What is a non-law-invariant risk measure?

Someone asked me recently, what is a risk-measure which is not law-invariant? Admittedly, the law invariance property seems so natural that any meaningful risk measure should have this property. Of course we can struggle to construct risk measures which are not law invariant, but are there any naturally occurring risk measures which do not have this property? The answer is yes, but let us take things from the start.A risk measure is a mapping: $\rho:\mathcal{Z}\to\mathbb{R}$ where $\mathcal{Z} = \mathcal{L}(\Omega, \mathcal{F}, \mathrm{P})$. Typically, risk measures are required to satisfy four regularity axioms; such risk measures are called coherent [1].

Law invariance. We say that a risk measure $\rho$ is law-invariant if we can compute $\rho[Z]$ by knowing only the probability distribution of $Z$ and not the function $Z(\omega)$, i.e., we do not need to know how this risk is produced. This requirement is usually formally written as

\begin{aligned} Z_1 \approx Z_2 \Rightarrow \rho[Z_1] = \rho[Z_2], \end{aligned}

where the notation $Z_1 \approx Z_2$ means that they have the same distribution.

Equal distributions. It is very easy to construct random variables which have equal distributions, but are almost never equal. Take for example $Z_1\sim \mathcal{N}(0, \sigma^2)$ and $Z_2 = -Z_1$. Yet another example: Take $Z_1$ to be the outcome of a fair coin toss (head/tail), and $Z_2$ to be the side of the coin facing down; these are never equal.

Non-law-invariant risk measures. Typical risk measures are law-invariant: the expectation, the average value-at-risk, the mean upper semi-deviations, the expectiles are all law-invariant. We may construct some wacky risk measure like $\rho[Z]=Z(\omega_1)$, but this will be of no use in practice. So, all widely used risk measures are law-invariant. However, take a look at the following example on a product probability space. Let $Z=(Z_1, Z_2)$ and define the risk of $Z$ to be $\rho(Z_1+Z_2)$; that is, we define a new risk measure $\bar{\rho}:\mathcal{Z}\times\mathcal{Z}\to\mathbb{R}$ defined as

$\bar{\rho}(Z) = \rho(Z_1 + Z_2)$

Then, this risk measure cannot be shown to be law invariant (unless we draw very restrictive assumption on $\rho$ such as $\rho=0$ or unless $\rho=\mathbb{E}$).

Indeed, take $Z=(Z_1, Z_2), Y=(Z_2, Z_1)$ in $\mathcal{Z}\times \mathcal{Z}$ so that $Z_1, Z_2$ have different distributions. Then $Z$ and $Y$ have the same multistage distributions, but yield the same risk. For the above example to be meaningful, $Z_1$ and $Z_2$ must both be $\mathcal{F}_1$-measurable.

It is interesting that multistage risk-averge stochastic optimisation problems which are formulated using nested conditional risk measures as in [1], suffer from this paradoxical property.

[1] A. Shapiro, D. Dentcheva and A. Ruszczyński, Lectures on Stochastic Programming: Modeling and Theory (2nd Edition), SIAM editions, 2014.

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