Why Standard Deviation Does Not Satisfy Risk Monotonicity

Summary

The document raises a conceptual issue about treating standard deviation as a coherent risk measure. It states the monotonicity condition: if one position pays at least as much as another in every state, its risk measure should be no larger. The question observes that a position can dominate another state by state while still having higher standard deviation, and asks how to reconcile this with the definition.

The example highlights that standard deviation measures dispersion around the mean, not downside risk alone. Adding a positive constant to every outcome leaves standard deviation unchanged, while changing payoffs across states can increase dispersion even when all outcomes improve relative to another position. Thus standard deviation generally fails monotonicity as a risk measure in the coherent-risk sense. The document is framed as a question and supplies no worked resolution or alternative risk measure, so it is most useful as a prompt to distinguish variability from loss-focused risk concepts.

Key ideas

• Monotonicity requires a position that dominates another in every state to have no greater measured risk.
• Standard deviation captures dispersion around the mean rather than losses alone.
• A statewise higher payoff can still have greater dispersion than a lower payoff.
• The question exposes why standard deviation is not generally a coherent risk measure.

Tags

# Standard Deviation and Monotonicity property


# Standard Deviation and Monotonicity property












I just read that standard deviation is a coherent risk measure, and therefore it should satisfy the monotonicity property:

$X_1 \geq X_2 \implies \rho(X_1) \leq \rho(X_2)$ where $X_1,X_2$ are asset positions.

We could define $X_1$ being greater than $X_2$ for every $\omega \in \Omega$, but $X_1$ could still have a greater standard deviation than $X_2$. Am I missing something with the definition of monotonicity?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.