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Tight Approximations of Dynamic Risk Measures
- Source :
- Mathematics of Operations Research. 40:655-682
- Publication Year :
- 2015
- Publisher :
- Institute for Operations Research and the Management Sciences (INFORMS), 2015.
-
Abstract
- This paper compares two different frameworks recently introduced in the literature for measuring risk in a multi-period setting. The first corresponds to applying a single coherent risk measure to the cumulative future costs, while the second involves applying a composition of one-step coherent risk mappings. We summarize the relative strengths of the two methods, characterize several necessary and sufficient conditions under which one of the measurements always dominates the other, and introduce a metric to quantify how close the two risk measures are. Using this notion, we address the question of how tightly a given coherent measure can be approximated by lower or upper-bounding compositional measures. We exhibit an interesting asymmetry between the two cases: the tightest possible upper-bound can be exactly characterized, and corresponds to a popular construction in the literature, while the tightest-possible lower bound is not readily available. We show that testing domination and computing the approximation factors is generally NP-hard, even when the risk measures in question are comonotonic and law-invariant. However, we characterize conditions and discuss several examples where polynomial-time algorithms are possible. One such case is the well-known Conditional Value-at-Risk measure, which is further explored in our companion paper [Huang, Iancu, Petrik and Subramanian, "Static and Dynamic Conditional Value at Risk" (2012)]. Our theoretical and algorithmic constructions exploit interesting connections between the study of risk measures and the theory of submodularity and combinatorial optimization, which may be of independent interest.
- Subjects :
- Mathematical optimization
General Mathematics
media_common.quotation_subject
Management Science and Operations Research
Composition (combinatorics)
Entropic value at risk
Asymmetry
Measure (mathematics)
Upper and lower bounds
Computer Science Applications
FOS: Economics and business
Optimization and Control (math.OC)
Bounding overwatch
Risk Management (q-fin.RM)
Coherent risk measure
Metric (mathematics)
FOS: Mathematics
Mathematics - Optimization and Control
Quantitative Finance - Risk Management
Mathematics
media_common
Subjects
Details
- ISSN :
- 15265471 and 0364765X
- Volume :
- 40
- Database :
- OpenAIRE
- Journal :
- Mathematics of Operations Research
- Accession number :
- edsair.doi.dedup.....cd835a821e0e71f923c09c457a5d55a3