en>Dicklyon: /* Assigning attributes to implicit k-d tree-nodes */ take out spurious hyphen

2012-06-12T18:25:38Z

Assigning attributes to implicit k-d tree-nodes: take out spurious hyphen

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{{Confusing|date=July 2007}}
{{Refimprove|date=May 2012}}

'''Discounted maximum loss''', also known as '''worst case [[risk measure]]''', is the [[present value]] of the worst case scenario for a financial [[portfolio (finance)|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.

==Definition==
Given a finite state space <math>S</math>, let <math>X</math> be a portfolio with profit <math>X_s</math> for <math>s\in S</math>. If <math>X_{1:S},...,X_{S:S}</math> is the [[order statistic]] the discounted maximum loss is simply <math>-\delta X_{1:S}</math>, where <math>\delta</math> is the [[discount factor]].

Given a general [[probability space]] <math>(\Omega,\mathcal{F},\mathbb{P})</math>, let <math>X</math> be a portfolio with discounted return <math>\delta X(\omega)</math> for state <math>\omega \in \Omega</math>. Then the discounted maximum loss can be written as <math>-\operatorname{ess.inf} \delta X = -\sup \delta \{x \in \mathbb{R}: \mathbb{P}(X \geq x) = 1\}</math> where <math>\operatorname{ess.inf}</math> denotes the [[essential infimum]].<ref name="Schied Notes">{{cite web|author=Alexander Schied|title=Risk measures and robust optimization problems|url=http://people.orie.cornell.edu/~schied/PueblaNotes8.pdf|format=pdf|accessdate=May 18, 2012}}</ref>

==Properties==
* The discounted maximum loss is the [[expected shortfall]] at level <math>\alpha = 0</math>. It is therefore a [[coherent risk measure]].
* The worst-case risk measure <math>\rho_{\max}</math> is the most conservative (normalized) [[risk measure]] in the sense that for any risk measure <math>\rho</math> and any portfolio <math>X</math> then <math>\rho(X) \leq \rho_{\max}(X)</math>.<ref name="Schied Notes"/>

== 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%):
{| class="wikitable"
|-
! 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 <math>80\times0.8=64</math>

== References ==
{{Reflist}}

[[Category:Financial risk]]
[[Category:Mathematical finance]]

Implicit k-d tree - Revision history

en>Dicklyon: /* Assigning attributes to implicit k-d tree-nodes */ take out spurious hyphen