Idiosyncratic risk premium and firm characteristics in Europe: theoretical and empirical issues

Transcription

1 Idiosyncratic risk premium and firm characteristics in Europe: theoretical and empirical issues Marco Vulpiani 1 Abstract We investigated a possible relationship model between the idiosyncratic risk premium (IRP) and company characteristics. Beginning with decomposing the total risk premium into its two components of systematic risk premium and IRP, we found a theoretical expression to model the latter component. Afterwards, through an empirical analysis for Europe, we identified the characteristics of companies that most influence the IRP as quantified in the theoretical model. Our results can be used as a basis for prediction models of IRP as a function of both private and public companies characteristics. Keywords: idiosyncratic risk premium; systematic risk premium; specific risk premium; company characteristics; idiosyncratic risk premium modelling JEL classification code: G1 General Financial Markets, G3 Corporate Finance and Governance 1 Adjunct Professor of Advanced Corporate Finance LUISS Guido Carli, University of Rome Viale Pola 12, Rome, Italy Telephone: Mobile: address: mvulpiani@luiss.it Acknowledgements: The author gratefully acknowledges Prof. Gianluca Mattarocci for his fundamental and continuous review. He also wishes to thank Prof. Raffaele Oriani for his review and his valuable suggestions. Finally, he also wishes to thank Gennaro Aprile and Federico Grassi for their continuous and fundamental help in the entire process. Naturally, the author assumes full responsibility for any eventual errors. Funding: This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

2 1 Introduction Although idiosyncratic risk typically accounts for most of a share s total risk, it can theoretically be diversified away and therefore has long been considered to not be compensated in the estimation of the cost of capital. In particular, the Capital Asset Pricing Model ( CAPM ) theory (Sharpe, 1964) and other modern portfolio theories are based on the assumption that investors hold a portfolio of stocks with sufficient diversification to eliminate idiosyncratic risk. The appropriateness of considering only the systematic risk of common stocks in the estimation of cost of capital has recently received a great deal of attention in the finance and accounting literature. In particular, the assumption that investors hold fully diversified stock portfolios is often unrealistic. In real markets, there is direct evidence of investor under-diversification, and therefore, investors typically hold some idiosyncratic risk in their portfolios. As outlined by Goetzmann and Kumar (2008) based on a sample of more than 62,000 U.S. investors in the period, more than 25% of investor portfolios contained only one stock, more than 50% contained between one and three stocks, more than 70% of households held fewer than five stocks, while only 5%-10% of investor portfolios contained more than ten stocks. A significant strand of the literature has focused on the relationship between idiosyncratic risk and stock returns. For example, Goyal and Santa-Clara (2003) found a significantly positive correlation between average stock variance (largely idiosyncratic) and market returns, while Guo and Savickas (2006) reported a significant relationship between market average idiosyncratic risk and returns on a market index. Additionally, Vulpiani (2014) showed that stock returns are not fully explained by the capital asset price model ( CAPM ) in most markets analysed. Ang et al. (2006, 2009) and Fu (2009) demonstrated a connection between idiosyncratic risk and returns in a cross-section of securities. Although to date, there remains no consensus on the exact nature of the relationship between return and idiosyncratic risk, research now is more inclined to imply that idiosyncratic volatility is relevant in investment analysis (Vozlyublennaia, 2013). 2

3 From the literature relevant to the relationship between returns and idiosyncratic risk, several authors highlight that various factors not just systematic ones affect returns. Several theories have been developed to better understand the relationship between idiosyncratic volatility and firm characteristics. Barberis and Huang (2001), for instance, found a significant relationship between expected returns and idiosyncratic risk and asserted that stock returns are driven not by one but by many different factors, making it much more difficult to reduce the strategy s risk, even with many stocks. Numerous recent papers have investigated the driving factors of stock returns and proposed that certain company characteristics can explain the observed dynamics and cross-sectional differences in idiosyncratic risk. This literature includes but is not limited to Brandt et al. (2010), Morck et al. (2000), Irvine and Pontiff (2009), Brown and Kapadia (2007), and Malkiel and Xu (1999). On the basis of results in the literature, which confirm company characteristics as good indicators of the differences in idiosyncratic risk across securities, it seems reasonable to assume that characteristics are related to the risk premium required by investors for bearing such risk In the two strands of literature cited (concerning the relationship between idiosyncratic risk and stock returns and the relationship between idiosyncratic risk premium (IRP) and company characteristics) there are some limits in the models utilised and investigated from a predictive point of view. In fact, in all these analyses, the authors generally measure idiosyncratic risk as the variance of the residuals in regression models used to measure returns on securities (mostly the CAPM and the threefactor Fama-French model). Campbell et al. (2001) ( CLMX ), for example, referred to the single-factor CAPM of Sharpe and Leitener in their analysis, while Vozlyublennaia, 2013 referred to the Fama-French three-factor model (Fama and French, 1992). However, Fu (2009), Fink et al. (2012), Foster and Nelson (1994), Bekaert et al. (2012), Wei and Zhang (2004), Bali and Cakici (2008), Jiang et al. (2006), and Brown and Kapadia (2007) referred to either the single-factor model or to the three-factor model, while Cao et al. (2008) utilised five different ways to measure idiosyncratic volatility based on the single-factor 3

