Herein c denotes the p-quantile of the standard normal distribution. hh is a scaling factor that adapts the time
horizon of the volatility to the length of the holding period h. The problems that may arise when using such a scaling
factor are discussed in section 2.4. In the case of a portfolio that consists of n assets the volatility of the portfolio
return is calculated according to:
0.5
I ПП 1
cp = I ΣΣwi ■ wt ■ σij I (5)
к i=1 j=1 J
wi and Wj are the weights of assets i and j and Cij. is the covariance of their returns.
An apparent advantage of the VCM is its ease of computation. If the normality assumption holds, VaR figures can
be simply translated across different holding periods and confidence levels. Moreover, time-varying volatility
measures can be incorporated and what-if-analyses are easy to conduct. On the other hand, the normality assumption
is frequently criticized. There is empirical evidence that return distributions are fat tailed and in that case the VCM
will underestimate the VaR for high confidence levels. Further problems occur, if the portfolio return depends in a
nonlinear way on the underlying risk factors, which is typically the case with options.
Monte Carlo simulation
With this method the entire distribution of the value change of the portfolio is generated and VaR is measured as an
appropriate quantile from this relative frequency distribution. The simulation involves the following steps:
• Selection of distributions for the changes of the relevant market factors (e.g. commodity prices) and estimation
of the appropriate parameters, in particular variances and correlations
• simulation of random paths for the market factors
• evaluation of the portfolio for the desired forecast horizon ("mark-to-market")
• calculation of the gains or losses related to the current portfolio value
• repetition of the three aforementioned steps until a sufficient accuracy is gained
• ordering of the value changes in ascending order and determination of the frequency distribution
The main advantage of the Monte Carlo simulation is the ability to handle different return distributions. Harmful are
the high costs of computation in the case of complex portfolios.
Historical simulation
Historical simulation (HS) resembles the Monte Carlo simulation regarding the iteration steps. The difference is that
the value changes of the portfolio are not simulated by means of a random number generator, but directly calculated
from observed historical data. That means, VaR estimates are derived from the empirical profit-and-loss-