Portfolio Allocation and Model Uncertainty

Portfolio asset allocation problem

The realisation that investors are interested in not only maximising return but also minimising variance of their portfolio led to the development of mean-variance model due to Markowitz (1959). Let us consider the following problem for given

where , is the expected return vector of the set of assets being considered, is the covariance matrix of the returns, is the portfolio weights to be optimally determined and is the feasible set of these weights, including restrictions specified by the investor. can be the covariance of the return forecast errors or the historical covariance of the returns. As increases from zero, the investor becomes more risk-averse and put more emphasis on covariance.

Problem of model uncertainty

The main difficulty with the above is in estimating the correct value of and , and the inherent inaccuracy of these estimates is well known. Typically, one would regress against historical data for estimation of the variables. As with many estimation techniques based on historical data, the quality of the estimates or forecasts are therefore critically dependent on the applicability of the given historical data for forecasting over the given period, which is difficult to determine, if not impossible. Worse still, even if we assume we chose a "good'' set of historical data, the estimation techniques will be bound to further uncertainty.

This misforecasting problem has been tackled through forecast pooling (see, for example, Fuhrer and Haltmaier (1986), Granger and Newbold (1977), Lawrence et al. (1986) and Makridakis and Winkler (1983)). This pooling has been often achieved using stochastic programming approaches where the probability of each scenario is considered and their weighted average is evaluated. (See Kall and Wallace (1994), Pardalos and Sandstrom (1994)). An alternative is discussed by Becker et al. (1986) and Rustem et al. (2000) where the robust pooling is computed using a minimax approach.

[Model uncertainty in action......]

The above figure illustrate the effect of the uncertainty arising from different forecasts. It compares the performance of an optimal portfolio for a given future scenario against its worst possible realisation should an alternative future scenario materialises. In the figure, if an investor constructs her portfolio based on the future scenario 1 and either the future scenario 0 or 2 materialises, her portfolio may deliver -0.1\%, instead of 0.6\%, given the portfolio variance of 10.

Rustem et al. (2000) proposed a discrete minimax model that guarantees the performance in view of worst-case scenarios. The
approach is robust in the sense that the best decision is considered in the presence of rival forecasts and determined simultaneously with the worst-case scenario. Consider the following problem

where there are scenarios, denoting the objective function for the th scenario. This is an extension of the intuitive, but potentially suboptimal, approach based on determining the optimal strategy corresponding to each scenario and adopting the least damaging strategy in view of all the scenarios, Chow (1979)). Optimality in the above problem is no longer based on a single scenario, but on all the scenarios simultaneously. Rustem et al. 2000 proves the robustness of the above problem, provides the algorithm and support the theoretical work through extensive numerical experiments.

[Robustness of a portfolio based on minimax]

The above figure illustrates the robustness of a portfolio based on minimax. A portfolio was constructed taking into account eight plausible future scenarios simultaneously and this portfolio provides a guaranteed minimum return in view of those future scenarios, while retaining a lot of upside potential for the return should a scenario other than the worst-case materialises.

These conflicting forecasts often stem from using different models or using different estimates for the parameters given a particular model. Therefore we can interpret rival scenarios as possible realisations based on different models (or a single model with different estimates for its parameters).

While both model risk and parameter risk contribute to model uncertainties, they differ in an important way - there is a finite number of models one can choose from, while parameters often have an infinite number of candidate values. This difference in dimensionality means that one must treat them accordingly. This is potentially, however, an issue to the approach of Rustem et al. (2000), especially when the portfolio consists of a large number of assets, say, 100, which is not too uncommon in the real world. This is because the number of forecasts grows exponentially as the number of assets increases. For example, if a fund manager has two return forecasts for each asset with two hundreds assets in her portfolio, the number of possible forecasts will be .