4 CAPM of Sharpe and Leitener (like CLMX) and using both the unconditional and conditional versions of the Fama-French three-factor model with and without a factor related to momentum. In all these different approaches, idiosyncratic risk is defined and measured as the variance of the error in regression models for securities returns. While this approach is certainly fundamental to analyse idiosyncratic risk and returns in the markets or, more generally, the characteristics of idiosyncratic risk, this kind of modelling has some limitations. The first and in our opinion, the most important limitation of the approach based on residuals is the consequently impossible quantification of idiosyncratic risk from a predictive point of view. In fact, residuals can be derived only from real, actual returns on stocks. Therefore, they can be utilised for the analysis of market-measured IRP but cannot be utilised to predict the expected IRP. When searching for theoretical models that are useful for predicting or quantifying the specific risk premium on the basis of company characteristics, we found a gap in the literature. The second limit of the models found is that because the idiosyncratic risk is defined and measured as the variance of the error in regression models for securities returns, it is dependent on the kind of regression model chosen. According to Bali et al. (2008), computing idiosyncratic volatility on the basis of the residuals from the one-factor capital asset pricing model (CAPM) or from the Fama-French three-factor model makes the estimation of idiosyncratic risk model-dependent. According to the authors, in general, using residuals from different models for the quantification of systematic risk can produce different estimations of idiosyncratic risk. The third limitation concerns the relationship found in the literature between company characteristics and idiosyncratic risk, and it is related to the first and second limits. Additionally, these kinds of analyses are based on quantification of IRP as the variance of the residuals in regression models. Therefore, if from one side, the studies results demonstrate the relationship between company 4

5 characteristics and idiosyncratic risk and from the other side are based on residuals, they cannot be utilised to develop prediction models, and as shown by several authors, the results are model-dependant. The fourth limitation is that most of these studies have been developed with reference to companies listed in the largest markets (mostly American and Asian markets), where large companies are predominant. There are some limits in the literature in terms of the evidence of IRPs and its relationship with company characteristics in the European market, where small and medium enterprises are predominant. Considering that this dimension is one of the drivers of idiosyncratic risk (the lower the first, the higher the second), this implication could limit the extension of the conclusions found in other markets to Europe. The purpose of this work was to formulate IRP as a basis for a predictive model of IRP as a function of the company characteristics that is not model-dependent and can be empirically analysed in the European market. With this purpose, starting from the general definition of TRP, a theoretical formulation for the IRP was found in the first phase, as described in the following paragraph. In the second phase, as described in the following paragraph, an empirical analysis on the relationship between company characteristics and IRP as theoretically formulated in the first phase was carried out for the European market. This paper contributes to the previous literature by preliminarily finding a general, theoretical formulation for the IRP different from the most commonly measured residuals. By definition, this formulation (because it is not derived by a regression formula) is not model-dependent and can represent a basis for predicting IRP. Then, it systematically links the found IRP to firm-specific characteristics through an empirical analysis of European markets. 5

6 The empirical analysis brings together all the firm-specific variables identified by previous studies and also defines new ones. Moreover, it analyses the European context, which has been understudied in the previous analysis. In this process, this study provides new results that may be used as a basis for a predictive model of IRP. Section two proposes a theoretical model for the expression of the systematic risk premium (SRP) and the IRP as a function of the stock s volatility. Section three describes an empirical analysis of Europe that aimed to find the relationship between the main characteristics and the IRP (as defined by the proposed theoretical formula) of companies. Section four discusses the main research findings and policy implications. 2 The theoretical decomposition of TRP in its Systematic and Idiosyncratic components 2.1 Introduction In the first phase of the work, we took a possible definition of TRP ( TRP ) and decomposed into the two components of SRP ( SYRP ) and IRP ( IRP ). In particular, we began by modelling the risk premium of a complete undiversified portfolio (i.e., composed of just one stock) to represent TRP and then, relating it to the risk premium of a diversified portfolio, we decomposed TRP in its two components, SRP and IRP, in a theoretical analysis. In this way, we identified a formula for each of these two components of TRP. 6

