INDONESIAN JOURNAL OF BUSINESS AND ECONOMICS Printed ISSN 2621-6167, e-ISSN 2621-4466

Vol. 3 Issue 2, Desember 2020 https://journal.uniku.ac.id/index.php/ijbe

Stock Investment Portfolio Analysis with Single Index Model

Yasir Maulana1
1
Faculty of Economics, Universitas Kuningan
yasir@uniku.ac.id

ABSTRACT

In order to evaluate an optimal portfolio, an important step that investors or investment

managers is portfolio analysis. In stock portfolio analysis, methods that can be used include
the Markowitz approach and the Single Index Model. This study aims to apply the Single Index
Model in finding the beta value of an efficient portfolio line, so that investors can determine
the stocks and the proportion of funds needed to form an optimal portfolio. In this study, the
data sources used were 1) market share price index that represents market factor or market
data, 2) SBI interest rates that represents risk free (rf) and 3) The share prices of PT Ace
Hardware Indonesia Tbk, PT Indocement Tunggal Perkasa Tbk and PT Matahari Putra Prima
Tbk. The weight of each share in the active portfolio (Wi0) at Active Pf A 1.0000 is ACES of
0.1729, INTP of 0.0460 and MPPA of 0.7811. Then the alpha of the ACES active portfolio is
0.0051, INTP is 0.0002 and the MPPA is 0.0184. Then the calculation results show the residual
variance in the active ACES portfolio is 0.0041, INTP is 0.0001 and MPPA is 0.0147. The variance
of the Optimal Risky Portfolio of the variance index portfolio and the residual variance of the
active portfolio is 0.1054.

Keyword: Investment Portfolio, Single Index Model

JEL Clasification: G10, G11, G17

INTRODUCTION portfolio as a diversification step in

Investment decisions must have a relevant minimizing investment risk.
basis in order to achieve the goal of In order to form an optimal portfolio, an
maximizing profits and minimizing risk, important step that investors or investment
where this investment decision can be made managers must take is portfolio analysis. In
by two parties, namely investors or stock portfolio analysis, methods that can
investment managers. Investors or be used include the Markowitz approach and
investment managers who invest in shares in the Single Index Model.
the capital market are important to The Markowitz portfolio model is a portfolio
consider several factors, including the optimization method introduced by Hary
amount of capital to be invested, the Markowitz in the article Portfolio Selection
investment period, the level of risk that will in the Journal of Finance in 1952. Markowitz
arise, and the amount of return that will be model states that portfolios can be formed
obtained. The level of risk factor and the in two ways, namely minimizing variance or
factor of the amount of return (return) are maximizing expected return. The Markowitz
the main factors that form the basis of procedure has several weaknesses, firstly
making investment decisions, where one of this model requires a very large number of
the steps to achieve the objectives of these estimates to fill the covariance matrix,
two factors is to have an investment these two models cannot provide direction
for forecasting the risk premium of

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Stock Investment Portfolio Analysis with Single Index Model

securities which is fundamental to forming Is a return that has occurred. Return is

