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Monte Carlo
simulation is a statistical
measure. This means we can estimate the required number of runs to reach a
level of confidence in the quality of the results generated to measure our
risks.
In a Monte Carlo simulation the
standard error of the mean of the distribution is:
where
z
is to the confidence multiplier
of a TWO tailed normal
distribution.
For a 95% confidence, z=2. With 99%
z=3, etc.
= is the portfolio’s standard deviation.
Runs is the number of runs
or simulations performed in the Monte Carlo
simulation.
From
the above, one can see that the
error term can be reduced
either:
by
decreasing
the
numerator
or
by
increasing
the
denominator.
In the first case, we try to reduce the numerator by improving the accuracy of
the distribution's volatility estimate.
This approach is usually carried out either by choosing a better quality random
number generator or by improving the distribution of returns through importance
sampling, stratified sampling, variance reduction and other numerical
techniques.
In the second
case, we increase the denominator simply by the sheer number of runs. This
approach is akin to brute
force. since no effort is
made to harness the distribution. In certain circumstances, especially when the
distribution is not well understood, this might be the only alternative !
Unfortunately,
the problem with the
latter is that
accuracy only improves
as the square root of the
ratio of the number of additional Runs !
So, if we are running 1000 simulations and we want to reduce the error
term by 10 we must actually increase the number of simulations by 100!
In this case this means we must runs
100000 simulations instead of 1000 in order to achieve an improvement of
ten (one order of magnitude)!
For example, if the portfolio has
a standard deviation of 15% and we are running 1000 simulations, we have
95% chances that the true mean of the distribution lies within 1% of
our estimate with 1000 runs.
(2*0.15/1000^0.5)=0.9045 %.
Now, if we increase the number of
runs by one order of magnitude (i.e. 10), our error term will now be (2*0.15/10000^0.5)=0.3 %,
which indeed reduces the error term by 0.9045/0.3
(1000/10000)^0.5=10^0.5=3.16.
Measurements
in Practice . Estimate the Real Impact.
A practical estimate usually
starts at one standard deviation of the risk measure.
We then move to two standard deviations with two different portfolios in order
to estimate stability. Once this desirable property has been obtained we can
compute "Our" optimal number of runs for the portfolio under analysis.
Note:
Convergence is very
different from back testing.
This
approach only describes convergence of the portfolio's distribution mean under Monte Carlo simulation, not
the results themselves.
The procedure which is usually
carried out quarterly or semi-annually begins with the selection of two
(or more) portfolios that are well distributed across asset classes and
instruments.
If we are running credit risk and market risk combined we must also ensure they are well
distributed across counterparties, master agreements and countries.
The first portfolio should
include a smaller sample size than the second. You can obviously proceed with
more than two portfolios, provided coverage across asset classes and products is
different.
The portfolio is then run with,
say 1000, 2000, 5000 & 10000 runs for Market Risk. (A minimum number
of 50 000 simulations is recommended for Credit Risk).
Each Simulation is then run 20-30
times.
For each Risk Factor, accumulate
the mean “risk factor” and then compute the volatility in percent terms. You
then plot the results for each portfolio with one and two standard deviations.
Once you have plotted results,
you can identify rapidly the outliers.
For each estimate, if the mean of
the larger sample portfolio is not within two standard deviations of the smaller
portfolio, your portfolio is not
stable enough to draw any acceptable conclusion.
In this case, the largest
portfolio is either not well distributed across asset classes and
products, counterparty,
etc or you do not have enough positions in your
portfolio. If this is the case,
you must start over until results present sufficient stability.
Once mean and standard
deviations of the risk measures are
stable enough, you can proceed to compute the required number of runs.
Finding the appropriate
number of runs for your own organization is extremely simple:
For each portfolio, plot
the standard deviation from the mean by connecting the points between each
number of runs.
The numbers on your graph should show a
clear relationship
between a decrease in standard deviations and an increase in number of runs.
In
this case we assume standard deviations and numbers of runs can be interpolated
linearly.
From this you should be able to pinpoint immediately the necessary
number of runs needed to reach the percentage error you are seeking!
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