A risk measure, as defined in the Wikipedia page, is a function that maps random variables to real numbers and satisfies the normalized, translative, and monotone properties. There are many other functions of the same type that are common in quantitative finance but are not risk measures. I'm looking for definitions or terminologies that encompass such functions.
- Is there a standard formal notion that encompasses functions such as expected return, and probability of exceeding some return threshold? I.e., functions one would want to maximize, say, "reward measures"?
- As above, but for "reward-to-risk ratio measures" such as Sharpe ratio and Sortino ratio?
- Is there a standard formal notion for a broader class of functions that encompasses both risk measures and "reward measures", and possibly also "reward-to-risk ratio measures"? I will admit this is perhaps too broad for a formal definition to seem useful (since the definition might merely be a function that maps a random variable to a real number), however my interest is mainly to do with categorization.
Where there are no such formal notions, what is the quantitative finance nomenclature that gets used to categorize these kinds of functions?