  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. Assuming Gaussian returns and Gaussian dynamic weights
or signals, (e.g., linear filters of past returns, such as simple moving
averages, exponential weighted moving averages, forecasts from ARIMA 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
modelling the interaction of strategy weights with returns, we demonstrate that
positive skewness and excess kurtosis are essential components of all positive
Sharpe dynamic strategies, which is generally observed empirically; demonstrate
that total least squares (TLS) or orthogonal least squares is more appropriate
than OLS for maximizing the Sharpe ratio, while canonical correlation analysis
(CCA) is similarly appropriate for the multi-asset case; derive standard errors
on Sharpe ratios which are tighter than the commonly used standard errors from
Lo; and derive standard errors on the skewness and kurtosis of strategies,
apparently new results. We demonstrate these results are applicable
asymptotically for a wide range of stationary time-series.
