Gaussian Dynamic Trading: Return Moments and Sharpe Estimation

Summary

This paper derives the first four moments of returns for dynamic strategies when asset returns and trading signals or weights are jointly Gaussian. It covers signals such as linear filters of past returns and forecasts from time-series models, expressing strategy outcomes through correlations between signals and future returns.

The analysis argues that positive Sharpe dynamic strategies necessarily have positive skewness and excess kurtosis under its assumptions. For choosing weights to maximize Sharpe, it favors total least squares over ordinary least squares, and canonical correlation analysis for multiple assets. It also develops standard errors for Sharpe ratios and for strategy skewness and kurtosis, and states that the Sharpe estimates are tighter than a commonly used alternative. The results are extended asymptotically to many stationary time series. These conclusions depend on the Gaussian setup and its assumptions; the document provides no specific dataset, numerical performance results, or practical trading costs.

Key ideas

• The paper models dynamic strategy returns using Gaussian returns and Gaussian signals or weights.
• It derives the first four return moments from correlations between signals and future returns.
• Under the stated assumptions, positive Sharpe strategies have positive skewness and excess kurtosis.
• Total least squares is proposed for single-asset Sharpe optimization, with canonical correlation analysis for multiple assets.
• The paper derives standard errors for Sharpe ratios, skewness, and kurtosis.

Tags

# Optimal Dynamic Strategies on Gaussian Returns


# Optimal Dynamic Strategies on Gaussian Returns









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.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.