7 2.2 The theoretical analysis To determine the decomposition of TRP ( TRP ), defined as the difference between the expected return on stock i R i and the risk-free rate R f,, in its SYRP and IRP components, we made reference to the general portfolio theory (Markowitz, 1952). Investors were assumed to be indifferent between an investment in a diversified portfolio (Market Portfolio) with the highest risk premium per unit of risk and an investment in a portfolio composed of the single stock i, which provides an equal return per unit of risk (McConaughy and Covrig, 2007). The Sharpe Ratio, or the ratio of TRP (R i R f) of the specific stock i to its volatility as measured by its standard deviation σ i can be used as a measure of risk premium per unit of risk (Sharpe, 1966; Sharpe, 1975). On the basis of the above assumption of equal returns per unit of risk of the two portfolios, the Sharpe Ratio of a portfolio composed of only one stock can be assumed to be equal to that of a fully diversified portfolio. Therefore, the following equivalence between the Sharpe Ratio for stock i and the Sharpe Ratio for the market portfolio is assumed: R R i i f R m R m f [1] where: R i: Return on Stock i R f: Risk-Free Rate R m: Market Return σ i: Price Volatility of Stock i Price (Standard Deviation) σ m: Price Volatility of the Market Index (Standard Deviation) Solving [1] for R i we find 7

8 R i R f i ( Rm R f m ) [2] Considering that the difference Ri-Rf represents the premium required for a portfolio composed of only one stock (and therefore in the full absence of diversification), the second term on the right-hand side represents the TRP (TRP) of the portfolio (sum of SRP and IRP): Total Risk Premium TRP i ( Rm R f m ) [3] This formula, representing the TRP for investment i over the risk-free rate R f. for the portfolio composed of only one stock i, relates the TRP to the Market Risk Premium, to the volatility (standard deviation) σ i of investment i, to the volatility (standard deviation) σ m of the market and to the market risk premium (R m-r f). To decompose TRP into its IRP and SYRP components, the correlation ρ im between the volatility of the specific investment σ i and the volatility of the market σ m is introduced. We add and subtract the term ρ im TRP in the right-hand side of formula [2]: R R TRP TRP TRP R TRP ( 1 TRP [4] i f im im f im im) From the definition of correlation and its relationship with covariance, we find that im i m cov( Ri, R i m m ) i m cov( Ri, R var( R ) m m ) i [5] The right-hand side of formula [5] therefore represents the beta coefficient of the CAPM formulation (Sharpe, 1964) and ρ im TRP in formula [4] represents the SRP: i imtrp im ( Rm R f ) i ( Rm R f ) [6] m In this way, SYRP and IRP can be isolated in the TRP. In fact, the first two terms on the right-hand side of formula [4] represent compensation for the systematic risk component or cost of equity K e, according to the Capital Asset Pricing Model (Sharpe, 1964): 8

9 K e R R R ) [7] f i ( m f With the second term of equation [4] representing the premium for systematic risk: SYRP i ( Rm R f ) [8] Considering that the right-hand side of equation [4] defines the TRP for stock "i" in a portfolio with no diversification, and that, according to the above equation, the second term represents the SRP, the third term of equation [4] represents IRP. Therefore, this equation can be rewritten by expressing total risk compensation as the separate contribution of the two components for systematic risk (undiversifiable risk) and for idiosyncratic risk (diversifiable risk): R i R ( R R ) (1 ) TRP R SYRP IRP [9] f m f im f With the following formula expressing the IRP: IRP i 1 im) TRP (1 im) ( Rm R ) [10] ( f m As described in the following chapter, to analyse the behaviour of IRP with the companies characteristics, we carried out an empirical investigation of the relationship between such formulations of IRP and the main characteristics of the companies. 3 The Empirical Analysis: Companies Characteristics and IRP 3.1 Introduction The theoretical formula obtained, as described in the previous paragraph, for the quantification of IRP is unrelated to company characteristics, and because it is based on the volatility of the stock price of reference, it is applicable only to listed companies. 9

10 To make up for this limitation, in the second phase of the research, we further investigate the relationship between the IRP as defined by the proposed formula and the main measurable characteristics of companies. Beginning with results in the existing literature, we developed a cross-sectional regression model of the IRP with some of the company characteristics found by scholars to explain idiosyncratic volatility. We performed an empirical analysis on the stocks of the most important companies in Europe (part of Eurostoxx 600). With a panel data analysis (fixed effects), we analysed the relationship between the main characteristics and the IRP (as defined by the proposed formula) of the companies in the crosssection of the securities in the period. We found a regression formula applicable to listed companies and, with respect to characteristics similar to those identified for listed companies, one applicable to private companies. The regression formulas determined here can represent a basis on which to develop prediction models for the quantification of the IRP of a private or public company based on its characteristics. 3.2 Data and methodology To analyse the relationship between the IRP as formulated by the theoretical computation described in the previous chapter and company characteristics, market data relevant to stocks traded in Europe were analysed. The analysis was focused on Europe for the reduced availability in the literature of this kind of analysis for the European market and for the highest presence of small and medium enterprises (which should be characterised by a highest idiosyncratic risk) in this market. In particular, a database (hereafter the Database ) was built with data relevant to the stocks of the listed companies included in Eurostoxx 600 Index for the period. The analysis was conducted by collecting from Bloomberg all data needed to compute IRP according to formula [10]. We then calculated IRP for all the companies in the Eurostoxx 600 in the period (the Market IRP ) on the basis of formula [10] and the data collected in the Database. 10

