# What is the meaning of subadditivity in a risk measure?

$\rho(X_1+X_2) \leq \rho(X_1) + \rho(X_2)$

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.

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.

• @Chang, upon browsing your other questions I've noticed that you appear quite familiar with coherent risk measures. I apologize if my answer is too basic for you in its treatment of such measures; hopefully it will still benefit others. – jlowin Aug 27 '12 at 17:41

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.

• Just a minor comment: Unless $\rho$ is positively homogeneous (or $1/2$-subhomogeneous), subadditivity does not imply the inequality you stated above. Vise versa, without assuming any (sub)homogeneity for $\rho$, your inequality does not imply subadditivity. – Pantelis Sopasakis May 3 '18 at 9:54