Top

Journal of Business Economics

Published in:

01-10-2015 | Original Paper

Mean reversion adjusted betas used in business valuation practice: a research note

Authors: F. Echterling, B. Eierle

Published in: Journal of Business Economics | Issue 7/2015

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

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.

previous article The patenting activity of German Universities

next article New insights on CEO charisma attribution in companies of different sizes and ownership structure: the role of prior company performance

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Journal of Business Economics

From January 2013, the Zeitschrift für Betriebswirtschaft (ZfB) is published in English under the title Journal of Business Economics (JBE). The Journal of Business Economics (JBE) aims at encouraging theoretical and applied research in the field of business economics and business administration, promoting the exchange of ideas between science and practice.

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Available only for authorised users

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).

For the early tests see Black et al. (1972), Miller and Scholes (1972), Fama and MacBeth (1973), and Blume and Friend (1973).

For the (second-pass) cross section regression see Fama and MacBeth (1973), Black et al. (1972), and e.g. Levy (2012, pp. 191–192), and Goyal (2012, pp. 9–12).

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 \(\gamma _{0}=1-\gamma _{1}\).

In order to minimise spurious errors apart from the theoretical discussion of Sect. 2.2 we use a high number of stocks (Mooney 1997).

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.

Furthermore, this pattern confirms the suggestion of Lo and MacKinlay (1990) and Liang (2000) that sorting biases (CAPM) tests if variables used for grouping portfolios (here, the beta) are correlated with the coefficients being tested.

The correspondent 7 years variance of market returns for the first regression window, \(T=1\), is \(\sigma _{R_{M}}^{2}=.001248\).

Superior to adjusting portfolio betas would be choosing portfolio betas of period \(T=2\) as a starting point for further tests (as suggested by Black et al. 1972) or forming portfolios according to a random sampling approach (as suggested by Liang 2000).

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.

See e.g. approaches of Claus and Thomas (2001), Gebhardt et al. (2001), Gode and Mohanram (2003), Easton (2004), Ohlson and Juettner-Nauroth (2005), Daske et al. (2006), and Nekrasov and Ogneva (2011).

Barnett AG, Pols JCvd, Dobson AJ (2004) Regression to the mean: what it is and how to deal with it. Int J Epidemiol 34(1):215–220CrossRef

Bauer C (1992) Das Risiko von Aktienanlagen, Reihe Finanzierung, Steuern, Wirtschaftsprüfung, vol 15. Müller Botermann, Köln

Berk JB (2000) Sorting out sorts. JFinance 55(1):407–427

Black F (1972) Capital market equilibrium with restricted borrowing. J Bus 45(3):444–455CrossRef

Black F, Jensen MC, Scholes M (1972) The capital asset pricing model: some empirical tests. In: Jensen MC (ed) Studies in the theory of capital markets. Praeger, New York

Blume ME (1970) Portfolio theory: a step toward its practical application. J Bus 43(2):152–173CrossRef

Blume ME (1971) On the assessment of risk. J Finance 26(1):1–10CrossRef

Blume ME (1975) Betas and their regression tendencies. J Finance 30(3):785–795CrossRef

Blume ME (1979) Betas and their regression tendencies: some further evidence. J Finance 34(1):265–267CrossRef

Blume ME, Friend I (1973) A new look at the capital asset pricing model. J Finance 28(1):19–33CrossRef

Botosan CA (2006) Disclosure and the cost of capital: what do we know? Account Bus Res 36:31–40CrossRef

Breuer W, Fuchs D, Mark K (2014) Estimating cost of capital in firm valuations with arithmetic or geometric mean—or better use the Cooper estimator? Eur J Finance 20(6):568–594

Butler JS, Schachter B (1989) The investment decision: estimation risk and risk adjusted discount rates. Financial Manag 18(4):13–22CrossRef

Claus J, Thomas J (2001) Equity premia as low as three percent? Evidence from analysts’ earnings forecasts for domestic and international stock markets. J Finance 56(5):1629–1666CrossRef

Cooper I (1996) Arithmetic versus geometric mean estimators: setting discount rates for capital budgeting. Eur Financ Manag 2(2):157–167CrossRef

Daske H, Gebhardt G, Klein S (2006) Estimating the expected cost of equity capital using analysts’ consensus forecasts. Schmalenbach Bus Rev 58(1):2–36

Dimson E, Marsh PR (1983) The stability of UK risk measures and the problem of thin trading. J Finance 38(3):753–783CrossRef

Dimson E, Mussavian M (1999) Three centuries of asset pricing. J Bank Finance 23(12):1745–1769CrossRef

