2015-04: The Temporal Dimension of Drawdown

Multi-period measures of risk account for the path that the value of an investment portfolio takes. The most widely used such path-dependent indicator of risk is drawdown, which is a measure of decline from a historical peak in cumulative returns. In the context of probabilistic risk measures, the focus has been on one particular dimension of drawdown, its magnitude, and not on its temporal dimension, its duration. In this paper, the concept of temporal path-dependent risk measure is introduced to capture the risk associated with the temporal dimension of a stochastic process. We analyze drawdown duration, which measures the length of excursions below a running maximum, and liquidation stopping time, which denotes the first time drawdown duration exceeds a subjective liquidation threshold, in the context of coherent temporal path-dependent risk measures and show that they, unlike drawdown magnitude, do not satisfy any of the axioms for coherent risk measures. Despite its non-coherence, we illustrate through an empirical example some of the insights gained from analyzing drawdown duration in the investment process and discuss the challenges of path-dependent risk estimation in practice.