# Drawdown: from practice to theory and back again

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## 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.Acerbi, C., Tasche, D.: Expected shortfall: a natural coherent alternative to value at risk. Econ. Notes
**31**(2), 379–388 (2002a)CrossRefGoogle Scholar - 2.Acerbi, C., Tasche, D.: On the coherence of expected shortfall. J. Bank. Financ.
**26**(7), 1487–1503 (2002b)CrossRefGoogle Scholar - 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.Anderson, R,M., Bianchi, S.W., Goldberg, L.R.: Determinants of levered portfolio performance. Forthcom. Financ. Anal. J. (2014)Google Scholar
- 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.Burghardt, G., Duncan, R., Liu, L.: Deciphering drawdown. Risk Mag. 16–20 (2003)Google Scholar
- 7.Carr, P., Zhang, H., Hadjiliadis, O.: Maximum drawdown insurance. Int. J. Theor. Appl. Financ.
**14**, 1195–1230 (2011)MathSciNetCrossRefzbMATHGoogle Scholar - 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.Chekhlov, A., Uryasev, S., Zabarankin, M.: Drawdown measure in portfolio optimization. Int. J. Theor. Appl. Financ.
**8**(1), 13–58 (2005)MathSciNetCrossRefzbMATHGoogle Scholar - 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.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.Cvitanic, J., Karatzas, I.: On portfolio optimization under drawdown constraints. IMA Lect. Notes Math. Appl.
**65**, 77–88 (1995)zbMATHGoogle Scholar - 13.Denault, M.: Coherent allocation of risk capital. J. Risk
**4**(1), 7–21 (2001)CrossRefGoogle Scholar - 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.Föllmer, H., Schied, A.: Convex measures of risk and trading constraints. Financ. Stoch.
**6**, 429–447 (2002)MathSciNetCrossRefzbMATHGoogle Scholar - 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.Föllmer, H., Schied, A.: Stochastic Finance: An Introduction in Discrete Time. Walter de Gruyter, Berlin (2011)CrossRefzbMATHGoogle Scholar
- 18.Goldberg, L.R., Menchero, J., Hayes, M., Mitra, I.: Extreme risk analysis. J. Perform. Meas.
**14**(3), 17–30 (2010)Google Scholar - 19.Grossman, S.J., Zhou, Z.: Optimal investment strategies for controlling drawdowns. Math. Financ.
**3**, 241–276 (1993)CrossRefzbMATHGoogle Scholar - 20.Hadjiliadis, O., Vecer, J.: Drawdowns preceding rallies in the brownian motion model. Quant Financ.
**6**, 403–409 (2006)MathSciNetCrossRefzbMATHGoogle Scholar - 21.Kalkbrener, M.: An axiomatic approach to capital allocation. Math. Financ.
**15**(3), 425–437 (2005)MathSciNetCrossRefzbMATHGoogle Scholar - 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.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.Lehoczky, J.P.: Formulas for stopped diffusion processes with stopping times based on the maximum. Ann. Probab.
**5**, 601–607 (1977)MathSciNetCrossRefzbMATHGoogle Scholar - 25.Madhavan, A.: Exchange-traded funds, market structure, and the flash crash. Financ. Anal. J.
**68**(4), 20–35 (2012)Google Scholar - 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.McNeil, A.J., Frey, R., Embrechts, P.: Quantitative Risk Management: Concepts, Techniques, and Tools. Princeton University Press, Princeton (2005)Google Scholar
- 28.Menchero, J., Poduri, V.: Custom factor attribution. Financ. Anal. J.
**64**(2), 81–92 (2008)CrossRefGoogle Scholar - 29.Mijatovic, A., Pistorius, M.R.: On the drawdown of completely asymmetric lévy processes. Stoch. Process. Appl.
**122**, 3812–3836 (2012)CrossRefzbMATHGoogle Scholar - 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.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.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.Rockafellar, R.T., Uryasev, S.: Optimization of conditional value-at-risk. J. Risk
**2**, 21–41 (2000)CrossRefGoogle Scholar - 34.Rockafellar, R.T., Uryasev, S.: Conditional value-at-risk for general loss distributions. J. Bank. Financ.
**26**, 1443–1471 (2002)CrossRefGoogle Scholar - 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.Rockafellar, R.T., Uryasev, S.P., Zabarankin, M.: Generalized deviations in risk analysis. Financ. Stoch.
**10**(1), 51–74 (2006)MathSciNetCrossRefzbMATHGoogle Scholar - 37.Sekine, J.: Long-term optimal investment with a generalized drawdown constraint. SIAM J. Financ. Math.
**4**, 457–473 (2013)MathSciNetCrossRefzbMATHGoogle Scholar - 38.Tasche, D.: Risk contributions and performance measurement. Technical report, Research paper, Zentrum Mathematik (SCA) (2000)Google Scholar
- 39.Tasche, D.: Expected shortfall and beyond. J. Bank. Financ.
**26**, 1519–1533 (2002)CrossRefzbMATHGoogle Scholar - 40.Taylor, H.M.: A stopped brownian motion formula. Ann. Probab.
**3**, 234–246 (1975)MathSciNetCrossRefzbMATHGoogle Scholar - 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.Zabarankin, M., Uryasev, S.: Statistical Decision Problems: Selected Concepts and Portfolio Safeguard Case Studies, vol. 85. Springer, Berlin (2014)Google Scholar
- 43.Zhang, H.: Occupation time, drawdowns, and drawups for one-dimensional regular diffusion. Adv. Appl. Probab.,
**47**(1), 210–230 (2015)Google Scholar - 44.Zhang, H., Hadjiliadis, O.: Drawdowns and rallies in a finite time-horizon. Methodol. Comput. Appl. Probab.
**12**, 293–308 (2010)MathSciNetCrossRefzbMATHGoogle Scholar - 45.Zhang, H., Leung, T., Hadjiliadis, O.: Stochastic modeling and fair valuation of drawdown insurance. Insur. Math. Econ.
**53**, 840–850 (2013)MathSciNetCrossRefzbMATHGoogle Scholar