Easton PD (2004) PE ratios, PEG ratios, and estimating the implied expected rate of return on equity capital. Account Rev 79(1):73–95CrossRef

Elgers PT, Haltiner JR, Hawthorne WH (1979) Beta regression tendencies: statistical and real causes. J Finance 34(1):261–263CrossRef

Elsner S, Krumholz HC (2013) Corporate valuation using imprecise cost of capital. J Bus Econ 83(9):985–1014

Elton EJ, Gruber MJ, Urich TJ (1978) Are betas best? J Finance 33(5):1375–1384

Ernstberger J, Haupt H, Vogler O (2011) The role of sorting portfolios in asset-pricing models. Appl Financ Econ 21(18):1381–1396CrossRef

Eskew RK (1979) The forecasting ability of accounting risk measures: some additional evidence. Account Rev 54(1):107–118

Eubank AAJ, Zumwalt JK (1979a) An analysis of the forecast error impact of alternative beta adjustment techniques and risk classes. J Finance 34(3):761–776

Eubank AAJ, Zumwalt JK (1979b) How to determine the stability of beta values. J Portf Manag 5(2):22–26CrossRef

Fama EF (1977) Risk-adjusted discount rates and capital budgeting under uncertainty. J Financ Econ 5(1):3–24CrossRef

Fama EF, French KR (1997) Industry costs of equity. J Financ Econ 43(2):153–193CrossRef

Fama EF, French KR (2004) The capital asset pricing model: theory and evidence. J Econ Perspect 18(3):25–46CrossRef

Fama EF, MacBeth JD (1973) Risk, return, and equilibrium: empirical tests. J Polit Econ 81(3):607CrossRef

Fisher L (1966) Some new stock-market indexes. J Bus 39(1):191–225CrossRef

Friedman M (1992) Do old fallacies ever die? J Econ Lit 30(4):2129–2132

Friend I, Blume M (1970) Measurement of Portfolio performance under uncertainty. Am Econ Rev 60(4):561–575

Galton F (1869) Hereditary genius. Macmillan and Co, LondonCrossRef

Galton F (1889) Natural inheritance. Macmillan, LondonCrossRef

Gebhardt WR, Lee CMC, Swaminathan B (2001) Toward an implied cost of capital. J Account Res 39(1):135–176CrossRef

Gode D, Mohanram P (2003) Inferring the cost of capital using the Ohlson–Juettner model. Rev Account Stud 8(4):399–431CrossRef

Goyal A (2012) Empirical cross-sectional asset pricing: a survey. Financ Mark Portf Manag 26(1):3–38CrossRef

Graham JR, Harvey CR (2001) The theory and practice of corporate finance: evidence from the field: complementary research methodologies: the interplay of theoretical, empirical and field-based research in finance. J Financ Econ 60(2–3):187–243CrossRef

Hawawini GA, Vora A (1983) Is adjusting beta estimates an illusion? J Portf Manag 10(1):23–26CrossRef

Hawawini GA, Michel PA, Corhay A (1985) New evidence on beta stationarity and forecast for belgian common stocks. J Bank Finance 9(4):553–560CrossRef

Hotelling H (1933) Review of the triumph of mediocrity in business, by Horace Secrist. J Am Stat Assoc 28(184):463–465CrossRef

Jacobs A, Z’graggen P (1996) Varianz der Eigenkapitalkosten von Schweizer Aktiengesellschaften. Swiss J Econ Stat 132(I):87–108

Jensen JLWV (1906) Sur les fonctions convexes et les inégalités entre les valeurs moyennes. Acta Mathematica 30(1):175–193CrossRef

Klemkosky RC, Martin JD (1975) The adjustment of beta forecasts. J Finance 30(4):1123–1128CrossRef

Levy H (2012) The capital asset pricing model in the 21st century, 1st edn. Cambridge Univ Press, Cambridge and Mass

Liang B (2000) Portfolio formation, measurement errors, and beta shifts: a random sampling approach. J Financ Res 23(3):261–284CrossRef

Lintner J (1965) The valuation of risk assets and the selection of risky investments in stock Portfolios and capital budgets. Rev Econ Stat 47(1):13–37CrossRef

Lo AW, MacKinlay AC (1990) Data-snooping biases in tests of financial asset pricing models. Rev Financ Stud 3(3):431–467CrossRef

Mantripragada KG (1980) Beta adjustment methods. J Bus Res 8(3):329–339CrossRef

Miller MH, Scholes M (1972) Rates of return in relation to risk: a re-examination of some recent findings. In: Jensen MC (ed) Studies in the theory of capital markets. Praeger, New York

