Risk Measures for Returns with Non-Normal Distributions
Summary
The document asks whether standard deviation remains useful when returns have heavier tails than a normal distribution and surveys alternatives. Standard deviation can serve as a first approximation when risk means deviation from expected return and variance is finite, but it does not describe the full shape of most return distributions. Value at risk and expected shortfall are presented as distribution-general measures, with value at risk noted as relatively easy to interpret and backtest despite concerns about its coherence.
Expected shortfall, also called conditional value at risk, averages losses in the tail and is described as satisfying coherence properties such as subadditivity. Spectral measures generalize this approach by assigning different weights to losses. The discussion stresses that no single measure is universally appropriate: choices depend on investor preferences and risk aversion. It also cites research on preference-dependent ranking of non-normal investments, but supplies no empirical comparison or detailed formulas.
Key ideas
- Standard deviation is a limited risk summary when returns are non-normal, even if variance is finite.
- Value at risk is accessible and backtestable but has been criticized for lacking coherence.
- Expected shortfall summarizes tail losses and is presented as satisfying coherence properties.
- Spectral measures allow different weights across losses in the tail.
- The suitable risk measure depends on investor preferences and risk aversion.
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Full text
# Appropriate measure of risk if return are not normally distributed # Appropriate measure of risk if return are not normally distributed Normally standard deviation of an assets is used as an proxy for the risk in the financial market. In reality distribution of return is more peaked at the center and higher mass in the tail as predicted by the normal distribution. If this is true than extreme observations will occur more frequently than predicted by normal distribution. So my questions are : - If return are not normally distributed still then standard deviation can be used as an proxy for risk? - Alternative measure of risk if returns are not normally distributed ? ## Answer by user9403 (score 3) https://quant.stackexchange.com/a/23141 What is risk? If one defines risk heuristically as deviation from expectation, then (assuming returns have finite variance) standard deviation can be considered a first approximation for risk. For most distributions the mean and variance do not fully parameterize the distribution. Some standard measures of risk for general distributions include Value at risk and Expected shortfall. Value at risk in particular has come under scrutiny for lacking "coherence" but it has the benefit of being easy to understand and backtest. But in general the "appropriate" measure of risk will be an individual decision since everyone has unique utility functions and aversions to risk. The industry infatuation with the "best" risk measure is, when seen in this light, rather Sisyphean. ## Answer by LorenzQF (score 1) https://quant.stackexchange.com/a/23253 According to literature ADEH(1999) standard deviation (& VaR aswell) do not respect "properties" for coherent risk measures. (Sub-additivity etc.) An appropriate and easy risk measure could be the Expected Shortfall (CVaR) which instead do respect these properties. An alternative could be using the so called Spectral risk measures, which similarly to CVar are a weighted average of the worst losses (CVAr weights all them equally, instead here you can choose different weights. And like user9403 said you can relate these measures to your utility function or your risk aversion). Sorry for my english, hope it helps. Lorenz ## Answer by phdstudent (score 0) https://quant.stackexchange.com/a/23254 You should take a look at this paper: "A Sharper Ratio: A General Measure for Correctly Ranking Non-Normal Investment Risks". The authors prove closed form solution to rank alternative investments even when the underlying is not normally distributed under a very general utility specification. In their own words, they derive a generalized ranking measure that correctly ranks risks relative to the original investor problem for a broad utility-and-probability space. Like the Sharpe ratio, the generalized measure maintains wealth separation for the broad HARA utility class. The generalized measure can also correctly rank risks following different probability distributions, making it a foundation for multi-asset class optimization. This paper also explores the theoretical foundations of risk ranking, including proving a key impossibility theorem: any ranking measure that is valid for non-Normal distributions cannot generically be free from investor preferences.
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