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 aim is the optimization either by maximizing total return (or minimizing risk), in keeping with the constraints of normalization for a given risk (or return). Thus Lagrange method is perfectly suited for the job. For minimizing risk, respect to a desired total return , one must consider the function of total risk and minimize the lagrangian with constrain over the desired return and sum to one of weights.