Random processes for long-term market simulations

For long term investments, model portfolios are defined at the level of indexes, a setup known as Strategic Asset Allocation (SAA). The possible outcomes at a scale of a few decades can be obtained by Monte Carlo simulations, resulting in a probability density for the possible portfolio values at the investment horizon. Such studies are critical for long term wealth plannings, for example in the financial component of social insurances or in accumulated capital for retirement. The quality of the results depends on two inputs: the process used for the simulations and its parameters. The base model is a constant drift, a constant covariance and normal innovations, as pioneered by Bachelier. Beyond this model, this document presents in details a multivariate process that incorporate the most recent advances in the models for financial time series. This includes the negative correlations of the returns at a scale of a few years, the heteroskedasticity (i.e. the volatility' dynamics), and the fat tails and asymmetry for the distributions of returns. For the parameters, the quantitative outcomes depend critically on the estimate for the drift, because this is a non random contribution acting at each time step. Replacing the point forecast by a probabilistic forecast allows us to analyze the impact of the drift values, and then to incorporate this uncertainty in the Monte Carlo simulations.