  Value at risk (VaR) and expected shortfall (ES) are common high
quantile-based risk measures adopted in financial regulations and risk
management. In this paper, we propose a tail risk measure based on the most
probable maximum size of risk events (MPMR) that can occur over a length of
time. MPMR underscores the dependence of the tail risk on the risk management
time frame. Unlike VaR and ES, MPMR does not require specifying a confidence
level. We derive the risk measure analytically for several well-known
distributions. In particular, for the case where the size of the risk event
follows a power law or Pareto distribution, we show that MPMR also scales with
the number of observations $n$ (or equivalently the length of the time
interval) by a power law, $\text{MPMR}(n) \propto n^{\eta}$, where $\eta$ is
the scaling exponent. The scale invariance allows for reasonable estimations of
long-term risks based on the extrapolation of more reliable estimations of
short-term risks. The scaling relationship also gives rise to a robust and
low-bias estimator of the tail index (TI) $\xi$ of the size distribution, $\xi
= 1/\eta$. We demonstrate the use of this risk measure for describing the tail
risks in financial markets as well as the risks associated with natural hazards
(earthquakes, tsunamis, and excessive rainfall).
