OR Spectrum

, Volume 32, Issue 1, pp 49–60 | Cite as

On the non-existence of conditional value-at-risk under heavy tails and short sales

  • Günter BambergEmail author
  • Andreas Neuhierl
Regular Article


Value-at-Risk (VaR) and conditional value-at-risk (CVaR) are important risk measures. Especially VaR is very popular and widespread in risk management and banking supervision. However, VaR has some unwelcome properties which are not shared by CVaR. Therefore CVaR is preferable from a theoretical point of view. Both VaR and CVaR are discussed for long and short positions. It is pointed out that short positions and heavy tails are incompatible with a finite CVaR.


Conditional value-at-risk Value-at-risk Heavy tails 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Aas K and Haff IH (2006). The generalized hyperbolic skew student’s t-distribution. J Financ Econ 4: 275–309 Google Scholar
  2. Acerbi C and Tasche D (2002a). Expected shortfall: a natural coherent alternative to value at risk. Review of banking, Finance Monetary Econ 31: 379–388 Google Scholar
  3. Acerbi C and Tasche D (2002b). On the coherence of expected shortfall. J Finance 26: 1487–1503 Google Scholar
  4. Adler RJ, Feldman RE, Taqqu MS (eds) (1998). A practical guide to heavy tails: statistical techniques and applications. Birkhäuser, Basel Google Scholar
  5. Akgiray V and Booth GG (1988). The stable-law model of stock returns. J Bus Econ Stat 6: 51–57 CrossRefGoogle Scholar
  6. Akgiray V, Booth GG and Loistl O (1989). Statistical models of German stock returns. J Econ 50: 17–33 Google Scholar
  7. Albrecht P and Maurer R (2005). Investment- und riskomanagement, 2nd edn. Schäfer-Poeschel, StuggartGoogle Scholar
  8. Artzner P, Delbaen F, Eber JM and Heath D (1999). Coherent measures of risk. Math Finance 9: 203–228 CrossRefGoogle Scholar
  9. Asai M (2008). Autoregressive stochastic volatility models with heavy-tailed distributions: a comparison with multifactor volatility models. J Empir Finance 15: 332–341 CrossRefGoogle Scholar
  10. Asmussen S (2003). Applied probability and queues, 2nd edn. Wiley, New York Google Scholar
  11. Bamberg G and Dorfleitner G (2002). Is traditional capital market theory consistent with fat-tailed log returns?. Zeitschrift Betriebswirtschaft 72: 860–873 Google Scholar
  12. Basle Committee on Banking Supervision (1995a) An internal model-based approach to market risk capital requirements. Tech. rep., Basle Committee on Banking Supervision, BasleGoogle Scholar
  13. Basle Committee on Banking Supervision (1995b) Planned supplement to the capital accord to incorporate market risks. Tech. rep., Basle Committee on Banking Supervision, Basle, SwitzerlandGoogle Scholar
  14. Bawa VS, Elton EJ and Gruber MJ (1979). Simple rules for optimal portfolio selection in stable paretian markets. J Finance 34: 1041–1047 CrossRefGoogle Scholar
  15. Beeck H, Johannig L and Rudolph B (1999). Value-at-risk-Limitstrukturen zur Steuerung und Begrenzung von Marktrisiken im Aktienbereich. OR Spectrum 21: 259–286 CrossRefGoogle Scholar
  16. Bertsimas D, Lauprete GJ and Samarov A (2004). Shortfall as a risk measure: properties, optimization, and applications. J Econ Dyn Control 28: 1353–1381 CrossRefGoogle Scholar
  17. Blattberg RC and Gonedes NJ (1974). A comparison of the stable and student distributions as statistical models for stock prices. J Bus 47: 244–280 CrossRefGoogle Scholar
  18. Bollerslev T (1987). A conditionally heteroskedastic time series model for speculative prices and rates of return. Rev Econ Stat 69: 542–547 CrossRefGoogle Scholar
