IDEAS home Printed from https://ideas.repec.org/p/bie/wpaper/474.html
   My bibliography  Save this paper

The Foster-Hart measure of riskiness for general gambles

Author

Listed:
  • Riedel, Frank

    (Center for Mathematical Economics, Bielefeld University)

  • Hellmann, Tobias

    (Center for Mathematical Economics, Bielefeld University)

Abstract

Foster and Hart proposed an operational measure of riskiness for discrete random variables. We show that their defining equation has no solution for many common continuous distributions including many uniform distributions, e.g. We show how to extend consistently the definition of riskiness to continuous random variables. For many continuous random variables, the risk measure is equal to the worst-case risk measure, i.e. the maximal possible loss incurred by that gamble. We also extend the Foster-Hart risk measure to dynamic environments for general distributions and probability spaces, and we show that the extended measure avoids bankruptcy in infinitely repeated gambles.

Suggested Citation

  • Riedel, Frank & Hellmann, Tobias, 2014. "The Foster-Hart measure of riskiness for general gambles," Center for Mathematical Economics Working Papers 474, Center for Mathematical Economics, Bielefeld University.
    • Handle: RePEc:bie:wpaper:474
    as

    Download full text from publisher

    File URL: https://pub.uni-bielefeld.de/download/2674036/2674038
    File Function: First Version, 2013
    Download Restriction: no
    ---><---

    Other versions of this item:

