Introduction

Harry Markowitz laid down the foundation of the Modern Portfolio theory in 1951. Although his model was theoretically sound, it had certain limitations which were later rectified by his own protégé William Sharpe through his single-index modelFootnote 1. The original model given by Markowitz was based on the premise that there are gains from diversification. He propounded through his theory, a method for diversification of securities. Grouping securities with negative relationships given by covariance or correlation coefficients was imperative to his optimisation method (Markowitz, 1952). This model required a large number of inputs. For n securities, the number of inputs according to the Markowitz model would be \(\frac{{n^2 - n}}{2}\). So, if the portfolio manager wanted to select a portfolio from the total number of listed securities in the Indian markets like the NSEFootnote 2 (having around1800 listed securities), or the BSE (with around 4000 listed securities), the number of estimates would run into millions. The quality of these inputs would also affect the optimum portfolio. For example, a classic failure of the model would be when there was a portfolio with three securities A, B, and C with weights 1, 1, and −1 and each having a standard deviation of 20 percent. In this situation, the portfolio variance would be −200—an absurd result as risk can never be a negative number. Another problem was the concept of selecting securities having negative covariances. In the real world, securities tend to move together and hence are found to have positive covariances. Since markets are often moved by sentiments, securities tend to move together in one direction. Empirical evidence also showed that portfolios with securities having positive covariances outperformed Markowitz’s optimum portfolios (Sharpe, 1963). Owing to the limitations discussed and the contrary empirical evidence, the need for a simpler method for portfolio selection had become imperative. William Sharpe was a doctoral student at UCLA majoring in economics and finance. When the time came for Sharpe to write his thesis, Fred Weston suggested that he should meet Markowitz. Thus, Markowitz became Sharpe’s unofficial thesis advisor. Markowitz put him to work and asked him to find a simpler method for portfolio selection and optimisation. Sharpe simplified the model which we now know as the ‘market model’ or the single-index model (Varian, 1993). Sharpe said that instead of comparing each security with another security and trying to find negative covariances and correlations between individual securities for portfolio selection, securities should be compared to some common index. This gave birth to the concept of the market index. Sharpe reasoned that common economic factors such as business cycles, interest rates, technology changes, cost of labour, raw material, inflation, weather conditions, etc., affected the performance of all firms. Unexpected changes in these variables would cause unexpected changes in the prices and returns of all the stocks in the market. Sharpe proposed that all the economic factors could be summarised by one macroeconomic indicator which would move the entire market. Further, it was assumed that all other uncertainties in stock returns were firm-specific, i.e., there was no other form of correlation between the securities. Firm-specific events such as profits, management quality, new inventions, etc., would only affect the fortunes of individual firms and not the whole market or the broad economy in any significant way. Thus, Sharpe proposed the concept of a single market index as the surrogate for all the other individual securities in the market (Sharpe, 1963). Markowitz and Sharpe were awarded the Nobel Prize for their contributions to Modern Portfolio theory.

Review of literature

The Markowitz model has stood out to be a substantive contribution to the theory of individual asset demand under uncertainty. However, it does have limitations at the axiomatic level. Tobin’s extension of the introduction of the risk-free rate to the model given by Markowitz made it even more convincing. Its limitations were resolved to a large extent by the improvisations of Sharpe (1963), Lintner (1975), and Mossin (1966). Harris (1980) pointed out that the model had several applications and had been subjected to rigorous empirical testing which led to important propositions about the risk and its effect on the pricing of the assets. The CAPM and the single-index model received more validation in 1982 when an article published in the Harvard Business Review validated that the model provided a methodology for quantifying risk and translating that risk into estimates of expected return on equity (Mullins, 1982). Kaplan and Seigel in the year 1995 defended the mean-variance concept of Markowitz and brought out that even though the model might not be very practical, it should be understood in terms of a broad perspective. Later, Kaplan went on to create a very generalised functional model based on the pillars of the Markowitz model which he aptly renamed Markowitz 2.0. In Kaplan’s model, the user could select a measure of return and a measure of risk and have a wide choice of return distribution models (2017). Despite getting worldwide acclaim as a breakthrough model, the problems of real-world investment constraints such as cardinality and floor-ceiling drawbacks, meta-heuristic techniques were used by many economists to build optimal portfolios. However, the mean-variance model has usually been the base model which would then be modified to build further models with better optimisation such as the mean-semivariance portfolio selection model (Yahaya, 2010). Studies selecting portfolios based on the fundamentals of stocks such as book-to-market ratio or certain investment indices have also been used by investors to build optimal portfolios (Khatwani, 2021). However, these portfolios again were based on stock variables like balance sheet data and not flow variables.

