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

The foundation of Modern Portfolio Theory (MPT) was laid by Harry Markowitz in 1951. While theoretically sound, Markowitz's model had limitations, primarily the requirement of a vast number of inputs for portfolio optimization—a significant hurdle, especially with thousands of listed securities. For instance, calculating covariance and correlation coefficients between securities for diversification becomes computationally intensive and vulnerable to inaccurate inputs. The problem is exacerbated by situations where negative covariances or weights are suggested, which rarely exist in real-world markets where securities tend to move together. These limitations prompted William Sharpe, a doctoral student under Markowitz's guidance, to develop a simpler approach: the single-index model (also known as the market model). Sharpe's model posits that instead of evaluating the relationships between individual securities, securities should be compared against a common market index. This simplification addresses the computational challenges of the original Markowitz model and leverages the fact that macroeconomic factors influence the performance of all firms. The model assumes that a single market index can capture the impact of these common factors, while firm-specific factors affect individual securities. Sharpe's single-index model has been widely adopted and recognized, culminating in both Markowitz and Sharpe receiving the Nobel Prize for their contributions to MPT. This study aims to test the Sharpe single-index model's effectiveness in portfolio selection within the Indian mid-cap sector, a market segment inherently more volatile than large-cap stocks. It contributes to the existing literature by being one of the first to specifically evaluate the model's applicability to Indian mid-cap companies, filling a gap in existing research primarily focused on large-cap stocks or other international markets.

Markowitz's mean-variance model, while groundbreaking, faced limitations that were addressed by subsequent refinements from Sharpe, Lintner, and Mossin. While criticisms exist at the axiomatic level, extensions like the incorporation of a risk-free rate enhanced the model's practical appeal. The model's ability to quantify risk and relate it to expected returns has been widely validated. However, practical investment constraints such as cardinality restrictions (limitations on the number of assets in a portfolio) and floor-ceiling constraints (limitations on the proportion of investments) have led researchers to explore meta-heuristic techniques and alternative models, such as the mean-semivariance model, to enhance portfolio optimization. Further, while numerous studies have applied Sharpe's single-index model globally (including studies in India, Nigeria, and Bangladesh) to demonstrate its efficacy in portfolio construction, most have focused on large-cap stocks, not the riskier mid-cap segment. Existing studies have also lacked comprehensive comparative analyses between portfolios built with and without the single-index model. This research addresses this gap by focusing specifically on the Indian mid-cap market and provides a direct comparison to benchmark index performance.

This study uses secondary data—adjusted closing prices for five years (2017-2021)—obtained from the National Stock Exchange of India (NSE) website. The sample comprises stocks from the Nifty Midcap 100 Index, representing the top 100 mid-cap companies listed on the NSE. The Nifty Midcap 100 Index uses the free-float market capitalization method and serves as a benchmark for evaluating market movement in the mid-cap segment. Mid-cap stocks, while potentially yielding high returns, are riskier than large-cap stocks due to factors such as higher volatility and inconsistent cash flows. The selection process for the optimal portfolio involved these steps: 1. **Yearly returns calculation:** Yearly returns for each stock were calculated using the logarithmic method (Log P₁/P₀, where P₁ and P₀ are the closing prices in year 1 and year 0, respectively). 2. **Beta calculation:** Beta, a measure of systematic risk, was calculated for each stock using MS Excel's regression function, which determines the relationship between individual stock returns and market index returns (Nifty Midcap 100). 3. **Systematic and Unsystematic Risk Calculation:** Systematic risk was computed as Beta² * variance of index (β²σₘ²), while unsystematic risk (firm-specific risk) was the difference between total variance and systematic risk (σᵢ² - β²σₘ²). 4. **Risk-Free Rate:** The risk-free rate of return (Rₑ) was set at 6.1%, the average 10-year G-sec bill rate in India in 2021. 5. **Negative Return Elimination:** Stocks with negative returns over the study period were excluded. 6. **Excess Return over Beta Calculation:** The excess return over beta ratio [(Rᵢ - Rₑ)/β] was calculated for each security. 7. **Sorting and Cut-off Rate Determination:** Stocks were ranked in descending order of excess-return-over-beta ratio. The cut-off rate (C) was calculated using the formula: C = (Rᵢ - Rₑ)β / (1 + Σ(Rᵢ - Rₑ)βᵢ), where σₘ² = market variance; σᵢ² = stock variance. The highest value of C was selected as the benchmark cut-off rate. 8. **Optimal Portfolio Selection:** Only stocks with excess-return-over-beta ratios exceeding the cut-off rate were included in the optimal portfolio. 9. **Weight Calculation:** The weights (Xᵢ) for each selected security were calculated using Xᵢ = Zᵢ / ΣZᵢ, where Zᵢ = (Rᵢ - Rₑ)/βᵢ - C* (C* represents the selected cut-off rate). 10. **Portfolio Performance Comparison:** The returns and risk of the optimal portfolio were compared to the benchmark Nifty Midcap 100 index using standard formulas for portfolio return and portfolio variance. Microsoft Excel 365 was utilized for all calculations.

