scholarly journals Minimizing spectral risk measures applied to Markov decision processes

2021 ◽  
Author(s):  
Nicole Bäuerle ◽  
Alexander Glauner
Keyword(s):  
Minimization Problem ◽  
Risk Measures ◽  
Risk Measure ◽  
Sufficient Conditions ◽  
Planning Horizon ◽  
Insurance Company ◽  
Existence Of A Solution ◽  
Discounted Cost ◽  
Spectral Risk Measures ◽  
Markov Decision

AbstractWe study the minimization of a spectral risk measure of the total discounted cost generated by a Markov Decision Process (MDP) over a finite or infinite planning horizon. The MDP is assumed to have Borel state and action spaces and the cost function may be unbounded above. The optimization problem is split into two minimization problems using an infimum representation for spectral risk measures. We show that the inner minimization problem can be solved as an ordinary MDP on an extended state space and give sufficient conditions under which an optimal policy exists. Regarding the infinite dimensional outer minimization problem, we prove the existence of a solution and derive an algorithm for its numerical approximation. Our results include the findings in Bäuerle and Ott (Math Methods Oper Res 74(3):361–379, 2011) in the special case that the risk measure is Expected Shortfall. As an application, we present a dynamic extension of the classical static optimal reinsurance problem, where an insurance company minimizes its cost of capital.

scholarly journals Restricted Coherent Risk Measures and Actuarial Solvency

2012 ◽  
Vol 2012 ◽  
pp. 1-18
Author(s):  
Christos E. Kountzakis
Keyword(s):  
Minimization Problem ◽  
Risk Measures ◽  
Insurance Company ◽  
Coherent Risk Measures ◽  
Dual Representation ◽  
Coherent Risk ◽  
Representation Form ◽  
Solvency Capital

We prove a general dual representation form for restricted coherent risk measures, and we apply it to a minimization problem of the required solvency capital for an insurance company.

OPTIMAL REPLACEMENT POLICIES UNDER ENVIRONMENT-DRIVEN DEGRADATION

2012 ◽  
Vol 26 (3) ◽  
pp. 405-424 ◽  
Author(s):  
M. Yasin Ulukus ◽  
Jeffrey P. Kharoufeh ◽  
Lisa M. Maillart
Keyword(s):  
Process Model ◽  
Planning Horizon ◽  
Replacement Policy ◽  
Optimal Threshold ◽  
Discounted Cost ◽  
State Dependent ◽  
Optimal Thresholds ◽  
Degradation Rates ◽  
Markov Decision ◽  
Degrading System

We examine the problem of optimally maintaining a stochastically degrading system using preventive and reactive replacements. The system's rate of degradation is modulated by an exogenous stochastic environment process, and the system fails when its cumulative degradation level first reaches a fixed deterministic threshold. The objective is to minimize the total expected discounted cost of preventively and reactively replacing such a system over an infinite planning horizon. To this end, we present and analyze a Markov decision process model. It is shown that, for each environment state, there exists an optimal threshold-type replacement policy. Additionally, empirical evidence suggests that, when the environment process is monotone, and the state-dependent degradation rates are totally ordered, the optimal threshold is monotone. Lastly, we derive closed-form bounds on the optimal thresholds.

scholarly journals Tail Variance Premium with Applications for Elliptical Portfolio of Risks

2006 ◽  
Vol 36 (2) ◽  
pp. 433-462 ◽  
Author(s):  
Edward Furman ◽  
Zinoviy Landsman
Keyword(s):  
Risk Measures ◽  
Risk Measure ◽  
Insurance Company ◽  
Risk Capital ◽  
Normal Distributions ◽  
Risk Manager ◽  
Tail Conditional Expectation ◽  
Variance Risk ◽  
Risk Managers ◽  
Variance Risk Measure

In this paper we consider the important circumstances involved when risk managers are concerned with risks that exceed a certain threshold. Such conditions are well-known to insurance professionals, for instance in the context of policies involving deductibles and reinsurance contracts. We propose a new premium called tail variance premium (TVP) which answers the demands of these circumstances. In addition, we suggest a number of risk measures associated with TVP. While the well-known tail conditional expectation risk measure provides a risk manager with information about the average of the tail of the loss distribution, tail variance risk measure (TV) estimates the variability along such a tail. Furthermore, given a multivariate setup, we offer a number of allocation techniques which preserve different desirable properties (sub-additivity and fulladditivity, for instance). We are able to derive explicit expressions for TV and TVP, and risk capital decomposition rules based on them, in the general framework of multivariate elliptical distributions. This class is very popular among actuaries and risk managers because it contains distributions with marginals whose tails are heavier than those of normal distributions. This distinctive feature is desirable when modeling financial datasets. Moreover, according to our results, in some cases there exists an optimal threshold, such that by choosing it, an insurance company minimizes its risk.

