The Temporal Dimension of Risk

Abstract: Multi-period measures of risk account for the path that the value of an investment portfolio takes. In the context of probabilistic risk measures, the focus has traditionally been on the magnitude of investment loss and not on the dimension associated with the passage of time. In this paper, the concept of temporal path-dependent risk measure is mathematically formalized to capture the risk associated with the temporal dimension of a stochastic process. We discuss the properties of temporal measures of risk and show that they can never be coherent. We then study the temporal dimension of investment drawdown, its duration, which measures the length of excursions below a running maximum. Its properties in the context of risk measures are analyzed both theoretically and empirically. In particular, we show that duration captures serial correlation in the returns of two major asset classes. We conclude by discussing the challenges of path-dependent temporal risk estimation in practice.