Subadditivity and Diversification in Coherent Risk Measures

Summary

The discussion clarifies the direction of subadditivity: a portfolio risk measure should assign the combined positions risk no greater than the sum of their separate risks. This inequality captures the possibility that diversification reduces measured risk, while allowing for cases where combining positions produces no diversification benefit.

The answer identifies subadditivity as one property of a coherent risk measure and contrasts measures that meet coherence conditions with those that do not. It notes that Value at Risk is generally not coherent, while Expected Shortfall is an example of a coherent measure. The exchange is conceptual rather than empirical and does not give calculations or conditions for particular portfolios. Its brief comparison should be read as a distinction between measure properties, not as a guarantee that diversification always lowers risk in every portfolio.

Key ideas

• Subadditivity requires combined portfolio risk to be no greater than the sum of individual risks.
• This property formalizes how a risk measure can reflect diversification.
• Subadditivity is one condition in the definition of coherent risk measures.
• Value at Risk may violate coherence, while Expected Shortfall is presented as coherent.

Tags

# Is this comment right about subadditivity?


# Is this comment right about subadditivity?












I found this comment in a book I bought about risk management: Risk Management in Banking by Joel Bessis.

> This is the well-known rule that states that the sum of individual risks is less than the risk of the sum, or, that risks should be sub-additive. Risks do not add up algebraically because of diversification.

Doesn't he really mean the opposite, namely that the risk of the sum is less than the sum of the individuals:

$$ \rho(A+B) \leq \rho(A) + \rho(B) $$

Or is his wording just odd?

## Answer by SRKX (score 3)

https://quant.stackexchange.com/a/4915

You're right, I hope he meant exactly the opposite, and the formula you provided is indeed part of the definition of a coherent risk measure.

In fact, I would say that the risk of the sum is less than or equal to the sum of the individuals as in some cases you would like your model to accept no diversification effect.

As John mentioned in his comment, Value at Risk typically is not coherent, just as Volatility, but other measures such as the Expected Shortfall are.

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