  19. Brachinger HW and Weber M (1997). Risk as a primitive: a survey of measures of perceived risk. OR Spectr 19: 235–250 Google Scholar
  20. Bryson MC (1982) Heavy-tailed distributions. In: Johnson NL, Kotz S, Read CB (eds) Encyclopedia of statistical sciences, vol 3, Wiley, New York, pp 598–601Google Scholar
  21. Dorfleitner G (2002). Stetige versus diskrete Renditen. Überlegungen zur richtigen Verwendung beider Begriffe in Theorie und Praxis. Kredit Kapital Heft 2: 216–241 Google Scholar
  22. Dorfleitner G (2003). Why the return notion matters. Int J Theor Appl Finance 6: 73–86 CrossRefGoogle Scholar
  23. Fama EF (1965a). Portfolio analysis in a stable paretian market. Manage Sci 11: 404–419 CrossRefGoogle Scholar
  24. Föllmer H and Leukert P (2000). Efficient hedging: cost versus shortfall risk. Finance Stochastics 4: 117–146 CrossRefGoogle Scholar
  25. Fortin I and Kuzmics C (2002). Tail-dependence in stock-return pairs. Int J Intell Syst Account Finance Manage 11: 89–107 CrossRefGoogle Scholar
  26. Giacomini R, Gottschling A, Haefke C, White H (2008) Mixtures of t-distributions for finance and forecasting. J Financial Econom (forthcoming)Google Scholar
  27. Goldie CM, Klüppelberg C (1998) Subexponential distributions. In: Adler RJ, Feldman RE, Taqqu MS (eds) A practical guide to heavy tails: statistical techniques and applications, Birkhäuser, Boston, pp 435–459Google Scholar
  28. Guthoff A, Pfingsten A, Wolf J (1998) Der Einfluss einer Begrenzung des Value at Risk oder des Lower Partial Moment One auf die Risikoübernahme. In: Oehler A (ed) Credit risk and value-at-risk alternativen, Schäffer-Poeschel, Stuttgart, pp 111–153Google Scholar
  29. Huschens S (1999) Anmerkungen zur value-at-risk definition. In: Bol G, Nakhaeizadeh G, Vollmer KH (eds) Datamining and computational finance. Physica, Heidelberg, pp 29–41Google Scholar
  30. Ibragimov IA, Linnik YV (1971) Independent and stationary sequences of random variables. Wolters-Noordhoff, GroningenGoogle Scholar
  31. Jorion P (2001). Value at risk: the new benchmark for managing financial risk, 2nd edn. McGraw-Hill, New York Google Scholar
  32. Lau AHL, Lau HS and Wingender JR (1990). The distribution of stock returns: new evidence against the stable model. J Bus Econ Stat 8: 217–23 CrossRefGoogle Scholar
  33. Lux T (1996). The stable Paretian hypothesis and the frequency of large returns: an examination of major German stocks. Appl Financial Econ 6: 463–475 CrossRefGoogle Scholar
  34. Mandelbrot B (1963). The variation of certain speculative prices. J Bus 36: 394–419 CrossRefGoogle Scholar
  35. Mandelbrot B and Taylor HM (1967). On the distribution of stock price differences. Oper Res 6: 1057–1062 CrossRefGoogle Scholar
  36. Oehler A and Unser M (2002). Finanzwirtschaftliches Risikomanagement, 2nd edn. Springer, Heidelberg Google Scholar
  37. Samuelson PA (1967). Efficient Portfolio Selection for Pareto–Levy investments. J Financial Quant Anal 2: 102–122 Google Scholar
  38. Schulmerich M and Trautmann S (2003). Local expected shortfall–hedging in discrete time. Euro Finance Rev 7: 75–102 CrossRefGoogle Scholar
  39. Shao QM, Yu H and Yu J (2001). Do stock returns follow a finite variance distribution?. Ann Econ Finance 2: 467–468 Google Scholar
  40. Szegö G (2005). Measures of risk. Euro J Opera Res 163: 5–9 CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Institut für Statistik und Mathematische WirtschaftstheorieUniversität AugsburgAugsburgGermany
  2. 2.Institut für Statistik und Mathematische Wirtschaftstheorie und Atacama Capital GmbHMunichGermany

Personalised recommendations