This paper presents a comprehensive review of two foundational portfolio models in modern finance: the Markowitz mean-variance model and the Sharpe single-index model. The study systematically examines their theoretical frameworks, computational requirements, empirical performance, and practical applicability. The Markowitz model, as a model-free optimisation framework, aims to achieve theoretically optimal risk-return trade-offs by fully considering pairwise covariances among assets. However, its high sensitivity to estimation errors and intensive data requirements often limit its empirical usefulness, particularly for large portfolios. In contrast, the index model simplifies covariance estimation by attributing asset return variations to a common market factor, thereby reducing parameter dimensionality and enhancing computational efficiency. Multi-factor extensions further improve explanatory power while maintaining practical feasibility. By reviewing existing literature and empirical evidence, this paper highlights the strengths, limitations, and appropriate application scenarios of both models. The findings provide guidance for investors, portfolio managers, and researchers in selecting or integrating portfolio models to balance theoretical rigor with empirical robustness and operational efficiency.
This research paper discusses the issue of out-of-sample robustness of portfolio optimization within a complex market environment. Within the mean-variance framework, we compare four methods for covariance estimation and weighting: a baseline model based on sample covariances, the Ledoit-Wolf shrinkage estimator, Garber's robust covariance estimator, and a minimum-variance model incorporating an L2 regularization term. This study uses data from 60 American stocks from 2019 to 2021, with four rebalancing cycles (5, 20, 60, and 100 days) were established, and a moving-window backtesting method was applied to evaluate the performance of the various methods in terms of returns, risk, and out-of-sample stability. The empirical results indicate that the L2 normalization model generates higher cumulative returns and Sharpe ratios during most rebalancing periods, demonstrating effective stabilization of the weights; Garbner's robust covariance technique demonstrates consistent resilience to downturns in high-volatility market environments; And Reduit-Wolf's contraction estimates also show a relatively stable improvement in sample skewness compared to the benchmark model. These results suggest that including a covariance noise or normalization term in the traditional expected value and variance-based approach to portfolio management helps reduce the impact of estimation error and market volatility on portfolio performance. The period of analysis for this paper is specified. Furthermore, it covers the observation period, highlights limitations related to market comprehensiveness and parameter selection, and provides guidance for future research by incorporating financial data from diverse sources and more advanced robust optimization techniques.
Zichun Fu· Journal of Innovation and De...· 0 citations
In mean-variance portfolio analysis, the efficient frontier represents the optimal trade-off between expected return and risk, assuming stable underlying parameters. This paper investigates portfolio fragility: the instability of optimal weights, risk-adjusted performance, and diversification when model inputs and construction choices are jointly perturbed. Combining constrained Markowitz optimization with variance-based global sensitivity analysis (Sobol indices), we map out how input uncertainty and portfolio-construction choices propagate along the target-return dimension. Using an empirical universe of multi-asset exchange-traded funds (ETFs), we find a distinct transition in the sensitivity structure: in the baseline experiment, lower target returns are dominated by l2 regularization, whereas aggressive return requirements become increasingly sen- sitive to the weight cap and expected-return perturbations. This shift coincides with a sharp drop in effective diversification and a rise in weight dispersion. We extend the analysis to a multi-universe fragility atlas, showing that under a com- mon absolute concentration rule, the smallest universe is weight-cap-driven in the aggressive return region, while larger sampled universes remain more often regularization-driven. The fragile frontier serves as a direct diagnostic tool to evaluate the structural robustness of constrained optimizers without altering the underlying allocation rule.
