The complexity of planning and managing investments for private clients has lead to increasing use of numerical simulation methods within the financial community.  There are a variety of areas where techniques such as Monte Carlo simulations have become increasingly popular.  Among these are asset-liability and spending policy analysis for retirement planning, stress testing in risk assessment, tax deferral strategies and “portfolio opportunity distribution” methods in performance measurement.  Numerical methods are also employed to address parameter uncertainty in portfolio optimization exercises and to improve the quality of statistical analysis of markets when higher moments are observable in data samples.

While numerical methods are growing in popularity, we should remind everyone of the big limitation of numerical simulations:  Our conception of “what if?” is wholly dependent on experience to date.  We can’t simulate events we haven’t considered.  It should also be remembered that numerical methods are most suitable for situations wherein the analytical problem is too complex to allow tractable algebraic solution.   Alternative approaches are also available for consideration of this class of problem.

Most simulation methods are based on the “Monte Carlo” approach where we assume the parameters of the probability distribution of a variable, and then take a series of random draws from that probability distribution.  The assumed distribution can be simple (e.g. normal) and have independent and identically-distributed (IID) time series properties, where each member has the same probability distribution and the members are mutually independent.  Or it can have complex features such as skew, kurtosis, time-varying parameters, and even jumps.  We can also have systems of multiple correlated random variables and take random “vector” draws.  The downside is that if there are lots of moving pieces (variables), the number of draws needed to get a sufficient sample is large and hence time consuming.  It is tempting to overly simplify the problem to cut computation times

An alternative to Monte Carlo techniques is “bootstrap” methods.  Bootstrap simulations are based on real data samples rather than the parameters of probability distributions.   We’ve only lived through the path of history that actually happened. What else might have happened even if the distribution of possible events was similar?  Let’s look at an example:

• Imagine you have 180 monthly observations in a return time series for asset class X
• Pick a random number R from 1 to 180. Think of numbers selected in a “bingo” game
• Make the “Rth” observation from the data series, the first observation in a new series
• Repeat the random pick another 179 times, each time adding the Rth observation from the original series to the new series

The order will be different in the new series and some original observations might be omitted, while others may appear more than once, so parameters like mean and standard deviation of data series will change.  Using bootstrap methods, you can create as much simulated data as you want to explore “what might have happened”.

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