[R-sig-finance] Monte Carlo and Portfolio Optimisation

[david.jessop@ubs.com]

Hi,

As Patrick Burns said, you have various problems here.  

Resampling is a technique popularised by Richard Michaud and is described quite well in hid book which from memory is called Efficient Portfolio Management (or something like that).  The basic idea is that you assume the return / covariance structure you have is the real structure and then generate lots of samples using this as the distribution, calculate the efficient frontiers and then average them. 

There are numerous problems with this, and many are rehearsed in Bernd Scherer's book (again from memory Portfolio Construction and RiskBudgetting). These include the fact that your covariance matrix isn't the true one - it is a sample itself; log returns aren't normally distributed; and that there will be an upward bias to the weights of stocks with a high standard deviation of returns assuming you have the constraint that the weights in the portfolio will be positive.

One way round some of these problems was implemented (I don't know who did it first) by Robert Rice (www.occamsrazor.com I think - or google his name and I'm sure you'll find it), which is if you have the original time series used to build the covariance matrix then just use bootstrapping on this to generate your various covariance matrices.  

However, all these methods try and fix the problem by averaging everything, yet there is a relatively old academic paper (which as I'm not at home I can't find the reference) which shows that it is errors in the returns that explain virtually all the problems the original author was getting - so "messing around" with the covariance matrix probably doesn't add much. 

This leads to the ideas in the Black-Litterman model, where you condidion your return forecasts on the covariance matrix, explicitly admitting that your forecasts are distributions, not single points.

I hope that helps,

David Jessop
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David Jessop
Head of European Quantitative & Derivative Research
UBS Investment Research

+44 20 7567 9882

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