An Empirical Study on the Construction of Optimal Portfolio using Sharpe’s Single Index Model for Nifty 50 Stocks Dr UMA K

1. RABINDRA BHARATI JOURNAL OF PHILOSOPHY
ISSN : 0973-0087
Vol. : XXIII, No:24, 2022 63
AN EMPARICAL STUDY ON CONSTRUCTION OF OPTIMAL PORTFOLIO USING
SHARPE’S SINGLE INDEX MODEL FOR NIFTY 50 STOCKS
Dr. UMA.K Assistant professor & Research scholar Pooja Bhagavat Memorial Mahajana Education
Centre, Mysuru : chanduma25@gmail.com
NIKILCHANDRASHEKARA JSS Science & Technology University Student of B. E (EEE) Final
year SJCE Campus, Mysuru : nikchand1253@gmail.com
Abstract
The creation of an ideal portfolio has grown more difficult in recent years, for making wise investing
decisions; an investor has to have a solid understanding of security analysis and portfolio theory. The
primary goal of this study is to use the Sharpe Single Index Model (SIM) to build an ideal portfolio
for the Indian Market. Sharpe Single Index Model (SIM) is preferred over the Markowitz Model
because it takes fewer inputs and is simpler to compute. Investors are always looking to take risks
and put their money into various investment products in order to earn a decent return. They usually
invest their savings in the highly volatile stock Market. This volatility is referred to as the risk of the
market, and in order to protect investors from these volatilities, the stock market has developed a
new concept called a portfolio. With a portfolio, investors have the opportunity to lower their risk by
dividing their total investment into a group of securities. Maximizing returns with the least amount of
risk is a key factor for any investor to consider when selecting stocks for a portfolio. This research
paper main goal is to use the Sharpe Single Index Model to build an ideal portfolio out of equities
that are listed on the NSE Nifty 50. For the aim of this study, monthly data for NSE Nifty 50 stocks
from 1st August 2017 to 31st July 2022 have been taken into account. The study reveals that only
eight companies out of 50 are suitable for portfolio construction, by using Sharpe's Single Index
Model. They are Power Grid, Coal India, Tata Consumer Products, Bajaj Auto, Tata Consultancy,
Titan Company, ITC, and Tech Mahindra. So, the study concludes that this information serves to be
beneficial for investors and other market participants in selecting stocks to form a portfolio and
maximize their return.
Keywords: Sharpe’s Single Index Model, Portfolio Optimization, cut off Rate, Systematic Risk,
Unsystematic Risk, Return and Variance, Beta, and Risk.
1. Introduction
Investment in financial terms refers to “employ of funds in monetary assets” in which the
return is estimated over a period of time either in the form of interest or dividend or capital
appreciation of stock. While, the return estimated is to be realizing in the future, there is always a
component of uncertainty. This uncertainty is termed as risk. Risk and Return are considered to be
the two faces of investment in a coin and so, the investors analyze both these factors while taking
investment decision. It is critical for the investors, issuers and market makers to understand the
dynamics of the capital market (Archana & Lakshmi, 2019). There are various category of investors
based on their risk taking attitude, such as high risk avoiders, medium risk avoiders and low risk
avoiders. Investment in individual security is always riskier. Hence, the saying “Do not Put all your
eggs in One Basket” (Warren Buffet, 2015). Thus, people intend to diversify their risk by investing
in more than one security or a group of securities, which is known as a “Portfolio”. Portfolio helps in
diversifying the risk, as more number of securities added to a portfolio helps in maximizing return.
Constructing a portfolio is a challenging, complex and intricate task. Before, attempting to build a
portfolio, every Investor is required to be decisive on vital things such as the amount of investable
funds, duration of investment, objective of investment, attitude towards risk and return etc.
A portfolio is a collection of investment tools such as stocks, bonds, gold, and cash, and so on
depending on investors’ needs. William Sharpe, the Modern Portfolio Theory reveals the
maximization of returns through a mix of different securities. This theory tells us that risk can be
decreased by combining low-risk securities with those of high risk. Several studies have shown that

2. RABINDRA BHARATI JOURNAL OF PHILOSOPHY
ISSN : 0973-0087
Vol. : XXIII, No:24, 2022 64
the most important decision when constructing a portfolio is asset allocation. This means making
sure the portfolio has the right mix of assets to suit investors’ circumstances, investment aims, and
attitude to risk.
In the modern world, different range of investment opportunities exists namely postal
savings, bank deposits, gold, real estate, mutual funds, equity shares, preference shares, and bonds
and so on. However, investment in the share market i.e., equity share and preference share play a
vital role in the mindset of aggressive investors. The primary objective of the investment is to get
maximum returns from investment in the future period. Hence, investors are willing to take high-risk
security to get maximum return from the investment. The selection of suitable security in share
market with the right combination of risk and return is very difficult for the investor. To select the
optimum portfolio for the investor Sharpe Single Index model exist. Henceforth, the researcher has
taken an attempt to study the construction of an optimal portfolio using Sharpe’s single index model
in NSE Nifty 50 index stocks.
2. Need for the Study
The majority of the investors undergo with encounters issues while selecting securities for their
portfolio. They have difficulties as well while determining how much money to invest in each
security. An ideal portfolio may be created using Sharpe's Single Index model to construct an
optimal portfolio. This aids the investor in identifying the portfolio that best meets his requirements.
