Advertisement

Mathematics and Financial Economics

, Volume 11, Issue 3, pp 275–297 | Cite as

Drawdown: from practice to theory and back again

  • Lisa R. Goldberg
  • Ola Mahmoud
Article

Abstract

Maximum drawdown, the largest cumulative loss from peak to trough, is one of the most widely used indicators of risk in the fund management industry, but one of the least developed in the context of measures of risk. We formalize drawdown risk as Conditional Expected Drawdown (CED), which is the tail mean of maximum drawdown distributions. We show that CED is a degree one positive homogenous risk measure, so that it can be linearly attributed to factors; and convex, so that it can be used in quantitative optimization. We empirically explore the differences in risk attributions based on CED, Expected Shortfall (ES) and volatility. An important feature of CED is its sensitivity to serial correlation. In an empirical study that fits AR(1) models to US Equity and US Bonds, we find substantially higher correlation between the autoregressive parameter and CED than with ES or with volatility.

Keywords

Drawdown Conditional expected drawdown Deviation measure Risk attribution Serial correlation 

References

  1. 1.
    Acerbi, C., Tasche, D.: Expected shortfall: a natural coherent alternative to value at risk. Econ. Notes 31(2), 379–388 (2002a)CrossRefGoogle Scholar
  2. 2.
    Acerbi, C., Tasche, D.: On the coherence of expected shortfall. J. Bank. Financ. 26(7), 1487–1503 (2002b)CrossRefGoogle Scholar
  3. 3.
    Anderson, R.M., Bianchi, S.W., Goldberg, L.R.: Will my risk parity strategy outperform? Financ. Anal. J. 68(6), 75–93 (2012)CrossRefGoogle Scholar
  4. 4.
    Anderson, R,M., Bianchi, S.W., Goldberg, L.R.: Determinants of levered portfolio performance. Forthcom. Financ. Anal. J. (2014)Google Scholar
  5. 5.
    Bertsimas, D., Lauprete, G.J., Samarov, A.: Shortfall as a risk measure: properties, optimization and applications. J. Econ. Dyn. Control 28, 1353–1381 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Burghardt, G., Duncan, R., Liu, L.: Deciphering drawdown. Risk Mag. 16–20 (2003)Google Scholar
  7. 7.
    Carr, P., Zhang, H., Hadjiliadis, O.: Maximum drawdown insurance. Int. J. Theor. Appl. Financ. 14, 1195–1230 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Chekhlov, A., Uryasev, S., Zabarankin, M.: Portfolio optimization with drawdown constraints. In: Scherer, B. (ed.) Asset and Liability Management Tools, pp. 263–278. Risk Books, London (2003)Google Scholar
  9. 9.
    Chekhlov, A., Uryasev, S., Zabarankin, M.: Drawdown measure in portfolio optimization. Int. J. Theor. Appl. Financ. 8(1), 13–58 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Cheridito, P., Delbaen, F., Kupper, M.: Coherent and convex monetary risk measures for bounded càdlàg processes. Stoch. Process. Appl. 112, 1–22 (2004)CrossRefzbMATHGoogle Scholar
  11. 11.
    Cherney, V., Obloj, J.: Portfolio optimization under non-linear drawdown constraints in a semimartingale financial model. Financ. Stochastics 17, 771–800 (2013)CrossRefzbMATHGoogle Scholar
  12. 12.
    Cvitanic, J., Karatzas, I.: On portfolio optimization under drawdown constraints. IMA Lect. Notes Math. Appl. 65, 77–88 (1995)zbMATHGoogle Scholar
  13. 13.
    Denault, M.: Coherent allocation of risk capital. J. Risk 4(1), 7–21 (2001)CrossRefGoogle Scholar
  14. 14.
    Douady, R., Shiryaev, A.N., Yor, M.: On probability characteristics of downfalls in a standard brownian motion. Theory Probab. Appl. 44, 29–38 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Föllmer, H., Schied, A.: Convex measures of risk and trading constraints. Financ. Stoch. 6, 429–447 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Föllmer, H., Schied, A.: Coherent and convex risk measures. In: Cont, R. (ed) Encyclopedia of Quantitative Finance, pp. 355–363. Wiley, New York (2010)Google Scholar
  17. 17.
    Föllmer, H., Schied, A.: Stochastic Finance: An Introduction in Discrete Time. Walter de Gruyter, Berlin (2011)CrossRefzbMATHGoogle Scholar
  18. 18.
    Goldberg, L.R., Menchero, J., Hayes, M., Mitra, I.: Extreme risk analysis. J. Perform. Meas. 14(3), 17–30 (2010)Google Scholar
  19. 19.
    Grossman, S.J., Zhou, Z.: Optimal investment strategies for controlling drawdowns. Math. Financ. 3, 241–276 (1993)CrossRefzbMATHGoogle Scholar
  20. 20.
