Digital Portfolio Theory: Portfolio Size, Versus Alpha, Beta, and Horizon Risk

Abstract

Digital Portfolio Theory (DPT) permits investors to control their risk exposure with respect to multiple investment time horizons. DPT is a theoretical enhancement for estimating efficient portfolios that drops the normal distribution and zero autocorrelation assumptions of Modern Portfolio Theory (MPT) and allows effects of unconditional mean reversion risk at multiple horizons. The time horizon composition of single period risk creates hedging demands based on investor holding period and expectations without regard to changes in expected returns, changes in risk, or changes in state variables. DPT is a mean-variance-autocovariance portfolio optimization paradigm that allows investors to strategically control the risk that patterns occur but includes no information about what patterns other than their length. DPT incorporate more of the structure of the risk process contained in historical information compared to MPT, or CAPM representations. Academic financial research provides little specific portfolio recommendations suitable for long-term horizons. The MPT model assumes investors make decisions myopically in a single period framework. The Merton continuous-time model assumes investors make decisions dynamically based on intertemporal hedging over multiple periods. The DPT model assumes investor's make dynamic decisions based on longer holding periods in a static single period model. DPT analysis provides precise quantitative advice for short, intermediate, and long holding periods.Quadratic MPT does not permit users to specify the size in solution portfolios. An integer constraint added to linear DPT is useful to meet portfolio size requirements. This paper extends DPT to control portfolio size and empirically examines feasible size regions and in-sample optimal allocations for alternative levels of systematic, unsystematic and horizon length risk. The empirical test examines long only DPT solutions from a universe of 73 industries for a 16-year period that includes the October 1987 stock market crash. Optimal portfolios and feasible sizes are examined for three strategies; low versus high total risk, active versus passive, and short versus long horizon risk. Low-risk optimal portfolios and active optimal portfolios are smaller with fewer feasible size possibilities. Optimal industry allocations change with holding period and expectations. Long-term investors should hold smaller portfolios. The majority of allocations are not myopic; however a buy-and-hold strategy is appropriate for some sectors. Gold, oil, gas, and tobacco are included in solutions over multiple holding periods. This study contributes to the theory and practice of portfolio selection and optimization as well as to the question of holding period and horizon risk.