## 2 Theoretical Basis about Risk
### The Risk Management Process Referred to the Economic Evaluation of Projects  
Risk Management is well integrated with decision theory. On the topic, [PERSON], [PERSON] and [PERSON] (2016) and [PERSON] (2018) highlight not only that the phases of the decision-making process are in line with those of the risk analysis process, but also that the risk field provides multiple inputs that are useful to support analysts during decision phase, among others: the conceptualization and characterization of the problem in terms of goals, criteria, risks, uncertainty, knowledge and priorities; the hierarchization of the problem in order to understand its key principles; the analysis of statistical data to identify the elements of risk; risk assessment, including quantitative, of potential alternatives; and the perception of risk by the various actors involved in the decision-making process. So, risk management becomes critical when it is necessary to express a judgment of economic convenience on investments. This is particularly true when the uncertainty concerning sensitive project variables makes it difficult to express the outcome of the Cost Benefit Analysis (CBA) in deterministic terms, as in the case of major interventions or when the data needed for the analysis is incomplete or difficult to find (2018). This leads the evaluator to conduct the analysis in stochastic terms. In this case, it is needed to:  
1. Identify the sensitive variables of the system, i.e., those that have a significant influence on the final value of the economic performance indicator. According to the European Commission's Guide to Cost-Benefit Analysis, \"critical\" variables are considered to be those for which a change of \(\pm 1\%\) in the estimated value results in a variation of more than \(\pm 1\%\) of the Net Present Value (NPV) (2018);
2. Describe the sensitive variables in stochastic terms, that is, derive the probability distribution of the risky parameters of the analysis, identified in point (1);
3. Estimate the probability distributions of the investment performance indicators. This step returns the stochastic description of the profitability indicator as a function of the probability distribution associated with the risky parameters in point (2). If we consider the Internal Rate of Return (IRR) as an indicator of economic performance, then: \[\sum_{t=0}^{n}\frac{B_{t}-C_{t}}{\left(1+IRR\right)^{t}}=0,\] (3) where: * \(B_{t}\) are the benefits at time t and are a function of both probabilistic \(\left(B_{p1},\ \ldots B_{pn}\right)\) and deterministic \(\left(B_{d1},\ \ldots B_{dn}\right)\) terms;
* \(C_{t}\) represent the costs at time t described in both probabilistic \(\left(C_{p1},\ \ldots C_{pn}\right)\) and deterministic \(\left(C_{d1},\ \ldots,C_{dn}\right)\) terms;
* \(IRR\) is the Expected Internal Rate of Return. In fact, it is by analyzing the IRR probability density function and the cumulative probability distribution of the IRR that it is possible to derive important information on the riskiness of the project (2018). From the punctual probability distribution of the profitability indicator--shown in Figure 4a--it is possible to read all the possible values of the indicator and the probability of each of them occurring. For this reason, the expected value or mean value of the variable is also derived from this distribution.