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Risk bounds for factor models

Author

Listed:
  • Carole Bernard

    (Grenoble Ecole de Management
    Vrije Universiteit Brussel)

  • Ludger Rüschendorf

    (University of Freiburg)

  • Steven Vanduffel

    (Vrije Universiteit Brussel)

  • Ruodu Wang

    (University of Waterloo)

Abstract

Recent literature has investigated the risk aggregation of a portfolio X = ( X i ) 1 ≤ i ≤ n $X=(X_{i})_{1\leq i\leq n}$ under the sole assumption that the marginal distributions of the risks X i $X_{i} $ are specified, but not their dependence structure. There exists a range of possible values for any risk measure of S = ∑ i = 1 n X i $S=\sum_{i=1}^{n}X_{i}$ , and the dependence uncertainty spread, as measured by the difference between the upper and the lower bound on these values, is typically very wide. Obtaining bounds that are more practically useful requires additional information on dependence. Here, we study a partially specified factor model in which each risk X i $X_{i}$ has a known joint distribution with the common risk factor Z $Z$ , but we dispense with the conditional independence assumption that is typically made in fully specified factor models. We derive easy-to-compute bounds on risk measures such as Value-at-Risk ( VaR $\mathrm{VaR}$ ) and law-invariant convex risk measures (e.g. Tail Value-at-Risk ( TVaR $\mathrm{TVaR}$ )) and demonstrate their asymptotic sharpness. We show that the dependence uncertainty spread is typically reduced substantially and that, contrary to the case in which only marginal information is used, it is not necessarily larger for VaR $\mathrm{VaR}$ than for TVaR $\mathrm{TVaR}$ .

Suggested Citation

  • Carole Bernard & Ludger Rüschendorf & Steven Vanduffel & Ruodu Wang, 2017. "Risk bounds for factor models," Finance and Stochastics, Springer, vol. 21(3), pages 631-659, July.
  • Handle: RePEc:spr:finsto:v:21:y:2017:i:3:d:10.1007_s00780-017-0328-4
    DOI: 10.1007/s00780-017-0328-4
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    1. Di Lascio, F. Marta L. & Giammusso, Davide & Puccetti, Giovanni, 2018. "A clustering approach and a rule of thumb for risk aggregation," Journal of Banking & Finance, Elsevier, vol. 96(C), pages 236-248.
    2. Rüschendorf, L., 2019. "Analysis of risk bounds in partially specified additive factor models," Insurance: Mathematics and Economics, Elsevier, vol. 86(C), pages 115-121.
    3. Ansari Jonathan & Rüschendorf Ludger, 2021. "Sklar’s theorem, copula products, and ordering results in factor models," Dependence Modeling, De Gruyter, vol. 9(1), pages 267-306, January.
    4. Edgars Jakobsons & Steven Vanduffel, 2015. "Dependence Uncertainty Bounds for the Expectile of a Portfolio," Risks, MDPI, vol. 3(4), pages 1-25, December.
    5. Ruodu Wang & Yunran Wei & Gordon E. Willmot, 2020. "Characterization, Robustness, and Aggregation of Signed Choquet Integrals," Mathematics of Operations Research, INFORMS, vol. 45(3), pages 993-1015, August.
    6. Stephan Eckstein & Michael Kupper & Mathias Pohl, 2020. "Robust risk aggregation with neural networks," Mathematical Finance, Wiley Blackwell, vol. 30(4), pages 1229-1272, October.
    7. Thibaut Lux & Antonis Papapantoleon, 2016. "Model-free bounds on Value-at-Risk using extreme value information and statistical distances," Papers 1610.09734, arXiv.org, revised Nov 2018.
    8. Stephan Eckstein & Michael Kupper & Mathias Pohl, 2018. "Robust risk aggregation with neural networks," Papers 1811.00304, arXiv.org, revised May 2020.
    9. Ansari, Jonathan & Rüschendorf, Ludger, 2021. "Ordering results for elliptical distributions with applications to risk bounds," Journal of Multivariate Analysis, Elsevier, vol. 182(C).
    10. Bernard, Carole & Kazzi, Rodrigue & Vanduffel, Steven, 2020. "Range Value-at-Risk bounds for unimodal distributions under partial information," Insurance: Mathematics and Economics, Elsevier, vol. 94(C), pages 9-24.
    11. Lux, Thibaut & Papapantoleon, Antonis, 2019. "Model-free bounds on Value-at-Risk using extreme value information and statistical distances," Insurance: Mathematics and Economics, Elsevier, vol. 86(C), pages 73-83.
    12. Cornilly, Dries & Vanduffel, Steven, 2019. "Equivalent distortion risk measures on moment spaces," Statistics & Probability Letters, Elsevier, vol. 146(C), pages 187-192.
    13. Tuitman, Jan & Vanduffel, Steven & Yao, Jing, 2020. "Correlation matrices with average constraints," Statistics & Probability Letters, Elsevier, vol. 165(C).
    14. Cornilly, D. & Rüschendorf, L. & Vanduffel, S., 2018. "Upper bounds for strictly concave distortion risk measures on moment spaces," Insurance: Mathematics and Economics, Elsevier, vol. 82(C), pages 141-151.
    15. Yuyu Chen & Liyuan Lin & Ruodu Wang, 2021. "Risk Aggregation under Dependence Uncertainty and an Order Constraint," Papers 2104.07718, arXiv.org, revised Oct 2021.

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    More about this item

    Keywords

    Factor models; Risk aggregation; Dependence uncertainty; Value-at-Risk;
    All these keywords.

    JEL classification:

    • C02 - Mathematical and Quantitative Methods - - General - - - Mathematical Economics
    • C10 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - General
    • G11 - Financial Economics - - General Financial Markets - - - Portfolio Choice; Investment Decisions

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