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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.
The usual answer is that most risk assets tend to exhibit left-skew, with correlations ->1 into the left tail (ie diversification breaks down). And so positively skewed assets have attractive portfolio features, over and above their own intrinsic fundamentals.
The more formal answer is that for two distributions with the same mean and standard deviation, the one with left-skew will generate lower log-expected wealth. Simplistically imagine a classic normal curve, with the innovation of a 1% chance of 100% gain versus a 1% chance of 100% loss. The existence of this left-tail has changed the risk-reward significantly, without changing the mu, sigma, or skew... changing the skew to bias the left tail only makes it even more obvious and profound.
It's a little simplistic to say that positive skew is better, you could for example have a return distribution which is negatively skewed but has a mean of 10%, versus a positively skewed one with a mean of 5%. That said, negative skew has a serious downside when it comes to risk and estimation. At any given point in time, you will generally only have a finite sample from the distribution in question, and for a skewed distribution a single rare event in the tail of the distribution could meaningfully affect your estimate of the mean. You could be sampling from that negatively skewed distribution for a while, thinking that it's mean is 10%, where in reality it is -5%, and you won't know until one of those tail events hits. This is the classic reason for the fear around "selling puts" as a strategy- it oftentimes follows this rough profile.
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.
Suppose for simplicity that expectation is zero. If the distribution exhibits negative skew, then the percentile becomes more negative and, hance, the distribution is associated with higher VaR.
