CVaR and a lemma from Rockafellar & Uryasev

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In this previous post, I introduced conditional value-at-risk (CVaR), a risk measure used in mathematical finance. \alpha-CVaR is the expected value of the loss conditional on the loss being greater than the \alpha-VaR. If X is the random variable for the loss and q_\alpha (X) is the \alpha-VaR (i.e. \alpha-quantile of X), then

\begin{aligned} \alpha\text{-CVaR} = \mathbb{E} \left[ X \mid X \geq q_\alpha (X) \right] .\end{aligned}

Lemma from Rockafellar & Uryasev

It turns out that the CVaR can be thought of as the solution to a particular minimization problem. This formulation first appeared in Rockafellar & Uryasev (2000) (Reference 1), and I’ll present a simplified version here.

Denote the \alpha-CVaR of X by \phi_\alpha(X), and assume that X has a probability density function p(\cdot). (Reference 1 makes a similar assumption, but this assumption is more for mathematical simplicity rather than being a substantial blocker.) Let x_+ = \max(0, x) denote the ReLU function.

Lemma (Theorem 1 of Rockafellar & Uryasev (2000)). \phi_\alpha (X) satisfies

\begin{aligned} \phi_\alpha (X) = \min_{x \in \mathbb{R}} \left[ x + \dfrac{1}{1-\alpha}\int_{-\infty}^\infty (y - x)_+ p(y) dy \right]. \end{aligned}

While this lemma seems obscure, there have been applications. Reference 1 itself uses this formulation for minimizing CVaR, and see Reference 2 for a more modern application.

Where does this problem come from?

While interesting, the minimization problem seems to come out of nowhere. How does one even think to minimize such an expression as the one of the RHS?

I spent some time thinking about it and I think I have a plausible explanation. First, let’s rewrite the expression for CVaR:

\begin{aligned} \phi_\alpha (X) &= \mathbb{E} \left[ X \mid X \geq q_\alpha (X) \right] \\ &= \dfrac{1}{\mathbb{P}\{X \geq q_\alpha (X)\}} \int_{q_\alpha(X)}^\infty y p(y) dy  \\ &= \dfrac{1}{1-\alpha} \int_{q_\alpha(X)}^\infty [q_\alpha(X) + (y - q_\alpha(X))] p(y) dy  \\ &= \dfrac{1}{1-\alpha}\left[ q_\alpha (X) (1-\alpha) + \int_{q_\alpha(X)}^\infty (y - q_\alpha(X)) p(y) dy \right] \\ &= q_\alpha (X) + \frac{1}{1-\alpha} \int_{-\infty}^\infty (y- q_\alpha(X))_+ p(y) dy. \end{aligned}

This is exactly the RHS of the lemma, except that instead of minimizing over x, we just plug in q_\alpha(X) for x.

This explains why we might think about the expression on the RHS, but why should \alpha-CVaR be the solution where we replace q_\alpha (X) with x, then minimize over x? The key to that mystery is that quantiles can be expressed as the solution of a minimization problem. In particular, if we define \rho_\alpha (y) = y(\alpha - 1\{ y < 0 \}), then the \alpha-quantile of X, q_\alpha (X), satisfies

\begin{aligned} q_\alpha(X) &= \underset{x \in \mathbb{R}}{\text{argmin}} \;\mathbb{E}[\rho_\alpha(X - x)]. \end{aligned}

(This previous post presents this idea but for the empirical distribution of a sample. Note that if we define L(y,z) = \rho_\alpha(y - z), then L is the pinball loss associated with quantile regression.) Let’s rewrite the expectation on the RHS more explicitly:

\begin{aligned} \mathbb{E}[\rho_\alpha(X - x)] &= \mathbb{E}[(X - x)(\alpha - 1\{ X - x < 0 \})] \\ &= \int_{-\infty}^\infty (y - x)(\alpha - 1\{ y < x \}) p(y)dy \\ &= \int_{-\infty}^\infty (y-x)\alpha p(y) dy + \int_{-\infty}^\infty (y - x)(- 1 + 1 \{ y \geq x \}) p(y)dy \\ &= \int_{-\infty}^\infty (y-x)(\alpha-1) p(y) dy + \int_{-\infty}^\infty (y - x)1 \{ y \geq x \} p(y)dy \\ &= (\alpha-1) \mathbb{E}[X] - x(\alpha-1) + \int_{-\infty}^\infty (y-x)_+ p(y) dy \\ &= (\alpha - 1) \mathbb{E}[X] + (1 - \alpha) \left[ x + \frac{1}{1-\alpha} \int_{-\infty}^\infty (y-x)_+ p(y) dy \right] \end{aligned}

We recognize the expression inside the square brackets as the expression we are trying to minimize over to get the \alpha-CVaR! In summary,

\begin{aligned} \underset{x \in \mathbb{R}}{\text{argmin}} \;\mathbb{E}[\rho_\alpha(X - x)] &= q_\alpha(X), \\ \underset{x \in \mathbb{R}}{\min} \;\mathbb{E}[\rho_\alpha(X - x)] &= (1-\alpha) \left\{ \phi_\alpha (X) - \mathbb{E}[X] \right\}. \end{aligned}

Minimizing the expression in the lemma to get the \alpha-CVAR is equivalent to minimizing a linear transformation of the pinball loss to get the \alpha-quantile (or \alpha-VaR).

Credit: I learnt of CVaR and this lemma through a talk that Stefan Wager gave recently at a reading group. One of his students, Roshni Sahoo, used this lemma as the basis for a new method for learning from a biased sample (see Reference 2).

References:

  1. Rockafellar, R. T., and Uryasev, S. (2000). “Optimization of conditional value-at-risk.“
  2. Sahoo, R., et al. (2022). “Learning from a Biased Sample.“

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