11 Once the Market IRP was calculated, a multivariate regression analysis (Panel Data) of the Market IRP with the characteristics of the companies was conducted. For different purposes, several authors defined different characteristics of the companies as determinants of the idiosyncratic volatility of companies (see, for example, Bennett and Sias, 2006; Brandt et al., 2010; Brown and Kapadia, 2007; Campbell et al., 2001; Cao et al., 2008; Fink et al., 2006; Harvey and Siddique, 2000; Irvine and Pontiff, 2009; Malkiel and Xu, 2003; Pástor and Veronesi, 2003; Schwert, 2002; Vozlyublennaia, 2013; Wei and Zhang, 2006). To select the characteristics for the analysis, we first considered all the characteristics found in the literature related to idiosyncratic volatility (approximately thirty-four different characteristics). Then, for each characteristic, we analysed the frequency of its presence in the literature, defined as the number of articles in which the variable was considered as a determinant of idiosyncratic volatility out of the total number of articles found. Table 1 in the Appendix presents this frequency analysis. We selected the characteristics with a frequency higher than 10%, that is, those that were considered to be related to idiosyncratic volatility by at least two articles. We apply this constraint to select the characteristics shared among at least two authors. With this approach, the following variables were selected ( Main Characteristics ): 1. Size: Market capitalisation; 2. BM: Market-to-book ratio; 3. Lev: Financial leverage; 4. Age: Number of days since the company s incorporation; 5. Turn: Share turnover; 6. Institutional ownership: Institutional shares as a percentage of total shares outstanding; 7. ROA: EBITDA-to-total-assets ratio; 8. Dividend Dummy: Dummy variable equal to 1 if the firm paid any dividends; 9. VolROE: ROE volatility; 11

12 10. ROE: Return on equity; 11. Volatility: Total stock volatility; 12. EPS: Earnings per share. Afterwards, we carried out a multivariate analysis to find the potential regression between the Market IRP at time t and the Main Characteristics at time t-1 with a panel data technique with fixed effects, according to the following formula: IRP = β 1X 1 + β 2X β 10X 10 + d d 21 + α [11] With: βi: Coefficient i X i: Main characteristic i dj: dummy for the year j α: constant term. Dummies for the year variable were included in the analysis to improve the quality of the yearly regression. With the purpose of giving a research contribution, we tried to find additional variables that could be considered determinants of IRP but have not been contemplated in the literature, with a stepwise backward approach applied to an increased sample of generic characteristics. We focused on variables that could be representative of the level of development of the company business and relevant idiosyncratic risk, such as the level of the marginal growth rate. With the aim of providing a research contribution, we tried to extend the application of the above approach to private companies. With this objective, we sought possible substitutes of some characteristics available only for listed companies with others that had a similar meaning but were also available for private companies. In particular, we made the following substitutions and removals: Size: Total assets instead of market capitalisation; 12

13 Book-to-market ratio: Book value of equity rather than the ratio between the book value of equity to market value ( BV ); Volatility: we removed the volatility of the stock price because the formula already includes another variable of volatility (volatility of ROE); Turnover: we removed the turnover of shares because they were not applicable. 3.3 Results discussion multivariate analysis for listed companies The results of the analysis are reported in the following Table 2: [TABLE 2 ABOUT HERE] Most of the analysed characteristics (nine out of twelve) were significant and showed a behaviour consistent with that reported in the literature. A stepwise backward procedure was carried out to select only the characteristics with an acceptable significance (i.e., with p<0.1). With a stepwise backward approach, several variables were removed because they were not considered significant enough (i.e., with p>0.1). The final results are reported in the following Table 3: [TABLE 3 ABOUT HERE] In this way, we found the following relationship applicable to listed companies: IRP = β1*age + β2*bm+ β3size+β4*volroe + β5*turn + β6*volatility+β7*roa + β8*eps + β9*dividend Dummy + β10*g+ Σidi + α [12] The relative β i coefficients, the dummies d i coefficients and the constant α are reported in Table 4 in the Appendix. 13