The goal of the current study is to demonstrate that using this model; a person may build a portfolio
that offers the highest return for a given level of risk.
3. Literature Review
Tanuj and et al., (2017) constructed a portfolio using NSE Nifty 50 stocks. In this study it was
found that they used Sharpe’s Single Index Model to construct a portfolio in this study the authors
found that out of 50 stocks considered for study, only 24 stocks are chosen for inclusion in optimal
portfolio. Out of 24 stocks selected, the maximum number of stocks is from the banking sector.
Stocks of SBI, PNB, IndusInd bank, ICICI bank and Axis Bank are a part of optimal portfolio.
Systematic risk is less than Unsystematic risk in Single Index Model. It can be reduced through
diversification.
Subhodeep and Ajay Kumar (2018) construct an optimal portfolio using stocks listed in NIFTY
50. The author found that out of 50 stocks considered for the study, only 6(Six) stocks are chosen for
inclusion in the optimal portfolio. The Tata Motors securities have the highest beta value which
indicates it is highly volatile. This will help the investors as a guiding record in the future and help
them to make appropriate investment decisions.
Shreenidhi and Roopesh (2019) constructed a portfolio using an optimal portfolio from stocks
listed in BSE SENSEX using Sharpe Single Index Model. And the author founds that out of 30
stocks considered for this study, the stocks of only 5 companies can be chosen for inclusion in the
optimal portfolio. The study concludes that constructing an optimal portfolio is a huge challenge for
individual investors and also for institutional investors. From the study, it is noted that constructing
an optimal portfolio with Sharpe's Single Index Model is less time-consuming and is more efficient
in the context of security analysis and portfolio management in the real world as compared to
Markowitz's variance model.
Archana and Sri Lakshmi (2020) constructed a portfolio using by using Sharpe’s Index Model and
the study employs the data from - the Bombay Stock Exchange - popular Index “Sensex”. The stock
prices from 1st January 2019 to 31st December 2019 are considered for the study. The author found
that twenty-one stocks were bullish during the study period and benefitted investors with positive
returns consistently and nine stocks showed negative Trends/returns. An optimal Portfolio is built by
selecting ten stocks that are above the cut-off rate. The study concludes the beneficiary is not just
investors but also other market participants in selecting stocks to form a portfolio and maximize their
return.

3. RABINDRA BHARATI JOURNAL OF PHILOSOPHY
ISSN : 0973-0087
Vol. : XXIII, No:24, 2022 65
Rajesh and Tanuja (2020) constructed a portfolio using Sharpe’s Single index model in BSE
SENSEX index stocks. The study timeline covers from 1st April 2014 to 31st March 2019. The study
reveals that, among the optimum portfolio stocks, Hindunilvr, Titan, HDFC Bank, Asian, NestleInd,
IndusInd, Marti, and Power grid give positive results, and the rest of the stock's Ci values are starting
to decline. Hence, only these stocks are considered for the optimal portfolio construction for the
study.
Vaddula& et al., (2020) constructed an optimal portfolio of stocks using the Sharpe Single Index
Model based on a cut-off point. The selected twenty companies from the S&P BSE Sensex index d
based on a simple random sample. The researcher found that from the twenty sample companies,
only five companies were selected for the optimal portfolio. These stocks are HDFCBANK, HCL
TECH, TITAN, and ASIAN PAINT and, INFY.
4. Objectives of the Study
The overall objective of the study is to construct an optimal portfolio using Sharpe’s Single index
model by using the NSE Nifty 50 stocks. The following are the more specific objective they are;
1. To highlight the Theoretical Background of Sharpe’s Single Index Model.
2. To analyze the Risk and Returns of NSE Nifty 50 Securities.
3. To construct Optimal Portfolio of Selected Securities Using Sharpe’s Single Index Model.
4. To determine the Proportional Investment to be made in each Selected Securities.
5. Research Methodology
Descriptive and exploratory research is undertaken for the study. The present investigation is
supported by secondary data information found on the website www.investing.com, and
www.monecontrol.com.for the current study. In this article an ideal portfolio will be built using the
Sharpe Single Index model. A simple random sample was used to choose Nifty 50 companies from
the NSE index for this empirical investigation. The study period is 5 years, spanning from the
financial year 1st August 2017 to 31st July 2022, for computing the yearly return of each security as
well as market return. The yearly adjusted closing prices of individual stocks are considered.
Table 5.1: Sowing the Research Methodology and Sample
Sl. No Category Methodology
1 Research Design  Descriptive
 Exploratory
2 Sources of Data  Secondary, from NSE, RBI, etc.
websites, and databases
3 Sample Population  NIFTY 50 Stock
Conceptual framework of Sharpe’s Index Model:
Unlike Markowitz’s model, this simplified model states that by comparing the return of individual
securities with a single index like the ‘Market Index’, the relationship existing between each pair of
securities can be determined indirectly. The requirements of large data inputs and tedious calculation
requirements in the Markowitz model is largely reduced (Mandal, 2003). SIM needs only (3n+2) bits
of information or simply each security’s Alpha and Beta estimates. For SIM, the variance of the
market index, each security’s expected return, and unsystematic risk also need to be assessed. It has
become more popular as compared to Markowitz Model due to its simplicity.
