Is the inf-convolution of law-invariant preferences law-invariant?

Abstract

We analyze the question of whether the inf-convolution of law-invariant risk functionals (preferences) is still law-invariant. In other words, we try to understand whether the representative economic agent (after risk redistribution) only cares about the distribution of the total risk, assuming all individual agents do so. Although the answer to the above question seems to be affirmative for many examples of commonly used risk functionals in the literature, the situation becomes delicate without assuming specific forms and properties of the individual functionals.We illustrate with examples the surprising fact that the answer to the main question is generally negative, even in an atomless probability space. Furthermore, we establish a few very weak conditions under which the answer becomes positive. These conditions do not require any specific forms or convexity of the risk functionals, and they are the richness of the underlying probability space, and monotonicity or continuity of one of the risk functionals. We provide several examples and counter-examples to discuss the subtlety of the question on law-invariance.

Inf-convolution and optimal risk sharing with countable sets of risk measures

“…However, the right-hand term can be strictly greater than the arbitrary ǫ > 0 in the definition of uniform continuity. Nonetheless, Lemma 1 in (Liu et al, 2020) shows that uniform continuity is preserved when I is finite, whereas Example 6 in that paper provides a case where ordinary continuity is not preserved.…”

“…(ii) Liu et al (2020) showed that, in the usual case of finite I, law invariance is preserved under other continuity conditions that primarily rely on the existence of countable iid uniform random variables, which is always the case for atomless spaces. However, we are unable to extend their result, as in our framework, arbitrarily many such random variables would be required.…”

“…Beyond the usual approach of convex risk measures, some studies have been concerned with inf-convolution in relation to specific properties, as in (Acciaio, 2007), (Grechuk et al, 2009), (Grechuk and Zabarankin, 2012), (Carlier et al, 2012), (Mastrogiacomo and Rosazza Gianin, 2015), and (Liu et al, 2020), particular risk measures, as the recent quantile risk sharing in (Embrechts et al, 2018b), (Embrechts et al, 2018a), (Weber, 2018), (Wang and Ziegel, 2018), and (Liu et al, 2019), or even specific topics, as in (Wang, 2016) and (Liebrich and Svindland, 2019). However, these studies, with or without convexity, are restricted to finite (or at most countable in rare cases) sets of risk measures.…”

Automatic Fatou Property of Law-invariant Risk Measures

“…As a matter of fact, most concrete risk measures (such as Value-at-risk, Expected shortfall, Haezendonck-Goovaerts risk measures) and special classes of risk measures (such as distortion and quantile-based risk measures) are indeed law invariant. Research on general law-invariant risk measures has been intense and has produced many profound results; see, e.g., Bellini et al [4,5], Chen et al [10], Filipović and Svindland [16,17], Gao et al [20], Jouini et al [22,23], Krätschmer et al [24], Kusuoka [25], Liu et al [27], Wang and Zitikis [35], and Weber [36].…”

Do law-invariant linear functionals collapse to the mean?

“…In the past few years, law-invariant risk measures have been of intense research interest in Financial Mathematics; see, e.g., [2,3,5,7,8,10,12,13,14,15,17,19,20,21,22]. Of particular interest to us are two recent papers [3,11] that investigate when a law-invariant risk measure collapses to the mean, i.e., being a scalar multiple of expectation.…”