14 As reported in the above Table 3, the overall R-squared of the regression was approximately 30.4%. The selected variables do not exhibit multicollinearity problems, as shown by the VIF analysis reported in Table 5 in the Appendix and by the analysis of correlations among them (Table 6 in the Appendix). Of the above selected characteristics, most were the same as those reported in the literature. Nevertheless, with the enlarged analysis cited before, we found a new, additional variable. In particular, we found the EBITDA (earnings before interest tax depreciation and amortisation) growth rate (hereafter also Growth Rate or simply G ), which was not considered in the analysed literature, with a significant value (p<0.018) as a determinant of the IRP. In terms of statistical significance, among the characteristics selected, five of ten (Age, VolROE, Volatility, EPS, and Dividend Dummy) are strongly significant (at p<0.01). Two (Size and EBITDA growth rate) are significant at p<0.02, and the remaining (BM, ROA, Turn) are significant at p<0.1. This model can represent a basis on which to develop models for the quantification and the prediction of IRP for listed companies on the basis of their characteristics. 3.4 Results discussion multivariate regression for private companies For private companies, as anticipated in the previous paragraph, we substituted several characteristics available only for listed companies with others that had a similar meaning and were available for private companies. We then carried out the panel data analysis (with fixed effects), which gave the following results: [TABLE 7 ABOUT HERE] 14

15 Additionally, in this case, most characteristics were strongly significant, with six of eight determinants with a p<.01 (Age, VolROE, EPS, ROA, Dividend Dummy, and EBITDA growth rate), one out of eight (Size) with p<.02 and only one with p>0.1 (BV). A stepwise backward procedure was carried out to select only the characteristics with an acceptable significance (p<.1). With a stepwise backward approach, one variable (BV) was removed because it was not considered significant enough (i.e., with p>0.1). The final results are reported in the following Table 8: [TABLE 8 ABOUT HERE] In conclusion, we found that the following relationship was also applicable to private companies: IRP = β1*age + β2*size+β3*volroe +β4*roa + β5*eps + β6*dividend Dummy + β7*g + Σidi + α [13] Where 1. Age: number of days since the company s incorporation 2. Size: natural logarithm of market cap 3. VolROE: ROE volatility of last 5 years 4. ROA: EBITDA/total assets 5. EPS: earnings per share 6. Dividend Dummy: Dummy variable equal to 1 if the firm paid any dividends 7. G: EBITDA growth rate The relative β i coefficients, the dummies d i coefficients and the constant α are reported in Table 9 in the Appendix. As reported in Table 8 above, the overall R-squared of the regression was approximately 27.5%. 15

16 Results for correlations are in line with correlations found by other authors in previous works on linear relationships between idiosyncratic volatility and company characteristics. For example, in their model, Brandt et al. (2010) found a correlation between a minimum of 6.3% and a maximum of 33.2%, while Brown and Kapadia (2007) found a correlation of 11.4%. 4 Conclusion This work aimed to find a formulation of IRP as a function of company characteristics that was not model-dependent, that could represent a basis for a predictive model of IRP, and that could be empirically analysed in the European market. Given that the literature on the relationship between idiosyncratic risk and stocks returns and the relationship between IRP and company characteristics is not boundless, there is gap from a predictive point of view. In fact, in all these analyses, the authors generally measure idiosyncratic risk as the variance of the residuals in regression models utilised to measure returns on securities that cannot be utilised as a basis for prediction models and that is model-dependent. This paper contributes to the previous literature by preliminarily finding a general theoretical formulation for the IRP that is different from the residuals usually measured, is not model-dependent and is systematically linked to firm-specific characteristics through an empirical analysis conducted for European markets. In more detail, the work began with the decomposition of TRP into the two components of systematic and idiosyncratic risk. The identification of IRP on the basis of the stock volatility and its correlation with the market is a first contribution, considering that in the literature, IRP is generally defined as the variance of the residuals in regression models utilised to measure the returns of securities. By definition, this formulation (because it is not derived by a regression formula) is not model-dependent. Afterwards, on the basis of characteristics identified in the literature as the most important determinants of idiosyncratic volatility, we conducted a multivariate analysis on a portfolio of European 16

17 stocks (companies part of Eurostoxx 600 index) designed to identify the relationship between the theoretical IRP, as defined in the first phase of the study, and company characteristics (considered in the literature as the main determinants of idiosyncratic volatility). In addition, other than the characteristics mostly cited in literature, we found that an additional characteristic (EBITDA growth rate) significantly affects the IRP. In a multivariate analysis with a panel data approach for data on listed companies in Europe in the period, we found a linear regression formula that expresses IRP as a function of the characteristics of listed companies and of private companies. The empirical analysis brings together all firm-specific variables identified by previous studies and also defines new ones. Moreover, it analyses the European context, which has been understudied in previous analyses. In this way, it provides new results that can represent a basis for developing predictive models to quantify IRP. 17

