Variance-Covariance

This is the method used in RiskMetrics
This relies heavily on matrices
It involves using in-house or published (from Riskmetrics) volatility and correlation data in the matrix calculations
The biggest assumption is that the returns of the assets are normally distributed.

Also known as linear model
Also known as Asset-Normal

VAR is the maximum expected loss of a portfolio over a given horizon and at a given confidence level
VAR is the most comprehensive approach to measuring the overall risk associated with a portfolio

This is the method used in RiskMetrics
Also known as the "mean variance" or "parametric approach"
This relies heavily on matrices
Not suitable for stock options
This method uses volatility and correlation data to construct a variance-covariance matrix (VCM)

Assumptions
Correlations between the assets are stable
Returns on the assets (in the portfolio) are normally distributed (mean, standard deviation)

Calculation:
The mean is assumed to be zero as this calculation is done over a single day.
The volatility (or standard deviation) needs to be calculated for the portfolio.
Real world distributions are more leptokurtic (fat tails, where extreme losses are more frequent) and therefore this calculation underestimates the true VAR
A mesokurtic distribution is a distribution with the same kurtosis as a normal distribution

Assumptions

The daily change in the price is lineraly related to the daily returns from market variables
The returns from the market variables are normally distributed