an efficient frontier of risky assets. calculated based on historical data.
In 1963, William Sharpe developed the Realized return is important because it is
Single Index Model which is a simplification used as a measure of company
of the Markowitz Model, the Single Index performance. This historical return is
Model provides an easier variance analysis also useful as a basis for determining the
solution when compared to the Markowitz expected return and risk in the future.
Model analysis which requires using The calculation of realized return here
Lagrange Multiplier analysis. The Single uses total return. Total return is the total
Index Model can also be used to calculate of an investment in a certain period.
expected return and portfolio risk. The 2) Return Expected
Single Index Model can be an alternative in Is a return that is used for making
forming an optimal portfolio that is easier investment decisions. This return is
for investors or investment managers. By important compared to historical returns
using the Single Index Model approach, we because the expected return is the
can determine the efficient set of portfolios expected return on the investment
more simply because the Single Index Model made. Expected return can be calculated
simplifies the amount and type of input using the expectation value method,
(data), as well as the analysis procedure to which is to multiply each future outcome
determine the optimal portfolio. The Single by its probability of occurrence and add
Index Model assumes that the correlation of up all the products of this product.
the returns for each stock occurs because of Understanding risk according to Keown
the security's response to changes in a (1999), risk is the possibilities that a
particular index. return will be different from the
The step that needs to be taken by using the expected rate of return. According to
Single Index Model is to find the beta value Jones (2002), there are two types of risk,
of the stocks that will be included in the namely:
portfolio, in finding the beta value, we can a. Systematic risk
use an assumption / judgment or can use Is a risk related to conditions that
historical beta to calculate past beta which occur on the market in general,
is used as an estimate beta in the future. namely interest rate risk, political
Historical beta provides important risk, inflation risk, exchange rate risk
information about future beta. This study and market risk. Also called the risk of
aims to apply the Single Index Model in not diversification.
finding the beta value of an efficient b. Non-systematic risk
portfolio line, so that investors can Is the risk associated with the
determine the stocks and the proportion of condition of the company that occurs
funds needed to form an optimal portfolio. individually, namely business risk,
leverage risk and liquidity risk. Also
LITERATURE REVIEW called diversification risk, residual
Return and Investment Risk risk, unique risk, or company-specific
According to Brigham et al (1999), stated risk. So, it can be concluded that risk
that Return is: "measure the financial is the possibility of a real deviation of
performance of an investment." return is the rate of return against the
used in an investment to measure the expected rate of return. The amount
financial results of a company / an of the risk value can be found by
investment. According to (Jogiyanto, 2008) calculating the standard deviation, or
returns can be divided into: by calculating the variance.
1) Return Realization

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Vol. 3 Issue 2, Desember 2020 https://journal.uniku.ac.id/index.php/ijbe

Portfolio Ri = return of the ith security

Investment portfolios, especially securities αi = the expected value of the security's
investment portfolios, are formed from return that is independent of market
various combinations of risky assets or a returns
combination of risky assets with non-risk βi = beta which is a coefficient that
assets in the capital market. The measures the change in Ri as a result of
combination of these assets can reach an change in RM
unlimited number, therefore there is a wide Rm = rate of return of the market index, also
selection of possible portfolios that a random variable
investors can choose from. With rational ei = residual error which is a random
assumptions, investors will choose the variable with the same expected value with
optimal portfolio. “The optimal portfolio zero or E (ei) = 0
can be determined using the Markowitz
model or a single index model. To DATA AND METHODOLOGY
determine the optimal portfolio with these Data
models, an efficient portfolio is first In this study, the data sources used were:
needed” (Jogiyanto, 2008). 1) Market share price index / JCI taken
A portfolio can be concluded, namely the from the datastream. JCI data
investment of various stocks which aims to represents market factor or market
make an efficient combination of investing data.
in these stocks so that investors can get high 2) SBI interest rates taken from
returns and can reduce the risk of these http://www.bi.go.id SBI data
investments. represents risk free (rf)
3) The share prices of three companies
Single Index Model were taken from
In 1963, William Sharpe developed a http://finance.yahoo.com
portfolio analysis model called the Single The selected company stock data are three
Index Model which simplifies the calculation companies, namely:
of the Markowitz model by providing the a) PT Ace Hardware Indonesia Tbk (ACES),
input parameters required in the calculation b) PT Indocement Tunggal Perkasa Tbk
of the Markowitz model. The Single Index (IMTP),
Model can also be used to calculate c) PT Matahari Putra Prima Tbk (MPPA).
expected return and portfolio risk. The company data mentioned above were
"The single index model is based on the selected with consideration by the level α
observation that the price of a security (Intercept)> 0, where the excess of the
fluctuates in the direction of the market portfolio return when the excess of the
price index" (Jogiyanto, 2010: 339). In market return is zero. this selection was
general, the observed stocks that most made to avoid bias. The data period is
stocks experience an increase in shares if monthly data from the prices of the three
the stock price index rises, and vice versa if shares of this company and also monthly
the stock price falls, most stocks experience data from the IHSG for 5 years (2010 to
a decrease in price. This illustrates that the 2014), so we get 60 data series. The monthly
returns of securities may be correlated due period is taken because the data from SBI
to a common reaction (common response) (risk free) are issued by Bank Indonesia on a
to changes in market value. The single index monthly basis.
model can be formulated as follows:
Methodology
Ri = αi + βiRm + ei (1) Single Index Model Calculation.
The steps in calculating the single Index
Where: Model are as follows:

