Measuring the impact of the stock market index return on stocks return using the stochastic approximation

Abstract

Value at Risk (VAR) is a risk measure frequently used in market finance. This tool gives an idea of the losses that may occur to a financial asset (share or option), but does not predict when these losses may occur.

The objective of our work is to propose a method that allows us to know the state of the economy necessary for Microsoft's performance to suffer extreme losses with alpha probability. This represents a significant asset for investors to consider before trading on the stock market. To answer this problem, we will try to explain the VAR using the ROBBINS-MONRO-JOSEPH procedure for the estimation of a percentile for binary variables, as a function of a systemic risk factor. We will also propose a method for applying the procedure without using real-time experiments. We are going to estimate the different parameters of the process, based on the history of the data available for statistical modelling, then analyse the results of the convergence of the process over the iterations, and finally see the impact of adding a random element to the binary variable  of the process on convergence. The final results are satisfying and seem to be in line with reality, but there is room for improvement, either by increasing the number of iterations needed to refine convergence, whereas the stochastic approximation aims to obtain a good estimate in a minimum number of iterations, or by applying other more recent procedures that show better results following simulations.