A characteristic of the last financial crisis was the contamination or contagion of extreme events. Even if their assets were supposedly diversified, the losses in most portfolios of pension funds were much larger than usual. The main problem came from that very notion of diversification, usually based on correlations, a poor measure of dependence outside the Gaussian realm.

Value at Risk (VaR) is a well-known measure of risk, often criticized, but it is easy to interpret. Recall that, in term of losses, the 99% VaR is the value so that there is 1% of chance of observing a loss greater than that number. Of course, one has to specify the time horizon. For example, if the 99% VaR for a time horizon of one year is 75 M, then it means that approximately once every 100 years, the losses of the portfolio for a year will be larger than 75M.

**VaR: Good news, bad news**

The VaR is very sensitive to extreme values of individual assets in the portfolio. The fatter the (right) tail of the distribution of these individual losses, the larger the VaR. In addition to extreme values, the dependence between these assets can explain unusual losses. For example, if two assets exhibit strong (positive) dependence, then observing a large loss in one will increase the probability of a large loss for the other. That is the contagion effect. So the VaR is also very sensitive to contagion of extreme values of the components of the portfolio.

Because of its sensitivity to extreme values and contagion, VaR is usually difficult to evaluate with precision. There are two main ways to compute VaR: Macro and micro evaluation. Macro evaluation means that only portfolio values are taken into account; the behaviour of the individual components is ignored, including their dependence. To compute the VaR in that case, parametric or semiparametric models are fitted to historical data of the portfolio. One advantage of that method is that one can get explicit expressions for any level of VaR. On the negative side, contagion effect is ignored, resulting in an underestimation of the real VaR in general.

To tackle that problem, I suggest using micro evaluation of the VaR. Contrary to the macro approach, micro evaluation uses information from all components of the portfolio, including the dependence between assets. Because of the complexity of these dependencies and the mix of different distributions for modelling individual assets, no explicit formula is available and one relies on Monte Carlo simulations. However, that approach is more flexible, more precise, and more realistic. It also allows for variable weights making optimization possible.

**More modeling**

The main difficulty is that it requires much more modeling work than the macro approach. In particular, modeling adequately dependence for a large number of assets without using the Gaussian framework is a challenging problem and research is very active in that field, in particular in transferring the theoretical results on dependence (e.g., hierarchical copula models) into something relatively easy to implement. The research of better dependence models is far from being an academic fantasy. It is also essential part of risk hedging or pricing of credit derivatives.

In conclusion, because VaR is very sensitive to extreme events and contagion, one should use a micro approach to compute VaR, modelling the behaviour of individual components of the portfolio together with their inter-dependencies. It is a tailor-made approach, contrary to black-box solutions where one model fits all. The micro evaluation approach is more intuitive and more realistic than modelling the portfolio, and should provide better estimation of the Var. It requires much more work but it’s worth the effort.

*Bruno Rémillard is Professor, Department of Management Sciences, HEC Montréal*