Are there optimal portfolio theories than instead of the expected value they were based on the Mode of distributions?
During my engineer student days I saw the Markowitz theory for portfolio selection and there is something that always bothers me, and it is that are based on maximizing the expected value... let me explain why:
If the price of assets follows a skewed LogNormal distribution (as is suggested by the Black-Scholes model), the expected value is going to be always above the Mode, meaning that the most probable value I will find if I look the asset at a random time (the Mode), will be different from the expected value.
Since when making a portfolio I bought assets where I don't really know when I am going to sell them, I should be trying to achieve the higher prices for the most possible times, so, if I decide to sell at a random time, their observed values are the higher probable ones.
But if I have the average value always, other will be asking the same amount of prize since is the expected value for that asset, but the most probable value I should see is going to be below this value so I will be in a bad situation.
So, instead of this strategy, I should be trying to be over that value to sell it with a higher margin.
Or thinking it on this other way... If I maximized the expected value, and I go to sell at some random time, since for LogNormal distributions the expected value is above the Mode, I am going to be below the market value of the asset the most of the time, so I will be losing money.
This is why my intuition says that a portfolio optimization strategy should be maximizing the Mode of the distribution instead of the expected value (constrained to other figures like minimizing the overall variance, as example, but for now I don't want to close the question to other possible figures).
This is why I want to know If there exists porfolio strategies that maximized the Mode instead of the Expected Value of the assets. Please any reference are welcome, since I didn't find any on Google (maybe because I am not using the right terms gived I am not a researcher on this area, neither a native English speaker).
Summarizing, in my opinion: the "most expected value" in a skewed probability distribution is the Mode and not the "Mean value" (it is just that in symmetrical distributions they match to be the same value, and because of of the wide use of the Normal distribution on physics, due it is the maximum entropy distribution for finite mean and power, the idea is stack in everybody minds), so I want to know about alternatives where the Mode instead of the Mean Value is used to characterize the variables (also, if I am right for some distributions where the mean value is undefined, the Mode does indeed exist anyway).
Beforehand, thanks you very much.
Don't know if this is something easy to do or not, but maybe fastly comparing portfolios, one made with classic Markowitz, an the other choosing portfolios with a modified efficient frontier made with Modes instead of Expected values... It will behave better or not?
2nd Added Later
Due the few numbers of comments and answers, I would like to ask also... Is my intuition wrong?... or this is an example of being this an understudied assumption?