Path-dependent Risk Measures - Theory and Applications

Abstract

viii porin and Lisi (2011), Haas Ornelas et al. (2012), Auer and Schuhmacher (2013), and Auer (2015). Although all of these strands focus on drawdown -either on drawdown constraints, drawdown processes, or drawdown-based performance ratios -they do not specifically address drawdown risk measures. Regarding risk measures, Chekhlov et al. (2005) seminally define the conditional drawdown (CDD), which constitutes a family of drawdown risk measures. It includes the maximum drawdown (MDD) and the average drawdown (ADD), which are arguably the most prominent drawdown risk measures to date. An additional family, the conditional expected drawdown (CED), has been defined recently in Goldberg and Mahmoud (2017). It is theoretically interesting but less applicable in practice because it requires knowledge of tail means of the MDD distribution. Properties of drawdown measures are almost exclusively studied for the MDD. Its distribution is analyzed by Magdon-Ismail et al. (2004), Cheridito et al. (2012), and Casati and Tabachnik (2013). Comparative statics are computed in van Hemert et al. (2020) to analyze how changes in return, volatility, length of time horizon, and autocorrelation affect the MDD. 5 Despite the aforementioned literature, drawdown measures remain much less well-studied compared to more conventional risk measures, such as value-at-risk and volatility (Goldberg 5 The work of van Hemert et al. ( 2020) probably comes closest to this dissertation's objective of analyzing properties of drawdown measures. It partly follows the approach used in the first paper of this dissertation, but there are notable differences. For example, their comparative statics do not control for higher moments, they use much coarser data, and they only consider the MDD. country or sector does not deviate drastically from the corresponding proportions in the MSCI World index. Each month, there is some rebalancing and adjustment for stocks leaving the index, and each new stock receives a weight between 0 and 2%. In the random setup, all fictitious managers have a hit ratio of 50%, i.e., their probability of picking future winners that have above median return is 0.5. In hindsight, however, some portfolio managers can be endowed with higher hit ratios by providing them with a higher likelihood of identifying future winners over future losers. Therefore, each time the skillful portfolio managers create a portfolio and add or drop stocks during rebalancing, the odds of picking a winning stock and dropping a losing stock are in their favor. Once 1,000 portfolio paths are simulated with and without skill, the portfolio managers are ranked using different drawdown measures, and these ranks are compared via correlation coefficients. Both for hit ratios of 0.5 and 0.6, the results are similar: All correlations are positive, ranging between 0.258 and 0.874. Thus, correlations between drawdown measures differ substantially. Especially with an eye toward eopDD, with which correlations are lowest, drawdown measures do not appear to be "all the same".Wh...