Mean-Variance Portfolio Selection in Long-Term Investments with Unknown Distribution: Online Estimation, Risk Aversion under Ambiguity, and Universality of Algorithms
Duy Khanh Lam
TL;DR
This work tackles long horizon mean-variance portfolio optimization under unknown future distributions by proposing online learning based sequential MV strategies that adapt as data arrives. The approach avoids strong distributional assumptions, proving that the dynamic portfolios asymptotically match the performance of MV portfolios computed with perfect distribution knowledge, including MV utility, Sharpe ratio, and growth rate, under broad conditions. It introduces fixed and adaptive risk aversion schemes, proves growth enhancements along the efficient frontier for normal returns, and demonstrates universality in stationary markets; it also shows that Bayesian strategies based on true conditionals do not necessarily outperform the proposed methods in time reversible settings. Overall, the paper offers robust, distribution-agnostic strategies for long term investment with theoretical guarantees and practical adaptive mechanisms.
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 follows 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 updates this coefficient over time, forming a dynamic strategy that approaches the optimal empirical Sharpe ratio or growth rate associated with the true coefficient. The performance of these proposed strategies can be universally guaranteed under stationary stochastic markets. Furthermore, in certain time-reversible stochastic markets, the so-called Bayesian strategy utilizing true conditional distributions, based on past market information during investment, 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.