Discounted maximum loss

Given a finite state space S, let X be a portfolio with profit X_s for s\in S. If X_{1:S},...,X_{S:S} is the order statistic the discounted maximum loss is simply -\delta X_{1:S}, where \delta is the discount factor. Given a general probability space (\Omega,\mathcal{F},\mathbb{P}), let X be a portfolio with discounted return \delta X(\omega) for state \omega \in \Omega. Then the discounted maximum loss can be written as -\operatorname{ess.inf} \delta X = -\sup \delta \{x \in \mathbb{R}: \mathbb{P}(X \geq x) = 1\} where \operatorname{ess.inf} denotes the essential infimum. ==Properties==

• The discounted maximum loss is the expected shortfall at level \alpha = 0. It is therefore a coherent risk measure. • The worst-case risk measure \rho_{\max} is the most conservative (normalized) risk measure in the sense that for any risk measure \rho and any portfolio X then \rho(X) \leq \rho_{\max}(X). == Example ==

As an example, assume that a portfolio is currently worth 100, and the discount factor is 0.8 (corresponding to an interest rate of 25%): In this case the maximum loss is from 100 to 20 = 80, so the discounted maximum loss is simply 80\times0.8=64 == References ==