How are an investors risk preferences related to $\alpha \in (0,1)$ in a mean CVaR optimization?

Would a risk averse investor choose a higher value of $\alpha$, and if so why?

My understanding is, yes. CVaR is the average value of the worst case scenarios (in this case, the portfolio returns). So by choosing a higher $\alpha$ (i.e. closer to one), the investor wants to essentially minimize CVaR (minimize losses) even for scenarios which are very likely to occur.

If someone could provide a second opinion on my understanding, and/or provide a different perspective that would be great. Thanks


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