Approximating the Multivariate Distribution of Time-Aggregated Stock Returns Under GARCH

Abstract

An approach to approximate the multivariate distribution of time-aggregated stock returns in the GARCH context is developed here. The approach yields a one time-step simulation procedure as opposed to a multiple time-step simulation required in such a context. For this purpose, the exact moment formulas for the time-aggregated return under a QGARCH process are combined with multivariate non-normal simulation procedures using as inputs, the first four moments and correlation structure of the unknown target distribution. Estimation and simulation results are presented for a portfolio of 30 stocks from the Dow Jones Industrial Average index. The results reveal that the proposed simulation method can generate random numbers with moments and correlations agreeing with the targets. Using value at risk computations for different horizons and probabilities, we show that the percentiles of portfolios return distributions computed with the proposed approach provide good approximations of benchmark values obtained from a multi-step simulation.

Keywords: GARCH, multivariate distribution, time-aggregated return, value at risk, Johnson distributions

JEL Classification: C58, G17