# What is better: A negatively skewed return or a positively skewed returns distribution?

I noticed that in certain literature, like in CFA level 1, the theory put forth is that someone should prefer positively skewed returns as mean > median > mode. But why is that?

Based on a simple graphical drawing (pardon the sloppiness): Wouldn't I prefer a negative skew? We could swap the numbers in the axis but even then, intuitively, the negative skew should give me higher returns over time.

Do enlighten me as I maybe be missing the numerical concept behind this.

• You should plot 2 densities with the same mean and standard deviation (the two most important factors in decision making). Then we can discuss the additional effect of the skew... Jun 1, 2021 at 15:19
• A density with negative skew and positive mean will have large negative (below the mean), tails/outliers, while having lots of small positive realizations. The opposite holds for positive skew. In investment terms, positive skews corresponds to frequent, but small losses, and infrequent but large positive gains. Most stock returns/index returns have negative skew. Jun 1, 2021 at 15:23
• "Wouldn't I prefer a negative skew?" - incentives / business model / institutional constraints are probably as important as 'the numerical concept' in this case. Jun 1, 2021 at 21:05

Consider the definition of VaR with respect to Jorion's FRM Handbook: $$\begin{equation} VaR_{\alpha} = \mathbb{E}_t[S_T] - Q_t(S_T,\alpha) \end{equation}$$ where $$S_T$$ is the value of portfolio/asset at time $$T$$, $$\mathbb{E}_t$$ is the conditional expectation of the process at time $$t$$, and $$Q_t(S_T,\alpha)$$ is the conditional $$\alpha$$th percentile of the process at time $$t$$. This risk measure indicates by how much the risk manager will underperform expectations with $$1-\alpha$$ level of confidence.