Scalar version of the problem: in order to build a portfolio we sort list of companies by some score and choose top 10. Where the score - a single non-negative number calculated by some algorithm, (let's say we use normalised profit of the company for the last year).
Now, if we extrapolate the profit to the future year - it will be a guess, and the score would became a distribution of some sort, not a number.
Question: how to choose top 10 companies with the distribution used as the score? I see here two problems.
First problem - what is the "best"? It seems there could be multiple definitions of the "best" and we can choose what we want, like:
- "best aggressive portfolio" - highest probability of the maximum profit.
- "best safe portfolio" - highest probability that the revenue from the portfolio won't be less than X at any moment.
Second problem - how to sort list of companies by distribution? (I guess it could be an approximated solution, you check all the possible combinations of 10 companies form the list and evaluate every of those combination against your definition of the "best").
P.S.
Is there any practical and short book/materials on this topic? How to work with such kind of distributions in a practical way with numeric software, ideally in finance world.
The first thing you should make yourself accustomed to is stochastic dominance. Wikipedia is a good start:
https://en.wikipedia.org/wiki/Stochastic_dominance
Ones you leave this realm you will need to think about what exactly it is that you want and try to put it into a utility function.