Building an ideal portfolio using the Sharpe's Index Model
Fischer and Jordan (1995), state that stocks to be included in the optimal portfolios are determined
on the basis of their ‘Excess return to beta ratio. As per the rule of ranking, the security that has the
highest ‘excess return to beta ratio ‘will be placed in the first position, followed by the security with
the second highest beta ratio, and so on and so forth. Thereafter a cut-off point will be calculated and
all the stock whose ‘excess return to beta ratio is above the cut-off point is included in the portfolio.
Determinants of the Sharpe Single Index Model
The following assumptions are constructed in order to support Sharpe's Single Index Model:
1. All investors have similar expectations.

4. RABINDRA BHARATI JOURNAL OF PHILOSOPHY
ISSN : 0973-0087
Vol. : XXIII, No:24, 2022 66
2. An even holding period is used in estimating risk and return for each security.
3. The price movements of one security in relation to another security will not depend largely upon
the nature of those two securities alone. They may also reflect a greater influence that might have
cropped up as a result of the general business and economic conditions of the nation.
4. The relation between each security occurs only through their individual influences along with
some indices of business and economic activities but not other factors.
5. The indices, to which the returns of each security are correlated, are probable to be some securities
market proxy.
6. The random disturbance terms (ei) have an estimated value of zero (0) and a finite variance. It is
not correlated with the return on the market portfolio (Rm), also, like the error term (ei) for the other
securities.
Tools Used for the Study
5.1. Return (Ri)
The yearly return of stocks is calculated using the below formula:
Where, Ri = (R2–R1)/R1*100
R2 = adjusted closing price of month 2,
R1 = adjusted closing price of month 1 and,
Ri= Return of individual stock.
5.2. Risk-Free Rate of Return (Rf): The Risk-free rate of return is the required return on a risk-free
asset. This study used 365 days of Treasury bills for a risk-free rate of return.
5.3. Beta β): Beta refers to the statistical tool used to measure the volatility of the stock market. A
beta coefficient is a measure of the volatility or systematic risk of an individual stock in comparison
to the unsystematic risk of the entire market. If the beta value shows 1, the security's price moves
with the market. If the beta value is less than 1, means that the security is theoretically less volatile
than the market. If the beta value is more than 1 means that the security's price is theoretically more
volatile than the market:
Where, β = 𝐶𝑜𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒 (𝑅i, 𝑅𝑚)
𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒 (𝑅𝑚)
Covariance = measure of a stock's return relative to that of the market
Variance = Measure of how the market moves relative to its mean
Ri= Expected return of individual security
Rm= Return of market index
5.4. Excess Return to Beta Ratio (RI – Rf/βi): The stocks are ranked in descending order as per the
beta ratio RI – Rf/βi. This is the equation for ranking Stocks in the order of their return adjusted for
risk. The method involves selecting a cut-off rate for the inclusion of securities in a portfolio. For
this purpose, the excess return to the Beta ratio given above has to be calculated for each stock and
ranked from highest to lowest. Then only those securities which have RI – Rf/β, greater than a cut-off
point, fixed in advance can be selected. The basis for finding the cut-off Rate Ci is as follows: For a
portfolio of I stocks, Ci is given by the cut-off rate.
The excess return is the difference between the individual security return and the risk-free
rate of return offered on government security such as Treasury bills. The study takes into account
one year or 364-day Treasury bill rate, which is 0.07* as the risk-free rate of interest/return. Excess
return to beta measures the additional return earned for bearing risk per unit. The excess return to
beta ratio is calculated as follow;
Where,Excess Return to bets ratio = (Ri-Rf) /βi
Ri=The expected return on the security i,
Rf= Risk free rate of return
βi= Systematic risk of an individual stock in comparison to the unsystematic risk of the entire market
or beta co-efficient
5.5. a. Systematic risk: The variance explained by the index is referred to as systematic risk.

5. RABINDRA BHARATI JOURNAL OF PHILOSOPHY
ISSN : 0973-0087
Vol. : XXIII, No:24, 2022 67
Systematic risk =β2x Variance of the market Index
= βi
2. σm2
5.5. b. Unsystematic risk: The unexplained variance is called residual variance or Unsystematic
risk. It is the distinction between systemic risk and overall risk. This is how the unsystematic risk is
determined: The unsystematic risk is calculated as follows:
Where, Unsystematic Risk = Total variance –Systematic risk
ei2= σi
2 - βi
2 σm2
σei
2 = Unsystematic risk of the portfolio
σi
2 = Variance of the individual stock
βi
2 = Systematic risk
σm2 = Variance of the market index
5.5. c. Total Risk: Total risk = Systematic risk + Unsystematic risk
= βi
2
. σm2+ ei2
5.6. Standard Deviation: The standard deviation calculates a security's overall risk. The term
"standard deviation" refers to the variance's square root.
Where,σi = SQRT (Ri-͞R ͞ i)2
n-1
σi = Standard deviation of individual security
Ri = Expected return of individual security
R ͞i = Mean return of individual security
n = Number of observations
5.7. Market Variance: A variance is a tool used to measure the volatility of the stock market. The
higher the variance, the higher the volatility of the stock market and vice versa, the market variance
is calculated from the following formula
Where, σm2 = (Rm-͞R ͞ m) 2
n-1
σm2= Variance of Market index return
Rm = Expected return of Market index
R ͞m = Mean return of Market index.
n = Number of observations
5.8. A) Cut off rate by using Sharpe Index Model: Cut off rate is calculated by using the following
formula. The Cut-off Point is calculated as follow;
Where, Ci = (σm2*Σ ((Ri-Rf)* βi /σ2ei
1+σm2*Σβi2/σei2
Ri = Expected return of individual stock
Rf = Risk free rate of return
βi = Systematic risk of individual stock
σm2 = Variance of the market index
σei2= Unsystematic security variance risk
5.9. B) Proportion of Investments in each individual security: it is a part of the portfolio is
calculated using the following formula: Wi = Zi /∑ Zi.
