**4. Discussion**

In this section, there is a brief explanation of how a policy maker could employ this model to design power-generation portfolios. In Figure 9, we draw one of our pollutant efficient frontiers, specifically model 2.e, with all the pollutant technologies and with technological constraints in an emission-risk coordinate axis. In this graph, the non-pollutant model portfolios will be all of them on the coordinate origin; they have no emission and, consequently, no emission risk.

A policy maker can choose any combination: any linear combination between any portfolio on the pollutant efficient frontier and on the non-pollutant efficient frontier which, from an emission point of view, lays on the coordinate origin. In Figure 9, the shaded area delimited by the coordinate origin and the pollutant efficient frontier of model 2.e represents these combinations. We can see that the area allows the policy maker to lower the risk for a given level of emission or the emission for a given level of risk. For instance, given the emission and risk values for GMV and GME portfolios of model 2.e shown in Table 6, it is easy to calculate the portfolio proportions needed to reach an emission risk of 0.10 kg/MWh. Obviously, the policy maker will prefer the lowest emission possible for that level of risk and so they will choose the pollutant GME portfolio for the combination, resulting in point A in Figure 9. Point A can be reached by combining the pollutant GME portfolio and the non-pollutant portfolio in a proportion of 39.39–60.61%. Also, the policy maker could want to set the emission of the resultant portfolio, say, at 30 kg/MWh. In this case, they would like to reach a minimum risk combination for that level of emission. For this reason, they will want to combine the pollutant GMV portfolio with the non-pollutant portfolio, resulting in point B, which corresponds to a 47.44–52.56% proportion of pollutant GMV and non-pollutant portfolios.

**Table 6.** Emission and risk of the global minimum variance (GMV) and global minimum emission (GME) portfolios of model 2.e.


In the last example, it is easy to see that the combination B' with a 46.75% GME portfolio and a 53.25% non-pollutant portfolio has the same risk value as combination B but with a lower emission of 25.15 kg/MWh. It does not seem reasonable to lose the opportunity to lower emissions without increasing the risk. This is why the lower segmen<sup>t</sup> of the shaded area is more e fficient in the sense used in this work than the rest of the points in the area when the aim is to adapt the generation portfolio to a predetermined risk. This insight is similar to the financial CML, but in our case and due to the convexity of the e fficient frontier, instead of having a tangency or market portfolio, we propose to use the corresponding GME portfolio instead.
