The issues are non-trivial. For example, given the following multi-asset class universe and specifying the upper and lower partial moments with thresholds of both 9% and degrees of both 0.5 (implying an S-shape utility function in the spirit of Kahneman/Tversky)...

...the resulting investment opportunity sets in return/volatility, return/LPM and UPM/LPM space look like this...

Note how the efficient frontiers in Return/LPM and UPM/LPM space are not very smooth.

If we increase both thresholds to 2%, the topography changes dramatically...

It would be interesting to use the above visuals in a Rorschach Test. Anyway...

As we are calculating the partial moments based on historical data with a limited number of 120 observations, one explanation for the results is estimation risk. But looking at the data above and below the thresholds, this this only part of the story...

We calculated the "exact" endogenous portfolio lower and upper partial moments, by the way.

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