When Mean and Volatility Do Not Describe Return Risk
Summary
The document considers whether financial models should use the full return distribution instead of relying on mean and standard deviation, and whether skewness and kurtosis add useful information. Its answer depends on the task and the assumed model. In a lognormal framework, mean and standard deviation specify the distribution; with jumps, additional parameters are needed to represent jump arrivals and sizes. Volatility models can also allow distribution parameters to change over time.
For portfolio optimization, the answer argues that higher moments matter when the objective accounts for them. Risk management may likewise need to represent tail events that a Gaussian model understates, with scenario analysis offering one way to specify outcomes directly. For empirical return distributions, it names filtered historical simulation as an approach. The document gives conceptual examples, not comparative tests or a universal ranking; the appropriate inputs depend on the model assumptions and the decision being made.
Key ideas
- Mean and standard deviation fully specify a return distribution only under particular assumptions, such as lognormality.
- Jump models require parameters describing jump arrivals and jump sizes.
- Portfolio optimization can use higher moments when its objective reflects them.
- Gaussian risk models can understate extreme events, making scenario analysis useful.
- Filtered historical simulation is one way to use the empirical distribution of past returns.
Tags
Full text
# Is it always better to use the entire distribution of a financial returns series, not just $\mu$ and $\sigma$? # Is it always better to use the entire distribution of a financial returns series, not just $\mu$ and $\sigma$? In finance models that use historical returns for inputs, including option pricing models, forecasting and portfolio optimization, only the statistical moments of the returns distribution, $\mu$ and $\sigma$ (expected value, or mean, and standard deviation), are used as inputs because the moments summarize a return series' probability distribution (pdf). How strong is the argument that the user would be better off, and would get more accurate results, in using the data's entire pdf, instead of only $\mu$ and $\sigma$? And would using the entire pdf also be better than models that try to extend to the third and fourth moments (skewness, kurtosis)? given that you could even create a distribution of the rolling skewness and rolling kurtosis of a return series, i.e. the distribution of each moment ## Answer by Bob Jansen (score 6, accepted) https://quant.stackexchange.com/a/55799 It depends. For example, if you're doing option pricing in the log normal world returns are completely described by the mean and standard deviation. If you add jumps, you would also need to parametrize the underlying Poisson process which is fully described by one parameter and the jump size. In other words, if you have a (log)normal distribution and the mean and standard deviation you're using the complete distribution. Of course, there is no need to keep the parameters fixed over the whole time horizon. This is what local volatility and stochastic volatility models are all about. If you're doing portfolio optimization taking into account more than mean and variance, you should definitely use higher moments and do so almost by definition. For risk management taking other moments into account is common place. In the Gaussian world, the number of extreme events is heavily underestimated after all. Performing scenario analysis can be seen as specifying the return distribution and you're not really considering the statistical mean or standard deviation in any case. If you want to use the empirical distribution of past returns, you can use filtered historical simulation.
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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.