Representation results for law invariant time consistent functions

Kupper, Michael; Schachermayer, Walter

doi:10.1007/s11579-009-0019-9

Cited by 113 publications

(87 citation statements)

References 30 publications

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“…Nevertheless, it appears that the solvency capital obtained can be very expensive if we do not take care of the confidence level of each conditional VaR and TVaR measures involved in the iteration scheme. This is linked to a result obtained in [13], which tells us that, under certain hypotheses, if we consider a great number of iterations, the risk measure obtained is close to either the conditional expectation of the risk or its essential supremum.…”

Section: Introductionsupporting

confidence: 52%

“…We consider here T iterations, i.e., a finite number of steps t ∈ T (see Corollary 1), while in ( [13] [Theorem 1.10]) they consider an infinite (but still countable) number of iterations, i.e., t ∈ N. However, depending on the choice of the dynamic risk measure and on the actual final net worth considered, a great number of iterations T could converge towards ( [13] [Theorem 1.10]), which is the case in [10][11][12].…”

Section: Compositions Of Conditional Risk Measuresmentioning

confidence: 99%

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Compositions of Conditional Risk Measures and Solvency Capital

Devolder

Lebègue

2016

Risks

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Abstract:In this paper, we consider compositions of conditional risk measures in order to obtain time-consistent dynamic risk measures and determine the solvency capital of a life insurer selling pension liabilities or a pension fund with a single cash-flow at maturity. We first recall the notion of conditional, dynamic and time-consistent risk measures. We link the latter with its iterated property, which gives us a way to construct time-consistent dynamic risk measures from a backward iteration scheme with the composition of conditional risk measures. We then consider particular cases with the conditional version of the value at risk, tail value at risk and conditional expectation measures. We finally give an application of these measures with the determination of the solvency capital of a pension liability, which offers a fixed guaranteed rate without any intermediate cash-flow. We assume that the company is fully hedged against the mortality and underwriting risks.

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Section: Introductionsupporting

confidence: 52%

Section: Compositions Of Conditional Risk Measuresmentioning

confidence: 99%

Compositions of Conditional Risk Measures and Solvency Capital

Devolder

Lebègue

2016

Risks

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“…Investigation of the use of entropic risk measures of the form suggested by Kupper and Schachermayer (2009) also bears further investigation. These risk measures are time consistent but not scale invariant.…”

Section: Modeling Considerations and Concluding Remarksmentioning

confidence: 99%

“…Kupper and Schachermayer (2009) have shown that a certain class of "entropic" risk measures are the only possible ones that are both time consistent and law invariant. While use of such risk measures definitely warrants further study, they are are not coherent, failing to meet the scale-invariance property of coherent risk measures (see below).…”

Section: Introductionmentioning

confidence: 99%

Multilevel Optimization Modeling for Risk-Averse Stochastic Programming

Eckstein

Eskandani

Fan

2016

INFORMS Journal on Computing

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Abstract. Coherent risk measures have become a popular tool for incorporating risk aversion into stochastic optimization models. For dynamic models in which uncertainty is resolved at more than one stage, however, using coherent risk measures within a standard single-level optimization framework becomes problematic. To avoid severe time-consistency difficulties, the current state of the art is to employ risk measures of a specific nested form, which unfortunately have some undesirable and somewhat counterintuitive modeling properties. For one thing, this technique requires increasing risk aversion as risks and reward recede in time. Further, it produces objective functions that cannot be law invariant with respect to the total incurred costs and rewards, meaning that two solutions with identical probability distributions of final wealth may be assigned different levels of risk, and the nested form of the objective function cannot be simplified. These properties deter practical acceptance of such models, and are particularly undesirable for situations with close final time horizons. This paper summarizes these issues and then presents an alternative multilevel optimization modeling approach that enforces a form of time consistency through constraints rather than by restricting the modeler's choice of objective function. This technique leads to models that are time-consistent even while using time-inconsistent risk measures, and can easily be formulated to be law invariant with respect to the final wealth if so desired. We argue that this approach should be the starting point for all multistage optimization modeling. When used with time-consistent objective functions, we show its multilevel optimization constraints become redundant and the associated models thus simplify to a more familiar single-objective form. Unfortunately, this paper also shows that its proposed approach leads to N Phard models, even in the simplest imaginable setting in which it would be needed: three-stage linear problems on a finite probability space, using the standard meansemideviation and average-value-at-risk measures. Finally, we show that for a simple but reasonably realistic test application, that the kind of models we propose, while being drawn from an N P-hard family and certainly more time consuming to solve than those obtained from the nested-objective approach, are readily solvable to global optimality using a standard commercial MILP solver. Therefore, there seems some promise of the modeling approach being useful despite its computational complexity properties.Acknowledgements: This work was supported by a grant from the Air Force Office of Scientific Research, award FA9550-11-1-0164. The authors would like to thank Andrzej Ruszczyński for his assistance and support.Note: Sections 4-7 of this report essentially duplicate material presented in the earlier reports RRR 7-2013 and (using different notation) RRR 17-2012. The remaining sections contain extensive new material and results.

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“…In continuous time such strongly time consistent versions do not even exist, cf. Kupper and Schachermayer (2009) and Delbaen (2006).…”

mentioning

confidence: 99%

An algorithm for sequential tail value at risk for path-independent payoffs in a binomial tree

Roorda

2010

Ann Oper Res

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We present an algorithm that determines Sequential Tail Value at Risk (STVaR) for path-independent payoffs in a binomial tree. STVaR is a dynamic version of Tail-Valueat-Risk (TVaR) characterized by the property that risk levels at any moment must be in the range of risk levels later on. The algorithm consists of a finite sequence of backward recursions that is guaranteed to arrive at the solution of the corresponding dynamic optimization problem. The algorithm makes concrete how STVaR differs from TVaR over the remaining horizon, and from recursive TVaR, which amounts to Dynamic Programming. Algorithmic aspects are compared with the cutting-plane method. Time consistency and comonotonicity properties are illustrated by applying the algorithm on elementary examples.

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Representation results for law invariant time consistent functions

Cited by 113 publications

References 30 publications

Compositions of Conditional Risk Measures and Solvency Capital

Compositions of Conditional Risk Measures and Solvency Capital

Multilevel Optimization Modeling for Risk-Averse Stochastic Programming

An algorithm for sequential tail value at risk for path-independent payoffs in a binomial tree

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