VaR and ES of the portfolio's cumulative return over horizon days, as
positive losses. One day with every asset normal: closed form, the
portfolio return N(Σwᵢμᵢ, wᵀH_{T+1}w). Otherwise simulated: each day draws
every asset's own innovations (unit variance), correlates them through the
Cholesky factor of R_t, and runs every recursion forward — each asset's
variance and Q. Under non-normal innovations the correlated draws keep their
correlation but not exactly each asset's marginal.
VaR and ES of the portfolio's cumulative return over
horizondays, as positive losses. One day with every asset normal: closed form, the portfolio return N(Σwᵢμᵢ, wᵀH_{T+1}w). Otherwise simulated: each day draws every asset's own innovations (unit variance), correlates them through the Cholesky factor of R_t, and runs every recursion forward — each asset's variance and Q. Under non-normal innovations the correlated draws keep their correlation but not exactly each asset's marginal.