Examining the superiority of the Sharpe single-index model of portfolio selection: A study of the Indian mid-cap sector

The paper situates its research within modern portfolio theory, noting that while Markowitz’s 1952 mean-variance framework established the benefits of diversification, it required an impractically large number of inputs and relied on assumptions (e.g., negative covariances) often inconsistent with real market behavior where securities tend to move together. To address these issues, William Sharpe proposed the single-index (market) model, comparing each security to a common market index that captures shared macroeconomic influences, thereby simplifying portfolio selection and optimization. The study’s purpose is to test whether an optimal portfolio constructed via Sharpe’s single-index model can outperform the benchmark market portfolio in India’s mid-cap segment, which is riskier and more volatile than large caps. The research question is whether the Sharpe single-index model yields a superior risk-return profile versus the Nifty Midcap 100 index over a defined period.

The review acknowledges Markowitz’s foundational contribution to asset selection under uncertainty and Tobin’s inclusion of a risk-free asset, while highlighting practical limitations later addressed by Sharpe, Lintner, and Mossin. Subsequent research validated aspects of CAPM and mean-variance methods (e.g., Harris, 1980; Mullins, 1982), with Kaplan (1995, 2017) defending and extending Markowitz’s framework. To overcome real-world constraints (e.g., cardinality, floor-ceiling), meta-heuristic and alternative risk measures (e.g., mean-semivariance) have been explored, and fundamentals-based portfolio construction has also been used. Numerous studies in India and abroad applied Sharpe’s single-index model, generally finding improved risk-return trade-offs across sectors and markets (e.g., Saravanan and Natarajan, 2012; Gupta, 2008; Mandal, 2013; Mahmud, 2019, 2020; Yahayah and Ikani, 2020; Lal and Rao, 2016; Sangeetha et al., 2021; Sen and Dutta, 2022). Some studies augmented the model by vetting fundamentals of selected securities (Yadav and Sharma, 2020). Contrarian evidence exists (Naqvi, 2000), arguing it may not always build an optimal portfolio, although the unobservability of the true market portfolio limits definitive rejection. Most prior work focused on large-cap stocks; the literature lacks studies on mid-cap portfolios in India, motivating this research to assess the model’s efficacy for the riskier mid-cap segment and to compare model-based portfolios with a benchmark index.

Design: Empirical construction of an optimal portfolio of Indian mid-cap stocks using Sharpe’s single-index (market) model and comparison to the benchmark Nifty Midcap 100 index. Data: Secondary data of adjusted closing prices from NSE India; annual data for five years (January 1, 2017 to December 31, 2021). Tools: Microsoft Excel 365 (including Data Analysis Toolpak regression). Sample: Constituents of Nifty Midcap 100 index. Key steps: - Compute annual logarithmic returns for all stocks and the index: log(P1/P0). - Estimate each stock’s beta via regression of stock returns on index returns. - Compute systematic risk of each stock as beta^2 times market variance; compute unsystematic risk as total variance minus systematic risk. - Use a constant risk-free rate Rf = 6.1% (average 10-year G-sec rate in 2021). - Exclude securities with negative returns over the study period. - Calculate excess return-to-beta ratio for each stock: (Ri − Rf)/βi. - Sort stocks by descending excess-return-over-beta. - Calculate and iterate the cut-off rate C; select the highest C as the benchmark (found to be 40.535%). - Select securities with (Ri − Rf)/βi greater than the cut-off rate. - Compute weights using Zi = (Ri − Rf)/βi − C* and Xi = Zi / ΣZi, where C* is the highest cut-off rate. - Portfolio metrics: Portfolio return Rp = Σ Xi Ri. Portfolio risk computed using variance-covariance aggregation across selected securities. Assumptions of the Sharpe single-index model: homogeneous expectations; uniform holding period; borrowing/lending at the risk-free rate; security price movements driven by common economic conditions; selected index as a market proxy. Study limitations acknowledged in design: constant beta and Rf assumptions; focus on quantitative factors; market uncertainty; results reflect the chosen period; investor-specific assumed values may vary.

- After screening and ranking by excess return over beta, 11 companies were selected into the optimal portfolio, spanning eight sectors (Healthcare; Information Technology; FMCG; Consumer Services; Capital Goods; Chemicals and Pharmaceuticals; Consumer Durables; Realty), indicating strong diversification. - Highest cut-off rate C identified was 40.535%. - Selected securities (per Table 4): Metropolis Healthcare Ltd.; Coforge Ltd.; Varun Beverages Ltd.; Trent Ltd.; Astral Ltd.; Aarti Industries Ltd.; Navin Fluorine International Ltd.; Dixon Technologies (India) Ltd.; Laurus Labs Ltd.; Godrej Properties Ltd. (11 firms total as per study). - Weighting (as summarized in the text): approximately 31.07% Metropolis Healthcare; 26.80% Varun Beverages; 17.11% Coforge; 12.02% Trent; 7.06% Aarti Industries; 3.89% Astral; small residual weights allocated among Navin Fluorine, Dixon Technologies, Laurus Labs, and Godrej Properties. - Performance comparison over Jan 2017–Dec 2021: • Optimal portfolio return: 50.76% (mean annual) • Benchmark (Nifty Midcap 100) return: 15.61% (mean annual) • Optimal portfolio risk (standard deviation): 12.861% • Benchmark portfolio risk (standard deviation): 23.02% - Conclusion from results: The Sharpe single-index model constructed portfolio delivered substantially higher returns with substantially lower risk than the market benchmark over the study period, suggesting superior risk-adjusted performance and benefits of targeted diversification versus broad index exposure.

The study frames the Nifty Midcap 100 index as a proxy for the market portfolio for the mid-cap segment. Under the Capital Market Line of CAPM, the market portfolio lies among efficient combinations of risk and return; portfolios to its right typically require leveraging. The Sharpe single-index model, however, enables construction of a higher-return, lower-risk portfolio relative to the index over the same period without short selling. The observed outperformance suggests that broad index portfolios may suffer from over-diversification, potentially diluting returns, whereas a model-driven, selective approach improves the reward-to-risk profile. The findings therefore endorse the practical utility of the single-index model for investors in the mid-cap segment, given appropriate security selection and diversification across sectors.

Within the fixed five-year period studied (2017–2021), an optimal mid-cap portfolio built with Sharpe’s single-index model outperformed the Nifty Midcap 100 benchmark, achieving markedly higher returns with lower risk. The resulting portfolio was well diversified across 11 securities and eight sectors, demonstrating that a simplified, index-based selection framework can yield superior outcomes versus a broad market index in the mid-cap space. Despite its simplifying assumptions and reliance on historical data, the model eases computational burdens inherent to mean-variance optimization and remains accessible to both individual investors and portfolio managers. Future applications could extend to small-cap portfolios, sectoral comparisons, and multi-asset portfolios (equity, debt, mutual funds), as well as testing robustness across different market conditions and incorporating dynamic betas or time-varying risk-free rates.

- Reliance on historical data; performance may not persist under extreme or changing market conditions. - Assumption of constant beta, though beta can vary over time. - Assumption of a constant risk-free rate, which can change with monetary policy. - Focus on quantitative risk-return factors; security prices are influenced by numerous qualitative factors not modeled. - Market conditions are uncertain; results reflect the specific 2017–2021 period. - Assumed values and parameters (e.g., inputs, constraints) may vary across investors. - The selection process is static, while markets are dynamic; this may affect out-of-sample performance.