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Fully Flexible Extreme Views

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Figure 1: View on CVaR: extensive search of minimum relative-entropy posteriorThe combination of subjective views within a broadly accepted risk model is one of the main challenges in quantitative portfolio management. Indeed, any risk model, be it based on historical scenarios, parametric fits, or Monte Carlo scenarios generated according to a given distribution, is subject to estimation risk and thus it is inherently flawed. Therefore, it is important to provide a framework that allows practitioners to overlay their judgement to any risk model in a statistically sound way.

The mainstream approach to perform this task is the celebrated model by Black and Litterman (1990). In this case the distribution of the risk factors is assumed normal and the views are processed using a Bayesian formalism. For non-normal markets, one can choose an ad-hoc parametrization and modify the Black-Litterman approach accordingly. For instance, this is the route chosen in Giacometti, Bertocchi, Rachev, and Fabozzi (2007).

To allow for fully general market distributions and for views on general features of such distributions, one can use the "Fully Flexible Views" approach in Meucci (2008), FFV in the sequel. FFV combines an arbitrary market model, which is referred to as the "prior" and fully general views or stress-tests on the underlying market. The output is a distribution, referred to as the "posterior", which incorporates all the inputs and which can be used for risk management and portfolio optimization.

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In FFV, the posterior is obtained by warping the prior distribution so that the views are fulfilled, in such a way that the least possible amount of spurious structure is imposed. Specifically, the posterior distribution minimizes the entropy relative to the prior, which is the natural measure of discrepancy between two distributions. As a final step, opinion pooling assigns different confidence levels to different views and users, leading to a mixture of distributions.

FFV is advantageous in that it allows for full flexibility in the specification of the views: indeed, not only views on expectations, but also views on volatility, or on value at risk (VaR), among others, are possible.

FFV can be implemented in three ways: analytical, non-parametric, and parametric. The analytical solution, which only applies to normally distributed markets, and the non-parametric approach are discussed in the original paper Meucci (2008). The parametric approach is studied in Meucci, Ardia, and Keel (2011).

In the non-parametric approach to FFV, the prior distribution of the generic risk factor, say a return, is represented in terms of a set of scenarios and their associated probabilities. This representation presents two drawbacks when processing extreme views on the tails.

First, this representation allows for views of many sorts, including VaR, but cannot accommodate views on the conditional value at risk (CVaR). Second, the scenarios in the original FFV article were generated by standard Monte Carlo. Monte Carlo scenarios are inadequate when expressing extreme views on the tails of the distribution, unless the number of scenarios is unrealistically large.

This article addresses these two issues of the non-parametric approach. In Section 2 we review FFV. In Section 3 we discuss a recursive algorithm to cover views on the CVaR. In Section 4 we replace the Monte Carlo scenarios with a deterministic grid and the respective non-equal probabilities in such a way that extreme views can be processed. In Section 5 we present a case study: we model the distribution of a hedge fund return non-parametrically and we process extreme views on its expectation and its CVaR. 

To continue reading, click here to download the full article and MATLAB code. 

Attilio Meucci, PhD, CFA, is the chief risk officer at Kepos Capital LP and an adjunct professor at Baruch College. He is also the author of Risk and Asset Allocation and a regular contributor to publications, including Risk Magazine and to GARP Risk Professional Magazine.

Excerpted from an upcoming article in the Journal of Risk.


Black, F., and R. Litterman, 1990, Asset allocation: combining investor views with market equilibrium, Goldman Sachs Fixed Income Research.

Giacometti, M., I. Bertocchi, T.S. Rachev, and F. Fabozzi, 2007, Stable distributions in the Black-Litterman approach to asset allocation, Quantitative Finance 7, 423—433.

Meucci, A., 2008, Fully flexible views: Theory and practice, Risk 21, 97—102.

———, D. Ardia, and S. Keel, 2011, Fully flexible views with parametric multivariate distributions, Working Paper.

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