumwalt (1979b), Eubank and Zumwalt (1979a), Dimson and Marsh (1983), Hawawini and Vora (1983), Winkelmann (1984), Hawawini et al. (1985), Ushman (1987), and Schultz and Zimmermann (1989). See further Mantripragada (1980) and Reeves and Wu (2013).
Where true beta changes with \(\beta _{i,t}=\beta _{i,t-1}+\varepsilon _{t}\) with \(\varepsilon _{t}\sim N( 0,\sigma _{\varepsilon _{t}}^{2})\) and \(\sigma _{\varepsilon _{t}}=.04\). This autoregressive movement is defined as being set to vary in the range of the grand market mean of one plus/minus three standard deviations with \(\beta _{i,t}\in [ -.2;2.2]\).
The regression window was changed from 7 to 5 years for illustrative purposes only, to reveal the convergence effect more strongly by taking more side by side adjacent regression windows into account.
NYSE Constituent List as provided by Thomson Reuters. Even though a resulting survivorship bias may be criticised, its effect is non-essential here since first, the market index as the counterpart for estimating beta is calculated on the basis of these remaining companies and second, there is no fundamental interpretation deduced from these numbers since they serve as an illustration of the theoretically discussed statistical effect.
Even though our approach is equal to the use of an equally weighted index resulting in a symmetric distribution of betas measured, which is a common use in research, business valuation practice commonly uses a value weighted index (e.g. S&P 500, DAX, etc.). Hence, returns of stocks with a lower market value are thus underrepresented in returns of a value weighted index. These stocks might, therefore, reveal a smaller correlation to that market index possibly resulting in a smaller beta measured since \(\pm \sqrt{R^{2}}=\rho _{R_{i},R_{M}}\Leftrightarrow R^{2}=\hat{\beta }_{i}\frac{\sigma _{R_{i},R_{M}}}{\sigma _{R_{i}}^{2}}\). As a result, with a value weighted index empirically measured betas might reveal a non symmetric distribution and analyses like those of Figs. 2 and 3 might not necessarily reveal an alleged mean reversion to a mean of one.
For a more detailed illustration of this approximation using the example of the slope, \(\gamma _{1}\), between the first, \(T=1\), and subsequent, \(T=2\), 7 years regression window of Fig. 1 we refer to "Appendix A".
The correspondent 7 years variance of market returns in \(T=1\) is \(\sigma _{R_{M}}^{2}=.001248\).
The sum squared error, \(SSE\), is calculated as the squared difference between the beta estimated and the true underlying beta scaled by the total number, \(n\), of stocks, \(SSE=\sum _{i}^{n}( \hat{\beta }_{i}-\beta _{i}) ^{2}n^{-1}\) (here \(n=10,000\)).
Figure 4 is provided without additional tables.
One may have noted that the reported mean estimated variance of single stocks, \(\hat{\sigma }_{\hat{\beta }_{i}}^{2}\), and the mean estimated variance of all stocks, \(\hat{\sigma }_{\hat{\beta }}^{2}\), of the Tables 2 and 4 are consistently higher than the theoretical ones of Table 1 (\(\sigma _{\beta _{i}}^{2}\) and \(\sigma _{\beta }^{2}\) respectively). This is due to the Jensen Inequality and the fact that a sample variance is chi square and not symmetrically distributed. Furthermore, the mean values of the estimated variance of all stocks, \(\hat{\sigma }_{\hat{\beta }}^{2}\), in Table 4 slightly differ from those in Table 2 since values in Table 4 are calculated over the cross section on a rolling window basis whereas values in Table 2 are calculated simultaneously over the pooled sample.
The adjustment with reference to Blume (1971), \(\beta _{i}^{adj.}=.371+.635\beta _{i}^{raw}\), represents the average coefficients of five subsequently performed regressions using a regression window of 7 years over the time period from 7/1926 to 6/1968. Even though our analysis is based on the S&P 500 (while Blume 1971 uses NYSE stocks) and we are only able to cover the time series covered by Blume (1971) partially, we achieve similar results to Blume (1971).
Where \(Var[ \hat{\beta }_{i}^{adj.}] =Var[ \gamma _{0}+\gamma _{1}\hat{\beta }_{i}] =\gamma _{1}^{2}\sigma _{\hat{\beta }_{i}}^{2}\).
We use a sixth degree Taylor Approximation here. See "Appendix C" for the derivation. For literature on how to adjust for effects arising from the Jensen Inequality due to using econometric estimators for determining cost of capital, we refer to Butler and Schachter (1989), Cooper (1996), Breuer et al. (2014) and Elsner and Krumholz (2013).
With Eq. (12) in the form of \(f_{V_{i}}\left( R_{E,i}\right) =\sum _{t=1}^{\infty }\frac{E\left( D_{t}\right) }{\left( 1+R_{E,i}\right) ^{-t}}=\left( R_{f}+\left[ E\left( R_{M}\right) -R_{f}\right] \beta _{i}\right) ^{-1}\) where \(E( D_{t}) =E\left( D\right) =1\).