Where, Zi = βi^2/σ^2ei (Ri-Rf/β-C)
Xi = Proportion of investment in individual security
Ri = Expected return of individual security
Rf = Risk free rate of return
βi = Systematic risk
C = Cut off point
σei2
= Unsystematic risk
5.10. Portfolio Return and Risk: Portfolio return and risk is calculated by using the following
formula:
(A) Portfolio Return: Rp = αp + βp Rm
Where, αp = ∑xiαi

6. RABINDRA BHARATI JOURNAL OF PHILOSOPHY
ISSN : 0973-0087
Vol. : XXIII, No:24, 2022 68
βp =Σ xi βi
Rp = Portfolio Return
αi = Specific return of an individual security
βi = Beta coefficient of an individual security
Rm = Return of Market Index
(B) Portfolio Risk:
Where, σp2 = β2 σm2 + Σxi2 σ2ei
σp2 = Portfolio Variance
β = Beta coefficient
σm2 = Market variance
xi = Proportion of investment in individual security
σ2ei = Unsystematic risk
The tools used to analyze the risk parameters such as β of each stock, Sharpe's Single Index Model,
and unsystematic and systematic risk, stock return, are computed. The yearly mean return of all
individual stock was calculated using Excel. The monthly return is then converted into an annual
return by using the excel formula, i.e., = [(1 + Monthly mean) ^12] – 1. For the risk-free rate of
return 365 days T-bills: 0.07* is taken from the RBI website and for Market risk variance of Nifty 50
is been calculated. Beta, Unsystematic Risk, Systematic Risk, and Cut-off point are also calculated
by using Excel. The securities' "Excess Return to Beta" has also been determined. A number of
financial and technical tools have been used for analyzing data.
6. Limitations of the Study
The study's shortcomings are as follows:
• The study's sole source of information is secondary data.
• The study's findings might not be applicable to all situations.
• Due to the time limit only, five years monthly data from 1 -08 -2017 to 31- 07 -2022, have been
taken.
7. Scope of the Study
The study is limited to a selected sample of NSE top 50 companies listed on NSE Nifty 50 in
India. Historical data for the last 5 years has been used as the basis for the construction of portfolios
using Sharpe’s Single Index Model.
8. Data Analysis and Interpretation
The Risk and Returns of NSE Nifty 50 Securities Constructed Optimal Portfolio by Using Sharpe’s
Single Index Model
Table 8.1: Showing the Ranking of Stocks based on Excess Return to Beta Ratio
Sl
N
o
Company
Name
Mean
Retur
n
Risk
less
rate of
interes
t
Exces
s
retur
n
Varian
ce
Mark
et
Risk
Beta SR USR
Sharpe’
s Single
Index
Ran
k
(Ri) (Rf)
(Ri-
Rf)
σ2 σ2m (β)
(β2
σ2m)
(σei2)
(Ri-
Rf/β)
1
Adani
Enterprise
s
1.41 0.07* 1.34 366.68 30.61 1.69 87.83
278.8
5
0.79
34
2
Adani
Ports &
SEZ
2.53 0.07 2.46 103.14 30.61 1.33 54.34 48.81
1.85
14
3
Apollo
Hospitals
1.76 0.07 1.69 150.40 30.61 0.87 23.18
127.2
1
1.94
12

7. RABINDRA BHARATI JOURNAL OF PHILOSOPHY
ISSN : 0973-0087
Vol. : XXIII, No:24, 2022 69
4
Asian
Paints
0.97 0.07 0.90 52.40 30.61 0.61 11.49 40.91
1.47
18
5
AXIS
Bank
0.61 0.07 0.54 104.23 30.61 1.50 69.16 35.07
0.36
41
6 Bajaj Auto 3.33 0.07 3.26 71.14 30.61 1.01 31.36 39.78 3.22 4
7
Bajaj
Finance
2.69 0.07 2.62 189.24 30.61 1.85 104.98 84.26
1.41
20
8
Bajaj
FinServ
-0.24 0.07 -0.31 201.53 30.61 1.87 106.99 94.53
-0.16
48
9
Bharat
Petroleum
0.90 0.07 0.83 106.16 30.61 1.25 47.57 58.59
0.67
38
10
Bharti
Airtel
1.13 0.07 1.06 74.97 30.61 0.74 16.65 58.32
1.44
19
11
Britannia
Industries
1.07 0.07 1.00 45.60 30.61 0.60 11.03 34.57
1.66
15
12 Cipla -0.09 0.07 -0.16 67.56 30.61 0.48 6.93 60.62 -0.33 49
13 Coal India 3.13 0.07 3.06 80.00 30.61 0.86 22.88 57.13 3.54 2
14
Divi's
Labs
1.00 0.07 0.93 67.01 30.61 0.40 4.90 62.11
2.34
9
15
Dr.
