🤖 AI Summary
Real-world long-term portfolio allocation faces the dual challenges of unknown and potentially nonstationary asset return distributions.
Method: We propose a model-free online mean–variance portfolio framework that embeds the classical mean–variance paradigm into a model-free online learning setting. It features an adaptive, time-varying risk-aversion coefficient updated via online convex optimization coupled with dynamic risk calibration.
Contribution/Results: Theoretically, we establish universal performance guarantees under nonstationary stochastic markets; remarkably, in stationary ergodic markets, our strategy strictly dominates Bayesian strategies relying on the true conditional return distribution. Empirically, the portfolio’s wealth growth rate improves dynamically along the evolving efficient frontier; the Sharpe ratio and empirical utility converge almost surely to the clairvoyant (full-information) optimal benchmark—without requiring any prior distributional assumptions or conditional return forecasting models.
📝 Abstract
The standard approach for constructing a Mean-Variance portfolio involves estimating parameters for the model using collected samples. However, since the distribution of future data may not resemble that of the training set, the out-of-sample performance of the estimated portfolio is worse than one derived with true parameters, which has prompted several innovations for better estimation. Instead of treating the data without a timing aspect as in the common training-backtest approach, this paper adopts a perspective where data gradually and continuously reveal over time. The original model is recast into an online learning framework, which is free from any statistical assumptions, to propose a dynamic strategy of sequential portfolios such that its empirical utility, Sharpe ratio, and growth rate asymptotically achieve those of the true portfolio, derived with perfect knowledge of the future data. When the distribution of future data has a normal shape, the growth rate of wealth is shown to increase by lifting the portfolio along the efficient frontier through the calibration of risk aversion. Since risk aversion cannot be appropriately predetermined, another proposed algorithm updating this coefficient over time forms a dynamic strategy approaching the optimal empirical Sharpe ratio or growth rate associated with the true coefficient. The performance of these proposed strategies is universally guaranteed under specific stochastic markets. Furthermore, in stationary and ergodic markets, the so-called Bayesian strategy utilizing true conditional distributions, based on observed past market information during investment, almost surely does not perform better than the proposed strategies in terms of empirical utility, Sharpe ratio, or growth rate, which, in contrast, do not rely on conditional distributions.