i

Saturday, 21 December 2013

Topography of UPM/LPM Space

Cumova/Nawrocki's 2013 "Portfolio Optimization in an Upside Potential and Downside Risk Framework" is an interesting summary of the upside and downside (pun intended) of portfolio construction based on partial moments.

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...



 
The opportunity spaces were generated with 20'000 long-only no-leverage random portfolios. Portfolio constituent weights are not equally distributed, but biased towards the vertexes in order to have enough points on the "efficient frontier", defined as convex hulls of the opportunity sets.

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.

No comments:

Post a Comment