Multiple tracking error constraints - is this problem convex?

Quantitative Finance Asked by cune on December 11, 2020

Let’s say I have a return forecast for each stock in the DAX index. I also have a covariance matrix for these 30 stocks.

I want to solve for the 30 weights by maximising the forecast portfolio return, subject to keeping the overall volatility at a certain target level, and subject to multiple tracking error constraints: the overall t.e. for the entire portfolio should be less than or equal to 1%, and tracking error for each sector to also be below or equal to 1% (relative to dax and each Dax sector, respectively).

So, the objective function is linear, but I have multiple quadratic constraints.

Is this a convex problem? Could there exist multiple local minima?

One Answer

The problem being convex depends on the structure of the quadratic constraints in this case, particularly if the quadratic part is positive semi-definite. So you need to write out the constraints in matrix form and do the algebra to check.

Answered by river_rat on December 11, 2020

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