IDEAS home Printed from https://ideas.repec.org/a/gam/jrisks/v7y2019i2p52-d227816.html
   My bibliography  Save this article

Spatial Risk Measures and Rate of Spatial Diversification

Author

Listed:
  • Erwan Koch

    (EPFL, Chair of Statistics STAT, EPFL-SB-MATH-STAT, MA B1 433 (Bâtiment MA), Station 8, 1015 Lausanne, Switzerland)

Abstract

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).

Suggested Citation

  • Erwan Koch, 2019. "Spatial Risk Measures and Rate of Spatial Diversification," Risks, MDPI, vol. 7(2), pages 1-26, May.
    • Handle: RePEc:gam:jrisks:v:7:y:2019:i:2:p:52-:d:227816
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-9091/7/2/52/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-9091/7/2/52/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Idris A. Eckley & Guy P. Nason & Robert L. Treloar, 2010. "Locally stationary wavelet fields with application to the modelling and analysis of image texture," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 59(4), pages 595-616, August.
    2. Ombao H. C & Raz J. A & von Sachs R. & Malow B. A, 2001. "Automatic Statistical Analysis of Bivariate Nonstationary Time Series," Journal of the American Statistical Association, American Statistical Association, vol. 96, pages 543-560, June.
    3. Anderes, Ethan B. & Stein, Michael L., 2011. "Local likelihood estimation for nonstationary random fields," Journal of Multivariate Analysis, Elsevier, vol. 102(3), pages 506-520, March.
    4. Dombry, Clément & Eyi-Minko, Frédéric, 2012. "Strong mixing properties of max-infinitely divisible random fields," Stochastic Processes and their Applications, Elsevier, vol. 122(11), pages 3790-3811.
    5. Christian Yann Robert & Erwan Koch & Clément Dombry, 2018. "A central limit theorem for functions of stationary max-stable random fields on R d," Post-Print hal-02006799, HAL.
    6. Gneiting, Tilmann, 2011. "Making and Evaluating Point Forecasts," Journal of the American Statistical Association, American Statistical Association, vol. 106(494), pages 746-762.
    7. Martin Schlather, 2003. "A dependence measure for multivariate and spatial extreme values: Properties and inference," Biometrika, Biometrika Trust, vol. 90(1), pages 139-156, March.
    8. Philippe Artzner & Freddy Delbaen & Jean‐Marc Eber & David Heath, 1999. "Coherent Measures of Risk," Mathematical Finance, Wiley Blackwell, vol. 9(3), pages 203-228, July.
    9. Padoan, S. A. & Ribatet, M. & Sisson, S. A., 2010. "Likelihood-Based Inference for Max-Stable Processes," Journal of the American Statistical Association, American Statistical Association, vol. 105(489), pages 263-277.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Jiandong Ren & Kristina Sendova & Ričardas Zitikis, 2019. "Special Issue “Risk, Ruin and Survival: Decision Making in Insurance and Finance”," Risks, MDPI, vol. 7(3), pages 1-7, September.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Erwan Koch, 2018. "Spatial risk measures and rate of spatial diversification," Papers 1803.07041, arXiv.org, revised Jun 2019.
    2. Erwan Koch, 2018. "Extremal dependence and spatial risk measures for insured losses due to extreme winds," Papers 1804.05694, arXiv.org, revised Dec 2019.
    3. Said Khalil, 2022. "Expectile-based capital allocation," Working Papers hal-03816525, HAL.
    4. Dimitriadis, Timo & Schnaitmann, Julie, 2021. "Forecast encompassing tests for the expected shortfall," International Journal of Forecasting, Elsevier, vol. 37(2), pages 604-621.
    5. Dingshi Tian & Zongwu Cai & Ying Fang, 2018. "Econometric Modeling of Risk Measures: A Selective Review of the Recent Literature," WORKING PAPERS SERIES IN THEORETICAL AND APPLIED ECONOMICS 201807, University of Kansas, Department of Economics, revised Oct 2018.
    6. Chao Wang & Richard Gerlach, 2021. "A Bayesian realized threshold measurement GARCH framework for financial tail risk forecasting," Papers 2106.00288, arXiv.org, revised Oct 2022.
    7. Qinyu Wu & Fan Yang & Ping Zhang, 2023. "Conditional generalized quantiles based on expected utility model and equivalent characterization of properties," Papers 2301.12420, arXiv.org.
    8. Maziar Sahamkhadam, 2021. "Dynamic copula-based expectile portfolios," Journal of Asset Management, Palgrave Macmillan, vol. 22(3), pages 209-223, May.
    9. Marcell Béli & Kata Váradi, 2017. "A possible methodology for determining the initial margin," Financial and Economic Review, Magyar Nemzeti Bank (Central Bank of Hungary), vol. 16(2), pages 119-147.
    10. Asimit, Alexandru V. & Bignozzi, Valeria & Cheung, Ka Chun & Hu, Junlei & Kim, Eun-Seok, 2017. "Robust and Pareto optimality of insurance contracts," European Journal of Operational Research, Elsevier, vol. 262(2), pages 720-732.
    11. Santos, Douglas G. & Candido, Osvaldo & Tófoli, Paula V., 2022. "Forecasting risk measures using intraday and overnight information," The North American Journal of Economics and Finance, Elsevier, vol. 60(C).
    12. Hirbod Assa & Liyuan Lin & Ruodu Wang, 2022. "Calibrating distribution models from PELVE," Papers 2204.08882, arXiv.org, revised Jun 2023.
    13. Onur Babat & Juan C. Vera & Luis F. Zuluaga, 2021. "Computing near-optimal Value-at-Risk portfolios using Integer Programming techniques," Papers 2107.07339, arXiv.org.
    14. Ruodu Wang & Yunran Wei, 2020. "Risk functionals with convex level sets," Mathematical Finance, Wiley Blackwell, vol. 30(4), pages 1337-1367, October.
    15. Hoga, Yannick, 2021. "The uncertainty in extreme risk forecasts from covariate-augmented volatility models," International Journal of Forecasting, Elsevier, vol. 37(2), pages 675-686.
    16. Xia Han & Liyuan Lin & Ruodu Wang, 2023. "Diversification quotients based on VaR and ES," Papers 2301.03517, arXiv.org, revised May 2023.
    17. Han, Xia & Lin, Liyuan & Wang, Ruodu, 2023. "Diversification quotients based on VaR and ES," Insurance: Mathematics and Economics, Elsevier, vol. 113(C), pages 185-197.
    18. Anthony Coache & Sebastian Jaimungal & 'Alvaro Cartea, 2022. "Conditionally Elicitable Dynamic Risk Measures for Deep Reinforcement Learning," Papers 2206.14666, arXiv.org, revised May 2023.
    19. Zhengkun Li & Minh-Ngoc Tran & Chao Wang & Richard Gerlach & Junbin Gao, 2020. "A Bayesian Long Short-Term Memory Model for Value at Risk and Expected Shortfall Joint Forecasting," Papers 2001.08374, arXiv.org, revised May 2021.
    20. Mohammed Berkhouch & Fernanda Maria Müller & Ghizlane Lakhnati & Marcelo Brutti Righi, 2022. "Deviation-Based Model Risk Measures," Computational Economics, Springer;Society for Computational Economics, vol. 59(2), pages 527-547, February.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jrisks:v:7:y:2019:i:2:p:52-:d:227816. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.