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
Article
  • 250 Downloads

Abstract

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 \(\mathrm{TVaR}\).

Keywords

Factor models Risk aggregation Dependence uncertainty Value-at-Risk 

Mathematics Subject Classification (2010)

97K50 60E05 60E15 62P05 

JEL Classification

C02 C10 G11 

References

  1. 1.
    Bäuerle, N., Müller, A.: Stochastic orders and risk measures: consistency and bounds. Insur. Math. Econ. 38, 132–148 (2006) MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bernard, C., Denuit, M., Vanduffel, S.: Measuring portfolio risk under partial dependence information. J. Risk Insur. (2017), forthcoming. doi: 10.1111/jori.12165 Google Scholar
  3. 3.
    Bernard, C., Jiang, X., Wang, R.: Risk aggregation with dependence uncertainty. Insur. Math. Econ. 54, 93–108 (2014) MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bernard, C., Rüschendorf, L., Vanduffel, S.: Value-at-risk bounds with variance constraints. J. Risk Insur. (2017), forthcoming. doi: 10.1111/jori.12108 Google Scholar
  5. 5.
    Bernard, C., Rüschendorf, L., Vanduffel, S., Yao, J.: How robust is the value-at-risk of credit risk portfolios? Eur. J. Finance 23, 507–534 (2017) CrossRefGoogle Scholar
  6. 6.
    Bernard, C., Vanduffel, S.: A new approach to assessing model risk in high dimensions. J. Bank. Finance 58, 166–178 (2015) CrossRefGoogle Scholar
  7. 7.
    Bignozzi, V., Puccetti, G., Rüschendorf, L.: Reducing model risk via positive and negative dependence assumptions. Insur. Math. Econ. 61, 17–26 (2015) MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Burgert, C., Rüschendorf, L.: Consistent risk measures for portfolio vectors. Insur. Math. Econ. 38, 289–297 (2006) MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Carhart, M.M.: On persistence in mutual fund performance. J. Finance 52, 57–82 (1997) CrossRefGoogle Scholar
  10. 10.
    Chamberlain, G., Rothschild, M.: Arbitrage, factor structure, and mean-variance analysis on large asset markets. Econometrica 51, 1281–1304 (1983) MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Connor, G., Korajczyk, R.A.: A test for the number of factors in an approximate factor model. J. Finance 48, 1263–1291 (1993) CrossRefGoogle Scholar
  12. 12.
    Cont, R., Deguest, R., Scandolo, G.: Robustness and sensitivity analysis of risk measurement procedures. Quant. Finance 10, 593–606 (2010) MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Daníelsson, J., Jorgensen, B., Mandira, S., Samorodnitsky, G., de Vries, C.G.: Subadditivity re-examined: the case for value-at-risk. Discussion paper, Financial Markets Group, London School of Economics and Political Science (2005). Available online at http://eprints.lse.ac.uk/24668/
  14. 14.
    Deelstra, G., Diallo, I., Vanmaele, M.: Bounds for Asian basket options. J. Comput. Appl. Math. 218, 215–228 (2008) MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Dhaene, J., Vanduffel, S., Goovaerts, M., Kaas, R., Tang, Q., Vyncke, D.: Risk measures and comonotonicity: a review. Stoch. Models 22, 573–606 (2006) MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Embrechts, P., Puccetti, G., Rüschendorf, L.: Model uncertainty and VaR aggregation. J. Bank. Finance 37, 2750–2764 (2013) CrossRefGoogle Scholar
  17. 17.
    Embrechts, P., Puccetti, R., Rüschendorf, L., Wang, R., Beleraj, A.: An academic response to Basel 3.5. Risks 2(1), 25–48 (2014) CrossRefGoogle Scholar
  18. 18.
    Embrechts, P., Wang, B., Wang, R.: Aggregation-robustness and model uncertainty of regulatory risk measures. Finance Stoch. 19, 763–790 (2015) MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Emmer, S., Kratz, M., Tasche, D.: What is the best risk measure in practice? A comparison of standard measures. J. Risk 18(2), 31–60 (2015) CrossRefGoogle Scholar
  20. 20.
    Engle, R.F., Ng, V.K., Rothschild, M.: Asset pricing with a factor-ARCH covariance structure: empirical estimates for treasury bills. J. Econom. 45, 213–237 (1990) CrossRefGoogle Scholar
  21. 21.
    Fama, E.F., French, K.R.: Common risk factors in the returns on stocks and bonds. J. Financ. Econ. 33, 3–56 (1993) CrossRefMATHGoogle Scholar
  22. 22.
    Föllmer, H., Schied, A.: Stochastic Finance. An Introduction in Discrete Time, 2nd revised and extended edn. de Gruyter, Berlin (2004) MATHGoogle Scholar
  23. 23.
    Gneiting, T.: Making and evaluating point forecasts. J. Am. Stat. Assoc. 106, 746–762 (2011) MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Gordy, M.B.: A comparative anatomy of credit risk models. J. Bank. Finance 24, 119–149 (2000) CrossRefGoogle Scholar
