The need for a practical long-term portfolio management decision model is increasing. In addition, the need for more diversification than is recommended by portfolio optimization models has become apparent. Considerable research focuses on short-term conditional volatility models, or suggests using a string of short-term volatility models in multiple periods. Digital portfolio theory (DPT) [2] is a long-term portfolio selection model since it includes non-overlapping mean-reversion variances for different horizons. DPT is a normative model for portfolio optimization that includes the importance of holding period and horizon risk in single period solutions. This paper extends the research of Jones by clarifying the horizon structure of DPT and examines portfolio size constraints to control diversification.

Solutions to the maximum flow, risk constrained portfolio network have focused on using Modern portfolio theory (MPT) [1] to construct mean-variance efficient portfolios. The nonlinear MPT model provides a solution to the generalized portfolio network by adding a parametric quadratic variance side-constraint to the maximize flow objective function. In practice, MPT solutions are myopic, ill-conditioned, unstable, and are sometimes inappropriate and not intuitive. MPT solutions do not permit the user to specify the number of securities in solution portfolios. Adding integer constraints to quadratic MPT is NP-hard with computational effort that grows exponentially with universe size. Conventional portfolio optimization models do not have the capability to apply integer constraints. This is primarily because portfolio theory models are risk constrained optimization models and therefore nonlinear. Controlling portfolio size is not practical since solutions are NP-hard.

Digital portfolio theory (DPT) gives a solution to the risk constrained stochastic portfolio network by representing variance non-parametrically with orthogonal mean-reversion components. The DPT methodology solves mean-variance optimization using multiple linear constraints that constrain multiple long-term mean-reversion risks. The linear DPT solution can be solved quickly for large universes. DPT is non-myopic since it uses information about the risk of mean-reversions in return processes. It is a long-term investment model since it finds horizon dependent solutions. Long and short horizon mean-reversions contribute to single period risk. DPT finds single period mean-variance solutions to the long horizon problem. DPT only considers long horizon variance, volatility is not included. DPT represents return stochastic processes as digital signals and the power spectral density (PSD) describes the risk characteristics of the multipliers in the portfolio network. Solutions are more appropriate because they control exposure to long and short horizon variances of calendar and non-calendar length expected returns. The independent control of horizon based variances allow investors to find portfolios that satisfy long and short horizon risk requirements based on their holding periods. For a given holding period investors will have hedging demand for shorter horizon risk and speculative demand for longer horizon risk. Because DPT offers a time dimension to risk assessment, the holding period of the investor plays a significant role in the optimal decision. As holding periods shorten and as markets change, optimal portfolios must be rebalanced to adjust horizon risk levels, to satisfy hedging and speculative demands for mean-reversion risk.

Recent research suggests that the relatively small portfolios found using MPT, or DPT should be larger do the higher volatility in the markets today. In the 1970s 20 randomly selected stocks could eliminate unsystematic risk while in the 1990s it required 50 stocks to eliminated unsystematic risk. An investor’s subjective estimate of the number of securities that should be held in a portfolio may differ markedly from the number recommended by an optimization model. Small portfolios may be subject to individual idiosyncratic (active) risk while very large portfolios may approach an indexing strategy with high passive risk. The larger the portfolio, the larger the brokerage fees required to keep it rebalanced. The holding period risk tolerance will have a bearing on the number of assets to be held by a particular investor. Active portfolio managers may prefer concentrated portfolios with fewer securities to capitalize on forecasts. Alternatively passive investors may have longer holding periods, trade less frequently, and prefer larger portfolio sizes.

This paper presents a mean-variance-autocovariance portfolio selection decision support application that allows the number of securities in the optimal portfolio to be pre-specified by the investor. Zero-one variables are used to control portfolio size in the stochastic portfolio network. Size constrained optimal portfolios can be used to meet investors’ diversification objectives. Portfolio managers may have strong convictions regarding the size of their portfolios. Integer constraints can be used to find larger optimal portfolio solutions resulting in more diversification. Alternatively smaller portfolios can be found to benefit from special situations and still achieve optimal diversification. Zero-one integer side-constraints and DPT allow control of optimal portfolio size, turnover, and diversification. Integer constraints can also be used to include fixed trading costs in optimal portfolio solutions [3].

The paper defines and tests portfolio optimization solutions assuming the stochastic portfolio network model is the appropriate representation of the problem and DPT gives the best solution when expected returns are timevarying. In the DPT problem non-integer variables are used to solve for maximum portfolio return while constraining calendar and non-calendar length mean-reversion risk. Integer variables are used to control portfolio size. The optimal solution will depend on risk profile and portfolio size preference. The risk profile will depend on the investors holding period and hedging and speculative demands for mean-reversion risk. The mixed integer DPT solution finds optimal portfolios based on the investors preferences with respect to portfolio size, horizon risk, systematic risk, and unsystematic risk. We test the DPT model with zero-one constraints and find it effective in identifying size constrained optimal asset allocations. Low risk investors are constrained to small portfolios while high risk investors can hold large portfolios with small reduction in return.

Figure 1 shows the one-period generalized portfolio network [4,5]. The multipliers represent return and are uncertain or stochastic. When the input to the portfolio at node 0 is constrained to be one the output of the portfolio network from node 1 is the stochastic portfolio return. In this portfolio network we are assuming no negative flows (short sales) and no arcs from right to left representing borrowing or leverage. Jones [6] has examined these cases in the stochastic portfolio network framework. The maximum flow network is the natural structure for the return maximization portfolio selection problem. The arc flows, w_j, are constrained to represent percentages of the initial investment.