Mean-variance optimisation, in the form developed by Markowitz and his successors is a very powerful tool, but like all such tools requires careful use. In particular, it assumes that the returns and risks being analysed are known with sufficient certainty that any estimation risk can be ignored. To cover situations where this assumption is unsafe, a number of solutions have been proposed over the years. A non-exhaustive summary will be found in the bibliography on our Resources page.

One solution (see, for example, Ledoit 1996 and Ledoit and Wolf 1997) is to shrink the covariance matrix; a facility to achieve this is provided in **POW! Frontier**. Alternatively, Jorion 1986 suggests we could use more explicitly Bayesian techniques. For a rather different implementation of such techniques, see **POW! Bayes**. One might also leave the covariance matrix unadjusted, and simply increase the level of risk aversion (ter Horst et al 2002).

In their second 1990 article, Bey, Burgess and Cook proposed a radically different approach: let us assume, they said, that the covariance matrix is normal and use Monte Carlo to resample it, producing a series of alternative matrices, each of which is different from what was actually observed but statistically consistent with it. Then, suggested Jorion 1992, take these matrices one by one and feed them into the optimiser, generating a series of efficient portfolios which (if they share a common characterisic - say return or risk aversion) can then be combined to provide an average solution. This technique was developed further by Michaud (1998) who, instead of averaging by common characteristic, divides each resulting frontier into a given number of equally spaced points, and averages by reference to these points. This development is not an improvement, since when applied in the context of two-way optimisation, that is, by reverse optimisation (qv) followed by forward optimisation, the original *ex hypothesi* efficient portfolio cannot be guaranteed to lie on the efficient frontier.

But the disadvantage shared by all 3 of the Bey/Jorion/Michaud approaches is that they force to assume that the covariance matrix is normal, or at least some other specific shape which will enable us to do the resampling. In a world of hedge funds, fat tails and skewed distributions this is unattractive, but luckily it can be circumvented by the technique first devised by Julian Simon as much as a teaching aid as a research tool, and known as **Bootstrapping**. Its research applications were developed by Bradley Efron (1979 on), and were adopted in the context of optimisation by Liang et al in their groundbreaking 1996 article.

The idea of Bootstrapping is to go back to the original time-series (or other data set) and resample from that directly, developing a series of new sets of observations, each of which can be used to calculate a new covariance matrix and then fed into the optimiser. It is often said that bootstrapping assumes resampling with replacement, but in fact the most common kind, and the one we prefer for most purposes, is the balanced bootstrap, which ensures that each observation occurs an equal (or, in the presence of weighting, a controlledly unequal) number of times. This is achieved by placing the desired number of instances of each observation into the statistical urn, and then resampling from it the required number of times *without* replacement.

Bootstrapping does not require us to assume a particular shape for the distribution, but it does require us to assume that they are i.i.d. If autocorrelation is present, this can be tackled by means of the block bootstrap or by removing the autocorrelation and then replacing it. Neither of these is entirely effective, and improved solutions are the subject of current research.

Returning to our bootstrap optimisation, the resulting portfolios can either be averaged by reference to return (as proposed by Liang et al) or by risk aversion. The latter is our preference, as it allows us to develop a fully parametric bootstrap frontier: see Rice and Daniel 2003.

This is the technique offered by POW! **Robust**. We regret that we are unable for the time being to make it available in the United States of America or any of its territories or possessions.

For more information please contact us