Zoo of models

 We study the following 5 different models for generating the forecast distribution p_t over a holding period of 1 day:

1.

Historical simulation with 250 day memory

2.

250 day rectangular moving average

3.

J. P. Morgan RiskMetrics - Exponential moving average

4.

GARCH(1,1)

5.

Tail emphasized GARCH(1,1)

Model 1 is non-parametric and models 2 and 3 are pre-defined and do not require any optimization. Models 4 and 5 are optimized in an in-sample period of 1250 days (250 days for build up and 1000 days for optimization). The latter in-sample period precedes a 1000 day out-of-sample in which the performance of the models is gauged in the context of risk management.

The determination of the conditional distribution p_t^(k) forecast by a model for each of the series k is of course an entirely univariate exercise. The multivariate aspect comes through the covariance matrix $\Sigma_t$which is determined by bivariate analysis - for all but model 1. In the case of models 2 and 3 the covariance is measured in a straightforward - intuitively obvious - bivariate generalization of the variance computation (details in sections  and ). In models 4 and 5 however - a more sophisticated approach is presented - wherein the dynamics of the bivariate sum are also fit to the candidate model and the implied covariance is used (details in section ). Another innovative feature (described in section ) is introduced in model 5 - wherein the optimization of the model is performed using a specialized method which emphasizes optimization of the residual tails. This optimization methodology turns out to have dramatic impact when using these models in the context of risk-management - as the performance measures will show.