scholarly journals Spectral risk measure of holding stocks in the long run

2020 ◽  
Vol 295 (1) ◽  
pp. 75-89
Author(s):  
Zsolt Bihary ◽  
Péter Csóka ◽  
Dávid Zoltán Szabó
Keyword(s):  
Stock Price ◽  
Risk Measures ◽  
Risk Measure ◽  
Stable Process ◽  
Important Special Case ◽  
Finite Moment ◽  
Holding Period ◽  
Long Run ◽  
Spectral Risk Measures ◽  
Spectral Risk Measure

AbstractWe investigate how the spectral risk measure associated with holding stocks rather than a risk-free deposit, depends on the holding period. Previous papers have shown that within a limited class of spectral risk measures, and when the stock price follows specific processes, spectral risk becomes negative at long periods. We generalize this result for arbitrary exponential Lévy processes. We also prove the same behavior for all spectral risk measures (including the important special case of Expected Shortfall) when the stock price grows realistically fast and when it follows a geometric Brownian motion or a finite moment log stable process. This result would suggest that holding stocks for long periods has a vanishing downside risk. However, using realistic models, we find numerically that spectral risk initially increases for a significant amount of time and reaches zero level only after several decades. Therefore, we conclude that holding stocks has spectral risk for all practically relevant periods.

scholarly journals Spatial Risk Measures and Rate of Spatial Diversification

Risks ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 52 ◽  
Author(s):  
Erwan Koch
Keyword(s):  
Random Fields ◽  
Value At Risk ◽  
Risk Measures ◽  
Risk Measure ◽  
Natural Extension ◽  
Sufficient Conditions ◽  
Value Theory ◽  
Spatial Homogeneity ◽  
Spatial Risk ◽  
Spatial Diversification

An accurate assessment of the risk of extreme environmental events is of great importance for populations, authorities and the banking/insurance/reinsurance industry. Koch (2017) introduced a notion of spatial risk measure and a corresponding set of axioms which are well suited to analyze the risk due to events having a spatial extent, precisely such as environmental phenomena. The axiom of asymptotic spatial homogeneity is of particular interest since it allows one to quantify the rate of spatial diversification when the region under consideration becomes large. In this paper, we first investigate the general concepts of spatial risk measures and corresponding axioms further and thoroughly explain the usefulness of this theory for both actuarial science and practice. Second, in the case of a general cost field, we give sufficient conditions such that spatial risk measures associated with expectation, variance, value-at-risk as well as expected shortfall and induced by this cost field satisfy the axioms of asymptotic spatial homogeneity of order 0, −2, −1 and −1, respectively. Last but not least, in the case where the cost field is a function of a max-stable random field, we provide conditions on both the function and the max-stable field ensuring the latter properties. Max-stable random fields are relevant when assessing the risk of extreme events since they appear as a natural extension of multivariate extreme-value theory to the level of random fields. Overall, this paper improves our understanding of spatial risk measures as well as of their properties with respect to the space variable and generalizes many results obtained in Koch (2017).

IMPRECISE PREVISIONS FOR RISK MEASUREMENT

2003 ◽  
Vol 11 (04) ◽  
pp. 393-412 ◽  
Author(s):  
RENATO PELESSONI ◽  
PAOLO VICIG
Keyword(s):  
Value At Risk ◽  
Risk Measures ◽  
Risk Measure ◽  
Sufficient Conditions ◽  
Risk Measurement ◽  
Random Numbers ◽  
Coherent Risk Measure ◽  
Coherent Risk ◽  
Consistency Properties ◽  
Special Case

In this paper the theory of coherent imprecise previsions is applied to risk measurement. We introduce the notion of coherent risk measure defined on an arbitrary set of risks, showing that it can be considered a special case of coherent upper prevision. We also prove that our definition generalizes the notion of coherence for risk measures defined on a linear space of random numbers, given in literature. Consistency properties of Value-at-Risk (VaR), currently one of the most used risk measures, are investigated too, showing that it does not necessarily satisfy a weaker notion of consistency called 'avoiding sure loss'. We introduce sufficient conditions for VaR to avoid sure loss and to be coherent. Finally we discuss ways of modifying incoherent risk measures into coherent ones.