18 References Ang, A., Hodrick, R.J., Xing, Y., & Zhang, X. (2006). The cross-section of volatility and expected returns. Journal of Finance 61, Ang, A., Hodrick, R.J., Xing, Y., & Zhang, X. (2009). High idiosyncratic volatility and low returns: international and further U.S. evidence. Journal of Financial Economics 91, Bali, T.G., & Cakici, N. (2008). Idiosyncratic volatility and the cross-section of expected returns. Journal of Financial and Quantitative Analysis 43, Bali, T.G., Cakici, N., & Levy, H. (2008). A model-independent measure of aggregate idiosyncratic risk. Journal of Empirical Finance 15, Barberis, N., & Huang, M. (2001). Mental accounting, loss aversion, and individual stock returns. Journal of Finance 56, doi: / Bekaert, G., Hodrick, R.J., & Zhang, X. (2012). Aggregate idiosyncratic volatility. The Journal of Financial and Quantitative Analysis Vol. 47, No. 6, pp doi: /s Bennett, J.A., & Sias, R.W. (2006). Why company-specific risk changes over time. Financial Analysts Journal 62, Brandt, M.W., Brav, A., Graham, J.R., & Kumar, A. (2010). The idiosyncratic volatility puzzle: time trend or speculative episodes? Review of Financial Studies 23, Brown, G., & Kapadia, N. (2007). Firm-specific risk and equity market development. Journal of Financial Economics 84, Butler, P., & Pinkerton, K. (2006). Company-specific risk - a different paradigm: a new benchmark. Business Valuation Review 25, Butler, P., & Pinkerton, K. (2009). There is a new beta in town and it s not called total beta for nothing. 15:3 Business Valuation Update

19 Campbell, J.Y., Lettau, M., Malkiel, B.G., & Xu, Y. (2001). Have individual stocks become more volatile? An empirical exploration of idiosyncratic risk. The Journal of Finance 56, Canefield, D., Kruschwitz, L., & Löffler, A. (2014). Why total beta produces arbitrary valuations: a violation of the "no-arbitrage" principle. Business Valuation Review 33, Cao, C., Simin, T., & Zhao, J. (2008). Can growth options explain the trend in idiosyncratic risk? Review of Financial Studies 21, Douglas, J.W. (1969). Risk in the equity markets: an empirical appraisal of market efficiency. Yale Economic Essays, 9, Fama, E.F., & French, K.R. (1992). The cross-section of expected stock returns. The Journal of Finance 47, Fama, E.F., & MacBeth, J.D. (1973). Risk, return, and equilibrium: empirical tests. Journal of Political Economy 81, Fink, J., Grullon, G., Fink, K., & Weston, J.P. (2006). Firm age and fluctuations in idiosyncratic risk. SSRN Electronic Journal. doi: /ssrn Fink, J.D., Fink, K.E., & He, H. (2012). Expected idiosyncratic volatility measures and expected returns. Financial Management 41, Foster, D.P., & Nelson, D.B. (1994). Continuous record asymptotics for rolling sample variance estimators. NBER technical working paper no National Bureau of Economic Research. doi: /t0163. Fu, F. (2009). Idiosyncratic risk and the cross-section of expected stock returns. Journal of Financial Economics 91, Goetzmann, W.N., & Kumar, A. (2008). Equity portfolio diversification. Review of Finance 12, Goyal, A., & Santa-Clara, P. (2003). Idiosyncratic risk matters! The Journal of Finance 58, Guo, H., & Savickas, R. (2006). Understanding stock return predictability. Federal Reserve Bank of St. Louis, FRB of St. Louis Working Paper No B. 19

20 Harvey, C.R., & Siddique, A. (2000). Conditional skewness in asset pricing tests. The Journal of Finance 55, Helfenstein, S.B. (2009). Revisiting total beta. Business Valuation Review 28, doi: / Irvine, P.J., & Pontiff, J. (2009). Idiosyncratic return volatility, cash flows, and product market competition. Review of Financial Studies 22, Jiang, X., & Lee, B-S. (2006). The dynamic relation between returns and idiosyncratic volatility. Financial Management, Vol. 35, issue 2, pp Kasper, L.J. (2008). The Butler Pinkerton model for company-specific risk premium: a critique. Business Valuation Review 27, doi: / Lehmann, B.N. (1990). Fads, martingales, and market efficiency. The Quarterly Journal of Economics 105, doi: / Levy, H. (1978). Equilibrium in an imperfect market: a constraint on the number of securities in the portfolio. American Economic Review 68, Lintner, J. (1965). The valuation of risk assets and the selection of risky investments in stock portfolios and capital budgets. Review of Economics and Statistics 47, Malkiel, B.G., & Xu, Y. (1997). Risk and return revisited. Journal of Portfolio Management 23, Malkiel, B.G., & Xu, Y. (1999). The structure of stock market volatility. Princeton: Economics Department, Princeton University. Malkiel, B.G., & Xu, Y. (2002). Idiosyncratic risk and security returns. Dallas: University of Texas at Dallas. Malkiel, B.G., & Xu, Y. (2003). Investigating the behavior of idiosyncratic volatility. Journal of Business 76, Markowitz, H. (1952). Portfolio selection. The Journal of Finance 7,

