Monte Carlo may overestimate individual asset risks by up to two thirds

Injudicious use of Monte Carlo simulations could overestimate individual asset risks by two thirds if strict normality is assumed and the shape of the underlying distributions is ignored. Similarly total returns and Sharpe Ratios could be overstated by 20% or more when assuming strict normality. Individual assets that have positively skewed distributions may have their returns under-estimated by 13% or more.

Most investors and software do not consider the asymmetry of returns and risks when conducting Monte Carlo simulations of future portfolio returns and simply rely on the central limit theorem to assume normality. Whilst the CLT certainly holds it does so only over the long-term. In the short-run things may be very different indeed.

This month we look at the three main methods of generating correlated random deviates for the purposes of performing Monte Carlo simulations, Namely;

• The Cholesky Method

• The Spectral ( SVD ) Method and;

• The Inverse method

As  you may know, Monte Carlo simulations were invented by Stanislaw Ulam and John von Neumann during their work on the Manhattan project. The requirement for significant computing power meant that the method remained the preserve of think tanks and large universities and corporations until around the the 1980's and the advent of the Personal Computer. Since then the use of the method has exploded and it has become ubiquitous in finance and financial planning even to the detriment of stopping people from trying to find closed form solutions for some problems which may have them.

Thanks to Moore’s law most of us now have sufficient computing power to be able to conduct simulations on even fairly large portfolios. Unfortunately, as is often the case methodology and common usage have not kept pace with the available technology and there are now many people using the method without a sufficient degree of understanding as to what goes on under the hood.

The method still most widely in use is the generation of correlated random numbers by using the Cholesky decomposition of the Correlation matrix.

However, the Cholesky decomposition requires that the Correlation matrix be 

-  Square (number of columns equals number of rows)

-  Symmetric (elements in upper diagonal matrix equal elements in lower diagonal matrix)

-  The main diagonal consists of ones

-  All elements are between -1 and 1

-  The matrix is positive semi definite (matrix determinant is equal to or larger than zero)

Any violation of the above fairly strict requirements can result in the matrix being non-invertible and a Cholesky factorisation not being able to be found by the standard algorithm.

Typical issues which might cause the matrix not to be positive semi definite include multicolinearity which may arise when for instance two time series are almost identical which often occurs with fixed income time series or different classes of the same fund or if they are perfectly negatively correlated. In such cases the Cholesky factorisation will fail possibly leading to missing or incorrect values being included in your simulation.

Fortunately the Spectral method which utilises the Singular Value Eigen Decomposition approach does not suffer from these defects. The matrix does not even have to be square. For this reason this approach is rapidly gaining favour over the traditional Cholesky approach.

Neither the Cholesy or Spectral approach explicitly require normality of returns although the correlation matrix embeds only the information contained in the first two moments and also imposes symmetry which may not be present in reality. However the use of random standard normal deviates as the seed values upon which to impose the correlation typically does make this assumption.

An actuarial approach most likely due to Iman and Conover, known as the Inversion method, overcomes this limitation. By generating uniform random numbers and then mapping these to any desired probability distribution through the inverse function any arbitrary distribution may be simulated - providing an inverse form is available.  

The method is fairly complex but can now even be performed in an Excel spreadsheet with some effort. Obviously replacing one distributional assumption with another that is likely to prove equally false as the normal assumption out-of-sample is a problem but the main benefit of being able to simulate arbitrary distributions and impose some dependence / correlation structure on them is that more realistic fat tailed distributions can be used and correlation matrices can be varied to be represent more stressed periods.

In the simple 3 asset  example which follows I show that for a small and fairly perverse case

( Asset A and C are reciprocals of one another ) imposing strict normality on a simulation and then relaxing that assumption to allow for the best fitting distributions to be used we can see the degree to which the normal assumptions lead to an overestimation of returns at the portfolio level and an overestimation of downside risks and underestimation of returns for the one very positively skewed holding.

Many other issues remain concerning Monte Carlo simulations and it is an important field. Not least among these is the generation of random numbers by deterministic methods which Von Neumann himself equated to being in a state of sin. The random number generator within Excel for instance has come in for some particularly harsh criticism and there are many who refuse to use it at all. Fortunately many good substitutes exist including the highly regarded NTRand which uses the Mersenne Twister algorithm.

Anyone who considers arithmetical methods of producing random digits is, of course, in a state of sin.--John Von Neumann (1951)