Discounted maximum loss

Given a finite state space ${\displaystyle S}$, let ${\displaystyle X}$ be a portfolio with profit ${\displaystyle X_{s}}$ for ${\displaystyle s\in S}$. If ${\displaystyle X_{1:S},...,X_{S:S}}$ is the order statistic the discounted maximum loss is simply ${\displaystyle -\delta X_{1:S}}$, where ${\displaystyle \delta }$ is the discount factor.

Given a general probability space ${\displaystyle (\Omega ,{\mathcal {F}},\mathbb {P} )}$, let ${\displaystyle X}$ be a portfolio with discounted return ${\displaystyle \delta X(\omega )}$ for state ${\displaystyle \omega \in \Omega }$. Then the discounted maximum loss can be written as ${\displaystyle -\operatorname {ess.inf} \delta X=-\sup \delta \{x\in \mathbb {R}$ :\mathbb {P} (X\geq x)=1\}} where ${\displaystyle \operatorname {ess.inf} }$ denotes the essential infimum.[1]

• The discounted maximum loss is the expected shortfall at level ${\displaystyle \alpha =0}$. It is therefore a coherent risk measure.
• The worst-case risk measure ${\displaystyle \rho _{\max }}$ is the most conservative (normalized) risk measure in the sense that for any risk measure ${\displaystyle \rho }$ and any portfolio ${\displaystyle X}$ then ${\displaystyle \rho (X)\leq \rho _{\max }(X)}$.[1]

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 ${\displaystyle 80\times 0.8=64}$