Several studies in India and abroad (Saravanan and Natarajan, 2012; Dharmalingam and Gurunathan, 2021; Mahmud, 2020; Mandal, 2013; Mohith et al., 2017) have embraced the Sharpe single-index model to prove portfolio efficacy as the model is versatile and it can accommodate changes based on the study being conducted (Lal and Rao, 2016). It is a very simple model as compared to the Markowitz model because it requires very few inputs (Bodie et al., 2020). Based on a study conducted in the Indian pharma sector, an optimal portfolio of pharma companies was created using the Sharpe single-index model. It was found that this portfolio outperformed the Nifty index. A portfolio of 12 stocks was created, and the intrinsic value of the shares selected in the portfolio was calculated. The final portfolio was selected of stocks that showed progressive intrinsic value (S. Sangeetha et al., 2021). Another Indian study that used the mean-variance design to optimise portfolios using Sharpe, Sortino, and Calmar Ratio took samples from six important sectors from the NSE of India. The study, which took place between January 2017 and December 2020, sought to discover the ratio that produced the highest cumulative returns for both the training and test periods for the majority of industries (Sen and Dutta, 2022). A similar Indian study compared 11 sectors using the Sharpe single-index model. This was a novel study wherein the Sharpe model was used to give sectoral weights in a portfolio rather than finding the weights of individual stocks. This study reinforces the versatility of the model (Lal and Rao, 2016). The sectors with the highest cumulative returns for the same ratio were also determined. A Nigerian study used the Sharpe single-index model to find an optimal portfolio of five stocks which decreased the risk of the original 20-stock portfolio (Yahayah and Ikani, 2020). Another such study conducted in Bangladesh for 178 companies listed on the Dhaka Stock Exchange helped build an optimal portfolio of 54 companies and this portfolio had a better risk-return combination than the index as well as individual companies and outperformed both (Mahmud, 2019). An interesting approach to the Sharpe model was brought about by two Indian researchers who analysed the fundamentals of the securities selected by the model. The model basically selected securities based on the yields/returns. But at times the high yields could be on account of bubbles due to insider trading. A more refined approach was adopted by checking the stock fundamentals of the model-selected portfolio (Yadav and Sharma, 2020). Several studies across India examined the efficacy of the single-index model for the stocks listed on the BSE. Nalini (2014) studied 15 Indian stocks and selected four stocks to form an optimal portfolio using the single-index model from the S&P BSE index. Gupta (2008) examined daily market data from April 1997 to April 2007 on a sample of ten industry sectors chosen at random and discovered that investors could significantly improve their reward to risk when compared to market returns. The Sharpe ratio of the optimised portfolio rose from 0.527 to 0.994 (for the S&P Nifty index). Many such sectoral studies have also been conducted for Indian companies listed on the BSE and NSE (Ahuja, 2017; Anithadevi and Mallikharjunarao, 2017; R. and Reddy, 2022; Shriguru and Bagrecha, 2022).

Just as there are enough studies to prove that Sharpe’s model works, there have been studies that prove the contrary. One such study by the Lahore School of Economics has empirically shown that Sharpe’s model does not actually build an optimal portfolio. However, the final verdict of the same study was that since in reality the true market portfolio cannot be observed, it is impossible to disregard the model (Naqvi, 2000).

