Discounted maximum loss

Discounted maximum loss, also called the worst case risk measure, is the present value of the worst case scenario for a financial portfolio.

An investor must consider all possible alternatives for the value of his investment. How he weights the different alternatives is a matter of preference. One might require a pension fund never to go bankrupt. If this is the case, the manager of its portfolio must consider the worst alternative as the benchmark. Finally, as the investment takes place today he must evaluate the alternatives in their present value, hence the discounting.

Definition

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%):

probability	value
of event	of the portfolio
40%	110
30%	70
20%	150
10%	20

In this case the maximum loss is from 100 to 20 = 80, so the discounted maximum loss is simply $80\times0.8=64$