Optimal Investing: Revisiting the Original Arguments of Samuelson and Merton

Sammanfattning

Consider an investor who maintains a portfolio consisting of a risky and risk-free asset. Suppose the investor not only wishes to optimize the allocation of their portfolio but also wishes to consume from the portfolio. Economists Paul A. Samuelson and Robert C. Merton tackled this problem in discrete and continuous time. Since their work, advances have been made in the field of financial mathematics. Here, we revisit their original papers on what we call the portfolio problem. Our main aim is to solve the optimal allocation and consumption processes an investor should adopt for discrete and continuous time. In discrete time, we follow and explain the arguments made by Samuelson and solve the case for independent, identically distributed Bernoulli random variables with a power utility function. We find our optimal weight to be constant for each period and similarly our optimal consumption to be proportional to the portfolio value, where the proportion depends on time. For the continuous model, we solve our optimal weight and consumption using modern methods for a risky asset following geometric Brownian motion and a power utility function. We then begin to investigate some of the claims made by Merton in his paper. We find analogous solutions to the discrete model; namely, a constant optimal weight and an optimal consumption proportional to the portfolio value; however, the proportion is fixed in the continuous solution. The main claim by Merton that we investigate is the convergence of his discrete model to the solution of his proposed SDE. We find that an approximation of his discrete model closely resembles a Milstein scheme, which we support with a numerical approximation of the convergence rate.