The current study was taken up to check the efficacy of this model for mid-cap stocks which are riskier than large-cap stocks. Usually, it is more difficult to build a mid-cap stock portfolio than a large-cap stock portfolio due to the volatility of the securities as well the irregularity in the cashflows of companies. Several studies have been found on the stocks listed on the Bombay Stock Exchange and the National Stock Exchange in the Indian context. However, the studies which were reviewed for this paper mostly focussed on large-cap companies (Saravanan and Natarajan, 2012; Ahuja, 2017; Anithadevi and Mallikharjunarao, 2017; Dharmalingam and Gurunathan, 2021; Gupta, 2008; Lal and Rao, 2016; Mandal, 2013; Mohith et al., 2017; Nalini, 2014; S. Sangeetha et al., 2021; Sen and Dutta, 2022; Shriguru and Bagrecha, 2022). The studies conducted in other countries such as Bangladesh also focussed on A group (large cap) companies listed on the Dhaka and the Chittagong stock exchanges (Mahmud, 2019, 2020). Similarly, the study conducted in Nigeria also focussed on large-cap companies listed on the Nigerian stock exchange (Yahayah and Ikani, 2020). Hence, in reviewing the literature, no studies on the mid-cap sector could be identified in India or abroad. Therefore, this study becomes important for investors who might have a preference for investing in the mid-cap sector in India. This study was taken up to incorporate companies listed on the NSE. Many studies have used the model to build optimal portfolios, but the comparative analysis of portfolios constructed with and without the use of the single-index model was seen to be lacking in the studies. This study was taken up to build an optimum portfolio in the mid-cap sector and then compare the risk and return of that portfolio with the risk and return of the benchmark index i.e., the NSE mid-cap 100 index.

Methodology

The study is based on using the Sharpe single-index model to build an optimal portfolio of mid-cap stocks. Since it was established that diversification is a valid strategy for return optimisation, every investor would want to build a portfolio of investments instead of holding individual stocks. This diversification could be across instruments/assets, industries, or even economies. However, retail investors would want to build diversified portfolios across industries and companies. This study has tried to evaluate if a mid-cap stock portfolio constructed using Sharpe’s single-index model outperforms the benchmark index. The study was based on secondary data of adjusted closing prices collected from the website of the NSE of India. Yearly data for 5 years were considered for the study. The stocks were selected from the Nifty mid-cap index of the top 100 mid-cap companies listed on the NSE. The NIFTY mid-cap 100 Index is useful for understanding market movement in the mid-cap segment. It includes 100 stocks listed for trading on the National Stock Exchange (NSE) of India. The index is calculated using the free float market capitalisation method, with the level of the index reflecting the total free float market value of all the stocks in the index relative to a specific base market capitalisation value. The NIFTY mid-cap 100 index can be used for a variety of purposes, including fund portfolio benchmarking, the launch of index funds, ETFsFootnote 3, and structured products (NSE India, 2022). Mid-cap stocks carry greater risk than large-cap stocks. However, they have the potential to yield extremely high returns. Selection of good mid-cap stocks at the right time can be compared to sitting on an undiscovered gold mine. A unique feature of mid-cap stocks is that they would be re-rated if they came under the radar of institutional buyers. That is the time when the value of these stocks could skyrocket. Good mid-cap stocks would certainly outperform large-cap stocks as well as the market index in the bull run. However, they could also plummet very fast during the bear run. Nevertheless, with the proper stock-selection skills and investment disposition, these stocks have the potential to generate very attractive returns. Despite the fact that mid-caps have a long history of strong returns, not all mid-caps are profitable investments. Therefore, an investor seeking to profit from mid-caps must exercise extreme caution. Usually, the value of mid-cap stocks is derived from their growth potential, but this growth potential does not always materialise. This can result in significant value loss. Compared to large-cap stocks, mid-cap stocks represent relatively young companies. Mid-cap investors must be extremely patient, as these companies are typically in the early stages of a business cycle and can take a long time to realise their full potential (10 Important Facts about Indian Mid-Cap Stocks, 2022). However, with the right stock selection, investors could amass a modest fortune by investing in mid-cap stocks. Therefore, it becomes even more important to construct an optimal portfolio with mid-cap stocks.

Owing to these unique characteristics of mid-cap companies, an optimal portfolio was created using Sharpe’s single-index model. Microsoft Excel 365 was used to analyse the data. Microsoft Excel is a spreadsheet programme developed by Microsoft Inc. that is available for Windows, macOS, Android, and iOS. It includes calculating or computation skills, graphing tools, pivot tables, and Visual Basic for Applications, a macro programming language (VBA). Excel is a part of the Microsoft Office software suite.