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A. Calculate the monthly return of each 𝛼𝐴 = ∑𝑛𝑖=1 𝜔𝑖 𝛼𝑖 (6)

company stock index and IHSG.
4) Calculate the residual variance
Return of stock i is: of the active portfolio
𝑅𝑖 =
𝑆𝑡𝑜𝑐𝑘𝑅𝑒𝑡𝑢𝑟𝑛𝑡 −𝑆𝑡𝑜𝑐𝑘𝑅𝑒𝑡𝑢𝑟𝑛𝑡−1
(2) 𝜎 2 (𝑒𝐴) = ∑𝑛𝑖=1 𝜔𝑖 2 𝛼𝑖 2 (7)
𝑆𝑡𝑜𝑐𝑘𝑅𝑒𝑡𝑢𝑟𝑛𝑡−1

5) Calculate the initial position in

the active portfolio
B. Calculating the Excess Return of each 𝛼𝐴
(𝑒𝐴)
portfolio index by subtracting the risk- 𝜔𝐴0 = 𝜎2
(8)
𝐸(𝑅𝑀)
free return of each index. 𝜎2𝑀

Excess return (t) = return index (t) – risk 6) Calculate the beta of the active
free (t) portfolio
(3) 𝛽𝐴 = ∑𝑛𝑖=1 𝜔𝑖 𝛽𝑖 (9)

C. Compute the average and standard 7) Adjust the initial position in the
deviation of Excess Returns active portfolio
D. To regress the stock excess return (Ri) to 0
𝜔𝐴
the market (Rm) 𝜔𝐴∗ = 0 (10)
(1+(1−𝛽𝐴 )𝜔𝐴
E. Perform a portfolio analysis
a) Calculate the annualized Risk 8) The current weights of the
Parameters of the Investable optimal risky portfolio
Universe 𝜔𝑀∗
= 1 − 𝜔𝐴∗ ; 𝜔𝑖∗ = 𝜔𝐴∗ 𝜔𝑖 (11)
b) Correlation of Residuals
c) Macro Forecast and Forecasts of 9) Calculate the risk premium of
Alpha Values the optimal risky portfolio from
d) Calculate the Optimization the risk premium of the index
Procedure as follows: portfolio and the alpha of the
1. Calculate annualized Risk active portfolio
Parameters of the Investable ∗
𝐸(𝑅𝑝 ) = (𝜔𝑀 + 𝜔𝐴∗ 𝛽𝐴 )𝐸(𝑅𝑀 ) + 𝜔𝐴∗ 𝛼𝐴 (12)
Universe
2. Correlation of Residuals 10) Calculate the variance of the
3. Macro Forecast and Forecasts of optimal risky portfolio from the
Alpha Values variance of the index portfolio
4. Calculate Optimizing Procedure and the residual variance of the
as follows: active portfolio
𝜎𝑃 = (𝜔𝑀 + 𝜔𝐴∗ 𝛽𝐴 )2 + [𝜔𝐴∗ 𝜎(𝑒𝐴 )]2
2 ∗
(13)
1) Calculate the initial position of
each security in the active 11) Calculate sharpe ratio from
portfolio portofolio.
𝛼𝑖
𝜔𝑖0 = 2 (𝑒𝑖) (4)
𝜎
2) Scale the initial positions so that
the portfolio weights to sum to 1 RESULT
using Compute the average and standard
𝜔𝑖0 deviation of Excess Returns
𝜔𝑖 = 𝑛
∑𝑖=1 𝜔10
(5)
Here the result computation of standard
3) Calculate the alpha of the active deviation of market index (IHSG), ACES,
portfolio INTP and MPPA. Form the result we can see

