Optimal Lag in Dynamical Investments

A portfolio of different stocks and a risk-less security whose composition is dynamically maintained stable by trading shares at any time step leads to a growth of the capital with a nonrandom rate. This is the key for the theory of optimal-growth investment formulated by Kelly. In presence of transaction costs, the optimal composition changes and, more important, it turns out that the frequency of transactions must be reduced. This simple observation leads to the definition of an optimal lag between two rearrangement of the portfolio. This idea is tested against an investment in a risky asset and a risk-less one. The price of the first is proportional to NYSE composite index while the price of the second grows according to the American Discount Rate. An application to a portfolio of many stochastically equivalent securities is also provided.