Mooney CZ (1997) Monte Carlo simulation, Sage university papers series. In: Quantitative applications in the social sciences, vol 7–116. Sage Publications, Thousand Oaks and Calif

Morningstar (2013) Ibbotson SBBI valuation yearbook 2012: market results for stocks, bonds, bills, and inflation 1926–2011, Ibbotson Associates

Mossin J (1966) Equilibrium in a capital asset market. Econometrica 34(4):768–783CrossRef

Nekrasov A, Ogneva M (2011) Using earnings forecasts to simultaneously estimate firm-specific cost of equity and long-term growth. Rev Account Stud 16(3):414–457CrossRef

Nissim D, Penman SH (2001) Ratio analysis and equity valuation: from research to practice. Rev Account Stud 6(1):109–154CrossRef

Ohlson JA, Juettner-Nauroth BE (2005) Expected EPS and EPS growth as determinants of value. Rev Account Stud 10(2–3):349–365CrossRef

Penman SH (2013) Financial statement analysis and security valuation, 5th edn. McGraw-Hill, New York

Pratt SP, Grabowski RJ (2010) Cost of capital, 4th edn. Wiley, Hoboken

Quah D (1993) Galton’s Fallacy and tests of the convergence hypothesis. Scand J Econ 95(4):427–443CrossRef

Reeves JJ, Wu H (2013) Constant versus time-varying beta models: further forecast evaluation. J Forecast 32(3):256–266CrossRef

Reichling P (1995) Warum ist die Wertpapierkennlinie zu flach? Finanzmarkt und Portf Management 9(1):96–110

Ross SA (1978) The current status of the capital asset pricing model (CAPM). J Finance 33(3):885–901CrossRef

Schultz J, Zimmermann H (1989) Risikoanalyse schweizerischer Aktien: Stabilität und Prognose von Betas. Finanzmarkt und Portf Manag 3(3):196–209

Secrist H (1933) The triumph of mediocrity in business. Northwestern Univ, Evanston

Shanken J (1992) On the estimation of beta-pricing models. Rev Financ Stud 5(1):1–55CrossRef

Sharpe WF (1964) Capital asset prices: a theory of market equilibrium under conditions of risk. J Finance 19(3):425–442

Sharpe WF (1999) Investments, 6th edn. Prentice Hall international editions. Prentice Hall, Upper Saddle River

Stambaugh RF (1982) On the exclusion of assets from tests of the two-parameter model: a sensitivity analysis. J Financ Econ 10(3):237–268CrossRef

Stein C (1956) Inadmissibility of the usual estimator for the mean of a multivariate normal distribution. In Neyman J (ed) Proceedings of the third Berkeley symposium on mathematical statistics and probability. University of California Press, Berkeley, pp 197–206

Stein C (1961) Estimation with quadratic loss. In: Neyman J (ed) Proceedings of the fourth Berkeley symposium on mathematical statistics and probability. University of California Press, Berkeley, pp 361–380

Stigler SM (1996) The history of statistics in 1933. Stat Sci 11(3):244–252CrossRef

Stigler SM (1997) Regression towards the mean, historically considered. Stat Methods Med Res 6(2):103–114CrossRef

Subrahmanyam A (2010) The cross-section of expected stock returns: what have we learnt from the past twenty-five years of research? Eur Financ Manag 16(1):27–42CrossRef

Ushman NL (1987) A comparison of cross-sectional and time-series beta adjustment techniques. J Bus Finance Account 14(3):355–375CrossRef

Vasicek OA (1973) A note on using cross-sectional information in bayesian estimation of security betas. J Finance 28(5):1233–1239CrossRef

Winkelmann M (1984) Aktienbewertung in Deutschland, Quantitative Methoden der Unternehmungsplanung, vol 19. Hain, Königstein/Ts

Zimmermann P (1997) Schätzung und Prognose von Betawerten: Eine Untersuchung am deutschen Aktienmarkt, vol 7. Uhlenbruch, Bad Soden/Ts

Title: Mean reversion adjusted betas used in business valuation practice: a research note
Authors: F. Echterling
B. Eierle
Publication date: 01-10-2015
Publisher: Springer Berlin Heidelberg
Published in: Journal of Business Economics / Issue 7/2015
Print ISSN: 0044-2372
Electronic ISSN: 1861-8928
DOI: https://doi.org/10.1007/s11573-014-0750-4

Springer Professional

Abstract

Please log in to get access to your license.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Journal of Business Economics

Springer Professional "Wirtschaft"

Premium Partner