Abstract
Portfolio selection models using variance, Value-at-Risk (VaR) and expected shortfall measures of risk are analysed, assuming differing underlying return distributions. The expected shortfall approach provides advantages relative to the VaR approach in terms of lower portfolio downside risks. Furthermore, using the extreme value distribution provides more insights for the investor relative to the empirical distribution. Analysing portfolio selection using these differing risk measures around the recent sub-prime mortgage problem period provides topical insights into the asset allocation process for the investor.
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Notes
Adapted mean-variance procedures for non-normal VaRs are not appropriate since the objective function is not convex and multiple solutions can occur. Accordingly, we employ a search algorithm across portfolio combinations (see the Optimisation Procedures section).
Further details of the approach adopted in this paper are available from the authors on request.
This algorithm is also used in estimating the optimal portfolios for VaR p - and ES p -based risk measures when the normal and historical distributions are assumed.
We do not make any assumptions about the correlations between the assets, since the correlation structure is embedded in the portfolio return series when constructed given one set of weights.
Hill-plots are useful tools for the preliminary selection of the threshold. The optimisation procedures, however, in this paper involve 5,151 simulation runs of fitting GPD to portfolio returns, and, as a consequence, it would be rather time consuming to apply Hill-plots in each simulation.
Of course, mean-variance optimisations themselves can still be conducted even without normally distributed returns since all that such optimisations require is that expected portfolio returns (variances) are linear (quadratic) in the weights.
Weekly returns were also analysed. The results obtained were similar to those obtained using daily returns and are available from the authors upon request.
Chopra and Ziemba (1993) have demonstrated the sensitivity of portfolio optimisation results to the mean returns vector. In this study our focus is on the impacts of differing risk measures for a particular mean returns vector.
Effectively maximising return/standard deviation will be exactly equivalent to maximising a scalar times return/standard deviation, which is the case for the normal distribution.
As would be anticipated ES values are greater than their corresponding column VaR values.
The number of observations is reduced to 19 when a high confidence level of 99.5 per cent is used, with the consequence that the solution may be less accurate due to the lesser degrees of freedom.
The results at the 90, 95 and 99 per cent confidence levels are available from the authors upon request.
The lower and upper bounds of the estimated tail index increase as the threshold value rises, which reflects that the fatter tails that arise as more of the ‘middle’ returns are removed from the observation set. These results provide insights into the stability of estimation results under these differing threshold values.
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2holds a doctorate from MIT and has lectured on courses in the quantitative and risk areas at Birmingham Business School for a number of years. He has wide consulting/teaching experience and has published in the Journal of Derivatives, Journal of Futures Markets and Japanese Financial Engineering.
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Ho, Lc., Cadle, J. & Theobald, M. Portfolio selection in an expected shortfall framework during the recent ‘credit crunch’ period. J Asset Manag 9, 121–137 (2008). https://doi.org/10.1057/jam.2008.15
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DOI: https://doi.org/10.1057/jam.2008.15