Convex risk measures

Question

What is the intuitive explanation for convex risk measures represented as: $$\rho(X)=\sup_{P\in Q}\{E_{P}(-X)+\alpha(P)\}$$ where $\alpha(P)$ is a penalty function depending on the plausibility of P. I understand that the function measures the maximum expected loss and the higher the value of X, the lesser the risk but this looks like a concave function. Assist with any intuition about the supremum.

Answer 1

You can show that it is convex in the following way.

Note that the supremum is sub-additive in the sense that the supremum of a sum is less than or equal to the sum of the supremums, that is,

$$\sup_{P\in Q}\left(f(P)+g(P)\right)\leq \sup_{P\in Q} f(P) + \sup_{P\in Q} g(P),$$ where $f$ and $g$ are some (possible stochastic) functions of the measure $P$, and where $Q$ is your collection of probability measures.

To show that your risk measure $\rho$ is convex, take some $\lambda\in [0,1]$, and take some random variables/risks/portfolios $X$ and $Y$. You want to show that $$\rho(\lambda X + (1-\lambda) Y) \leq \lambda\rho(X)+(1-\lambda)\rho(Y),$$ because this is what it means to be convex. Starting from the left, using linearity of the expected value, using the trick $\alpha(P) = \lambda\alpha(P)+(1-\lambda)\alpha(P)$, and using the sub-additivity of the supremum, you get:

$$ \begin{align*} \rho(\lambda X + (1-\lambda)Y) &= \sup_{P\in Q}\left[\mathbb{E}_P(-(\lambda X + (1-\lambda)Y))+\alpha(P)\right]\\ &= \sup_{P\in Q}\left[\lambda\mathbb{E}_P(-X) + (1-\lambda)\mathbb{E}_P(-Y)+\lambda\alpha(P)+(1-\lambda)\alpha(P)\right]\\ &= \sup_{P\in Q}\left[\lambda(\mathbb{E}_P(-X)+\alpha(P)) + (1-\lambda)(\mathbb{E}_P(-Y)+\alpha(P))\right]\\ &\leq \sup_{P\in Q}\left[\lambda(\mathbb{E}_P(-X)+\alpha(P))\right] + \sup_{P\in Q}\left[(1-\lambda)(\mathbb{E}_P(-Y)+\alpha(P))\right]\\ &= \lambda\sup_{P\in Q}\left[\mathbb{E}_P(-X)+\alpha(P)\right] + (1-\lambda)\sup_{P\in Q}\left[\mathbb{E}_P(-Y)+\alpha(P)\right]\\ &= \lambda\rho(X) + (1-\lambda)\rho(Y). \end{align*} $$

Answer 2

An intuitive explanation is that for every portfolio value X, the risk measure evaluates possible loss and the larger the loss, -X, the larger the risk...this traces a convex function or curve. The mathematical proof given below makes this point more clear.