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Vol. 3 Issue 2, Desember 2020 https://journal.uniku.ac.id/index.php/ijbe

that all stock Standard deviation of excess highest value and INTP is the lowest value
return are above the market index Standard among all stock and index.
deviation of excess return. MPPA is the

Table 1. Standard Deviation of Excess Return

Stock SD of excess return
IHSG 0.1558
ACES 0.3868
INTP 0.2824
MPPA 0.4738

Perform a regression of stock excess return

(Ri) against the market (Rm), so that the
following results are obtained:

Table 2. ACES Regression Statistics

Multiple R 0,29123
R Square 0,084815
Adjusted R Square 0,068759
Standard Error 0.10774083
Observations 59
Coefficients Std. Error
Intercept 0,029719 0,014202
IHSG 0,722756 0,314465

Table 3. INTP Regression Statistics

Multiple R 0,480847042
R Square 0,231213878
Adjusted R Square 0,217726402
Standard Error 0.072090384
Observations 59
Coefficients Std. Error
Intercept 0,00353935 0,009503
IHSG 0,87118649 0,210412

Table 4. MPPA Regression Statistics

Multiple R 0,32737389
R Square 0,107173664
Adjusted R Square 0,091510044
Standard Error 0.130352338
Observations 59
Coefficients Std. Error
Intercept 0,023585759 0,017182878
IHSG 0,99519683 0,380461761

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Stock Investment Portfolio Analysis with Single Index Model

Table 5. Risk Parameters of the Investable Universe (annualized)

SD of SD of
SD of Correlation
excess Beta Systematic
Residual with IHSG
return component
IHSG 0.1558 1.00 0.1558 0 1
ACES 0.3868 0.72 0.1126 0.3700 0.29
INTP 0.2824 0.87 0.1358 0.2476 0.48
MPPA 0.4738 1.00 0.1551 0.4476 0.33

Table 6. Correlation of Residuals

ACES INTP MPPA

ACES
1 0.004 -0.04
INTP
0.004 1 -0.13
MPPA
-0.04 -0.13 1

Table 7. The Index Model Covariance Matrix

IHSG ACES INTP MPPA

Beta 1.00 0.72 0.87 1.00

JKSE 1.00 0.0243 0.0176 0.0212 0.0242
ACES 0.72 0.0176 0.1496 0.0153 0.0175
INTP 0.87 0.0212 0.0153 0.0797 0.0211
MPPA 1.00 0.0242 0.0175 0.0211 0.2244

To minimize firm spesific risk, a protfolio table 7 we can see that all covariances has
must consist of positif covariances, from no negative value.

Table 8. Macro Forecast and Forecasts of Alpha Values

IHSG ACES INTP MPPA
Alpha 0 0.0297 0.0035 0.0236
Risk
0.10 0.1020 0.0907 0.1231
premium

Table 8 shows that INTP risk premium is All stock forecast of alpha values are higher
below the market index but ACES and MPPA than market index alpha values.
are higher risk premium than market index.

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Vol. 3 Issue 2, Desember 2020 https://journal.uniku.ac.id/index.php/ijbe

CONCLUSION

Table 9. Computation of the Optimal Risky Portfolio

Active Overall
IHSG ACES INTP MPPA
Pf A Pf
σ2(e) 0.1369 0.0613 0.0241
a/σ2(e) 1.2554 0.2171 0.0577 0.9805
0
W (i) 1.0000 0.1729 0.0460 0.7811
[W0(i)]2 0.0299 0.0021 0.6101
aA 0.0237 0.0051 0.0002 0.0184
σ2(eA) 0.0189 0.0041 0.0001 0.0147
W0 0.3049
*
W 0.7004 0.2996 0.0518 0.0138 0.2340
Beta 1 0.9424 0.1250 0.0401 0.7773 0.9827
Risk
0.1000 0.1180 0.0176 0.0042 0.0962 0.1054
premium
SD 0.1558 0.2012 0.1586
Sharpe
0.6417 0.5864 0.6645
Ratio

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