## 4 Threshold Values for the Investment Risk in the Construction Sector in Europe  
The methodology described in Section 3.2 is implemented to estimate the expected tolerable return \(r_{t}\) and the expected acceptable return \(r_{a}\) (Formulas (10) and (11)) for the \"Engineering--Construction\" sector in Europe. Therefore, it is necessary to estimate the following parameters: \(r_{f}\), \(ERP\), \(\beta_{a}\), \(\beta_{t}\). The risk-free rate \(r_{f}\) represents the minimum return that a shareholder expects for investing in a company. \(r_{f}\) can approximate the yield on sovereign bonds. While it is true that risk-free investments do not exist in reality, it is also clear that in sufficiently stable markets an investment in such bonds has zero risk. Since the market of interest is the European one, \(r_{f}\) is given by the average yield of government bonds in Europe with 10-year maturity. The data analyzed refer to the time period September 2004 to November 2020. (sources: European Central Bank; Bloomberg). The equity risk premium \(ERP\), i.e., the expected return that investors require in addition to the risk-free rate to compensate for the risk of an investment, is equal to the spread between the stock market return \(r_{m}\) and the risk-free bond return \(r_{f}\). In this study \(ERP\) is estimated as the average of the risk premium of the European countries (source: Damodaran, 2020). For the estimation of the \(\beta\) variable, a preliminary step is the definition of the panel of firms from which to estimate the systematic sector risk: these are 139 listed civil engineering-construction firms in Europe. For each of the selected civil engineering firms, we estimate the firm beta through Formulas (8) and (9), respectively: ratio between the covariance of expected stock returns to market returns and the variance of expected market returns; slope of the straight line interpolating investment returns to market returns. Specifically, the weekly returns over three and five years are analyzed for both the generic listed firm and the market. A linear regression of the weekly return of the listed firm against the local index--which is the most widely followed index in that market--yields the firm's beta over 2 years \(\beta_{x(2y)}\) and over 5 years \(\beta_{x(5y)}\). Finally, the beta of generic company \(x\) is calculated by weighing \(\
icefrac{{2}}{{3}}\) the estimated beta over two years and \(\
icefrac{{1}}{{3}}\) the estimated beta over five (Damodaran A., 2020):  
\[\beta_{x}=\
icefrac{{2}}{{3}}\cdot\beta_{x(2y)}+\
icefrac{{1}}{{3}}\cdot \beta_{x(5y)}. \tag{14}\]  
It follows that the \"tolerable\" systematic risk \(\beta_{t}\) is obtained by averaging the \(\beta_{x}\) of the 149 panel firms:  
\[\beta_{t}=\frac{\sum_{i=1}^{149}\beta_{xi}}{149}. \tag{15}\]  
The \"acceptable\" systematic risk \(\beta_{a}\) is obtained by averaging 25% of the highest betas in the panel. In other words, with the betas sorted in ascending order, \(\beta_{a}\) is the average of the 37 highest betas, i.e., betas ranging from firm 112 to firm 149. \[\beta_{a}=\frac{\sum_{i=112}^{149}\beta_{xi}}{37}. \tag{16}\]  
Table 1 summarizes the results. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Variable** & **Value** & **Source** \\ \hline \(r_{f}\) & 2.31\% & European Central Bank. \\ \(ERP\) & 7.51\% & [PERSON], Stern School of Business. \\ \(\beta_{t}\) & 1.13 & YahooFinance; [PERSON], Stern School of Business. \\ \(\beta_{a}\) & 1.48 & YahooFinance; [PERSON], Stern School of Business. \\ \hline \(r_{t}\) & 10.80\% & Formula (10) \\ \(r_{a}\) & 13.21\% & Formula (11) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: \(r_{t}\) and \(r_{a}\) estimation for Europe. In order to estimate the expected return considering both the equity and the capital invested by the companies, the tolerable Weighted Average Cost of Capital (WACC\({}_{t}\)) and the acceptable Weighted Average Cost of Capital (WACC\({}_{a}\)) are used, implementing (12) and (13) respectively.