Why Standard Deviation Is Subadditive Across Portfolio Returns

Summary

The document addresses whether standard deviation satisfies the subadditivity condition: the risk of a combined position should not exceed the sum of the risks of its parts. It derives the volatility of the sum of two returns from their individual volatilities and correlation, then compares that expression with the sum of the individual volatilities.

Because correlation cannot exceed one and standard deviations are nonnegative, the inequality holds for this calculation, including when the assets are positively correlated. This answers the specific subadditivity concern raised in the question. The discussion is limited to that property of standard deviation; it does not show that volatility is a coherent risk measure in full, which also depends on other axioms and on how losses and returns are defined.

Key ideas

• The variance of a sum includes a covariance term determined by the assets' correlation.
• The correlation bound ensures the volatility of a sum is no greater than the sum of the individual volatilities.
• Positive correlation does not by itself violate subadditivity.
• The response addresses subadditivity only and does not establish all conditions for a coherent risk measure.

Tags

# Is volatility really a coherent risk measure?


# Is volatility really a coherent risk measure?












Why people say that volatility is a coherent risk measure?

I don't see it clearly because what happen if the two assets are correlated positively? subadditivity would not be preserved.

That affirmation is in some papers online or even in this question

## Answer by SRKX (score 3, accepted)

https://quant.stackexchange.com/a/15666

Well, if you assume $X$ has volatility $\sigma_X$ and $Y$ has volatility $\sigma_Y$, then

$$\sigma_{X+Y} = \sqrt{ Var( X + Y) } = \sqrt{ \sigma_X^2+\sigma_Y^2 + 2 \sigma_X \sigma_Y \rho }$$

Then, you want to show

$$ \sigma_{X+Y} = \sqrt{ \sigma_X^2+\sigma_Y^2 + 2 \sigma_X \sigma_Y \rho } \leq \sigma_X + \sigma_Y $$

Squaring both sides:

$$\sigma_X^2+\sigma_Y^2 + 2 \sigma_X \sigma_Y \rho \leq \sigma_X^2 + \sigma_Y^2 + 2 \sigma_X \sigma_Y $$

Given the fact that, by definition, $\sigma_X \geq 0$, $\sigma_Y \geq 0$ and $\rho \in [ -1, 1]$, it looks to me that the property holds.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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