I am looking for references introducing the problem of portfolio optimization when the target characteristic is value at risk. A textbook treatment would be great. Surveys on the topic are also welcome.

(I checked for example the dedicated volume "Value-at-Risk Models" (2009) by Alexander and was surprised not to find a chapter on portfolio optimization w.r.t. VaR. Or did I happen to miss it? I also found some papers focusing on special cases, but these are not exactly where I would like to start.)

  • $\begingroup$ this is probably not exactly what you are after but "Pricing and Trading Interest Rate Derivatives" by Darbyshire has a chapter on VaR, based on variance-covariance, where he outlines the calculation of VaR minimisation trades of single instruments and of multiple instruments simultaneously, which is really just basic calculus and clever arrangement in Excel. $\endgroup$ – Attack68 Sep 25 '19 at 12:15
  • $\begingroup$ Are you looking at minimizing VaR or at optimizing expected return conditional on a VaR level? $\endgroup$ – raptor22 Sep 25 '19 at 18:59
  • $\begingroup$ @raptor22, either one would be interesting. I thought I would start with minimizing VaR, but if you have any points on maximizing expected return conditional on a VaR level, it could be helpful, too. $\endgroup$ – Richard Hardy Sep 25 '19 at 19:24

I would suggest to start with Euler capital allocation as a first step to dive into the subject, here is an example of introductory paper (Capital Allocation to Business Units and Sub-Portfolios: the Euler Principle, Dirk Tasche).

In general, I don't think that you will find any satisfactory theory on the subject for practical uses beyond the case of an aggregated normal distribution in your risk model (e.g. delta-normal).

For simulation based models, I generally do it by fixing a large number of simulations and then turning it into a numerical optmization problem with your specific constraints and fixed simulations. Note however that convergence of the VaR estimator can take some simulations and becomes difficult to look at theoretically, already with sums of lognormals it is a very challenging problem (see Tail behavior of sums and differences of log-normal random variables, ARCHIL GULISASHVILI and PETER TANKOV). So you need to put great care to be sure about the convergence of your estimator and to fix a high enough number of simulations in the first step, this can usually be done empirically.

However I give a warning here: it is not a good idea to directly optimize for VaR. The reason being that this risk measure is not a coherent risk measure. In particular, the sub-additivity axiom is not respected. For a risk measure $\rho : \mathcal{G} \mapsto \mathbb{R}$, the sub-additivity axiom writes:

$$\forall X_1, X_2 \in \mathcal{G}, \rho(X_1 + X_2) \leq \rho(X_1) + \rho(X_2),$$

and what it means is that putting all your eggs in one basket is riskier than diversifying. This has led to criticism from the academic community of the measure and of its use in risk-based financial regulations, because it can give institutions the wrong incentives for steering capital.

Here is the original paper abour coherent risk measures (COHERENT MEASURES OF RISK, Atzner, Delbean et al.). The expected shortfall is an example of a coherent risk measure.

I guess this does not exactly answer you question but I hope it gives some hints.

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  • $\begingroup$ ETL or CVaR is a better risk measure when optimization (either risk minimization or return maximization under risk constraints) is desired. The reason (related to coherence, referenced in this answer) is that optimization problems involving CVaR are well behaved (in theoretical terms and in practice); whereas those involving VaR are not (in theory or in practice). $\endgroup$ – Drew Sep 26 '19 at 11:52
  • $\begingroup$ See this post: quant.stackexchange.com/q/38095/11767 . $\endgroup$ – Drew Sep 26 '19 at 11:54
  • $\begingroup$ Or this one: quant.stackexchange.com/q/36953/11767 $\endgroup$ – Drew Sep 26 '19 at 11:56
  • $\begingroup$ Thank you raptor22 and @DrewSaunders! I will check out the references. I am also aware of noncoherence of VaR and it is of course a valid concern. $\endgroup$ – Richard Hardy Sep 26 '19 at 12:43

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