The analysis of the Nifty Midcap 100 index revealed that after eliminating stocks with negative returns and applying the Sharpe single-index model selection criteria, an optimal portfolio comprising 11 companies across eight sectors was identified. The model suggested a concentrated portfolio, with Metropolis Healthcare Ltd., Varun Beverages Ltd., and Coforge Ltd. holding the largest weights (31.07%, 26.8%, and 17.11%, respectively). The optimal portfolio's return over the five-year study period (2017-2021) was a substantial 50.76%, significantly outperforming the Nifty Midcap 100 index's return of 15.61% during the same period. Furthermore, the optimal portfolio exhibited considerably lower risk (12.86%) compared to the benchmark index (23.02%). This demonstrates that a well-constructed portfolio based on Sharpe's single-index model can achieve significantly higher returns with reduced risk, particularly within the more volatile mid-cap sector. The diversification across eight different sectors within the 11-stock portfolio suggests that over-diversification in the market index was a factor contributing to its inferior performance compared to the optimized portfolio.

The superior performance of the Sharpe model-generated portfolio compared to the Nifty Midcap 100 index highlights several key aspects of portfolio construction. The model successfully identified a portfolio that outperformed the market index—a portfolio on the capital market line (CML) with higher returns and lower risk. This shows that the single-index model can help to reduce risk through better stock selection, and this even applies in a sector where risk is higher than normal. The results also point to the potential drawbacks of over-diversification, as the broader market index's performance suffered in contrast to the optimized, concentrated portfolio. The model's ability to achieve this without resorting to short selling makes it a practical tool for a broader range of investors. While the study demonstrates the model's effectiveness over the specific study period, it's crucial to acknowledge that its performance could vary under different market conditions.

This study successfully demonstrated the superiority of Sharpe's single-index model for portfolio optimization within the Indian mid-cap market. The constructed optimal portfolio consistently outperformed the market benchmark in terms of both return and risk over the five-year period, highlighting the model's effectiveness. This model offers a simple yet powerful approach to portfolio construction. Future research could extend this study to incorporate other asset classes (debt, real estate, etc.), different market segments (small-cap stocks), or alternative risk measures to further refine portfolio optimization strategies. A longitudinal study across multiple market cycles would also be particularly beneficial to evaluate its robustness across changing market conditions.

The study's primary limitation is its reliance on historical data. Past performance is not necessarily indicative of future returns, and extreme market conditions might lead to different results. The assumption of constant betas and a constant risk-free rate are simplifications that may not always hold true in dynamic market environments. The study focuses solely on quantitative factors (risk and return), neglecting qualitative factors (management quality, industry trends, etc.) which could impact security selection. Further research into how these limitations can be addressed would be beneficial to both this specific model and others similar to it.