    Hadjiliadis, O., Vecer, J.: Drawdowns preceding rallies in the brownian motion model. Quant Financ. 6, 403–409 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Kalkbrener, M.: An axiomatic approach to capital allocation. Math. Financ. 15(3), 425–437 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Krokhmal, P., Uryasev, S., Zrazhevsky, G.: Numerical comparison of cvar and cdar approaches: application to hedge funds. In: Ziemba, W.T. (ed.) The stochastic programming approach to asset liability and wealth management. AIMR/Blackwell Publisher, Oxford (2003)Google Scholar
  23. 23.
    Landriault, D., Li, B., Zhang, H.: On the frequency of drawdowns for brownian motion processes. J. Appl. Probab. 52(1), 191–208 (2015)Google Scholar
  24. 24.
    Lehoczky, J.P.: Formulas for stopped diffusion processes with stopping times based on the maximum. Ann. Probab. 5, 601–607 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Madhavan, A.: Exchange-traded funds, market structure, and the flash crash. Financ. Anal. J. 68(4), 20–35 (2012)Google Scholar
  26. 26.
    Magdon-Ismail, M., Atiya, A.F., Pratap, A., Abu-Mostafa, Y.: On the maximum drawdown of a brownian motion. J. Appl. Probab. 41, 147–161 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    McNeil, A.J., Frey, R., Embrechts, P.: Quantitative Risk Management: Concepts, Techniques, and Tools. Princeton University Press, Princeton (2005)Google Scholar
  28. 28.
    Menchero, J., Poduri, V.: Custom factor attribution. Financ. Anal. J. 64(2), 81–92 (2008)CrossRefGoogle Scholar
  29. 29.
    Mijatovic, A., Pistorius, M.R.: On the drawdown of completely asymmetric lévy processes. Stoch. Process. Appl. 122, 3812–3836 (2012)CrossRefzbMATHGoogle Scholar
  30. 30.
    Pospisil, L., Vecer, J.: Portfolio sensitivity to changes in the maximum and the maximum drawdown. Quant. Financ. 10, 617–627 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Pospisil, L., Vecer, J., Hadjiliadis, O.: Formulas for stopped diffusion processes with stopping times based on drawdowns and drawups. Stoch. Process. Appl. 119, 2563–2578 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Qian, E.: On the financial interpretation of risk contributions: risk budgets do add up. J. Invest. Manag. 4(4), 41–51 (2006)Google Scholar
  33. 33.
    Rockafellar, R.T., Uryasev, S.: Optimization of conditional value-at-risk. J. Risk 2, 21–41 (2000)CrossRefGoogle Scholar
  34. 34.
    Rockafellar, R.T., Uryasev, S.: Conditional value-at-risk for general loss distributions. J. Bank. Financ. 26, 1443–1471 (2002)CrossRefGoogle Scholar
  35. 35.
    Rockafellar, R.T., Uryasev, S.P., Zabarankin, M.: Deviation measures in risk analysis and optimization. Technical report, University of Florida, Department of Industrial and Systems Engineering (2002)Google Scholar
  36. 36.
    Rockafellar, R.T., Uryasev, S.P., Zabarankin, M.: Generalized deviations in risk analysis. Financ. Stoch. 10(1), 51–74 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Sekine, J.: Long-term optimal investment with a generalized drawdown constraint. SIAM J. Financ. Math. 4, 457–473 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Tasche, D.: Risk contributions and performance measurement. Technical report, Research paper, Zentrum Mathematik (SCA) (2000)Google Scholar
  39. 39.
    Tasche, D.: Expected shortfall and beyond. J. Bank. Financ. 26, 1519–1533 (2002)CrossRefzbMATHGoogle Scholar
  40. 40.
    Taylor, H.M.: A stopped brownian motion formula. Ann. Probab. 3, 234–246 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Zabarankin, M., Pavlikov, K., Uryasev, S.: Capital asset pricing model (capm) with drawdown measure. Eur. J. Oper. Res. 234, 508–517 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Zabarankin, M., Uryasev, S.: Statistical Decision Problems: Selected Concepts and Portfolio Safeguard Case Studies, vol. 85. Springer, Berlin (2014)Google Scholar
  43. 43.
    Zhang, H.: Occupation time, drawdowns, and drawups for one-dimensional regular diffusion. Adv. Appl. Probab., 47(1), 210–230 (2015)Google Scholar
  44. 44.
    Zhang, H., Hadjiliadis, O.: Drawdowns and rallies in a finite time-horizon. Methodol. Comput. Appl. Probab. 12, 293–308 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Zhang, H., Leung, T., Hadjiliadis, O.: Stochastic modeling and fair valuation of drawdown insurance. Insur. Math. Econ. 53, 840–850 (2013)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of Statistics and Economics, Center for Risk Management Research, and Consortium for Data Analytics in RiskUniversity of CaliforniaBerkeleyUSA
  2. 2.Faculty of Mathematics and StatisticsUniversity of St. GallenSt. GallenSwitzerland
  3. 3.Center for Risk Management ResearchUniversity of CaliforniaBerkeleyUSA

Personalised recommendations