Euler Decomposition of VaR for G-and-H Portfolio Returns

Summary

The document raises how to allocate portfolio Value at Risk when returns are non-normal and modeled with a g-and-h distribution. It summarizes a 2013 result attributed to Doowoo Nam: after estimating the distribution’s four parameters, an individual asset’s VaR contribution can be obtained using Euler’s homogeneous function theorem, in a manner analogous to decomposing VaR for normally distributed returns. The author suggests this may be more reliable for strongly skewed returns than a Cornish-Fisher expansion, but provides no empirical comparison or worked example.

The central question is how a contribution based on an asset’s beta to the portfolio reflects co-skewness and co-kurtosis, rather than appearing to rely only on covariance. The excerpt does not resolve that question or provide the underlying derivation. It is therefore a brief pointer to a risk-allocation method and a modeling issue, not a complete treatment of estimating parameters, calculating contributions, or validating the approach.

Key ideas

• The cited method models non-normal portfolio returns with a g-and-h distribution.
• Euler’s theorem can be used to allocate portfolio VaR across assets after fitting the distribution parameters.
• The document suggests the method may handle strong skewness better than a Cornish-Fisher approximation.
• It questions how beta-based contributions incorporate co-skewness and co-kurtosis.
• The excerpt gives no derivation or empirical evidence resolving that concern.

Tags

# VaR decomposition of non-normal portfolio by g-and-h distribution


# VaR decomposition of non-normal portfolio by g-and-h distribution












According to Doowoo Nam (2013), VaR of non-normal portfolio returns approximated by g-and-h distribution can be decomposed pretty much in the same way as the VaR of a portfolio with normal returns. After the estimation of the four g-and-h parameters for the non-normal portfolio, Nam shows that VaR contribution of an individual asset can be determined using Euler's homogeneous function theorem. The result is interesting since it seems to provide more reliable way to estimate VaR for strongly skewed returns than the Cornish-Fischer expansion based approach.

However, I'm little confused how this decomposition, which is based on the Beta between the asset and the portfolio, takes into account the co-skewness and co-kurtosis of the individual asset and the portfolio. Rather it seems that the decomposition is only based on covariance. Could anyone elucidate this interesting result?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.