A note on optimal estimation from a risk management perspective under possibly mis-specified tail behavior

Many financial time-series show leptokurtic behavior, i.e., fat tails. Such tail behavior is important for risk management. In this paper I focus on the calculation of Value-at-Risk (VaR) as a downside-risk measure for optimal asset portfolios. Using a framework centered around the Student t distribution, I explicitly allow for a discrepancy between the fat-tailedness of the true distribution of asset returns and that of the distribution used by the investment manager. As a result, numbers for the over-estimation or under-estimation of the true VaR of a given portfolio can be computed. These numbers are used to rank several well-known estimation methods for determining the unknown parameters of the distribution of asset returns. Minimizing the absolute (percentage) mismatch between the nominal and actual or true VaR leads to the choice of a Gaussian maximum quasi-likelihood estimator, i.e., a least-squares type estimator. The maximum likelihood estimator has a less satisfactory behavior. Outlier robust estimators perform even worse if the required confidence level for the VaR is high. An explanation for these results is provided.