Merton portfolio allocation under stochastic dividends

Current methodologies for finding portfolio rules under the Merton framework employ hard-to-implement numerical techniques. This work presents a methodology that can derive an allocation à la Merton in a spreadsheet, under an incomplete market with a time-varying dividend yield and long-only constraints. The first step of the method uses the martingale approach to obtain a portfolio rule in a complete artificial market. The second step derives a closed-form optimal solution satisfying the long-only constraints, from the unconstrained solution of the first step. This is done by determining closed-form Lagrangian dual processes satisfying the primal-dual optimality conditions between the true and artificial markets. The last step estimates the parameters defined in the artificial market, to then obtain analytical approximations for the hedging demand component within the optimal portfolio rule of the previous step. The methodology is tested with real market data from 16 US stocks from the Dow Jones. The results show that the proposed solution delivers higher financial wealth than the myopic solution, which does not consider the time-varying nature of the dividend yield. The sensitivity analysis carried out on the closed-form solution reveals that the difference with respect to the myopic solution increases when the price of the risky asset is more sensitive to the dividend yield, and when the dividend yield presents a higher probability of diverging from the current yield. The proposed solution also outperforms a known Merton-type solution that derives the Lagrangian dual processes in another way.