Estimating allocations for Value-at-Risk portfolio optimization

Original Article

Abstract

Value-at-Risk, despite being adopted as the standard risk measure in finance, suffers severe objections from a practical point of view, due to a lack of convexity, and since it does not reward diversification (which is an essential feature in portfolio optimization). Furthermore, it is also known as having poor behavior in risk estimation (which has been justified to impose the use of parametric models, but which induces then model errors). The aim of this paper is to chose in favor or against the use of VaR but to add some more information to this discussion, especially from the estimation point of view. Here we propose a simple method not only to estimate the optimal allocation based on a Value-at-Risk minimization constraint, but also to derive—empirical—confidence intervals based on the fact that the underlying distribution is unknown, and can be estimated based on past observations.

Keywords

Value-at-Risk Optimization Portfolio Non-parametrics 

References

  1. Acerbi C, Tasche D (2002) On the coherence of expected shortfall. J Bank Financ 26: 1487–1503CrossRefGoogle Scholar
  2. Alexander G, Baptista A (2002) Economic implications of using a mean-VaR model for portfolio selection: a comparison with mean-variance analysis. J Econ Dyn Control 26: 1159–1193MATHCrossRefMathSciNetGoogle Scholar
  3. Artzner P, Delbaen F, Eber JM, Heath D (1999) Coherent measures of risk. Math Finance 9: 203–228MATHCrossRefMathSciNetGoogle Scholar
  4. Basak S, Shapiro A (2001) Value-at-risk management: optimzl policies and asset prices. Rev Financ Stud 14: 371–405CrossRefGoogle Scholar
  5. Bawa VS (1978) Safety-first, stochastic dominance, and optimal portfolio choice. J Financ Quant Anal 13: 255–271CrossRefGoogle Scholar
  6. Beirlant J, Goegebeur Y, Segers J, Teugel J (2006) Statistics of extremes. Wiley/Interscience, New YorkGoogle Scholar
  7. Charpentier A, Oulidi A (2007) Nonparametric quantile estimation. (submitted)Google Scholar
  8. Cohen JB, Zinbarg ED (1967) Investment analysis and portfolio management. Homewood, Ill.: Richard D. Irwin, Inc.Google Scholar
  9. Coles JL, Loewenstein U (1988) Equilibrium pricing and portfolio composition in the presence of uncertain parameters. J Financ Econ 22: 279–303CrossRefGoogle Scholar
  10. Cornish EA, Fisher RA (1937) Moments and cumulants in the specification of distributions. Rev Int Stat Inst 5: 307–320CrossRefGoogle Scholar
  11. Dowd K, Blake D (2006) After VaR : the theory, estimation, and insurance applications of quantile-based risk measures. J Risk Insur 73: 193–229CrossRefGoogle Scholar
  12. Duffie D, Pan J (1997) An overview of value at risk. J Deriv 4: 7–49CrossRefGoogle Scholar
  13. Embrechts P, Kluppelberg C, Mikosh T (1997) Modeling extremal events. Springer, BerlinGoogle Scholar
  14. Feller GW (1968) Generalized asymptotic expansions of Cornish-Fisher type. Ann Math Stat 39: 1264–1273CrossRefGoogle Scholar
  15. Fishburn PC (1970) Utility theory for decision-making. Wiley, New YorkMATHGoogle Scholar
  16. Föllmer H, Schied A (2004) Stochastic finance: an introduction in discrete time. Walter de Gruyter, New YorkMATHCrossRefGoogle Scholar
  17. Gaivoronski AA, Pflug G (2000) Value-at-Risk in portfolio optimization: properties and computational approach. Working Paper, 00-2, Norwegian University of Sciences & TechnologyGoogle Scholar
  18. Gustafsson J, Hagmann J, Nielsen JP, Scaillet O (2006) Local Transformation Kernel Density Estimation of Loss. Cahier de Recherche 2006-10, HEC GenveGoogle Scholar
  19. Harrel FE, Davis CE (1982) A new distribution free quantile estimator. Biometrika 69: 635–670CrossRefMathSciNetGoogle Scholar
  20. Hill GW, Davis AW (1968) Generalized asymptotic expansions of Cornish-Fisher type. Ann Math Stat 39: 1264–1273MATHCrossRefMathSciNetGoogle Scholar
  21. Hyndman RJ, Fan Y (1996) Sample quantiles in statistical packages. Am Stat 50: 361–365CrossRefGoogle Scholar
  22. Ingersoll Jonathan E (1987) Theory of financial decision making. Rowman & Littlefield, 496 pp. ISBN:0847673596–9780847673599Google Scholar
  23. Jorion P (1997) Value at risk : the new benchmark for controlling market risk. McGraw-Hill, New YorkGoogle Scholar
  24. Kast R, Luciano E, Peccati L (1998) VaR and optimization: 2nd international workshop on preferences and decisions. Proceedings of the conference, Trento, 1–3 July 1998Google Scholar
  25. Klein RW, Bawa VS (1976) The effect of estimation risk on optimal portfolio choice. J Financ Econ 3: 215–231CrossRefGoogle Scholar
  26. Kroll Y, Levy H, Markowitz HM (1984) Mean-variance versus direct utility maximization. J Financ 39: 47–61CrossRefGoogle Scholar
  27. Levy H, Sarnat M (1972) Safety first–an expected utility principle. J Financ Quant Anal 7: 1829–1834CrossRefGoogle Scholar
  28. Lemus G (1999) Portfolio optimization with quantile-based risk measures. Ph.D. Thesis. MITGoogle Scholar
  29. Litterman R (1997) Hot spots and edges II. Risk 10: 38–42Google Scholar
  30. Markowitz HM (1952) Portfolio selection. J Financ 7: 77–91CrossRefGoogle Scholar
  31. Merton RC (1992) Continuous time finance 2nd edn. Blackwell Publishers, OxfordGoogle Scholar
  32. Padgett WJ (1986) A kernel-type estimator of a quantile function from right-censored data. J Am Stat Assoc 81: 215–222MATHCrossRefMathSciNetGoogle Scholar
  33. Park C (2006) Smooth nonparametric estimation of a quantile function under right censoring using beta kernels. Technical Report (TR 2006-01-CP), Departement of Mathematical Sciences, Clemson UniversityGoogle Scholar
  34. Quiggin J (1993) Generalized expected utility theory: the rank-dependent expected utility model. Kluwer/Nijhoff, Dordrecht/The HagueGoogle Scholar
  35. Rockafellar RT, Uryasev S (2000) Optimization of conditional value-at-risk. J Risk 2: 21–41Google Scholar
  36. Roy AD (1952) Safety first and the holding of assets. Econometrica 20: 431–449MATHCrossRefGoogle Scholar
  37. Waart AW van der (1998) Asymptotic statistics. Cambridge University Press, LondonGoogle Scholar
  38. von Neumann J, Morgenstern O (1947) Theory of games and economic behaviour. Princeton University Press, PrincetonGoogle Scholar
  39. Wang S (1996) Premium calculation by transforming the layer premium density. ASTIN Bull 26: 71–92CrossRefGoogle Scholar
  40. Wirch J, Hardy M (1999) A synthesis of risk measures for capital adequacy. Insur Math Econ 25: 337–348MATHCrossRefGoogle Scholar
  41. Yaari M (1987) A dual theory of choice under risk. Econometrica 55: 95–115MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.CREM-Université Rennes 1Rennes cedexFrance
  2. 2.UCO/IMAAngersFrance

Personalised recommendations