This article discusses Monte Carlo and historical VaR measures. Exhibit 1 is reproduced from the overview article measuring value-at-risk. It indicates the three processes that are essential to all practical VaR measures. This article assumes familiarity with the concepts referenced in that exhibit.
A transformation procedure accepts two inputs:
The transformation procedure must somehow combine these to characterize the conditional distribution of the portfolio's value. Based upon that characterization, the transformation procedure values the desired VaR metric. If a portfolio mapping function is a linear or quadratic polynomial, then a linear transformation or quadratic transformation should generally be used. Both run rapidly and entail little or no error. Monte Carlo and historical transformations take longer to run and introduce the standard error common to all Monte Carlo estimators. For this reason, they are reserved for only those portfolios for which portfolio mapping functions are neither linear nor quadratic. Such portfolios generally contain derivatives, mortgage-backed securities or other complex non-linear instruments.
Let time 0 be the current time, and let time 1 be the end
of the VaR horizon (see the
notation conventions documentation). A
portfolio mapping
Based upon the characterization of the joint distribution
of
Because it depends upon the Monte Carlo method, a Monte Carlo transformation procedure entails standard error. The magnitude of the standard error depends upon many things, including:
A crude rule of thumb is that the standard error will be about 1% of calculated VaR for a typical portfolio if a quantile-of-loss VaR metric is used and the sample size is 10,000. Because valuing a portfolio mapping function 10,000 times can be an enormous computational load, most organizations don't use nearly as large a sample size. Sample sizes as low as 500 are commonly used. Because the standard error of a Monte Carlo analysis is proportional to the square root of the sample size, such low sample sizes introduce standard errors on the order of 4.5%. (I know of several software vendors who obfuscate this fact in their product literature.) A solution to this problem is to employ variance
reduction. Techniques based upon
control variates and
stratified sampling were
published by Cárdenas, et al. (1999). Both methods
they propose employ a quadratic
remapping
With the method of control variates,
With the method of stratified sampling,
Variance reduction is excellent for most portfolios and VaR metrics. To further improve variance reduction, the above two methodologies can be combined. Cárdenas, et al. (1999) also suggested a technique of selective valuation that can be used. While this is technically not a variance reduction technique, it has largely the same effect. See Holton (2003) for a detailed discussion of all these methodologies. Historical
transformations are identical to Monte Carlo transformations
except for one difference. Both employ the Monte Carlo method to construct
a histogram of realizations
Historical transformations were popular during the early 1990s because they were intuitively easy to explain to non-technical professionals—VaR was being calculated based on one-day profit or losses that the portfolio would have realized based on market movements that occurred during each of the past 500 or so trading days. There are, however compelling reasons to avoid historical transformations. First, because they are based on the Monte Carlo method, they entail exactly the same standard error as Monte Carlo transformations. While Monte Carlo transformations can minimize standard error by using large sample sizes, this is not possible for historical transformations. Their sample sizes are limited by the availability of relevant historical market data. It is rare that historical transformations have sample sizes grater than 1000, so standard errors is generally significant. Historical transformations are not amenable to many of the powerful methods of variance reduction available for Monte Carlo transformations. Finally, use of historical realizations introduces biases relating to conditional heteroskedasticity. See Holton (2003).
VaR measures are called
Monte Carlo VaR measures or
historical VaR measures if they
employ, respectively, Monte Carlo or historical transformations. With
Monte Carlo VaR measures, an inference procedure typically characterizes
the distribution of
|
|||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
a 
is obtained from a mapping procedure, where
is the random variable for the
portfolio's value at the end of the VaR horizon,
is the
portfolio mapping function, and
is the
for 
.
The histogram of these realizations
provides a discrete approximation for the distribution of
for 
of 1R are stratified into two regions:
exceeds the VaR of
_125.gif)
_125.jpg)
_125.gif)
