# Building a CapEx portfolio using mathematical optimization

Let's say you have a set of potential capital projects $$C$$, each defined by an up-front investment $$c_i$$ and random payoff (say, NPV) $$P_i(\omega)$$, where $$\omega \in \Omega$$ is a point in a sample space that is part of a probability triple $$(\Omega, \sigma(\Omega), \mathbb{P})$$. I may or may not know $$\mathbb{P}$$. I have a fixed budget $$B>\min{C}$$

I want to maximize the expected return on investments:

$$\text{maximize}_{i \in C} \sum_{1}^{|C|} z_iP_i(\omega) + \left(B-\sum_{1}^{|C|}z_ic_i\right)$$

Where I am explicitly accounting for "holding cash" as a possible "investment option".

We can think of these two terms as the (random) return from investment and the actual value of cash remaining.

We also need the necessary constraint:

$$\sum_{1}^{|C|} z_ic_i \leq B$$

And the domain constraint $$z_i \in \{0,1\}$$

It seems that there are different approaches based on our risk appetite.

Expected return --> replace $$P_i$$ with $$E_{\mathbb{P}}[P_i]$$ but for CapEx I don't see this as being useful since we cannot rely on LLN to assume convergence to this optimal value (think about the though experiment of the drunkard on a highway).

Risk control: Minimize CVaR (if we know $$\mathbb{P}$$) or use robust uncertainty sets $$\mathcal{U}$$ that force the binary vector $$z$$ to take values that ensure a minimum payoff or maximum loss (subject to some reasonable minimum for each project above 0).

Is this a sensible formulation or are there better/more practical applications in the literature that I am not aware of?