Stefano Pellegrino, Giulia Vannucci, R. Siciliano· 0 citations
The rapid growth of retail investors in Indonesia, from 2.48 million in 2019 to over 20 million by 2025, underscores an urgent need for empirically grounded portfolio optimization frameworks adoptable into practical tools such as robo-advisory systems. This study applies the Markowitz Mean-Variance model to construct and evaluate an optimal stock portfolio from the LQ45 index over January 2022 to December 2025, a post-pandemic period characterized by predominantly adverse risk-adjusted returns. Using daily closing price data for 27 consistently listed LQ45 stocks (958 trading days) from the Indonesia Stock Exchange (IDX), with sample consistency verified through official BEI constituent evaluation announcements, this study constitutes an ex-post empirical analysis designed to isolate the mathematical efficacy of the Mean-Variance model under adverse market conditions. The analysis encompasses individual return and risk profiling, construction of a 27x27 covariance matrix, efficient frontier derivation via constrained quadratic optimization using Microsoft Excel Solver (GRG Nonlinear), and Sharpe ratio-based performance evaluation against a risk-free rate of 5.3281% per annum (average BI Rate 2022-2025). The core finding is that 19 of 27 sample stocks (70.4%) generated negative Sharpe ratios, confirming the inadequacy of undiversified single-stock strategies in adverse markets. In contrast, the tangency portfolio achieved a Sharpe ratio that materially surpasses all 27 individual stocks, establishing that quantitative portfolio optimization delivers superior risk-adjusted outcomes precisely when markets are most challenging. These results validate the enduring relevance of Modern Portfolio Theory in Indonesia's capital market and provide a transparent, replicable framework for investors and practitioners.
Irfan Andi Pramudya, Intan Shaferi· The International Conference...· 0 citations
In the context of normalized volatility in the A-share market, traditional VaR models are unable to effectively capture the tail risk of extreme market conditions, while Expected Shortfall (ES) can make up for this deficiency. This paper selects 12 A-shares from two equally weighted stock portfolios within and outside the CSI 300 index, with a sample period from May 2022 to April 2026. It uses three methods—historical simulation, parametric method (normal), and GARCH (1,1)-t—to estimate VaR and ES, and conducts backtesting using the Kupiec POFF test, the Christoffersen conditional coverage test, and the Acerbi-Székely Z2 test. The results show that GARCH (1,1)-t can more accurately characterize tail risk at a 99% confidence level. The historical simulation method is generally robust, while the normal parameter method tends to underestimate systematic risk. ES is greater than VaR in all scenarios, which verifies its stronger ability to capture extreme losses. The conclusion remains unchanged after replacing the confidence level and portfolio weights. This paper recommends that A-share investors and regulatory agencies use ES as an important supplementary indicator to VaR, and prioritize the use of GARCH-t or historical simulation methods for risk measurement.
Yuhui Gao· Advances in Economics, Manag...· 0 citations
This paper proposes methods to estimate and compare asset pricing models in settings
with a large number of test assets. Models are specified through a linear stochastic discount
factor (SDF). We propose two regularization schemes to extend the Hansen-Jagannathan
distance to high-dimensional environments. In addition to stabilizing the inversion of the
covariance matrix, the proposed regularizations admit an economic interpretation as relaxing
the exact pricing restrictions, thereby accommodating market frictions.
We derive the asymptotic properties of the SDF parameter estimator under a double
asymptotic framework in which both the cross-sectional and time dimensions grow. These
results allow for inference on whether individual factors are priced. We further develop tests
for comparing competing asset pricing models under misspeci cation, providing a formal
procedure to identify the least misspecified model. The analysis covers both nested and
non-nested specifications.
An empirical application compares 4 models using a dataset of 647 test portfolios.
This paper develops a model-driven analytical framework for portfolio optimization under a multidimensional Black–Scholes model with time-varying parameters, where both the drift and volatility functions evolve linearly over time. Within this framework, explicit closed-form expressions are derived for the covariance matrix of normalized asset prices and subsequently incorporated into the classical Markowitz mean–variance framework to obtain analytical representations of the global minimum-variance portfolio, the mean–variance efficient portfolio, and the corresponding efficient frontier. The proposed methodology establishes a direct connection between continuous-time stochastic asset-price modeling and portfolio optimization through a model-implied covariance structure. Its practical implementation is investigated through both numerical experiments and an empirical study using daily stock price data from 20 constituents of the S&P 500 index over the period 2020–2024. Monte Carlo simulations demonstrate the finite-sample sensitivity of portfolio optimization to covariance estimation, while the empirical analysis illustrates how the estimated model parameters, obtained using the maximum likelihood framework of Aït-Sahalia for discretely sampled diffusion processes, can be incorporated into the analytical covariance matrix for constructing efficient frontiers under realistic market conditions. Overall, the proposed framework provides an analytically tractable methodology that integrates continuous-time asset pricing models with classical mean–variance portfolio optimization, offering a coherent model-based covariance representation for portfolio selection under time-varying market environments.
Touch Toem, S. Rujivan, Angelo E. Marasigan· Mathematics· 0 citations