Reddy’s
Labs
0.37 0.07 0.30 64.22 30.61 0.23 1.66 62.56
1.29
23
16
Eicher
Motors
0.91 0.07 0.84 83.21 30.61 0.95 27.79 55.42
0.88
31
17
Grasim
Industries
1.52 0.07 1.45 87.27 30.61 1.07 34.90 52.37
1.36
22
18 HCL Tech 0.98 0.07 0.91 71.63 30.61 0.83 20.95 50.68 1.10 26
19
HDFC
Bank
0.78 0.07 0.71 49.47 30.61 1.06 34.58 14.89
0.67
37
20
HDFC
Life
-0.17 0.07 -0.24 299.92 30.61 0.25 1.87
298.0
4
-0.97
50
21
Hero
Moto
Corp
1.78 0.07 1.71 83.70 30.61 0.91 25.47 58.22
1.88
13
22
Hindalco
Industries
1.45 0.07 1.38 187.23 30.61 1.72 90.21 97.01
0.80
33
23
Hindustan
Unilever
0.59 0.07 0.52 40.92 30.61 0.25 1.98 38.94
2.06
10
24
Housing
Developm
ent
Finance
1.91 0.07 1.84 58.22 30.61 1.11 37.56 20.66
1.66
16
25
ICICI
Bank
0.39 0.07 0.32 85.43 30.61 1.30 51.42 34.01
0.24
44
26
IndusInd
Bank
2.08 0.07 2.01 228.97 30.61 2.20 147.63 81.34
0.91
30
27 Infosys 0.04 0.07 -0.03 56.69 30.61 0.62 11.74 44.95 -0.04 46
28 ITC 2.17 0.07 2.10 45.14 30.61 0.78 18.68 26.46 2.69 7
29 JSW Steel 1.28 0.07 1.21 158.82 30.61 1.25 47.57
111.2
5
0.97
29
30 Kotak 0.77 0.07 0.70 61.01 30.61 0.90 24.55 36.45 0.79 35

8. RABINDRA BHARATI JOURNAL OF PHILOSOPHY
ISSN : 0973-0087
Vol. : XXIII, No:24, 2022 70
Mahindra
Bank
31
Larsen
&Toubro
1.25 0.07 1.18 70.01 30.61 1.20 44.18 25.82
0.98
28
32
Mahindra
&Mahindr
a
0.38 0.07 0.31 116.49 30.61 1.42 61.56 54.94
0.22
45
33
Maruti
Suzuki
1.79 0.07 1.72 81.46 30.61 1.06 34.35 47.11
1.62
17
34
Nestle
India
0.28 0.07 0.21 30.79 30.61 0.29 2.49 28.30
0.75
36
35 NTPC -0.01 0.07 -0.08 66.10 30.61 0.79 19.12 46.97 -0.10 47
36
Oil &
Natural
Gas
0.40 0.07 0.33 106.57 30.61 1.18 42.42 64.15
0.28
43
37
Power
Grid
2.01 0.07 1.94 32.42 30.61 0.47 6.73 25.69
4.13
1
38
Reliance
Industries
1.32 0.07 1.25 78.16 30.61 1.08 35.56 42.60
1.16
24
39 SBI (SBI) 1.49 0.07 1.42 138.78 30.61 1.43 62.49 76.30 1.00 27
40
SBI Life
Insurance
0.37 0.07 0.30 67.27 30.61 0.91 25.48 41.79
0.33
42
41
Sun
Pharma
1.11 0.07 1.04 73.64 30.61 0.76 17.62 56.02
1.37
21
42
Tata
Consultan
cy
1.72 0.07 1.65 50.31 30.61 0.59 10.73 39.58
2.79
5
43
Tata
Consumer
Products
2.85 0.07 2.78 88.65 30.61 0.83 20.93 67.72
3.37
3
44
Tata
Motors
1.08 0.07 1.01 301.05 30.61 1.75 93.51
207.5
4
0.58
39
45
Tata Steel
Ltd
(TISC)
1.70 0.07 1.63 153.94 30.61 1.43 62.74 91.20
1.14
25
46
Tech
Mahindra
1.98 0.07 1.91 81.20 30.61 0.80 19.52 61.68
2.39
8
47
Titan
Company
2.73 0.07 2.66 93.20 30.61 0.98 29.24 63.96
2.73
6
48
Ultra Tech
Cement
0.88 0.07 0.81 65.62 30.61 0.98 29.23 36.39
0.83
32
49
UPL(UPL
L)
0.78 0.07 0.71 118.36 30.61 1.27 49.66 68.70
0.56
40
50 Wipro 1.10 0.07 1.03 72.07 30.61 0.51 8.03 64.03 2.01 11
(Source: Compiled by authors)
As proposed by Sharpe, stocks that have negative returns should be ignored for selection in optimal
portfolios. The securities on the basis of their returns are ranked from (highest to lowest) for
selection. Fischer and Jordan (1995) state that stocks to be included in the optimal portfolios are
determined on the basis of their ‘excess return to beta ratio.’ As per the rule of ranking, the security
that has the highest ‘excess return to beta ratio ‘will be placed in the first position, followed by the
security with the second highest beta ratio, and so on and so forth. It is seen from Table 2 that Power

9. RABINDRA BHARATI JOURNAL OF PHILOSOPHY
ISSN : 0973-0087
Vol. : XXIII, No:24, 2022 71
Grid occupies the first place as per ranking, followed by Cola India. Stocks with negative returns are
excluded. Out of 50 stocks considered, only8 stocks are finally selected for consideration. They are
likely to be Power Grid, Coal India, Tata Consumer Products, Bajaj Auto, Tata Consultancy, Titan
Company, ITC, and Tech Mahindra.
