# 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

• You need a constraint $w^T e = 1$ (see Page 108) to force the sum of the weights to be 1, i.e. to make the "all weights equal zero" solution infeasible. Nov 18, 2017 at 20:25
• The hyperlink in your post is not working anymore. Do you recall what paper it linked to?
– arni
Oct 4, 2021 at 5:20
• Through wayback machine I found it as: Yamai, Yasuhiro, and Toshinao Yoshiba. "Comparative analyses of expected shortfall and value-at-risk: their estimation error, decomposition, and optimization." Monetary and economic studies 20.1 (2002): 87-121. Oct 4, 2021 at 17:02

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 $z_i$.