Journal of Investment Strategies

Optimal dynamic strategies on Gaussian returns

Nick Firoozye and Adriano S. Koshiyama

Dynamic trading strategies, in the spirit of trend-following or mean reversion, represent an only partly understood but lucrative and pervasive area of modern finance. By assuming Gaussian returns and Gaussian dynamic weights or "signals" (eg, linear filters of past returns, such as simple moving averages, exponential weighted moving averages and forecasts from autoregressive integrated moving average models), we are able to derive closed-form expressions for the first four moments of the strategy's returns in terms of correlations between the random signals and unknown future returns. By allowing for randomness in the asset allocation, and by modeling the interaction of strategy weights with returns, we demonstrate that positive skewness and excess kurtosis are essential components of all positive Sharpe dynamic strategies (as is generally observed empirically), and that total least squares or orthogonal least squares are more appropriate than ordinary least squares for maximizing the Sharpe ratio, while canonical correlation analysis is similarly appropriate for the multi-asset case. We derive standard errors on Sharpe ratios that are tighter than the commonly used standard errors from Lo, and derive standard errors on the skewness and kurtosis of strategies that are apparently new results. We demonstrate that these results are applicable asymptotically for a wide range of stationary time series. Possible future extensions of this work to normalized signals, to multi-period returns and to nonlinear transforms, together with extensions to multi-asset dynamic strategies, are discussed.

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