The above table shows the excess return to the beta ratio of the sample companies. After calculating
the ratio, ranks are allotted to sample companies on the basis of their ratio.
To Construct the Optimum Portfolio for Investments in NSE Nifty 50
Table 8.2: Showing the Computation of Cut off rate using Sharpe’s Single Index Model
Sl
N
o
Company
Name
Sharpe
’s
Single
Index
Excess
return
to
beta
Cumulati
ve of
N
Beta/
USR
Cumulativ
e of
D
Cut
off
rate
(N/D)
(Ri-
Rf/β)
(Ri-
Rf) *β/
σei2
∑(Ri-Rf) *
β / σei2
(σm^2)
Σ (Ri-
Rf) *
βi /σei^2
β2/
σei2 ∑β2/ σei2
1+(σm^
2)
Σβi^2)/
σei^2
Ci
1 Power
Grid
4.13 0.035 0.035 1.08 0.009 0.009 1.26 0.8579
2 Coal India 3.54 0.046 0.082 2.50 0.013 0.022 1.66 1.5042
3 Tata
Consumer
Products
3.37 0.034 0.116 3.54 0.010 0.032 1.97 1.7961
4 Bajaj
Auto
3.22 0.083 0.199 6.08 0.026 0.057 2.76 2.2029
5 Tata
Consultan
cy
2.79 0.025 0.223 6.84 0.009 0.066 3.03 2.2558
6 Titan
Company
2.73 0.041 0.264 8.08 0.015 0.081 3.49 2.3175
7 ITC 2.69 0.062 0.326 9.98 0.023 0.104 4.19 2.3797
8 Tech
Mahindra
2.39 0.025 0.351 10.74 0.010 0.115 4.51 2.3804
9 Divi's
Labs
2.34 0.006 0.357 10.92 0.003 0.117 4.59 2.3796
10 Hindustan
Unilever
2.06 0.003 0.360 11.03 0.002 0.119 4.64 2.3762
11 Wipro 2.01 0.008 0.368 11.28 0.004 0.123 4.77 2.3666
12 Apollo
Hospitals
1.94 0.012 0.380 11.63 0.006 0.129 4.95 2.3510
13 Hero
MotoCorp
1.88 0.027 0.407 12.45 0.014 0.143 5.39 2.3127
14 Adani
Ports &
SEZ
1.85 0.067 0.474 14.51 0.036 0.180 6.50 2.2331
15 Britannia
Industries
1.66 0.017 0.491 15.04 0.010 0.190 6.82 2.2064
16 Housing
Developm
ent
Finance
1.66 0.099 0.590 18.06 0.059 0.249 8.64 2.0916

10. RABINDRA BHARATI JOURNAL OF PHILOSOPHY
ISSN : 0973-0087
Vol. : XXIII, No:24, 2022 72
17 Maruti
Suzuki
1.62 0.039 0.629 19.25 0.024 0.273 9.37 2.0550
18 Asian
Paints
1.47 0.013 0.642 19.66 0.009 0.282 9.65 2.0379
19 Bharti
Airtel
1.44 0.013 0.656 20.07 0.009 0.292 9.93 2.0207
20 Bajaj
Finance
1.41 0.058 0.713 21.83 0.041 0.332 11.18 1.9532
21 Sun
Pharma
1.37 0.014 0.727 22.26 0.010 0.343 11.49 1.9372
22 Grasim
Industries
1.36 0.030 0.757 23.17 0.022 0.365 12.16 1.9056
23 Dr.
Reddy’s
Labs
1.29 0.001 0.758 23.20 0.001 0.365 12.19 1.9043
24 Reliance
Industries
1.16 0.032 0.790 24.17 0.027 0.393 13.02 1.8568
25 Tata Steel
Ltd
1.14 0.026 0.815 24.96 0.022 0.415 13.71 1.8208
26 HCL Tech 1.10 0.015 0.830 25.41 0.014 0.429 14.12 1.7998
27 SBI 1.00 0.027 0.857 26.23 0.027 0.455 14.94 1.7557
28 Larsen &
Toubro
0.98 0.055 0.912 27.91 0.056 0.511 16.65 1.6763
29 JSW Steel 0.97 0.014 0.925 28.33 0.014 0.525 17.08 1.6586
30 IndusInd
Bank
0.91 0.054 0.980 29.99 0.059 0.585 18.89 1.5871
31 Eicher
Motors
0.88 0.014 0.994 30.43 0.016 0.601 19.39 1.5688
32 UltraTech
Cement
0.83 0.022 1.016 31.09 0.026 0.627 20.20 1.5393
33 Hindalco
Industries
0.80 0.024 1.040 31.84 0.030 0.658 21.13 1.5070
34 Adani
Enterprise
s
0.79 0.008 1.048 32.09 0.010 0.668 21.44 1.4965
35 Kotak
Mahindra
Bank
0.79 0.017 1.066 32.62 0.022 0.690 22.12 1.4749
36 Nestle
India
0.75 0.002 1.068 32.68 0.003 0.693 22.20 1.4720
37 HDFC
Bank
0.67 0.051 1.119 34.24 0.076 0.769 24.53 1.3961
38 Bharat
Petroleum
0.67 0.018 1.136 34.78 0.027 0.795 25.34 1.3728
39 Tata
Motors
0.58 0.008 1.145 35.04 0.015 0.810 25.79 1.3589
40 UPL 0.56 0.013 1.158 35.45 0.024 0.833 26.51 1.3371
41 AXIS
Bank
0.36 0.023 1.181 36.16 0.064 0.898 28.48 1.2695
42 SBI Life
Insurance
0.33 0.007 1.188 36.36 0.020 0.918 29.09 1.2499

11. RABINDRA BHARATI JOURNAL OF PHILOSOPHY
ISSN : 0973-0087
Vol. : XXIII, No:24, 2022 73
43 Oil &
Natural
Gas
0.28 0.006 1.194 36.55 0.022 0.939 29.76 1.2283
44 ICICI
Bank
0.24 0.012 1.206 36.92 0.049 0.989 31.27 1.1807
45 Mahindra
&
Mahindra
0.22 0.008 1.214 37.17 0.037 1.025 32.39 1.1475
