Value at Risk on expected return is the lower bound for the loss in an investement respect to a probability.
Is a way to estimating losses probabilities in terms of value, instead of standard deviation. E.g. in the case of , means that the investor have a probability of loosing € or more, investing € on the portfolio .

Considering the system ergodic and thus all the variances of the titles summing up coherently to for a variance of the portfolio distributed within the normal distribution, this is simply calculated considering the quantile (the value of the distribution corresponding to a certain fraction of probability) .

(!) The <img src="/wilt/wp-content/plugins/latex/cache/tex_899857f2795e1307be64b0d9b2acac5c.gif/> of a portfolio is not a weighted average of the of its component, because but is given by the covariance matrix due to the correlation of assets .