Minimizing Portfolio Expected Shortfall with Joint Losses
Summary
The document explains why portfolio expected shortfall cannot generally be minimized by taking a weighted sum of the expected shortfalls of individual instruments. Portfolio risk depends on the distribution of the combined, weighted losses, so the objective must calculate expected shortfall on that portfolio loss distribution. This captures how assets’ losses occur together, rather than treating their standalone risk measures as additive.
For optimization, the answer points to a linear programming formulation that introduces auxiliary variables to represent the shortfall calculation. It also identifies portfolio constraints: weights must satisfy a budget or sum condition, and the portfolio must meet a target return. The method is presented as an operational approach associated with the Rockafellar–Uryasev formulation. The document does not specify the full distributional model, data estimation procedure, or constraint values, so an implementation must supply these and ensure that its loss scenarios and expected shortfall calculation match the intended assumptions.
Key ideas
- Portfolio expected shortfall is generally not the weighted sum of standalone expected shortfalls.
- The objective should compute expected shortfall from the combined weighted losses.
- Auxiliary variables can express the expected shortfall minimization as a linear program.
- The optimization requires portfolio weight and target return constraints.
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# Expected shortfall minimization as portfolio objective # Expected shortfall minimization as portfolio objective I'm trying to solve portfolio problem with minimising its Expected shortfall, assuming the returns follow a stable distribution. If I'm able to calculate MLE fit to the series, calculate expected shortfall of that instrument, then how would I optimise the portfolio? I've read for instance here it is then linear programming problem. I've tried to calculate ES of each of my instrument and then optimize the objective function `weights*ES`, but I always get minimum for vector weights 0, which makes sense, that's when its lowest. But that's of course not what I want, what is it that I'm missing? Is it good idea to take code which calculates this for normally distributed returns and just replace the function which calculates the actual ES? Thanks a lot ## Answer by Drew (score 3, accepted) https://quant.stackexchange.com/a/36958 The ES of the optimized portfolio is (except in trivial cases) not the same as the weighted sum to the ES of the individual instruments; your objective function should be the expected shortfall of the weighted sum of the losses, not the weighted sum of the expected shortfalls of the individual instruments. (So it is an integral over the weighted sum of instrument losses.) This is equation (20) or (21) in the paper you linked. Minimization is facilitated by breaking this into two pieces (shown as equations (22) and (23)) by introducing the auxiliary variables $z_i$. There are two other constraints. One is the constraint on the sum of the weights (mentioned by a Alex C above); this is equation (25). The other is the constraint that the portfolio delivers a desired level of return, equation (24). The optimization program you want is analogous to equations (22)-(25) in the doc you linked. It's really a pretty amazing thing that minimizing expected shortfall is numerically feasible. That insight is due to Rockafellar and Uryasev (and these papers are mostly available online link1); but getting from an intuitive description of the problem to the operational form of the programming problem isn't as pretty as you would like! On operationalization, however, I found these slides by Guy Yollin invaluable: http://www.r-programming.org/files/RFinance2009.pdf
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