Discounted maximum loss

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

In investment, in order to protect the value of an investment, one must consider all possible alternatives to the initial investment. How one does this comes down to personal preference, however, the worst possible alternative is generally considered to be the benchmark against which all other options are measured. The present value of this worst possible outcome is the 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.[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%):

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


  1. ^ a b Alexander Schied. "Risk measures and robust optimization problems" (pdf). Retrieved May 18, 2012.