Explain what skewness and kurtosis say about a return distribution
The idea
Skewness measures asymmetry. A right-skewed distribution has a long upper tail and a mean above its median; a left-skewed one has the long tail on the downside and a mean below the median. Kurtosis measures tail weight relative to the normal distribution: excess kurtosis above zero means fat tails, so extreme outcomes are more likely than a normal distribution implies. Asset returns are routinely left-skewed and leptokurtic, which matters because risk measures built on normality will then understate the frequency of large losses.
A practice question
A return distribution has a mean below its median. How is it described?
- Right-skewedWhy not: The relationship read backwards. Right skew has a long upper tail that pulls the mean *above* the median.
- Left-skewed — correct
- SymmetricWhy not: Symmetry puts the mean and the median in the same place. Any gap between them is evidence of skew.
A long lower tail drags the mean below the median, which is left or negative skew. For an investor this is the unwelcome kind: occasional large losses.
A harder one
A distribution has positive excess kurtosis. What does this imply?
- Extreme outcomes are more likely than under a normal distribution — correct
- The distribution is perfectly symmetricWhy not: Kurtosis is about the tails, not about balance. Skewness is the measure of asymmetry, and the two are independent.
- The standard deviation is understated by definitionWhy not: The standard deviation is whatever the data make it. The problem is not that it is miscalculated but that it describes a fat-tailed distribution badly.
Positive excess kurtosis means fatter tails and a more peaked centre than the normal. Models assuming normality will therefore assign too little probability to extreme moves.
The hardest tier
Why does left skew combined with fat tails concern a risk manager using a normal-distribution model?
- It makes the mean return unreliableWhy not: The mean is still the mean. What fails is the model's account of how far outcomes stray from it, and in which direction.
- It means volatility cannot be calculatedWhy not: It can be, and is. The issue is that one number cannot describe an asymmetric, fat-tailed distribution — not that the number is unobtainable.
- The model will understate both the frequency and the size of large losses — correct
Normality assigns the tails too little probability, and left skew puts the heavy tail on the loss side. Value-at-risk figures built on that assumption are optimistic exactly where it matters.
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