Subadditivity, Diversification, and Coherent Risk Measures
Summary
Subadditivity says that the risk assigned to a combined position should be no greater than the sum of the risks assigned to its components. The document explains the right-hand side as the separate risks of two positions, whether those positions are held in distinct accounts or simply compared as standalone portfolios. Rewriting the condition for average holdings also shows that it need not describe actual investors or accounts.
The discussion connects subadditivity to diversification and to one of the axioms of coherent risk measures. It uses volatility as an intuitive example and contrasts it with Value at Risk, which can fail the axiom because it measures a loss quantile. The text presents the condition as a desirable property imposed on a risk measure, not a universal fact about every possible measure. Its examples are conceptual; it does not provide portfolio data or calculations, and the volatility illustration should be understood as an intuition about portfolio risk rather than a general empirical claim.
Key ideas
- Subadditivity bounds combined portfolio risk by the sum of component risks.
- The component risks can be interpreted as standalone measures without assuming separate investors.
- The condition is one of the axioms used to characterize coherent risk measures.
- Volatility illustrates subadditivity, while Value at Risk can violate it.
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# What is the meaning of subadditivity in a risk measure?
# What is the meaning of subadditivity in a risk measure?
The subadditivity reads:
$\rho(X_1+X_2) \leq \rho(X_1) + \rho(X_2)$
What is the meaning of this condition? I can vaguely accept that one should diversify the investment portfolio. Or, I can understand that $\rho(X_1+X_2)$ describes the situation of two assets $X_1$ and $X_2$ held together. Then what is the meaning of $\rho(X_1) + \rho(X_2)$? One person holds $X_1$ and another person holds $X_2$? I am having difficulty to interpret the right-hand-side.
## Answer by jlowin (score 10)
https://quant.stackexchange.com/a/4015
As you inferred, this is related to the concept of diversification as a risk-mitigation tool.
In short, think of $\rho$ as representing some risk measure, and $\rho(x)$ as the risk of asset $x$ under that measure. If subadditivity holds, then the risk of holding assets 1 and 2 simultaneously must be less than or equal to the sum of their individual risks: $\rho(x_1 + x_2) \leq \rho(x_1) + \rho(x_2)$.
For example, volatility (standard deviation) is a subadditive risk measure. We know this intuitively from diversification: a portfolio is less volatile than the sum of its component volatilities.
As it relates to finance, subadditivity is one of the four axioms characterizing "coherent" measures of risk. This class of risk measures was introduced in Artzner et al, 1998, see the bottom of page 6. Think of these as risk measures with desirable properties that won't be subverted by strange-behaving portfolios. It's important to note that subadditivity is not a statement of fact -- it's easy to define risk measures that are not subadditive -- but rather an axiom that risk measures must satisfy in order to be coherent.
Artzner describes subadditivity nicely as the idea that "a merger does not create extra risk," and lists a number of practical points which follow from it. One interesting one is that if risk were not subadditive, then a person wanting exposure to asset 1 and asset 2 would be better off opening a separate account for each asset, as the (risk-based) margin requirement would be lower than if he held both in the same account. (Note this can be seen as a very literal interpretation of the right hand side of the equation.)
The most (in)famous risk measure that does not satisfy this axiom is VaR. The VaR of a portfolio of two assets can be greater than the sum of their individual VaRs. This is because VaR is a quantile-based measure; see the Artzner paper for examples.
## Answer by David Nehme (score 3)
https://quant.stackexchange.com/a/4014
Rewriting the condition as
$$\rho\left({X_1+X_2 \over 2}\right) \leq {\rho(X_1) + \rho(X_2) \over 2}$$
You can interpret it as a portfolio containing the average holdings of two other portfolios has at most the risk of the average risk of the two other portfolios. There is no need to have any concept of anyone actually holding any of the portfolios.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.