In this thesis we study the performance of a re-balanced portfolio strategy relative to the market portfolio in the presence of transaction costs. The strategy involves re-balancing to fixed weights at regular time steps. We consider an equity market with m Stocks. Our goal is to compare the asymptotic growth rate of such strategies to the market. With the application of an Ergodic theorem, we show that the problem can be transformed to computing the expectation of a functional of the market weights of the stocks. Expressing the gain in the re-balanced portfolio over the market portfolio as a functional of the market weights, we derive the condition under which the growth rate of the rebalanced strategy beats that of the market portfolio. We also show a method to compute the maximum transaction cost that can be paid in order for the rebalanced portfolio to beat the market portfolio. We discuss the result in the context of the Volatility-Stabilized model and the Geometric Brownian Motion model. In the second part of the thesis, we define the optimal re-balancing portfolio that maximizes the growth rate within the class of such fixed weight re-balancing strategies. We study the relationship of the optimum portfolio and optimal growth rate to the re-balancing time step and transaction cost coefficient. Finally, we look at the performance of such re-balancing strategies on real data sets obtained from Yahoo Finance. The study indicates that for small trading time steps, the re-balancing strategy under performs compared to the market in the presence of transaction costs.