  25. 25.
    Gordy, M.B.: A risk-factor model foundation for ratings-based bank capital rules. J. Financ. Intermed. 12, 199–232 (2003) CrossRefGoogle Scholar
  26. 26.
    Ingersoll, J.E.: Some results in the theory of arbitrage pricing. J. Finance 39, 1021–1039 (1984) CrossRefGoogle Scholar
  27. 27.
    Jorion, P.: Value-at-Risk: The New Benchmark for Managing Financial Risk. McGraw-Hill, New York (2006) Google Scholar
  28. 28.
    Jouini, E., Schachermayer, W., Touzi, N.: Law invariant risk measures have the Fatou property. In: Kusuoka, S., Yamazaki, A. (eds.) Advances in Mathematical Economics, vol. 9, pp. 49–71. Springer, Berlin (2006) CrossRefGoogle Scholar
  29. 29.
    Kaas, R., Dhaene, J., Goovaerts, M.J.: Upper and lower bounds for sums of random variables. Insur. Math. Econ. 27, 151–168 (2000) MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Krätschmer, V., Schied, A., Zähle, H.: Qualitative and infinitesimal robustness of tail-dependent statistical functionals. J. Multivar. Anal. 103, 35–47 (2012) MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Krätschmer, V., Schied, A., Zähle, H.: Comparative and qualitative robustness for law-invariant risk measures. Finance Stoch. 18, 271–295 (2014) MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Lewbel, A.: The rank of demand systems: theory and nonparametric estimation. Econometrica 59, 711–730 (1991) CrossRefGoogle Scholar
  33. 33.
    Meilijson, I., Nadas, A.: Convex majorization with an application to the length of critical paths. J. Appl. Probab. 16, 671–677 (1979) MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Puccetti, G., Rüschendorf, L.: Computation of sharp bounds on the distribution of a function of dependent risks. J. Comput. Appl. Math. 236, 1833–1840 (2012) MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Puccetti, G., Rüschendorf, L.: Asymptotic equivalence of conservative VaR- and ES-based capital charges. J. Risk 16(3), 1–19 (2014) CrossRefGoogle Scholar
  36. 36.
    Puccetti, G., Wang, B., Wang, R.: Complete mixability and asymptotic equivalence of worst-possible VaR and ES estimates. Insur. Math. Econ. 53, 821–828 (2013) MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Ross, S.A.: The arbitrage theory of capital asset pricing. J. Econ. Theory 13, 341–360 (1976) MathSciNetCrossRefGoogle Scholar
  38. 38.
    Rüschendorf, L.: Random variables with maximum sums. Adv. Appl. Probab. 14, 623–632 (1982) MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Rüschendorf, L.: The Wasserstein distance and approximation theorems. Z. Wahrscheinlichkeitstheor. Verw. Geb. 70, 117–129 (1985) MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    Santos, A.A., Nogales, F.J., Ruiz, E.: Comparing univariate and multivariate models to forecast portfolio value-at-risk. J. Financ. Econom. 11, 400–441 (2013) CrossRefGoogle Scholar
  41. 41.
    Sharpe, W.F.: Capital asset prices: a theory of market equilibrium under conditions of risk. J. Finance 19, 425–442 (1964) Google Scholar
  42. 42.
    Vanduffel, S., Shang, Z., Henrard, L., Dhaene, J., Valdez, E.A.: Analytic bounds and approximations for annuities and Asian options. Insur. Math. Econ. 42, 1109–1117 (2008) MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    Vanmaele, M., Deelstra, G., Liinev, J., Dhaene, J., Goovaerts, M.: Bounds for the price of discrete arithmetic Asian options. J. Comput. Appl. Math. 185, 51–90 (2006) MathSciNetCrossRefMATHGoogle Scholar
  44. 44.
    Wang, B., Wang, R.: The complete mixability and convex minimization problems with monotone marginal densities. J. Multivar. Anal. 102, 1344–1360 (2011) MathSciNetCrossRefMATHGoogle Scholar
  45. 45.
    Wang, B., Wang, R.: Extreme negative dependence and risk aggregation. J. Multivar. Anal. 136, 12–25 (2015) MathSciNetCrossRefMATHGoogle Scholar
  46. 46.
    Wang, B., Wang, R.: Joint mixability. Math. Oper. Res. 41, 808–826 (2016) MathSciNetCrossRefMATHGoogle Scholar
  47. 47.
    Wang, R.: Asymptotic bounds for the distribution of the sum of dependent random variables. J. Appl. Probab. 51, 780–798 (2014) MathSciNetCrossRefMATHGoogle Scholar
  48. 48.
    Wang, R., Peng, L., Yang, J.: Bounds for the sum of dependent risks and worst value-at-risk with monotone marginal densities. Finance Stoch. 17, 395–417 (2013) MathSciNetCrossRefMATHGoogle Scholar

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

Personalised recommendations