Markowitz and Van Dijk created a framework for active TAA analysis in which actions at time t are a function of the portfolio structure at time t and the ‘state of nature’ (‘state’) that is expected for time t+1.1 This economic ‘state’ is based on some kind of forecasting system and can be based on macro-economic, micro-economic, political and investment-related factors or a combination of them. We translate return-risk forecasts into 1 of 5 ‘states’ with state 0 being the top-quintile in terms of risk-adjusted return and state 4 being the bottom-quintile. Combining that information with ‘from-to’ state transition probabilities’ (i.e. if the current state is 0 what is the chance that next period’s state is again 0 or 1 etc?) we derive optimal portfolios for various sets of transaction costs, risk-aversion levels and interest rates. We both analyse scenarios with and without the opportunity to shift the asset balance by doing futures transactions. The optimisation formula for deriving the ideal portfolios is based on research done by Markowitz and Levy during the seventies. An investor’s utility level U is defined as:
o with m being the expected return on the portfolio, V the portfolio’s variance, C the total cost level to transform the current portfolio into the desired one and k being the risk-aversion factor. The optimal portfolio is the portfolio that optimizes the net present value of an investor’s expected utility. Space limitations urge us to present the results in a compact manner, but some of the general outcomes are rather fascinating:
There is surprisingly little action with respect to the portfolio of underlying assets. When transaction costs are at the 2% level (including average market impact costs) the equity allocation should be between 60-90% and remain stationary once it reaches that interval no matter what ‘state’ we predict. When transaction costs are at the 1% level the optimal equity allocation would be between 60-80%. This time the optimum is non-stationary, moving back and forth within the interval as a function of the predicted ‘state’.
With respect to the use of futures, the activity level for futures trading is very cost-dependent. With futures costs of 10 basis points or less most of the activity is in futures trading and not in modifications of the underlying asset mix. When futures are more expensive, modifications in the underlying asset portfolio are preferred. However, the number of portfolio shifts is relatively small and they only occur when the manager expects a very positive ‘state’ (state 0) or a very negative ‘state’ (state 4).
In general, even when futures costs are relatively low, trading activity is highly correlated with the level of exceptionality of ‘state’ forecasts. Expected equity return levels (on an annualised basis) should be either more than approximately 3% higher or 3% lower than in an average year. The only exception occurs when futures are extremely cheap (1 basis point or less).