Steps to calculate the optimal portfolio using the Sharpe single-index model:

  1. i.

    The yearly returns of all stocks in the Nifty 100 mid-cap index were calculated using the logarithmic method. This was done using the log function in MSFootnote 4 Excel given by Eq. (1).

    $${{{\mathrm{Log}}}}\,{{{P}}}_1/{{{P}}}_0$$
    (1)

    where P1 is the closing price of the stock in year 1 and P0 is the closing price of the stock in year 0.

  2. ii.

    The Betas of all the stocks were calculated. Beta represents the relationship between the risk of each stock and the market risk. Beta is a coefficient and a measure of systematic risk of security. Beta in this study shows the rate of change in the mid-cap stock due to a unit change in the benchmarked market index. Betas were calculated using the regression function in MS Excel. The regression function is a part of the Data analysis tool pack add-in available in MS Excel.

    The systematic risk and unsystematic risk of the stock were calculated using Eq. (2) as follows:

    $$\begin{array}{l}{{{\mathrm{Systematic}}}}\,{{{\mathrm{risk}}}}\,{{{\mathrm{of}}}}\,{{{\mathrm{mid}}}} - {{{\mathrm{cap}}}}\,{{{\mathrm{security}}}} = {{{\mathrm{Beta}}}}^2 \ast \\ {{{\mathrm{variance}}}}\,{{{\mathrm{of}}}}\,{{{\mathrm{index}}}} = \beta ^2 \ast \sigma _{{{\mathrm{m}}}}^2\end{array}$$
    (2)
  3. iii.

    The unsystematic risk of each stock in the market was calculated. The unsystematic risk is due to the firm-specific factors of the mid-cap stocks under consideration and is a random error term. It was calculated as a difference between the total risk of the security and the market-related risk. The following formula given by Eq. (3) was inserted in MS Excel to calculate unsystematic risk.

    $$\begin{array}{l}{{{\mathrm{Unsystematic}}}}\,{{{\mathrm{risk}}}}\,{{{\mathrm{of}}}}\,{{{\mathrm{mid}}}} - {{{\mathrm{cap}}}}\,{{{\mathrm{security}}}} = \\ {{{\mathrm{Total}}}}\,{{{\mathrm{variance}}}}\,{{{\mathrm{of}}}}\,{{{\mathrm{security}}}}\,{{{\mathrm{return}}}}-{{{\mathrm{Systematic}}}}\,{{{\mathrm{risk}}}}\\ \sigma_{{{\mathrm{ei}}}}^2 = \sigma_{{{\mathrm{i}}}}^2 - \beta ^2 \ast \sigma _{{{\mathrm{m}}}}^2\end{array}$$
    (3)

    where σi2 is the total risk of the mid-cap stock and σm2 is the variance of the NSE 100 mid-cap index.

  4. iv.

    The risk-free rate Rf was taken as 6.1 percent which was the average 10-year G-sec bill rate in India in the year 2021.

  5. v.

    The securities which had negative returns during the period of study were removed.

  6. vi.

    The market return was calculated using the closing price data of the Nifty 100 mid-cap index using the logarithmic method in MS Excel.

  7. vii.

    The excess return over beta ratio was calculated using the formula given by Eq. (4):

    $$\frac{{R_{\rm {i}} - R_{\rm {f}}}}{\beta }$$
    (4)

    where Ri is the security return and Rf is the risk-free return

  8. viii.

    The stocks were rearranged in descending order from highest to lowest values of excess-return-over-beta ratios. This was accomplished using the ‘sort’ function in MS Excel.

  9. ix.

    The market return for the period of five years from January 1, 2017 to December 31, 2021 was found to be 11.37 percent, and the market risk was calculated and found to be 9.81 percent.

  10. x.

    The cut-off rate was calculated using the formula for cut-off rate using Eq. (5):

$$C = \frac{{\sigma _{\rm {m}}^2\mathop {\sum}\nolimits_{t = 1}^j {\frac{{\left( {R_{\rm {i}} - R_{\rm {f}}} \right)\beta _i}}{{\sigma _{{\rm {ei}}}^2}}} }}{{1 + \sigma _{\rm {m}}^2\mathop {\sum}\nolimits_{t = 1}^j {\frac{{\beta _i^2}}{{\sigma _{{\rm {ei}}}^2}}} }}$$
(5)

where σm2 = market variance; σei2 = stock variance. The above formula was manually entered in MS Excel to get the cut-off rate.

