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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?

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