Rearrangement Algorithm in Risk Aggregation

Sometimes there’s no closed-form analytical solutions for the risk measure of aggregate losses representing, say, a company’s losses in each country or city it operates in, a portfolio of losses subdivided by investment, or claims made by clients to an insurance company. Assuming there’s enough data to assign a distribution to those losses, we examine the Rearrangement Algorithm’s ability to numerically compute the Expected Shortfall and Exponential Premium Principle/Entropic Risk Measure of aggregate losses. A more efficient discretization scheme is introduced and the algorithm is extended to the Entropic Risk Measure which turns out to have a smaller uncertainty spread than the Expected Shortfall at least for the cases that we examined.

Rearrangement Algorithm in Risk Aggregation

Sometimes there’s no closed-form analytical solutions for the risk measure of aggregate losses representing, say, a company’s losses in each country or city it operates in, a portfolio of losses subdivided by investment, or claims made by clients to an insurance company. Assuming there’s enough data to assign a distribution to those losses, we examine the Rearrangement Algorithm’s ability to numerically compute the Expected Shortfall and Exponential Premium Principle/Entropic Risk Measure of aggregate losses. A more efficient discretization scheme is introduced and the algorithm is extended to the Entropic Risk Measure which turns out to have a smaller uncertainty spread than the Expected Shortfall at least for the cases that we examined.

AN ERGODIC BSDE RISK REPRESENTATION IN A JUMP-DIFFUSION FRAMEWORK

We consider the representation of forward entropic risk measures using the theory of ergodic backward stochastic differential equations in a jump-diffusion framework. Our paper can be viewed as an extension of the work considered by Chong et al. (2019) in the diffusion case. We also study the behavior of a forward entropic risk measure under jumps when a financial position is held for a longer maturity.

SET-VALUED DYNAMIC RISK MEASURES FOR BOUNDED DISCRETE-TIME PROCESSES

In this paper, we study how to evaluate the risk of a financial portfolio, whose components may be dependent and come from different markets or involve more than one kind of currencies, while we also take into consideration the uncertainty about the time value of money. Namely, we introduce a new class of risk measures, named set-valued dynamic risk measures for bounded discrete-time processes that are adapted to a given filtration. The time horizon can be finite or infinite. We investigate the representation results for them by making full use of Legendre–Fenchel conjugation theory for set-valued functions. Finally, some examples such as the set-valued dynamic average value at risk and the entropic risk measure for bounded discrete-time processes are also given.

Risk management with Tail Quasi-Linear Means

We generalise Quasi-Linear Means by restricting to the tail of the risk distribution and show that this can be a useful quantity in risk management since it comprises in its general form the Value at Risk, the Conditional Tail Expectation and the Entropic Risk Measure in a unified way. We then investigate the fundamental properties of the proposed measure and show its unique features and implications in the risk measurement process. Furthermore, we derive formulas for truncated elliptical models of losses and provide formulas for selected members of such models.

DIFFERENTIABILITY OF BSVIEs AND DYNAMIC CAPITAL ALLOCATIONS

Capital allocations have been studied in conjunction with static risk measures in various papers. The dynamic case has been studied only in a discrete-time setting. We address the problem of allocating risk capital to subportfolios for the first time in a continuous-time dynamic context. For this purpose, we introduce a differentiability result for backward stochastic Volterra integral equations and apply this result to derive continuous-time dynamic capital allocations. Moreover, we study a dynamic capital allocation principle that is based on backward stochastic differential equations and derive the dynamic gradient allocation for the dynamic entropic risk measure.

REPRESENTATION OF BSDE-BASED DYNAMIC RISK MEASURES AND DYNAMIC CAPITAL ALLOCATIONS

In this paper, we provide a new representation result for dynamic capital allocations and dynamic convex risk measures that are based on backward stochastic differential equations (BSDEs). We derive this representation from a classical differentiability result for BSDEs and the full allocation property of the Aumann–Shapley allocation. The representation covers BSDE-based dynamic convex and dynamic coherent risk measures. The results are applied to derive a representation for the dynamic entropic risk measure. Our results are also applicable in a specific way to the static case, where we are able to derive a new representation result for static convex risk measures that are Gâteaux-differentiable.

Optimal Design of Dynamic Default Risk Measures

We consider the question of an optimal transaction between two investors to minimize their risks. We define a dynamic entropic risk measure using backward stochastic differential equations related to a continuous-time single jump process. The inf-convolution of dynamic entropic risk measures is a key transformation in solving the optimization problem.