Properties of a risk measure derived from the expected area in red

Abstract

To cite this version:Stéphane Loisel, Julien Trufin. Properties of a risk measure derived from the expected area in red. Insurance: Mathematics and Economics, Elsevier, 2014, 55, pp.191-199. hal-00870224 Properties of a risk measure derived from the expected area in red AbstractThis paper studies a new risk measure derived from the expected area in red introduced in Loisel (2005). Specifically, we derive various properties of a risk measure defined as the smallest initial capital needed to ensure that the expected time-integrated negative part of the risk process on a fixed time interval [0, T ] (T can be infinite) is less than a given predetermined risk limit. We also investigate the optimal risk limit allocation: given a risk limit set at company level for the sum of the expected areas in red of all lines, we determine the way(s) to allocate this risk limit to the subsequent business lines in order to minimize the overall capital needs.

Special Issue “Risk, Ruin and Survival: Decision Making in Insurance and Finance”

“…The time of the cumulative Parisian ruin is the first time the surplus process stays cumulatively below a critical level longer than the pre-determined grace period. Several dynamic risk measures, which are those based on ruin-theoretic quantities, have been studied by, e.g., Trufin et al (2011), Mitric and Trufin (2016), and Loisel and Trufin (2014). These results have, in turn, motivated Lkabous and Renaud (2018) to explore a VaR-type risk measure based on the cumulative Parisian ruin for the classical risk model.…”

Dynamic risk measures for stochastic asset processes from ruin theory

“…This is a risk measure in the actuarial sense; ε is a level that makes the company's position acceptable. This is motivated from Value-at-Risk (VaR); the concept is not quite new but is a natural extension of the VaR risk measure of ruin theory in Trufin et al (2011) or Loisel & Trufin (2014).…”

On the Capital Allocation Problem for a New Coherent Risk Measure in Collective Risk Theory

“…We can only cite, [30], where the author discusses the problem of capital allocation for risk measures defined on the space of cádlág processes. Or, [6] where the authors study the capital allocation problem for a new risk measure that, as it turned out, it does not satisfy an axiomatic definition of coherent risk measures defined on stochastic processes proposed in [8]. As a drawback, the resulting solution of the capital allocation problem, does not follow an axiomatic definition of capital allocation.…”

“…In fact, any meaningful risk management application, such as capital allocation, would be hard to implement using (5). Recently, other risk measures have been studied such as one based on the concept of area in red, which is a measure of how large the overall deficit of the company can be (see [6] for details). These new notions turn out to exhibit some interesting properties, yet the issue with these risk measures remains, they are very difficult to implement in a risk management problem such as capital allocation.…”