  1. i.

    The excess return over beta ratios was compared with the cut-off rates.

  2. ii.

    The securities which had excess return over beta ratio higher than the cut-off rate were selected in the portfolio and others were rejected.

  3. iii.

    The weights of individual securities were calculated using Eqs. (6) and (7), and an optimal portfolio was built.

$$X_{\rm {i}} = \frac{{Z_{\rm {i}}}}{{\mathop {\sum}\nolimits_{j = 1}^n {Z_j} }}$$
(6)
$$Z_{\rm {i}} = \frac{{\beta _{\rm {i}}}}{{\sigma _{{\rm {ei}}}^2}}\left( {\frac{{R_{\rm {i}} - R_{\rm {f}}}}{{\beta _{\rm {i}}}}} \right) - C^ \ast$$
(7)

Xi is the weight of each security; ƩZj is the summation of all the Zis; C* is the highest cut-off rate selected

The returns of the optimal portfolio over the five-year period were compared to the returns of the market portfolio for the same period.

Assumptions of the Sharpe single-index model:

  1. i.

    Investors have homogenous expectations with respect to return and risk.

  2. ii.

    A uniform holding period is considered for calculating the risk and return of every security.

  3. iii.

    Investors can borrow and lend at a risk-free rate of return.

  4. iv.

    Price movements of securities are influenced by prevailing economic conditions.

  5. v.

    The index selected is a proxy of the market.

Limitations of the study:

  1. i.

    The beta of individual securities is assumed to be constant but, it fluctuates daily.

  2. ii.

    Only the quantitative aspects in terms of risk and return are being considered but, security prices are affected by infinite reasons.

  3. iii.

    The risk-free rate of return is also assumed to be constant, but it could change with the review of monetary policy.

  4. iv.

    The market condition is always uncertain; the result reflects the market of that period.

  5. v.

    Assumed values would vary from one investor to another investor.

Results and findings

All the securities of the Nifty mid-cap 100 index were analysed and their risks, returns, and betas were calculated. The securities data is given in Table 1:

Table 1 Calculation of risk, return and beta.
Full size table

The companies with negative returns were removed from the list and the excess return over beta and the cut-off rate were calculated after arranging the companies in ascending order of excess return over beta. Table 2 shows the ranking of securities. The list has been furnished in Appendix 1.

Table 2 Ranking securities and calculation of cut-off rate.
Full size table

The cut-off rates were calculated, and the highest value of C was selected as the cut-off benchmark. The value of the cut-off was 40.535 percent. It has been emboldened in the above Table 2. All the values of C are depicted in Table 3.

Table 3 Selection of securities with excess return to beta higher than cut-off rate.
Full size table

The securities with an excess return over beta higher than the cut-off rate were selected to form the optimal portfolio according to the Sharpe single-index model. Table 4 shows the securities that were selected in the optimal portfolio.

Table 4 Securities selected for the optimal portfolio.
Full size table

Based on the criteria of selection, only eleven companies were selected as part of the optimal portfolio according to the single-index model. The portfolio of the 11 companies has included securities from various sectors such as Healthcare, Information Technology, Fast Moving Consumer Goods, Capital Goods, Consumer Services, Chemicals and Pharmaceuticals, Consumer Durables, and Realty. Hence, it can be said that the portfolio of mid-cap stocks selected by the model was fairly diversified.

The weights of individual securities selected were found and are depicted in Table 5.

Table 5 Calculation of weights for optimal portfolio.
Full size table

According to the model, it could be observed that the maximum investment of around 31.07 percent should be recommended in Metropolis Healthcare Ltd., followed by Varun Beverages at around 26.8 percent. 17 percent of the investment amount should be placed in Coforge Ltd., 12.02 percent in Trent Ltd., and around 7 percent in Aarti Industries Ltd. About 3.89 percent of the portfolio should comprise the stock of Astral Ltd. These six stocks would comprise around 97.9 percent of the portfolio. The remaining 2.1 percent is divided into the remaining five stocks namely Navin Fluorine International Ltd., Dixon Technologies, Laurus Labs Ltd., Godrej Properties Ltd., and Metropolis Healthcare Ltd.

