What is the use of auxiliary variable in optimization?
Answers
Simple Algorithm for CVaR Optimization
We start with a brief description around the technicalities of CVaR optimization. Let us first define an auxiliary variable, es, for each of s = 1,…, S scenarios.
It measures the excess loss of a portfolio consisting of i = 1,…, n securities with respective weights wi that pay off ris in scenario s. There are no restrictions on the distribution of scenarios. We can, for example, include nonlinear positions as they would typically arise from call or put options.4 This is a major advantage relative to mean–variance optimization that cannot deal with these instruments. Suppose VaR = −20% and We then get es = max[0, −20%– −25%] = 5%, i.e., a 5% excess. In practice, we will need many scenarios to approximate a continuous portfolio distribution from a grid of discrete scenarios. We will come back to this point when we discuss approximation error in CVaR optimization. CVaR will always be larger than VaR (because it represents the average of losses larger than value at risk)