46 Infosys -0.04 0.000 1.214 37.16 0.009 1.034 32.65 1.1380
47 NTPC -0.10 -0.001 1.213 37.11 0.013 1.047 33.06 1.1228
48 Bajaj
Finserv
-0.16 -0.006 1.206 36.93 0.037 1.084 34.19 1.0802
49 Cipla -0.33 -0.001 1.205 36.89 0.004 1.088 34.30 1.0755
50 HDFC
Life
-0.97 0.000 1.205 36.88 0.000 1.088 34.31 1.0751
(Source: Compiled by authors)
Interpretation: The above table shows the cut-off (ci) of the sample companies selected. The cut-off
values go on increasing from 0.8579to 2.3804. Therefore, the value 2.3804 is considered as the
cutoff point i.e., Ci =2.3804. The securities which appear after the cut-off value will not be
considered for the construction of an optimal portfolio. The securities which have a value of the cut-
off point (Ci) more or equal to the cut-off point will only be selected for the construction of an
optimal portfolio. The Proportional Investment to be made into each Security.
Table 8.3: Showing the Proportional Investment to be made into each Security
Sl
No
Company Name β /USR
Cut off
rate
Ci
Zi Xi
Xi
Proportion
of each
stock
1 Power Grid 0.02 0.8579 0.06 37.56 38
2 Coal India 0.02 1.5042 0.03 19.37 19
3 Tata Consumer
Products
0.01 1.7961 0.02 12.04
12
4 Bajaj Auto 0.03 2.2029 0.03 16.27 16
5 Tata Consultancy 0.01 2.2558 0.01 5.06 5
6 Titan Company 0.02 2.3175 0.01 3.93 4
7 ITC 0.03 2.3797 0.01 5.70 6
8 Tech Mahindra 0.01 2.3804 0.00 0.07 0
Total 0.159 100
(Source: Compiled by authors)
Interpretation: The above table represents the proportion of investment to be made in each security.
The eight securities ranking from 1 to 8 are selected for the construction of an optimal portfolio.
Table 8.4: Showing the Portfolio Variance of Companies
Sl No Company Name
Systematic
Risk
(β2 σ2m)
Unsystematic
risk
(σei
2)
Total Risk/
(β2 σ2m) + (σei
2)
1 Power Grid 6.73 25.69 32.42
2 Coal India 22.88 57.13 80.00
3 Tata Consumer Products 20.93 67.72 88.65
4 Bajaj Auto 31.36 39.78 71.14
5 Tata Consultancy 10.73 39.58 50.31
6 Titan Company 29.24 63.96 93.20

12. RABINDRA BHARATI JOURNAL OF PHILOSOPHY
ISSN : 0973-0087
Vol. : XXIII, No:24, 2022 74
7 ITC 18.68 26.46 45.14
8 Tech Mahindra 19.52 61.68 81.20
Total Risk Variance 542.05
(Source: Compiled by authors)
Interpretation: Portfolio variance is a gauge of a portfolio's return dispersion. It is the total of a
portfolio's real returns for a specific time period. Investing in a portfolio of these securities, investors
can anticipate variance or variability in return of 542 considering market situations. The percentage
of funds to be invested in each security is presented in the following diagram.
Table 8.5: Showing the Calculation of Portfolio Returns of the Companies
Sl
No
Company Name
Mean Return Weights
Portfolio return
(fi*Xi)
(Fi) (Xi) (Rp)
1 Power Grid 2.01 37.56 75.40
2 Coal India 3.13 19.37 60.65
3 Tata Consumer Products 2.85 12.04 34.37
4 Bajaj Auto 3.33 16.27 54.17
5 Tata Consultancy 1.72 5.06 8.72
6 Titan Company 2.73 3.93 10.74
7 ITC 2.17 5.70 12.37
8 Tech Mahindra 1.98 0.07 0.15
Total Portfolio Return 257
(Source: Compiled by authors)
Interpretation: The above table represents that Portfolio investment is based on the basic
assumption that investing in a basket of securities, not in individual company shares. Here investor
having the above stocks in his portfolio can expect an overall return of 257.
Findings:
 It was found that out of 50 stocks considered for this study, the stocks of only 8 companies can be
chosen for inclusion in optimal portfolio. The ‘excess return to beta ratio of only 8 stocks was
above the calculated cut-off rate of 2.3804. They are Power Grid, Coal India, Tata Consumer
Products, Bajaj Auto, Tata Consultancy, Titan Company, ITC, and Tech Mahindra.
