This paper studies the performance of self-financing constant proportions trading strategies, i.e. dynamic asset allocation strategies that keep a fixed constant proportion of wealth invested in each asset in all periods in time. We prove that any self-financing constant proportions strategy yields a strictly positive exponential rate of growth of investor's wealth in a financial market in which prices are described by stationary stochastic processes and the price ratios are non-degenerate. This result might be regarded as being counterintuitive because any such strategy yields no increase of wealth under constant prices. We further show that the result also holds under small transaction costs, which is important for the viability of this approach, since constant proportions strategies require frequent rebalancing of the portfolio.