Note that the continuously ascending \(R^{2}\) is caused by the evenly created dataset and the property of \(R^{2}=\rho _{R_{i},R_{M}}^{2}=\frac{\sigma _{R_{i},R_{M}}^{2}}{\sigma _{R_{M}}^{2}\sigma _{R_{i}}^{2}}=\hat{\beta }_{i}\frac{\sigma _{R_{i},R_{M}}}{\sigma _{R_{i}}^{2}}\).
For literature on how to correct for the effect resulting from the Jensen Inequality see footnote (40). Where \(E[ f_{V_{i}}( \hat{\beta }_{i})] \ge f_{V_{i}}( E[ \hat{\beta _{i}}]) =f_{V_{i}}( \beta _{i})\).
In order to stabilise results due to the exemplary risk free rate, \(R_{f}=3\,\%\), and the expected market risk premium, \(E\left( R_{M}\right) -R_{f}=5\,\%\), chosen, betas estimated up to \(-.4\) have been winsorised.
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Appendices
Appendix A: Progression of the slope at the portfolio level
Figure 7 highlights the slope progression of a regression for portfolio betas from the subsequent period, \(\hat{\beta }_{p,T=2}\), on portfolio betas from the first regression window, \(\hat{\beta }_{p,T=1}\), of Fig. 1 (where a 7 years’ regression window has been used), \(\hat{\beta }_{p,T=2}\sim \hat{\beta }_{p,T=1}\), dependent on the numbers of portfolios used in which to group estimated betas. In order to avoid any bias on the calculation of portfolio means caused by an uneven allocation of stocks to portfolios, the numbers, \(P\), used for portfolio formation (x-axis in Fig. 7; namely 2, 4, 5, 8, 10, 16, 20, 25, 40, 50, 80, 100, 125, 200, 250, 400, 500, and 625) guarantees a portfolio allocation with a balanced number of stocks.
As outlined in Sect. 3.3, the slope moves closer to the slope of a similar regression on a single beta level according to Eq. (6) (here: \(\gamma _{1}=0.622514\) of \(\hat{\beta }_{T=2}\sim \hat{\beta }_{T=1}\)), the more portfolios are used within which to sort the stocks. If the maximum number of portfolios is reached (here: 10,000), the portfolio regression is identical to a regression represented by the Eqs. (6) and (8).
Progression of the slope at the portfolio level. The estimated slope of the regression \(\hat{\beta }_{p,T=2}\sim \hat{\beta }_{p,T=1}\) (of Fig. 1) dependent on the numbers of portfolios used within which to group estimated betas
Appendix B: Sensitivity of \(\gamma _{0}\) and \(\gamma _{1}\)
In order to provide a sensitivity analysis for our values \(\hat{\gamma }_{0}=.374824\) and \(\hat{\gamma }_{1}=.622514\) in the first period, \(T=1\) (whereas the mean over our 50 years period as reported in Table 4 was \(\hat{\gamma }_{0}=.307447\) and \(\hat{\gamma }_{1}=.692847\); 7 years regression window example similar to Blume (1971) we use the corresponding first available variance of market index returns from 01/1964 to 12/1970 of the S&P 500 with \(\sigma _{R_{M}}^{2}=.001248\) and vary the values for the variance of the error term, \(\sigma _{\varepsilon _{i}}\), and for the distribution of true beta, \(\sigma _{\beta }\), which were explicitly assumed to be fix in the analysis before, using \(\sigma _{\varepsilon _{i}}=.1\) and \(\sigma _{\beta }=.4\). For varying values of \(\sigma _{\varepsilon _{i}}\) around \(.10\) and \(\sigma _{\beta }\) around \(.40\) we still receive similar results as those suggested by Morningstar (2013) with (Table 6) reference to Blume (1971) (\(\gamma _{0}\equiv .371\), \(\gamma _{1}\equiv .635\)).
Appendix C: Taylor expansion valuation example
If constant cost of equity capital is assumed with an exogenously given and constant market risk premium, \(E\left( R_{M}\right) -R_{f}\), and risk free rate, \(R_{f}\), Eq. (18) may be approximated by an \(N\)-th degree Taylor Series expansion, \(T_{N}f( R_{E,i};\mu _{R_{E,i}})\). For \(R_{E,i}\) as the beta dependent measurement the expansion notes in the form of Eq. (19) whereas \(R_{E,i}\) is decomposed to \(R_{E,i}\equiv \mu _{R_{E,i}}+\sigma _{R_{E,i}}e\) with \(e\sim N\left( 0,1\right)\) as a standard error.
For a perpetual dividend stream of \(E\left( D_{t}\right) =E\left( D\right) =1\) the expansion and, therefore, the expected company value determined by betas measured then simplifies to Eq. (20). Since for odd orders, \(n\), \(E\left[ e^{n}\right] =0\), a sixth order approximation notes as Eq. (21).
Whereas the first part of Eq. (21), \(\mu _{R_{E,i}}^{-1}\), represents the theoretical true company value, the rest represents the effect due to the Jensen Inequality.
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Echterling, F., Eierle, B. Mean reversion adjusted betas used in business valuation practice: a research note. J Bus Econ 85, 759–792 (2015). https://doi.org/10.1007/s11573-014-0750-4
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DOI: https://doi.org/10.1007/s11573-014-0750-4