Finance and Stochastics

, Volume 21, Issue 3, pp 631–659 | Cite as

Risk bounds for factor models

  • Carole Bernard
  • Ludger Rüschendorf
  • Steven Vanduffel
  • Ruodu Wang


Recent literature has investigated the risk aggregation of a portfolio \(X=(X_{i})_{1\leq i\leq n}\) under the sole assumption that the marginal distributions of the risks \(X_{i} \) are specified, but not their dependence structure. There exists a range of possible values for any risk measure of \(S=\sum_{i=1}^{n}X_{i}\), 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 \(X_{i}\) has a known joint distribution with the common risk factor \(Z\), 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 (\(\mathrm{VaR}\)) and law-invariant convex risk measures (e.g. Tail Value-at-Risk (\(\mathrm{TVaR}\))) 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 \(\mathrm{VaR}\) than for \(\mathrm{TVaR}\).


Factor models Risk aggregation Dependence uncertainty Value-at-Risk 

Mathematics Subject Classification (2010)

97K50 60E05 60E15 62P05 

JEL Classification

C02 C10 G11 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  • Carole Bernard
    • 1
    • 2
  • Ludger Rüschendorf
    • 3
  • Steven Vanduffel
    • 4
  • Ruodu Wang
    • 5
  1. 1.Grenoble Ecole de ManagementGrenobleFrance
  2. 2.Vrije Universiteit BrusselElseneBelgium
  3. 3.University of FreiburgFreiburgGermany
  4. 4.Vrije Universiteit BrusselBruxellesBelgium
  5. 5.University of WaterlooWaterlooCanada

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