VaR Portfolio Optimization: Simulation Methods and Coherence Limits
Summary
The discussion offers starting points and practical cautions for optimizing portfolios against value at risk (VaR). It recommends Euler capital allocation as an introduction, then describes a simulation approach: hold a large set of scenarios fixed and numerically optimize the portfolio subject to its constraints. This can make a simulation-based problem computationally manageable, but the quality of the result depends on whether the VaR estimate has converged; tail estimation can be difficult even for sums of lognormal variables, and adequacy may need empirical checking.
The answer warns against using VaR as the direct optimization target because it is not generally subadditive. In some cases, combining positions can therefore appear riskier than the separate positions, potentially undermining diversification incentives. Expected shortfall is mentioned as a coherent alternative. The response is introductory rather than a complete treatment: it does not specify an optimizer, constraints, confidence level, or convergence procedure, and its practical remarks about tractable theory are qualified rather than universal.
Key ideas
- Euler capital allocation is suggested as an introductory route into portfolio risk optimization.
- A fixed set of simulated scenarios can turn constrained VaR optimization into a numerical problem.
- Simulation-based VaR optimization requires careful empirical checks of estimator convergence, especially in the tails.
- VaR may violate subadditivity, so portfolio aggregation can produce counterintuitive risk assessments.
- Expected shortfall is identified as a coherent risk measure that may avoid this particular issue.
Tags
Full text
# Portfolio optimization w.r.t. value at risk: introductory or survey references
# Portfolio optimization w.r.t. value at risk: introductory or survey references
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.)
## Answer by raptor22 (score 4)
https://quant.stackexchange.com/a/48922
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.