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.

通过在线学习框架将原模型重新设计为一种动态策略，以在统计假设下不受限制地接近真实总结的组合的经验效用、夏普比率和增长率。

未知分布的长期投资中的均值方差组合选择：在线估计，不确定性下的风险厌恶与算法的普适性

Mean-Variance Portfolio Selection in Long-Term Investments with Unknown  Distribution: Online Estimation, Risk Aversion under Ambiguity, and  Universality of Algorithms

Portfolio optimization has been an area that has attracted considerable
attention from the financial research community. Designing a profitable
portfolio is a challenging task involving precise forecasting of future stock
returns and risks. This chapter presents a comparative study of three portfolio
design approaches, the mean-variance portfolio (MVP), hierarchical risk parity
(HRP)-based portfolio, and autoencoder-based portfolio. These three approaches
to portfolio design are applied to the historical prices of stocks chosen from
ten thematic sectors listed on the National Stock Exchange (NSE) of India. The
portfolios are designed using the stock price data from January 1, 2018, to
December 31, 2021, and their performances are tested on the out-of-sample data
from January 1, 2022, to December 31, 2022. Extensive results are analyzed on
the performance of the portfolios. It is observed that the performance of the
MVP portfolio is the best on the out-of-sample data for the risk-adjusted
returns. However, the autoencoder portfolios outperformed their counterparts on
annual returns.

本文通过比较三种投资组合设计方法 (均值 - 方差组合，分层风险平价组合和自编码器组合)，应用于国家证券交易所上列出的十个主题行业的股票历史价格，并在测试数据上进行了表现测试，结果发现 MVP 组合在风险调整回报上表现最佳，而自编码器组合在年收益率上表现最好

投资组合优化：比较研究

Portfolio Optimization: A Comparative Study

This paper presents a comparative analysis of the performances of three
portfolio optimization approaches. Three approaches of portfolio optimization
that are considered in this work are the mean-variance portfolio (MVP),
hierarchical risk parity (HRP) portfolio, and reinforcement learning-based
portfolio. The portfolios are trained and tested over several stock data and
their performances are compared on their annual returns, annual risks, and
Sharpe ratios. In the reinforcement learning-based portfolio design approach,
the deep Q learning technique has been utilized. Due to the large number of
possible states, the construction of the Q-table is done using a deep neural
network. The historical prices of the 50 premier stocks from the Indian stock
market, known as the NIFTY50 stocks, and several stocks from 10 important
sectors of the Indian stock market are used to create the environment for
training the agent.

通过比较三种投资组合优化方法的表现，本文提出了一种基于强化学习的组合设计方法，利用印度股市历史数据进行训练和测试，并使用年回报、年风险和夏普比率等指标进行了比较。

印度股市中使用均值 - 方差、层次风险平价和强化学习方法进行组合优化的比较分析

A Comparative Analysis of Portfolio Optimization Using Mean-Variance,  Hierarchical Risk Parity, and Reinforcement Learning Approaches on the Indian  Stock Market