WEIGHTED COMONOTONIC RISK SHARING UNDER HETEROGENEOUS BELIEFS

2020 ◽  
Vol 50 (2) ◽  
pp. 647-673
Author(s):  
Haiyan Liu
Keyword(s):  
At Risk ◽  
Value At Risk ◽  
Risk Sharing ◽  
Optimal Allocation ◽  
Risk Measures ◽  
Risk Measure ◽  
Sufficient Conditions ◽  
Heterogeneous Beliefs ◽  
Optimal Allocations ◽  
Necessary And Sufficient

AbstractWe study a weighted comonotonic risk-sharing problem among multiple agents with distortion risk measures under heterogeneous beliefs. The explicit forms of optimal allocations are obtained, which are Pareto-optimal. A necessary and sufficient condition is given to ensure the uniqueness of the optimal allocation, and sufficient conditions are given to obtain an optimal allocation of the form of excess of loss or full insurance. The optimal allocation may satisfy individual rationality depending on the choice of the weight. When the distortion risk measure is value at risk or tail value at risk, an optimal allocation is generally of the excess-of-loss form. The numerical examples suggest that a risk is more likely to be shared among agents with heterogeneous beliefs, and the introduction of the weight enables us to prioritize some agents as part of a group sharing a risk.

scholarly journals Tail Variance Premium with Applications for Elliptical Portfolio of Risks

2006 ◽  
Vol 36 (02) ◽  
pp. 433-462 ◽  
Author(s):  
Edward Furman ◽  
Zinoviy Landsman
Keyword(s):  
Risk Measures ◽  
Risk Measure ◽  
Insurance Company ◽  
Risk Capital ◽  
Normal Distributions ◽  
Risk Manager ◽  
Tail Conditional Expectation ◽  
Variance Risk ◽  
Risk Managers ◽  
Variance Risk Measure

In this paper we consider the important circumstances involved when risk managers are concerned with risks that exceed a certain threshold. Such conditions are well-known to insurance professionals, for instance in the context of policies involving deductibles and reinsurance contracts. We propose a new premium called tail variance premium (TVP) which answers the demands of these circumstances. In addition, we suggest a number of risk measures associated with TVP. While the well-known tail conditional expectation risk measure provides a risk manager with information about the average of the tail of the loss distribution, tail variance risk measure (TV) estimates the variability along such a tail. Furthermore, given a multivariate setup, we offer a number of allocation techniques which preserve different desirable properties (sub-additivity and fulladditivity, for instance). We are able to derive explicit expressions for TV and TVP, and risk capital decomposition rules based on them, in the general framework of multivariate elliptical distributions. This class is very popular among actuaries and risk managers because it contains distributions with marginals whose tails are heavier than those of normal distributions. This distinctive feature is desirable when modeling financial datasets. Moreover, according to our results, in some cases there exists an optimal threshold, such that by choosing it, an insurance company minimizes its risk.

scholarly journals The theory and application of spectral risk measures in Vietnam

2017 ◽  
Vol 24 (04) ◽  
pp. 29-45
Author(s):  
Hai Ho Hong ◽  
Hoa Nguyen Thi
Keyword(s):  
Risk Aversion ◽  
Portfolio Management ◽  
Value At Risk ◽  
Utility Theory ◽  
Risk Measures ◽  
Risk Measure ◽  
Coherent Risk ◽  
Spectral Risk Measures ◽  
Spectral Risk Measure ◽  
Current Risk

This paper aims to provide a new risk measure for portfolio management in Vietnam by incorporating investor’s risk aversion into current risk measures such as value at risk (VaR) and expected shortfall (ES). This measure shares several desirable characteristics with the coherent risk measures, as illustrated in Artzner et al. (1997). In Vietnam, our study makes the first attempt to utilize distortion theory, instead of utility theory, to facilitate the adoption of risk aversion level in the popular risk measures. We find that spectral risk measure is more flexible and effective to different groups of risk-adverse investors, compared to the more monotonic and conventional VaR and ES measures

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