2. Risk measures used in the financ

2. Risk measures used in the finance and insurance
2.1. Value-at-Risk
In the class of quantile-based risk measures, the most used is Value-at-Risk, which evaluates the maximal loss
that can occur in a time horizon with a given probability level. Let X : Ω → R a random variable defined on the
probability space (Ω, K, P), with cumulative distribution function FX(x) = P(X d x),  x  R.
Let D  (0, 1). The Value-at-Risk corresponding to a random variable X at the probability level D is given by:
VaRD )( inf^ ` R|P( xXxX ) td D .
If the random variable X has a continuous one-to-one cumulative distribution function, then VaRα(X) can be
computed as the unique solution of the equation:
VaR .)(F)( 1
D D  X X (1)
Value-at-Risk is used for setting the capital adequacy limits for banks and other financial institutions and plays an
important role in investment, risk management and regulatory control of financial institutions.
2.2. Conditional Tail Expectation
The limitations of the most common used risk measures (like variance, which is a symmetric me