A Comparison of the Three Methods
The VaR numbers derived from the three approaches produce a wide range of
results (Table 7). In this example, the historical simulation method which takes into
account the actual shape of the observed distribution of profits and losses (shown in Figure 3) yields the lowest risk estimates. The variance-covariance method’s
assumption of symmetry around a zero mean gives equal weight to both profits and
losses, resulting in VaR estimates which are slightly higher than those of the
historical simulation approach. The simulation of a bivariate t-distribution results in
VaR estimates which are much larger than the estimates given by the other
two methods. A t-distribution with the same mean and variance as a normal
distribution will have a greater proportion of its probability mass in the tails of the
distribution (in fact, in this case, the t-distribution also has longer tails than the
empirical distribution). The prime focus of a VaR model is the probability of tail
events, hence, the long tails of the t-distribution have a disproportionate effect on
the VaR estimate. It can be seen that this effect becomes more marked the higher
the confidence level. It should be noted that this ranking of results from the
three methods is dependent on the data and also the statistical distribution used
within the Monte-Carlo simulation technique. Other price series exhibiting different
mean, skew and tail characteristics may result in the relative sizes of the three
methods’ VaR estimates being quite different.