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**Variance vs downside risk: Is there really that much difference?**
*(English)*
Zbl 0935.91021

Summary: The popularity of downside risk among investors is growing and mean return-downside risk portfolio selection models seem to oppress the familiar mean-variance approach. The reason for the success of the former models is that they separate return fluctuations into downside risk and upside potential. This is especially relevant for asymmetrical return distributions, for which mean-variance models punish the upside potential in the same fashion as the downside risk.

The paper focuses on the differences and similarities between using variance or a downside risk measure, both from a theoretical and an empirical point of view. We first discuss the theoretical properties of different downside risk measures and the corresponding mean-downside risk models. Against common beliefs, we show that from the large family of downside risk measures, only a few possess better theoretical properties within a return-risk framework than the variance. On the empirical side, we analyze the differences between some US asset allocation portfolios based on variances and downside risk measures. Among other things, we find that the downside risk approach tends to produce on average – slightly higher bond allocations than the mean-variance approach. Furthermore, we take a closer look at estimation risk, viz. the effect of sampling error in expected returns and risk measures on portfolio composition. On the basis of simulation analyses, we find that there are marked differences in the degree of estimation accuracy, which calls for further research.

The paper focuses on the differences and similarities between using variance or a downside risk measure, both from a theoretical and an empirical point of view. We first discuss the theoretical properties of different downside risk measures and the corresponding mean-downside risk models. Against common beliefs, we show that from the large family of downside risk measures, only a few possess better theoretical properties within a return-risk framework than the variance. On the empirical side, we analyze the differences between some US asset allocation portfolios based on variances and downside risk measures. Among other things, we find that the downside risk approach tends to produce on average – slightly higher bond allocations than the mean-variance approach. Furthermore, we take a closer look at estimation risk, viz. the effect of sampling error in expected returns and risk measures on portfolio composition. On the basis of simulation analyses, we find that there are marked differences in the degree of estimation accuracy, which calls for further research.

### MSC:

91B28 | Finance etc. (MSC2000) |

### Keywords:

semi-variance; lower partial moments; portfolio optimization; estimation risk; downside risk
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\textit{H. Grootveld} and \textit{W. Hallerbach}, Eur. J. Oper. Res. 114, No. 2, 304--319 (1999; Zbl 0935.91021)

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