VaR Estimation: Covariance, Historical Simulation and Monte Carlo

Given the loss operator

l^{[t]}

from the previous lesson, the VaR estimation problem reduces to: find the distribution of

L_{t+1} = l^{[t]}(\mathbf{X}_{t+1})

. There are three families of approach, each making different tradeoffs between model risk, computational cost, and statistical efficiency.

Variance-covariance (delta-normal): assume $\mathbf{X}_{t+1} \mid \mathcal{F}_t \sim \mathcal{N}_d(\boldsymbol{\mu}_{t+1}, \boldsymbol{\Sigma}_{t+1})$ , use the linear loss operator, read off a Gaussian quantile. Fast, analytic, but misspecified for non-linear books and fat-tailed returns.
Historical simulation: apply the current loss operator to historical risk-factor changes. Distribution-free, no model assumption. But it is unconditional — slow to react to volatility spikes.
Dynamic (GARCH-filtered) historical simulation: standardise historical returns by GARCH volatility, rescale by tomorrow's forecast volatility. Gives a conditional loss distribution without full parametric modelling.
Monte Carlo: simulate from a fitted parametric model. Flexible, accurate — but requires a model for $\mathbf{X}_{t+1}$ and full portfolio revaluation per simulation path.

Learning objectives. By the end of this note you should be able to:

explain conditional vs unconditional loss distributions and give an example of when they diverge;
derive the VaR formula for the delta-normal model and extend it to multivariate $t$ innovations;
describe the historical simulation procedure and identify its key weaknesses;
outline the two approaches to dynamic historical simulation (univariate GARCH filtering and multivariate);
compare the three methods on computational cost, model risk, and responsiveness to changing volatility.

In this lesson

The full lesson, with worked examples and exercise solutions, is for members.