The Statistics of the Maximum Drawdown in Financial Time Series

The maximum drawdown (MDD) in financial time series plays an important role in investment management and has been widely studied in the literature. MDD is associated with standards of performance measures such as Calmar or Sterling ratios. Various forms of portfolio optimization based on MDD have been considered (see for example A. Chekhlov and Zabarankin (2005)). In addition, Leal and de Melo Mendes (2005) have proposed a coherent risk measure possessing the properties required by Artzner et al., (1999) similar to the conditional value-at-risk: the maximum drawdown-at-risk MDaRα, which is just a quantile with exceedance probability α of the distribution of the maximum drawdown. Despite the widespread use of maximum drawdown among practitioners, financial economists have not paid much attention to this concept. It provides an alternative or complement to the other commonly used risk measures such as value-at-risk, which is still used extensively by the industry and regulatory standards for the calculation of risk capital in banking and insurance despite its well-known shortcomings. The evaluation of both MDD’s expectation value and its probability density function (pdf) is of importance for various practical applications, especially when building a robust framework for risk management and capital allocation. This chapter is motivated by the need to gain insights into the statistical properties of the MDD for stochastic processes that, set aside from the academic example of the Brownian motion, are possibly closer to the stylized facts that characterize the real financial time series (see Rosario N. Mantegna, 2000; Cont, 2001; Bouchaud and Potters, 2003).