21 McConaughy, D.L., & Covrig, V. (2007). Owner s lack of diversification and the cost of equity capital for a closely held firm. Business Valuation Review, Vol. 26, No. 4, pp doi: / Merton, R.C. (1987). A simple model of capital market equilibrium with incomplete information. The Journal of Finance 42, Miller, M., & Scholes, M. (1972). Rates of return in relation to risk: A re-examination of some recent findings. Michael C. Jensen ed. Studies in the theory of Capital markets, pp Morck, R., Yeung, B., & Yu, W. (2000). The information content of stock markets: why do emerging markets have synchronous stock price movements? Journal of Financial Economics 58, Pástor, Ľ, & Veronesi, P. (2003). Stock valuation and learning about profitability. The Journal of Finance 58, Rojo Ramirez, A., Cruz Rambaud, S., & Alonso Cañadas J. (2011). Discount Rate and Cost of Capital: Some More about the Puzzle. Social Science Research Network, p Rubin, A., & Smith, D.R. (2011). Comparing different explanations of the volatility trend. Journal of Banking and Finance 35, Schwert, G.W. (2002). Stock volatility in the new millennium: how wacky is NASDAQ? Journal of Monetary Economics 49, doi: /s (01)00099-x. Sharpe, W.F. (1964). Capital asset prices: a theory of market equilibrium under conditions of risk. Journal of Finance 19, Sharpe, W.F. (1966). Mutual fund performance. Journal of Business 39, Sharpe, W.F. (1975). Adjusting for risk in portfolio performance measurement. Journal of Portfolio Management 1, Vozlyublennaia, N. (2013). Do firm characteristics matter for the dynamics of idiosyncratic risk? Journal of International Financial Markets, Institutions and Money 27, Vulpiani, M. (2014). Special cases of business valuation. McGraw Hill, pp

22 Wei, S.X., & Zhang, C. (2004). Idiosyncratic risk does not matter. Journal of Banking and Finance 29, doi: /j.jbankfin Wei, S.X., & Zhang, C. (2006). Why did individual stocks become more volatile? Journal of Business 79,

23 APPENDIX [TABLE 1, 4, 5, 6 AND 9 ABOUT HERE] 23

24 About the author Marco Vulpiani holds a Master s Degree in Aircraft Engineering from the University of Rome La Sapienza, a Master of Science in Business Administration from the University of Rome Tor Vergata and a Ph.D. in Business Administration from the University of Rome Tor Vergata. Marco is Adjunct Professor of Advanced Corporate Finance in the Master of Science course in Business Administration at Luiss Guido Carli University. He is also Adjunct Professor of Luiss Business School in the MBA, in the Executive Master in Finance and Markets and in the Master in Corporate Finance. He has approximately 25 years of professional experience in corporate finance. As an Equity Partner with Deloitte Financial Advisory S.r.l., he is the head of the Valuation and Business Modelling Services for Deloitte in Italy. Marco is the author of several publications (including the book Special Cases of Business Valuation McGraw-Hill 2014) and has been a speaker in many national and international conferences on issues of business valuation. He is also member of the Management Board of the Italian Valuation Standard Setter "Organismo Italiano di Valutazione (OIV)". 24

