General Acceptance Sets, Risk Measures and Optimal Capital Injections

2011 ◽  
Author(s):  
Walter Farkas ◽  
Pablo Koch Medina ◽  
Cosimo-Andrea Munari
Keyword(s):  
Risk Measures ◽  
General Acceptance ◽  
Optimal Capital ◽  
Capital Injections ◽  
Acceptance Sets

scholarly journals Optimal capital injections and dividends with tax in a risk model in discrete time

2020 ◽  
Vol 10 (1) ◽  
pp. 235-259
Author(s):  
Katharina Bata ◽  
Hanspeter Schmidli
Keyword(s):  
Discrete Time ◽  
Optimal Strategy ◽  
Bellman Equation ◽  
Value Function ◽  
Risk Model ◽  
Barrier Type ◽  
Optimal Capital ◽  
Capital Injections ◽  
The Value Function ◽  
Special Case

AbstractWe consider a risk model in discrete time with dividends and capital injections. The goal is to maximise the value of a dividend strategy. We show that the optimal strategy is of barrier type. That is, all capital above a certain threshold is paid as dividend. A second problem adds tax to the dividends but an injection leads to an exemption from tax. We show that the value function fulfils a Bellman equation. As a special case, we consider the case of premia of size one. In this case we show that the optimal strategy is a two barrier strategy. That is, there is a barrier if a next dividend of size one can be paid without tax and a barrier if the next dividend of size one will be taxed. In both models, we illustrate the findings by de Finetti’s example.

scholarly journals A unified approach to systemic risk measures via acceptance sets

2018 ◽  
Vol 29 (1) ◽  
pp. 329-367 ◽  
Author(s):  
Francesca Biagini ◽  
Jean-Pierre Fouque ◽  
Marco Frittelli ◽  
Thilo Meyer-Brandis
Keyword(s):  
Systemic Risk ◽  
Risk Measures ◽  
Unified Approach ◽  
Acceptance Sets

scholarly journals Surplus-Invariant Risk Measures

2020 ◽  
Vol 45 (4) ◽  
pp. 1342-1370 ◽  
Author(s):  
Niushan Gao ◽  
Cosimo Munari
Keyword(s):  
Systematic Study ◽  
Risk Measures ◽  
Orlicz Spaces ◽  
Capital Requirements ◽  
Vector Lattices ◽  
Dual Representations ◽  
Robust Model ◽  
Model Spaces ◽  
Acceptance Sets ◽  
Lattice Approach

This paper presents a systematic study of the notion of surplus invariance, which plays a natural and important role in the theory of risk measures and capital requirements. So far, this notion has been investigated in the setting of some special spaces of random variables. In this paper, we develop a theory of surplus invariance in its natural framework, namely, that of vector lattices. Besides providing a unifying perspective on the existing literature, we establish a variety of new results including dual representations and extensions of surplus-invariant risk measures and structural results for surplus-invariant acceptance sets. We illustrate the power of the lattice approach by specifying our results to model spaces with a dominating probability, including Orlicz spaces, as well as to robust model spaces without a dominating probability, where the standard topological techniques and exhaustion arguments cannot be applied.

scholarly journals INF-Convolution of Risk Measures on Rearrangement Invariant Spaces

2021 ◽  
Author(s):  
Shengzhong Chen
Keyword(s):  
Risk Measures ◽  
Model Space ◽  
Continuity Condition ◽  
Individual Risk ◽  
Risk Allocation ◽  
Fatou Property ◽  
Economic Agents ◽  
Research Areas ◽  
Optimal Capital ◽  
Invariant Spaces

The problem of optimal capital and risk allocation among economic agents, has played a predominant role in the respective academic and industrial research areas for decades. Typically as risk occurs in face of randomness the risks which are to be measured are identified with real-valued random variables on some probability space (Ω, F, P). Consider a model space X , and n economic agents with initial endowments X1, · · · , Xn ∈ X who assess the riskiness of their positions by means of law-invariant convex risk measures ρi : X → (−∞,∞]. In order to minimize total and individual risk, the agents redistribute the aggregate endowment X = X1 + · · · + Xn among themselves. An optimal capital and risk allocation Y1, · · · , Yn satisfies Y1 + · · · + Yn = X and ρ1(Y1) + · · · + ρ(Yn) = inf nXn i=1 ρi(Xi) : Xi ∈ X , i = 1, . . . , n, and Xn i=1 Xi = X o , (0.1) where n i=1ρi(X) = inf nPn i=1 ρi(Xi) : Xi ∈ X , i = 1, . . . , n, and Pn i=1 Xi = X o is the inf-convolution of ρ1, ..., ρn. In 2008, Filipovi´c and Svindland proved that if X is an L p (P) for some 1 ≤ p ≤ ∞ and ρi satisfy a suitable continuity condition (i.e. Fatou property), then Problem (0.1) always admits a solution. To reflect the fact of randomness of risk, we should consider the model space X chosen for risk evaluations to be as general as possible. The main contribution of this thesis is Theorem 4.10 has been published in [9]. It extends Filipovi´c and Svindland’s result from L p spaces to general rearrangement invariant (r.i.) spaces.

scholarly journals INF-Convolution of Risk Measures on Rearrangement Invariant Spaces

2021 ◽  
Author(s):  
Shengzhong Chen
Keyword(s):  
Risk Measures ◽  
Model Space ◽  
Continuity Condition ◽  
Individual Risk ◽  
Risk Allocation ◽  
Fatou Property ◽  
Economic Agents ◽  
Research Areas ◽  
Optimal Capital ◽  
Invariant Spaces

The problem of optimal capital and risk allocation among economic agents, has played a predominant role in the respective academic and industrial research areas for decades. Typically as risk occurs in face of randomness the risks which are to be measured are identified with real-valued random variables on some probability space (Ω, F, P). Consider a model space X , and n economic agents with initial endowments X1, · · · , Xn ∈ X who assess the riskiness of their positions by means of law-invariant convex risk measures ρi : X → (−∞,∞]. In order to minimize total and individual risk, the agents redistribute the aggregate endowment X = X1 + · · · + Xn among themselves. An optimal capital and risk allocation Y1, · · · , Yn satisfies Y1 + · · · + Yn = X and ρ1(Y1) + · · · + ρ(Yn) = inf nXn i=1 ρi(Xi) : Xi ∈ X , i = 1, . . . , n, and Xn i=1 Xi = X o , (0.1) where n i=1ρi(X) = inf nPn i=1 ρi(Xi) : Xi ∈ X , i = 1, . . . , n, and Pn i=1 Xi = X o is the inf-convolution of ρ1, ..., ρn. In 2008, Filipovi´c and Svindland proved that if X is an L p (P) for some 1 ≤ p ≤ ∞ and ρi satisfy a suitable continuity condition (i.e. Fatou property), then Problem (0.1) always admits a solution. To reflect the fact of randomness of risk, we should consider the model space X chosen for risk evaluations to be as general as possible. The main contribution of this thesis is Theorem 4.10 has been published in [9]. It extends Filipovi´c and Svindland’s result from L p spaces to general rearrangement invariant (r.i.) spaces.

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