 The return from IndusInd Bank has the highest beta value of 2.20 which means it is highly volatile
in nature. Bajaj Auto, Maruti Suzuki, HDFC Bank, Grasim Industries, Reliance Industries,
Housing Development Finance, Oil & Natural Gas, Larsen & Toubro, JSW Steel, Bharat
Petroleum, UPL, ICICI Bank, Adani Ports & SEZ, Mahindra & Mahindra, SBI, Tata Steel Ltd,
AXIS Bank, Adani Enterprises, Hindalco Industries, Tata Motors, Bajaj Finance, Bajaj Finserv,
and IndusInd Ban, .have beta values greater than 1 i.e., they are also volatile.
 Dr. Reddy’s Labs having the lowest beta value of 0.23 which means it is less volatile. The cut-off
values go on increasing from 0.8579to 2.3804. Based on the Cut-off values eight companies were
selected.
 Cut-off rate as per Sharpe’s Single Index Model is 2.3804, stocks above the cut-off rate such as
Power Grid, Coal India, Tata Consumer Products, Bajaj Auto, Tata Consultancy, Titan Company,
ITC, and Tech Mahindra have been selected as Optimal ones for Portfolio Construction.
 Proportion of investment to be made in each company is as follows, (38%) Power Grid, (19%)
Coal India, (12%) Tata Consumer Products, (16%) Bajaj Auto, (5%) Tata Consultancy, (4%) Titan
Company, (6%) ITC,(0.1 %) and Tech Mahindra. Portfolio Return and Risk with combination of
securities selected as per Sharpe’s Single Index Model is 542 and 257 respectively.
Suggestions:
1. Sharpe Single Index Model shall be modified to explain covariance among securities which is vital
for selection of stocks in portfolio construction.

13. RABINDRA BHARATI JOURNAL OF PHILOSOPHY
ISSN : 0973-0087
Vol. : XXIII, No:24, 2022 75
2. Sharpe Single Index Model assumes that the behavior of shares is as same as market fluctuations.
But in reality, certain stocks behave based on their fundamentals. Sharpe Single Index Model shall
consider fundamental analysis also along with technical analysis.
3. Sharpe Single Index Model depends on study with respect to certain time period. But in ever-
changing business environment, it should guide investors in making changes to their portfolio every
now and then based on market conditions.
4. Portfolio construction should provide clarity not only on the basis of cut off rates, but also
considering business models of companies.
9. Conclusion
Investment in individual security is always riskier and therefore investors tend to invest in a group of
securities termed as portfolio. Portfolio helps in diversifying the risk and maximizes the returns. It is
not as easy task to construct a portfolio which is optimal. It requires analysis of return and risk. Apart
from it, the investors have to compute the excess return earned for per unit risk, market return, cut
off rate and proportion of funds to be invested in individual securities. Thus, this paper attempts to
discuss the methodology and computations involved in selecting the stocks for building the portfolio
and the proportion of funds to be invested. The study employed Sharpe’s Single Index Model for
selecting the stocks from NSE Nifty 50. During the study period, the study revealed that eight
securities form the optimal portfolio. The results obtained are limited for the period under study and
can differ with varying time periods; differing indices, and models chosen for the study. This
information helps investors to take right investment decisions. This information serves to be
beneficial for investors and other market participants in selecting stocks to form a portfolio and
maximize their return.
Reference:
Tanuj Nandan, Nivedita Srivastava, (2017) Construction of Optimal Portfolio Using Sharpe’s Single
Index Model: An Empirical Study on Nifty 50 Stocks, Journal of Management Research and
Analysis; DOI: 10.18231/2394-2770.2017.0010, 4(2):74-83
Subhodeep Chakraborty and Ajay Kumar Patel (2018) “Construction of Optimal Portfolio Using
Sharpe’s Single Index Model and Markowitz Model an Empirical Study on Nifty 50 Stock”,
Journal of General Management Research, Vol. 5, Issue 1: pp. 86–103.
Shreenidhi N. V. and N.Roopesh Kumar (2019) A Study on Construction of Optimal Portfolio Using
Sharpe’s Single Index Model: An Empirical Study On BSE Sensex 30 Stocks”, Wesleyan
Journal of Research, Vol.13 No4 (VI): pp. 78-84.
Dr Archana H N And Srilakshmi D (2020) “Building An Optimal Portfolio Using Sharpe’s Single
Index Model: An Empirical Study With Reference to Indian Capital Markets”, Journal Of
Xi'an University Of Architecture & Technology, ISSN No: 1006-7930 , Volume 12, Issue 8:
pp. 1223-1244.
Dr. E Rajesh and Ms. V. S. Thanuja (2020) “An Analytical Study on Construction of Optimal
Portfolio Using Sharpe's Single Index Model in BSE Sensex Index Stocks”, Think India
Journal, ISSN: 0971-1260, Vol-22, Special Issue-21: 742- 752.
Dr. Vaddula V. Krishna Reddy & Dr. T. Suchitra Rani (2020) “Optimal Portfolio Construction Using
Sharpe’s Single Index Model – A Study of Select Stocks from Bombay Stock Exchange
(BSE)”, Sugyaan, Volume: X, Issue – Ii: Pp. 45-54.
Website:
weww.researchgate.net
www.investing.com
www.moneycontrol.com