    References listed on IDEAS

    as
    1. Dean P. Foster & Sergiu Hart, 2009. "An Operational Measure of Riskiness," Journal of Political Economy, University of Chicago Press, vol. 117(5), pages 785-814.
    2. Robert J. Aumann & Roberto Serrano, 2008. "An Economic Index of Riskiness," Journal of Political Economy, University of Chicago Press, vol. 116(5), pages 810-836, October.
    3. Hellmann, Tobias & Riedel, Frank, 2015. "A dynamic extension of the Foster–Hart measure of riskiness," Journal of Mathematical Economics, Elsevier, vol. 59(C), pages 66-70.
    4. Sergiu Hart, 2011. "Comparing Risks by Acceptance and Rejection," Journal of Political Economy, University of Chicago Press, vol. 119(4), pages 617-638.
    5. Turan G. Bali & Nusret Cakici & Fousseni Chabi-Yo, 2011. "A Generalized Measure of Riskiness," Management Science, INFORMS, vol. 57(8), pages 1406-1423, August.
    6. Amnon Schreiber, 2012. "An Economic Index of Relative Riskiness," Discussion Paper Series dp597, The Federmann Center for the Study of Rationality, the Hebrew University, Jerusalem.
    7. 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.
    8. Detlefsen, Kai & Scandolo, Giacomo, 2005. "Conditional and dynamic convex risk measures," SFB 649 Discussion Papers 2005-006, Humboldt University Berlin, Collaborative Research Center 649: Economic Risk.
    9. , P. & ,, 2013. "A wealth-requirement axiomatization of riskiness," Theoretical Economics, Econometric Society, vol. 8(2), May.
    10. Kai Detlefsen & Giacomo Scandolo, 2005. "Conditional and dynamic convex risk measures," Finance and Stochastics, Springer, vol. 9(4), pages 539-561, October.
    11. Homm, Ulrich & Pigorsch, Christian, 2012. "An operational interpretation and existence of the Aumann–Serrano index of riskiness," Economics Letters, Elsevier, vol. 114(3), pages 265-267.
    12. Hans Föllmer & Alexander Schied, 2002. "Convex measures of risk and trading constraints," Finance and Stochastics, Springer, vol. 6(4), pages 429-447.
    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. Niu, Cuizhen & Guo, Xu & McAleer, Michael & Wong, Wing-Keung, 2018. "Theory and application of an economic performance measure of risk," International Review of Economics & Finance, Elsevier, vol. 56(C), pages 383-396.
    2. Jiro Hodoshima & Toshiyuki Yamawake, 2021. "Sensitivity of Performance Indexes to Disaster Risk," Risks, MDPI, vol. 9(2), pages 1-22, February.
    3. Heller, Yuval & Schreiber, Amnon, 2020. "Short-term investments and indices of risk," Theoretical Economics, Econometric Society, vol. 15(3), July.
    4. Leiss, Matthias & Nax, Heinrich H., 2018. "Option-implied objective measures of market risk," Journal of Banking & Finance, Elsevier, vol. 88(C), pages 241-249.
    5. Hellmann, Tobias & Riedel, Frank, 2015. "A dynamic extension of the Foster–Hart measure of riskiness," Journal of Mathematical Economics, Elsevier, vol. 59(C), pages 66-70.
    6. Ehsani, Sina & Lien, Donald, 2015. "A note on minimum riskiness hedge ratio," Finance Research Letters, Elsevier, vol. 15(C), pages 11-17.
    7. Usategui, José M., 2017. "Riskiness in binary gambles: A geometric analysis," Economics Letters, Elsevier, vol. 159(C), pages 149-152.
    8. Chamorro Elosua, Arritokieta & Usategui Díaz de Otalora, José María, 2013. "A Note on Risk Acceptance, Bankruptcy Avoidance and Riskiness Measures," DFAEII Working Papers 1988-088X, University of the Basque Country - Department of Foundations of Economic Analysis II.
    9. Cui, Zhenyu & Wu, Cai & Zhu, Lingjiong, 2023. "Explicit solution to the economic index of riskiness," Economics Letters, Elsevier, vol. 232(C).
    10. Marilyn Pease & Mark Whitmeyer, 2024. "How to Make an Action Better," Papers 2408.09294, arXiv.org, revised Nov 2024.
    11. Jiro Hodoshima & Tetsuya Misawa & Yoshio Miyahara, 2020. "Stock Performance Evaluation Incorporating High Moments and Disaster Risk: Evidence from Japan," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 27(2), pages 155-174, June.
    12. Soo Hong Chew & Jacob S. Sagi, 2022. "A critical look at the Aumann-Serrano and Foster-Hart measures of riskiness," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 74(2), pages 397-422, September.
    13. Hodoshima, Jiro & Yamawake, Toshiyuki, 2022. "Temporal aggregation of the Aumann–Serrano and Foster–Hart performance indexes," International Review of Financial Analysis, Elsevier, vol. 83(C).
    14. Jiro Hodoshima & Toshiyuki Yamawake, 2022. "Comparing Dynamic and Static Performance Indexes in the Stock Market: Evidence From Japan," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 29(2), pages 171-193, June.
    15. Abhinav Anand & Tiantian Li & Tetsuo Kurosaki & Young Shin Kim, 2017. "The equity risk posed by the too-big-to-fail banks: a Foster–Hart estimation," Annals of Operations Research, Springer, vol. 253(1), pages 21-41, June.
    16. Anand, Abhinav & Li, Tiantian & Kurosaki, Tetsuo & Kim, Young Shin, 2016. "Foster–Hart optimal portfolios," Journal of Banking & Finance, Elsevier, vol. 68(C), pages 117-130.
    17. Yuval Heller & Amnon Schreiber, 2020. "Short-Term Investments and Indices of Risk," Papers 2005.06576, arXiv.org.

