Minimizing shortfall

This paper describes an empirical study of shortfall optimization using Barra fundamental factors. We compare minimum shortfall to minimum variance portfolios in the US, UK, and Japanese equity markets using Barra Style Factors (Value, Growth, Momentum, etc.). We show that minimizing shortfall generally improves performance over minimizing variance, especially during down-markets, over the period 1985-2010. The outperformance of shortfall is due to intuitive tilts towards protective factors like Value, and away from aggressive factors like Growth and Momentum. The outperformance is largest for the shortfall that measures overall asymmetry rather than the extreme losses.

Keywords:

• Downside risk
• Portfolio optimization
• Expected shortfall
• Portfolio construction
• Downside protection

JEL Classification::

• C1
• C6
• C61

Acknowledgements

We thank Carlo Acerbi, Valdislav Dubikovsky, Mark Horvath, Angelo Barbieri, Dan Stefek and Jyh-Huei Lee for their insights and support. We are grateful to two anonymous referees for a careful review that has led to substantial improvements of this article. We also thank the Editorial Staff at Quantitative Finance for their time and consideration.

Notes

§Conditional value at risk (CVar) and expected tail loss (ETL) are synonyms for expected shortfall.

†An empirical portfolio construction study that has substantial overlap with our study and incorporates forecasts of mean return is that of Bender et al. (2010).

‡Here we assume that N(1-p) is an integer. The general formula for continuous outcomes is conceptually similar; see Acerbi and Tasche (2002) for a detailed discussion.

§More specifically, any risk measure that is a function of the single-period return distribution.

†See, for example, Heyde and Kou (2004).

†More specifically, NN is independent of volatility when Normal expected shortfall is estimated using the same half-life as that used to normalize the FxR returns. Mathematically, NN of a factor is the same as NN of a factor times a positive constant.

‡Expected gain is analogous to expected shortfall. It is the average of the largest gains: those exceeding a specified confidence level.

§Confidence intervals are computed by bootstrapping the FxR scaled returns in each period.

†A factor portfolio return is equivalent to the Barra factor return; some Barra Style factor portfolios are also listed as MSCI indices.

†Trading Activity in UKE7 is a weighted average of monthly, quarterly, and annual share turnover, while in JPE3 it also includes recent growth in trading volume; however, this descriptor is weighted by only 3%; the remainder of the factor is defined analogously to the UKE7 factor.