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To split or not to split: Capital allocation with convex risk measures
Convex risk measures were introduced by Deprez and Gerber (1985). Here the problem of allocating risk capital to subportfolios is addressed, when aggregate capital is calculated by a convex risk measure. The Aumann-Shapley value is proposed as an appropriate allocation mechanism. Distortion-exponential measures are discussed extensively and explicit capital allocation formulas are obtained for the case that the risk measure belongs to this family. Finally the implications of capital allocation with a convex risk measure for the stability of portfolios are discussed
Risk measurement with the equivalent utility principles.
Risk measures have been studied for several decades in the actuarial literature, where they appeared under the guise of premium calculation principles. Risk measures and properties that risk measures should satisfy have recently received considerable at- tention in the financial mathematics literature. Mathematically, a risk measure is a mapping from a class of random variables defined on some measurable space to the (extended) real line. Economically, a risk measure should capture the preferences of the decision-maker. In incomplete financial markets, prices are no more unique but depend on the agents' attitudes towards risk. This paper complements the study initiated in Denuit, Dhaene & Van Wouwe (1999) and considers several theories for decision under uncertainty: the classical expected utility paradigm, Yaari's dual approach, maximin expected utility theory, Choquet expected utility theory and Quiggin rank-dependent utility theory. Building on the actuarial equivalent utility pricing principle, broad classes of risk measures are generated, of which most classical risk measures appear to be particular cases. This approach shows that most risk measures studied recently in the financial literature disregard the utility concept (i.e. correspond to linear utilities), causing some deficiencies. Some alternatives proposed in the literature are discussed, based on exponential utilities.Actuarial; Coherence; Decision; Expected; Market; Markets; Measurement; Preference; Premium; Prices; Pricing; Principles; Random variables; Research; Risk; Risk measure; Risk measurement; Space; Studies; Theory; Uncertainty; Utilities; Variables;
Data-driven satisficing measure and ranking
We propose an computational framework for real-time risk assessment and
prioritizing for random outcomes without prior information on probability
distributions. The basic model is built based on satisficing measure (SM) which
yields a single index for risk comparison. Since SM is a dual representation
for a family of risk measures, we consider problems constrained by general
convex risk measures and specifically by Conditional value-at-risk. Starting
from offline optimization, we apply sample average approximation technique and
argue the convergence rate and validation of optimal solutions. In online
stochastic optimization case, we develop primal-dual stochastic approximation
algorithms respectively for general risk constrained problems, and derive their
regret bounds. For both offline and online cases, we illustrate the
relationship between risk ranking accuracy with sample size (or iterations).Comment: 26 Pages, 6 Figure
A unified approach to generate risk measures.
Markov inequality; Premium; Premium principle; Principles; Probability; Recall; Risk; Risk measure;
Computational Dynamic Market Risk Measures in Discrete Time Setting
Different approaches to defining dynamic market risk measures are available
in the literature. Most are focused or derived from probability theory,
economic behavior or dynamic programming. Here, we propose an approach to
define and implement dynamic market risk measures based on recursion and state
economy representation. The proposed approach is to be implementable and to
inherit properties from static market risk measures.Comment: 16 pages, 3 figure
Robust Optimal Risk Sharing and Risk Premia in Expanding Pools
We consider the problem of optimal risk sharing in a pool of cooperative
agents. We analyze the asymptotic behavior of the certainty equivalents and
risk premia associated with the Pareto optimal risk sharing contract as the
pool expands. We first study this problem under expected utility preferences
with an objectively or subjectively given probabilistic model. Next, we develop
a robust approach by explicitly taking uncertainty about the probabilistic
model (ambiguity) into account. The resulting robust certainty equivalents and
risk premia compound risk and ambiguity aversion. We provide explicit results
on their limits and rates of convergence, induced by Pareto optimal risk
sharing in expanding pools
On the Neyman-Pearson problem for law-invariant risk measures and robust utility functionals
Motivated by optimal investment problems in mathematical finance, we consider
a variational problem of Neyman-Pearson type for law-invariant robust utility
functionals and convex risk measures. Explicit solutions are found for
quantile-based coherent risk measures and related utility functionals.
Typically, these solutions exhibit a critical phenomenon: If the capital
constraint is below some critical value, then the solution will coincide with a
classical solution; above this critical value, the solution is a superposition
of a classical solution and a less risky or even risk-free investment. For
general risk measures and utility functionals, it is shown that there exists a
solution that can be written as a deterministic increasing function of the
price density
Risk Measures
The present review of (financial) risk measures, prepared for the Encyclopaedia of Actuarial Science, first distinguishes two conceptions of risk. Risk of the first kind conceives risk as the magnitude of (one- or two-sided) deviations from a target, whereas risk of the second kind conceives risk as necessary capital or necessary premium, respectively. Some important axiomatic characterizations of risk measures are reviewed, including a characterization of a correspondence between risk measures of the first kind and risk measures of the second kind. Finally, a detailed overview of different risk measures of the first and second kind is presented.
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