## 3 ALARP and CAPM. A Model for the Investment Risk Management
### Phases of the Risk Assessment Model
where: \(\Delta\)C is the cost of the risk mitigation intervention net of benefits; \(\Delta\)R is the reduction of risk resulting from the implementation of the planned intervention. \(\Delta\)R is given by the difference between the E(IRR) before mitigation and the cumulative probability of E(IRR) after mitigation. It is evident that once a maximum cost is set that the investor is willing to bear to make the project initiative widely acceptable--the higher the \(\Delta\)R, the better the planned mitigation strategy performs. A Statistical Methodology for the Estimation of the Tolerability and Acceptability Thresholds of the Investment Risk  
Estimating investment risk thresholds is essential for the characterization of the model described in the previous paragraph. The objective is to define a methodology for establishing the minimum tolerable return and the minimum acceptable return for the investor. These returns depend on factors such as: the investment sector of the project; the risk appetite of the financing party; and the specific socio-economic conditions of the area in which the intervention is located. Thus, the theoretical reference to establish these limit values is the Capital Asset Pricing Model, as it allows the assessment of the risk-adjusted discount rate \(r(\beta)\), which can be interpreted as the minimum return expected from an investment project with a risk profile \(\beta\)[46]. In order to define the meaning of the minimum tolerable return \(r_{t}\) and the minimum acceptable return \(r_{a}\) for the investor on which the model is based, it is necessary to briefly recall the theoretical framework of the CAPM. The Capital Asset Pricing Model was introduced into financial economics by [PERSON] [47] and [PERSON] [48] to explain how the risk of a financial investment affects its expected return. It is an extension of the market portfolio model proposed by [PERSON] [49], also known as the mean-variance model, according to which investors are risk-averse and will therefore select a trading-off portfolio between risk and return for a certain investment period [50]. This means that investors will choose efficient portfolios that minimize the variance of the portfolio return for a specific level of expected return, or maximize the expected return, given the specific level of variance. The CAPM assumptions are specified below. They are the same as in the [PERSON] portfolio model, plus assumptions 2 and 3 added by [PERSON] [47] and [PERSON] [48]:  
1. All investors select efficient portfolios to maximize the utility of their investments. In other words, investors are risk-averse, so they maximize utility and focus only on their return (mean) and relative risk (variance);
2. Investors can borrow or lend funds at the risk-free rate of return;
3. All investors have homogeneous expectations, which means that they estimate the same distributions for future rates of return;
4. All investors hold investments for the same period of time;
5. Investors are able to buy or sell shares in any security or portfolio they hold;
6. There are no taxes or transaction costs on the purchase or sale of assets;
7. There is no inflation or any change in interest rates;
8. Capital markets are in equilibrium, and all investments are fairly priced, i.e., investors cannot influence prices [51]. The CAPM equation in accordance with the assumptions of [PERSON] and [PERSON] can be expressed as follows:  
\[\mathrm{E}(r_{i})=r_{f}+\beta\cdot[\mathrm{E}(r_{m})-r_{f}], \tag{7}\]  
where \(\mathrm{E}(r_{i})\) is the expected rate of return on investment \(i\), \(r_{i}\) and \(r_{m}\) represent respectively the gross return of the security in question and the return of the market portfolio. \(r_{f}\) is the return on a risk-free investment. The difference between the market return and the risk-free return gives the market risk premium, i.e., the remuneration for the risk taken by the investor.