25 TABLES Table 1 Frequency analysis for the identification of the Main Characteristics cited in the literature # Mnemonic Description Relative freq. 1 Size Market capitalisation 61.5% 2 BM Market-to-book ratio 53.8% 3 Lev Financial leverage 53.8% 4 Age Number of days since the company s incorporation 38.5% 5 Turn Share turnover 30.8% 6 Institutional ownership Institutional shares as a percentage of total shares outstanding 23.1% 7 ROA EBITDA-to-total-assets ratio 23.1% 8 Dividend Dummy Dummy variable equal to 1 if the firm paid any dividends 23.1% 9 VolROE ROE volatility 15.4% 10 ROE Return on equity 15.4% 11 Volatility Total stock price volatility 15.4% 12 EPS Earnings per share 15.4% 13 SASLS Operating leverage 7.7% 14 VOL Shares traded 7.7% 15 VOL2 Average monthly volume from CRSP 7.7% 16 Price Low-priced stock 7.7% 17 Industry turnover Proportion of industry market value that enters and exits an industry in a given time 7.7% period 18 Risky-to-safe Ratio of risky industries to safe industries 7.7% 19 Small-to-large Ratio of the number of small companies to the number of large companies 7.7% 20 Herfindahl Index Industry's Herfindahl Index 7.7% 21 Stock exchange market dummy Dummy for listing in Amex/Nasdaq 7.7% 22 Bubble dummy Dummy for tech bubble period of % 23 Crash dummy Dummy for crash of % 24 IPO dummies IPO listing group dummies 7.7% 25 Return Stock return 7.7% 26 Asset tangibility Property plant and equipment divided by total assets 7.7% 27 Variance - EPS Variance of EPS 7.7% 28 Variance - cash flow shock p Variance of cash flow shock per share 7.7% 29 Variance - sales shock per s Variance of sales shock per share 7.7% 30 Foreign market share Market share from foreign competitors: (imports)/(shipments exports) 7.7% 31 IGROWTH Inventory growth 7.7% 32 EGROWTH EPS growth 7.7% 33 DIVYLD Dividend yield 7.7% 34 SPREAD ( Bid - ask ) / price 7.7% 25

26 Table 2 Preliminary regression analysis results for listed companies Fixed-effects (within) regression Number of obs = 9,681 Group variable: company1 Number of groups = 461 R-sq: Obs per group: within = min = 21 between = avg = 21.0 overall = max = 21 F(32,9188) = corr(u_i, Xb) = Prob > F = IRP Coef. Std.Err. t P> t [95% Conf. Interval] Size BM Lev VolROE Volatility ROE ROA EPS Institutional Ownership Turn -4.86e e e e-10 Age -2.92e e e e-07 Dividend Dummy Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y α sigma_u sigma_e rho (fraction of variance due to u_i) F test that all u_i=0: F(460, 9188) = 3.52 Prob > F =

27 Table 3 Final regression analysis results for listed companies Fixed-effects (within) regression Number of obs = 9,681 Group variable: company1 Number of groups = 461 R-sq: Obs per group: within = min = 21 between = avg = 21.0 overall = max = 21 F(30,9190) = corr(u_i, Xb) = Prob > F = IRP Coef. Std.Err. t P> t [95% Conf. Interval] Age -2.87e e e e-07 BM Size VolROE Turn -5.03e e e e-10 Volatility ROA EPS Dividend Dummy G Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y α sigma_u sigma_e rho (fraction of variance due to u_i) F test that all u_i=0: F(460, 9190) = 3.54 Prob > F =

28 Table 4 βi coefficients, dummy coefficients and constant α of formula [12] for public companies Characteristics Coefficients β1-2.87e-07 β5-5.03e-09 β β β β β β β β Dummies Coefficients Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Constant alfa

29 Table 5 VIF analysis for the identified variables Variable VIF 1/VIF Dividend dummy BM ROA VolROE Volatility Lev EPS G Size Age Turn Mean VIF

30 Table 6 Correlation analysis for the identified variables IRP BM ROA VolROE Volatility Lev EPS G Size Age Turn IRP BM ROA VolROE Volatility Lev EPS G Size Age Turn

31 Table 7 Preliminary regression analysis results for private companies Fixed-effects (within) regression Number of obs = 9,681 Group variable: company1 Number of groups = 461 R-sq: Obs per group: within = min = 21 between = avg = 21.0 overall = max = 21 F(28,9192) = corr(u_i, Xb) = Prob > F = IRP Coef. Std.Err. t P> t [95% Conf. Interval] Age -2.89e e e e-07 Size BM -4.99e e e e-08 VolROE ROA EPS Dividend Dummy G Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y α sigma_u sigma_e rho (fraction of variance due to u_i) F test that all u_i=0: F(460, 9192) = 3.81 Prob > F =

32 Table 8 Final regression analysis results for private companies Fixed-effects (within) regression Number of obs = 9,681 Group variable: company1 Number of groups = 461 R-sq: Obs per group: within = min = 21 between = avg = 21.0 overall = max = 21 F(27,9193) = corr(u_i, Xb) = Prob > F = IRP Coef. Std.Err. t P> t [95% Conf. Interval] Age -2.92e e e e-07 Size VolROE ROA EPS Dividend Dummy G Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y α sigma_u sigma_e rho (fraction of variance due to u_i) F test that all u_i=0: F(460, 9193) = 3.80 Prob > F =

33 Table 9 βi coefficient, dummy coefficients and constant α of formula [13] for private companies Characteristics Coefficients β1-2.92e-07 β β β β β β Dummies Coefficients Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Constant alfa