Difference between revisions of "Value at Risk"

Revision as of 11:32, 18 March 2024

Definition

Value at Risk (VaR) is a Risk Measure that aims to capture the downside value risk of a Market Portfolio (a collection of financial instruments that can be marked-to-market).

Formula

VaR is a quantile Risk Measure and requires the specification of:

An aggregate (Portfolio) PnL (Profit and Loss) random variable that is constructed as the sum of potential individual market losses $L=\sum L_{i}$
A Confidence Level $\alpha$

Given the confidence level $\alpha\in(0,1)$ , the VaR of calculated portfolio loss $L$ at the confidence level $\alpha$ is the smallest number $K$ such that the Probability that the loss $L$ exceeds $K$ is at least $\alpha$ .

\operatorname{VaR}_\alpha(L)=-\inf\big\{l\in\mathbb{R}:F_L(l)>\alpha\big\} = F^{-1}_Y(1-\alpha).

@@ Line 1: / Line 1: @@
 == Definition ==
-'''Value at Risk''' (VaR) is a [[Risk Measure]] that aims to capture the downside [[Economic Value | value]] risk of a Market Portfolio.
+'''Value at Risk''' (VaR) is a [[Risk Measure]] that aims to capture the downside [[Economic Value | value]] risk of a Market [[Portfolio]] (a collection of financial instruments that can be marked-to-market).
 == Formula ==
-VaR is a quantile [[Risk Measure]] and requires the specification of
+VaR is a quantile [[Risk Measure]] and requires the specification of:
-* An aggregate Portfolio Profit and Loss variable constructed as the sum of potential individual market losses <math>L=\sum L_{i}</math>
+* An aggregate (Portfolio) PnL (Profit and Loss) random variable that is constructed as the sum of potential individual market losses <math>L=\sum L_{i}</math>
 * A [[Confidence Level]] <math>\alpha</math>
-Given a confidence level <math>\alpha\in(0,1)</math>, the VaR of calculated portfolio loss <math>L</math> at the confidence level <math>\alpha</math> is the smallest number <math>K</math> such that the probability that the loss<math>L</math> exceeds <math>K</math> is at least <math>\alpha</math>.
+Given the confidence level <math>\alpha\in(0,1)</math>, the VaR of calculated portfolio loss <math>L</math> at the confidence level <math>\alpha</math> is the smallest number <math>K</math> such that the [[Probability]] that the loss<math>L</math> exceeds <math>K</math> is at least <math>\alpha</math>.
+:<math>\operatorname{VaR}_\alpha(L)=-\inf\big\{l\in\mathbb{R}:F_L(l)>\alpha\big\} = F^{-1}_Y(1-\alpha).</math>
 == See also ==