Two Optimal Transportation approaches to measuring Risk.

Résumé

The evaluation and comparison of risks are basic tasks of risk analysis in Insurance and Finance. In this talk, we show how Optimal Transportation can be employed for this purpose. The first approach, based on {1], considers that risk is a relative notion between two distributions, quantifying an order relation. It introduces the notion of risk excess measure of one risk distribution $Q$ w.r.t. a benchmark risk $P$. More precisely, it measures the risk excess of $Q$ over $P$ by a hemi-metric $D_+( Q , P )$ on the space of probability measures. $D_+( Q , P )$ is a ``one-sided distance encoding an order'' on the space $(\mathcal M_1 ( E ),\prec)$ of probability measures, where $\prec$ is a given stochastic (pre)order. The stochastic order $\prec$ is related to the ordering $\le$ on the underlying space $E$. This allows to consider for a quantitative one-sided comparison of risks at the level of probability measures as an extension of the order and distance structure on $E$. We discuss several classes of risk excess measures $D+( Q , P )$ and consider the question when these are given as order extensions of hemi-distances $d_+$ on the underlying space $E$. Several relevant hemi-distances are induced by mass transportation and by function class induced orderings, thus giving access to natural interpretation. One particular extension is given by a version of the Kantorovich–Rubinstein theorem for hemi-distances. Our view towards measuring risk excess adds to the usually considered method to compare risks of $Q$ and $P$ by the values $\rho( Q )$ , $\rho( P )$ of a risk measure $\rho$. We argue that the difference $\rho(Q)-\rho(P)$ neglects relevant aspects of the risk excess which are adequately described by the new notion of risk excess measure. The second approach, based on {2], proposes another novel approach in the assessment of a random risk variable $X$ by introducing magnitude-propensity risk measures $(m_X , p_X)$. This bivariate measure intends to account for the dual aspect of risk, where the magnitudes $x$ of $X$ tell how high are the losses incurred, whereas the probabilities (propensities) $P(X = x)$ reveal how often one has to expect to suffer such losses. The basic idea is to simultaneously quantify both the severity $m_X$ and the propensity $p_X$ of the real-valued risk $X$. This is to be contrasted with traditional univariate risk measures, like $\text{VaR}$ or $\text{CVaR}$, which typically conflate both effects. In its simplest form, $(m_X , p_X)$ is obtained by mass transportation in Wasserstein metric of the law of X to a two- points $\{0, m_X \}$ discrete distribution with mass $p_X$ at $m_X$. The approach can also be formulated as a constrained optimal quantization problem. This allows for an informative comparison of risks on both the magnitude and propensity scales. Several examples illustrate the usefulness of the proposed approach. Some variants, (multivariate) extensions and applications (portfolio analysis) are also considered. [1] O.P. Faugeras, Rüschendorf, L. "Risk excess measures induced by hemi-metrics." Probability, Uncertainty and Quantitative Risk, Vol. 3, 6, 2018. https://doi.org/10.1186/s41546-018-0032-0} [2] O.P. Faugeras, G. Pagès. "Risk Quantization by Magnitude and Propensity." hal-03233068v2, Submitted, in revision, 2021.