Drawdown returns duration during speculative bubbles

Risk exposure in financial markets has been described through several measures (VaR, CVaR, etc.) based on statistical properties of data that do not consider the order of data sequences. In fact empirical mean, variance, as well as higher order moments of probability distributions are invariant under data shuffle. But long downward trends containing long lasting sequences of consecutive drawdown price movements could suggest investors to withdraw from the market, and they can quite force small investors to such a choice. Therefore, measures of risk based both on the duration of consecutive market drops and on the maximum drawdown can play a relevant role in driving investment strategies. Some attempt to give a model to the maximum drawdown has led to the definition of measures like Drawdown at Risk (DaR), MDaR (Maximum Drawdown at Risk) [14]. Much less attention has been payed to drawdown duration, although the time has a key role in the catastrophic events that undergo the name of large financial crashes, that show a drop of prices within short time intervals [2, 4, 3]. In this paper we focus on crashes due to the collapse of speculative bubbles due to endogenous causes [4]. We examine the structure of drawdown duration and we provide probability bounds on the maximum drawdown of returns that can be used for investment strategie