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Mean-Variance optimization trades off expected returns with portfolio variance. The idea is that excess variance is not desirable.

But what if you weren't averse to high variance and you wanted to maximize both expected returns and variance. Has there been any research done on this or concepts similar to this?

As an example of a situation where this might be the case, think of a paper trading competition where there are 100 participants. The winner receives $100, and everyone else gains nothing. Ideally, you'd want your portfolio to be high variance, because in order to win you need to outperform 99 others. If you maximized mean+variance (or mean+std), you would be essentially maximizing the odds that you get above some threshold. Compare this with mean-variance optimization, which might improve the chance you place in the top 10, but not necessarily maximize the chances you get first place.

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    $\begingroup$ You are right that contestants in paper competitions try to mazimize variance, but I am not sure it makes sense in any real investing or trading situation. $\endgroup$
    – nbbo2
    Commented Feb 3, 2023 at 16:47

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You would be risk loving. In a world with no trading frictions you would just take infinite leverage and invest in the tangency portfolio.

In a world where there are borrowing constraints you would take maximum leverage possible to invest in the tangency portfolio.

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  • $\begingroup$ Is there no other portfolio with the same mean but higher variance? Why not go inside the efficient frontier (to the right of the tangency portfolio) to find one? I think you would then have a superior solution. $\endgroup$
    – Richard Hardy
    Commented Feb 6, 2023 at 9:29
  • $\begingroup$ That starts depending on your coefficient of risk aversion. But the optimal point will still most likely be on the Capital Market Line. With infinite leverage for sure. With finite leverage, yes you probably will be inside the efficient frontier. $\endgroup$
    – phdstudent
    Commented Feb 6, 2023 at 15:01
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    $\begingroup$ Is it not pretty obvious that by moving to the right of the CML you gain variance without sacrifing mean? Therefore, a point on the CML is the worst, not the best solution for any given mean. The whole idea of the optimality of the CML is based on the assumption of risk aversion; no wonder it fails under risk seeking. $\endgroup$
    – Richard Hardy
    Commented Feb 6, 2023 at 16:22
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The problem is then a maximization of a convex criterion, which is not really interesting from the mathematical of economic viewpoint, at least not in portfolio optimization.

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I think the Kelly betting system might be a good place to start.

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  • $\begingroup$ In a markowitz type portfolio optimization problem where you want to maximize return, if you respect the constraints ( assuming there are just positivity constraints ), the portfolio will end up being made up of the highest returning stock. So, you need to impose some real position constraints in order to get a practical result. This is obviously related to the comment nbbo2 made. $\endgroup$
    – mark leeds
    Commented Feb 4, 2023 at 6:00

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