# Optimal Dynamic Strategies on Gaussian Returns

## TL;DR

This paper derives analytical formulas for the moments of Gaussian-based dynamic trading strategies, revealing the importance of skewness and kurtosis in positive Sharpe ratio strategies and proposing improved statistical methods.

## Contribution

It provides closed-form expressions for moments of Gaussian dynamic strategies and introduces new statistical techniques for analyzing their performance metrics.

## Key findings

- Positive skewness and kurtosis are key in profitable strategies.
- TLS and CCA are effective for multi-asset Sharpe maximization.
- New standard errors for Sharpe ratios, skewness, and kurtosis are derived.

## Abstract

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.