Variance/Covariance Approach: Theoretical Background 
Variance/Covariance Approach: Theoretical Background
The variance/covariance approach is an analytical procedure for determining the value at risk. It is based on the assumption of the distribution of price changes. In the variance/covariance approach, potential loss is calculated from the volatility of the risk factors. The volatility of the risk factors and their correlations can be estimated from the historical market data for the respective risk factors (with a statistics calculator), or imported from external sources (using a datafeed, or a market data file).
The resulting individual risks are aggregated using correlation matrices, taking any interdependencies into account.
Delta valuation:
In the delta approach, normal distribution of price factors is assumed and the elasticity of the price function is used to estimate the NPV change based on the various price-determining parameters (
delta positions). The portfolio changes are also normally distributed. The required quantile is determined by taking the inverses of the cumulated normal distribution function. The VAR arises as the product between the quantile and the variance of the NPV changes.
Delta-Gamma approach:
The assumption of normal distribution does not apply. Here, in contrast to the delta valuation, non-linear terms of the second order (gamma positions) are additionally taken into account at the risk factor level so that skewed distributions are also interpreted. The VAR is calculated from the four moments of the distribution (expected value, variance, skewness and kurtosis) using Cornish/Fisher approximation. It calculates the percentile of a normal distribution for a specified confidence level and for the first four moments of a distribution.
The calculation of the VAR then occurs in the same way as for the delta procedure.
The process used for both delta valuation and the delta-gamma approach is known as risk factor mapping.