    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. Steven Kou & Xianhua Peng, 2014. "On the Measurement of Economic Tail Risk," Papers 1401.4787, arXiv.org, revised Aug 2015.
    2. Heller, Yuval & Schreiber, Amnon, 2020. "Short-term investments and indices of risk," Theoretical Economics, Econometric Society, vol. 15(3), July.
    3. Steven Kou & Xianhua Peng, 2016. "On the Measurement of Economic Tail Risk," Operations Research, INFORMS, vol. 64(5), pages 1056-1072, October.
    4. Klaas Schulze, 2015. "General dual measures of riskiness," Theory and Decision, Springer, vol. 78(2), pages 289-304, February.
    5. Yuval Heller & Amnon Schreiber, 2020. "Short-Term Investments and Indices of Risk," Papers 2005.06576, arXiv.org.
    6. Moti Michaeli, 2014. "Riskiness for sets of gambles," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 56(3), pages 515-547, August.
    7. Tomer Siedner, 2015. "Risk of Monetary Gambles: An Axiomatic Approach," Discussion Paper Series dp682, The Federmann Center for the Study of Rationality, the Hebrew University, Jerusalem.
    8. Leiss, Matthias & Nax, Heinrich H., 2018. "Option-implied objective measures of market risk," Journal of Banking & Finance, Elsevier, vol. 88(C), pages 241-249.
    9. Kadan, Ohad & Liu, Fang, 2014. "Performance evaluation with high moments and disaster risk," Journal of Financial Economics, Elsevier, vol. 113(1), pages 131-155.
    10. Ji, Ronglin & Shi, Xuejun & Wang, Shijie & Zhou, Jinming, 2019. "Dynamic risk measures for processes via backward stochastic differential equations," Insurance: Mathematics and Economics, Elsevier, vol. 86(C), pages 43-50.
    11. Andreas H Hamel, 2018. "Monetary Measures of Risk," Papers 1812.04354, arXiv.org.
    12. Chamorro Elosua, Arritokieta & Usategui Díaz de Otalora, José María, 2013. "A Note on Risk Acceptance, Bankruptcy Avoidance and Riskiness Measures," DFAEII Working Papers 1988-088X, University of the Basque Country - Department of Foundations of Economic Analysis II.
    13. Zachary Feinstein & Birgit Rudloff, 2018. "Scalar multivariate risk measures with a single eligible asset," Papers 1807.10694, arXiv.org, revised Feb 2021.
    14. Rosazza Gianin, Emanuela, 2006. "Risk measures via g-expectations," Insurance: Mathematics and Economics, Elsevier, vol. 39(1), pages 19-34, August.
    15. Claudia Kluppelberg & Jianing Zhang, 2015. "Time-consistency of risk measures with GARCH volatilities and their estimation," Papers 1504.04774, arXiv.org, revised Feb 2016.
    16. Daniel Lacker, 2015. "Law invariant risk measures and information divergences," Papers 1510.07030, arXiv.org, revised Jun 2016.
    17. Chen, Yi-Ting & Ho, Keng-Yu & Tzeng, Larry Y., 2014. "Riskiness-minimizing spot-futures hedge ratio," Journal of Banking & Finance, Elsevier, vol. 40(C), pages 154-164.
    18. Fei Sun & Jingchao Li & Jieming Zhou, 2018. "Dynamic risk measures with fluctuation of market volatility under Bochne-Lebesgue space," Papers 1806.01166, arXiv.org, revised Mar 2024.
    19. Wing Fung Chong & Ying Hu & Gechun Liang & Thaleia Zariphopoulou, 2019. "An ergodic BSDE approach to forward entropic risk measures: representation and large-maturity behavior," Finance and Stochastics, Springer, vol. 23(1), pages 239-273, January.
    20. Ehsani, Sina & Lien, Donald, 2015. "A note on minimum riskiness hedge ratio," Finance Research Letters, Elsevier, vol. 15(C), pages 11-17.

    More about this item

    Keywords

    Operational; Risk Measures; Bankruptcy; Continuous Random Variable;
    All these keywords.

    JEL classification:

    • D81 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Criteria for Decision-Making under Risk and Uncertainty
    • G11 - Financial Economics - - General Financial Markets - - - Portfolio Choice; Investment Decisions

    NEP fields

    This paper has been announced in the following NEP Reports:

    Statistics

    Access and download statistics

    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:bie:wpaper:474. 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: Bettina Weingarten (email available below). General contact details of provider: https://edirc.repec.org/data/imbiede.html .

    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.