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Mean-Variance Optimization and Statistical Theory

October 3, 2013

dimensions and subspaceMean-variance optimization (MVO) was introduced by Markowitz as a means of compressing forecasts into an expression of portfolio weights for asset allocation. The theory and mathematical concepts have become central to modern finance. Views on MVO are highly polarized- some feel that it is worthless while others think it is the holy grail. Somewhere in the middle lies the truth. In reality the strengths and weaknesses of MVO lie rooted in statistical theory. MVO suffers from a lot of different weaknesses- many of which will not be addressed in this post  – for example MVO assumes that the data is normally distributed (model bias). In this post, the assumption is that MVO will use historical data rather than expected return/factor models. In this case, MVO requires the estimation of a large number of inputs for the return vector and the variance-covariance matrix. The number of different variables to estimate is:


where N= the number of assets in the optimization. The first term 2N= the return and risk estimates, and the second term 1/2(N^2-N) is the number of covariances to estimate.

For a modest sized portfolio of 10 assets, this is 65 different estimates. For the S&P500 this is 125,750 different estimates! If one is employing a dynamic optimization model with small amounts of data, this presents a major problem: the curse of dimensionality is a function of “combinatorial explosion”, where the volume of space increase so fast that the available data becomes sparse. In this context it becomes practically difficult to estimate variables. The amount of data required needs to increase exponentially commensurate with the degree of dimensionality.  This is a major problem since the biggest burden of computation lies with the estimation of the variance covariance matrix. In this situation, a simple equal weight portfolio is expected to perform comparatively well since it does not require any estimates. Research in fact demonstrates that an equal weight portfolio often outperforms mean-variance, minimum-variance and other heuristic methods especially on large data sets. This is especially true when factoring in transaction costs. Another factor is the use of historical returns which are notoriously difficult to estimate (while risk/volatility remains fairly predictable). In this case, historical returns represent the most unbiased estimator (which is good), but they also tend to be a simplistic/poor representation of time series data since it tends to contain a combination of trend, mean-reversion, and random noise. This leads to under-fitting since historical returns tend to dominate the weights of MVO. In statistical theory, this would place MVO as a “high-bias”/”low-variance” classifier.

There have been many different attempts to address these problems. The most frequent complaint about MVO is that small changes in the inputs lead to large changes in the output weights. Michaud- a pioneer in post-modern MVO- introduced re-sampling as a means of bagging/bootstrapping to improve upon out of sample performance and reduce estimation error. Re-sampled portfolios have greater diversification (their “transition maps” look better) but their out-of-sample performance is roughly equal if not sometimes inferior to traditional MVO. Where re-sampling seems to produce more benefit is with minimum variance portfolios and the potential reason will be addressed shortly. Bayesian methods that employ regularization attempt to reduce dimensionality explicitly by penalizing complexity. They “shrink” the observed data to a logical or well-structured prior estimate to prevent over-fitting. Ledoit-Wolf introduced a pioneering method to shrink the variance-covariance matrix method with this type of approach. Research shows that Ledoit-Wolf shrinkage often outperforms re-sampling, but tends to add value primarily in minimum variance portfolios rather than MVO.

In statistics, prediction error is a function of three factors (a good overview here): 1) bias- or the accuracy of the predictor/model (high bias is an overly simplistic predictor/under-fitting while low bias is a very close match/overfitting) 2) variance- this is the sensitivity of the estimate given different data (low variance means that a change in the data does not change the predictor very much, high variance means that a change in the data will change the required predictor) 3) noise- this is self-explanatory. High bias is associated with low variance, and low bias is associated with high variance. Essentially, a model that is overfit (low bias, high variance) is fragile if the future is unlike the past. In contrast, if a model is underfit (high bias, low variance) it is less fragile, but also fails to maximize predictive accuracy. Various methods such as bagging, boosting and regularization have been created to address these problems. All of these methods with the exception of boosting tend to reduce variance at the expense of raising bias. Variance reduction methods will improve out-of-sample performance when a predictor has high variance, and low bias, but they will have much less impact when a predictor has low variance and high bias.

In MVO, the historical return estimates can be considered high bias/low variance, the risk/volatility estimates are in the middle (since risk is a good linear predictor), and the covariances are low bias/high variance since there are so many to estimate. In terms of estimation error impact on the weights, the historical returns are the most important followed by risk and then by the covariances. This implies that MVO is primarily a high bias, low variance problem. Resampling is analogous to bagging/bootstrapping, which is primarily a means of reducing variance-it is less direct at trading off bias for lower variance than regularization/shrinkage and that is why it is less successful much of the time.  Since minimum variance optimization is partially a high variance/low bias problem, these methods have show success in improving out of sample performance. However both re-sampling and shrinkage are ill-equipped to adequately address MVO since the biggest driver is the return estimates which are high bias/low variance. Reducing variance at the expense of bias is the opposite of what needs to be done to improve MVO if returns are still being considered. This explains the failure for both methods to improve out of sample performance when used within MVO. Both re-sampling and shrinkage are adequate for only one component of MVO- the VCV (variance-covariance matrix). Improved estimates of returns using more complex models or more accurate predictors that can then have their variance reduced by the same methods is a more important direction. The same is true for risk/volatility estimates, although they are already so good using historical data that the margin for improvement is small.

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