The return of the market portfolio was then calculated using the weights of the selected securities given by Eq. (8).

$$R_{\rm {P}} = \mathop {\sum}\nolimits_{i = 1}^n {X_iR_i}$$
(8)

where Xi is the weight of security i and Ri is the return of security i.

The return of the optimum portfolio using the Sharpe single-index model was found to be 50.76 percent during the 5 years between January 2017 and December 2021. The return of the benchmark index Nifty mid-cap 100 index was found to be 15.61 percent during the same period.

The risk of both portfolios was also calculated using Eq. (9):

$$\begin{array}{l}\sigma _{{{\mathrm{P}}}} = \surd \left( {w_1^2\sigma _1^2 + w_2^2\sigma _2^2 + \ldots .W_n^2\sigma _n^2 + 2w_1w_2{\rm {Cov}}_{1,2}}\right. \\\qquad\,+\left. \ldots . 2w_{n - 1}w_n{\rm {Cov}}_{{n - 1,n}} \right)\end{array}$$
(9)

The risk of Sharpe’s optimal portfolio was 12.861 percent whereas the risk of the market portfolio was 23.02 percent.

Discussion

Market indices are benchmark portfolios that represent the performance of all the companies in the market. Hence, the index can be considered the proxy of the market portfolio which is the combination of all the risky securities in the market. In this case, the Nifty mid-cap 100 index represents the mid-cap segment of the NSE. According to the capital market line of the Capital Asset Pricing Model, the market portfolio is also a part of the risk-return line depicting all the fairly priced optimum portfolios of the market. This would mean that all the portfolios on the capital market line which lie to the left of the market portfolio would have a lower return and risk combination than the market portfolio and all the portfolios which lie to the right would have a higher risk-return combination than the market portfolio. However, the Sharpe single-index model helps the investor to identify a portfolio that has a higher return than the market portfolio with a lower risk than the market portfolio during the same period. The optimal portfolio using the Sharpe model has a mean return of 50.76 percent per year and a risk of 12.861 percent, whereas the index portfolio has a return of 15.61 percent and a very high risk of 23.02 percent. Hence, it can be inferred that investors would be better off investing in a portfolio suggested by the Sharpe model rather than investing in the market index made up of the top 100 mid-cap companies. It also gives a perspective on over-diversification. The Nifty mid-cap 100 index gave lower returns for a higher level of risk. One reason for that could be over-diversification which leads to a decrease in returns. Theoretically speaking, according to the CAPM, if an investor wants to move to a point to the right of the market index, he/she would only be able to do so by borrowing at the risk-free rate of return, which would mean short selling. However, the Sharpe single-index model can give an optimal portfolio without short sales.

The Sharpe model resolves most of the problems of the Markowitz mean-variance model. The calculation of the optimal portfolio becomes easy. The optimal portfolio in this study was found to be a well-diversified portfolio of 11 securities comprising eight sectors. However, the limitation of this model is that it is based on historical data. There would be many instances when the optimal portfolio does not perform as predicted. The portfolio selection by this model is a static process whereas no other market is as dynamic as the stock market. The assumption of constant beta is also a flawed one as is the assumption of a constant risk-free rate of return.

The study has been able to prove that within a fixed time period, Sharpe’s optimal portfolio outperforms the market index. In this study, the optimal portfolio has a mean return that far exceeds the index return. Hence, it can be inferred that diversification using a linear mathematical model such as the Sharpe single-index model can lead to higher returns at lower risk.

The same model can be applied to portfolio optimisation for small-cap companies or inter-sector comparison of portfolio risk and return. The Sharpe model can also be used to make a portfolio of different asset classes such as equity, debt, mutual funds, etc. This model can easily be used by all types of investors for designing optimal portfolios. Although the model is a few decades old, it is still in use in various modified forms. It can be said that the model will have significant use for portfolio managers and investors in the future.