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Portfolio Optimization for Several Industries among the U.S.
Stock Market
Yanru Li*
School of civil engineering, Central South University, Changsha, Hunan, China
*Corresponding author: 8210180731@csu.edu.cn
Abstract. Optimizing portfolio has been a popular topic since it was proposed because it can reduce
the investment risk. This study selected five active stocks from different industries, including Online
E-Commerce, Commercial Banks, Motor Vehicles, Mobile Communication Production, and
Telecommunications. Then they were allocated into five kinds of portfolios, which are tangency
portfolios and minimum variance portfolios under Capital Asset Pricing Model and Fama-French
three-factor Model, as well as the 1/N portfolio. The results found that ‘BAC’ and ‘T’ have the largest
weight and the lowest weight respectively in the two models. The comparison of the five portfolios
showed that the portfolio with the highest cumulative return is the tangency portfolio under FF3F
Model, and the 1/N portfolio also performed well. Only these two portfolios outperformed the SPDR
S&P 500EFT Trust. This research may help investors focused on the five sectors mentioned above
to have a better idea of how to allocate their capital.
Key words: Capital market, factor model, portfolio optimization.
1. Introduction
The American economics Harry Markowitz first derived a pathbreaking mathematical model for
portfolio selection in 1952, the mean-variance model, which is widely used to solve allocating risky
assets’ problems and impact profoundly on the global financial market [1]. In this model, the portfolio
risk is quantified as variance, and the investors prefer to minimize it because their capital is limited,
and they are afraid to lose the principal [2]. In general, the weaker the investors’ risk tolerance is, the
higher their loss aversion level is likely. Therefore, the optimizing portfolio method has been a hot
topic for numerous scholars. Many studies indicate that diversification can reduce risk to a certain
extent [3], and research on portfolio optimization have evolved considerably recently.
Previous studies often concentrated on the multivariate modeling of the market development trend
and analysis, or some new theories from the macro perspective under certain economic environment.
For example, Drake, Fabozzi, and Fabozzi proposed a portfolio theory and a capital market theory to
be as a framework of security’s risk and return measurement, and they also revealed the existing
relationship between expected return and risk [4]. Feng Yongfu and other scholars investigated the
volatility of the Chinese A-share market by mining data [5]. They also conducted a maximum
likelihood estimation of the returns and used a theoretical-empirical model which can calibrate the
volatility of the market well. Their model will contribute to simulating and forecasting, which can
help local governments regulate and the individuals invest. Procacci and Aste pointed out three
sources of error associated with the complex portfolio modeling system, including uncertainties
resulting from parameters’ sampling error, oversimplifying hypothesis, and intrinsic non-stationarity
of these systems. The last two can be quantified after analyzing parameters of the training sets of
various lengths. They also concluded the result that the optimal portfolio is related to its holding
period. Mo and Chen used the topological structure of the financial network to select stocks to build
the portfolio and then used the random matrix theory to improve the performance of the portfolio by
reducing its risk [7]. However, few studies aim to concentrate on industries like Online E-Commerce,
Commercial Banks, Motor Vehicles, Mobile Communication Production, and Telecommunications.
It is obvious that Online E-Commerce has become increasingly important in a strategic perspective
[8], as well as the other four industries, because they are an integral part of people’s lives.
This paper focuses on five representative companies selected from each industry mentioned before.
First, this paper collected the adjusted closing price of the five companies during the period between
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December 2016 and December 2021, and the data also was checked so that they matched with each
other in the time dimension. Second, different methods was used to analyze the portfolios’ efficiency,
which contributes to better decision-making for investors interested in certain fields. Arithmetic mean
returns and expected returns based on Capital Asset Pricing Model (CAPM) and Fama-French Three-
Factor Model (FF3F Model) are used to build up the tangency portfolios and the mean-variance
portfolios. This paper compared the changes of each portfolio’s weights and tried to explain the
reasons. Additionally, naïve strategy (1/N portfolio) was concerned as well, because Demiguel,
Garlappi, and Uppal indicate that the very simple portfolio performed better when comparing its
Sharpe ratio with those of other optimizing models [9]. Finally, all portfolios were compared to find
out which one has the highest cumulative return.
The remaining parts of this paper includes four sections. Section 2 introduces the chosen five
stocks and their data used in the future study of this article. Section 3 describes the methods of getting
the optimal portfolios. Section 4 and Section 5 present results with graphs and figures, as well as
conclusion and discussion.
2. Data
The data used later in this paper are from the most active stocks section in Yahoofinance
(https://finance.yahoo.com/most-active). This paper selected five representative companies,
including AMZN, BAC, F, NOK, NVDA, and T. These stocks are considered chosen because they
performed actively, and they belong to different industries in order to diversify. Their five years
monthly adjusted closing prices from December 2016 to December 2021 were downloaded for the
portfolio construction. The price data of SPDR S&P 500EFT Trust was also downloaded to represent
market sentiment. In this paper, prices were transferred into differential returns, and the processed
data were separated into a training set (2017.01-2020.12) and a test set (2020.01-2021.12). The first
set is used for constructing the portfolio, and the other one is for the performance evaluation of the
asset allocations. The stocks’ descriptive statistics of the training set are illustrated in the Table 1
below.
Table 1. Descriptive statistics of the selected stocks (Training Set)
AMZN BAC F NO
K
T
average return 3.44% 1.17% 0.19% 0.26% -0.01%
standard deviation 8.37% 8.28% 9.05% 9.83% 5.54%
maximum return 26.89% 18.82% 19.02% 20.58% 9.92%
minimum return -20.22% -25.51% -30.60% -27.87% -17.23%
It is illustrated in the graph that the ‘AMZN’ is the stock with the highest average return, while the
lowest is -0.01% from the ‘T’. Regarding standard deviation, the ‘NOK’ and the ‘T’ are the highest
and the lowest respectively. Additionally, the ‘T’ has the lowest maximum return and the highest
minimum return, while the highest maximum return is the ‘AMZN,’ and the lowest minimum return
is the ‘F’.
3. Method
3.1 Mean-Variance Analysis
A mathematical tool, the mean-variance analysis enables related investors to allocate assets
according to defined objectives. It is a useful approach to balancing return and risk and making trade
decisions rationally when constructing portfolios. In this framework, each asset is a random variable
obeying a probability distribution. The arithmetic mean and the variance of one asset’s return are
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given the economic meanings, which are the expected return and the risk respectively [10]. The
relevant formula are listed below:
𝑤
1
(1)
Where 𝑖 is the serial number of 𝑎𝑠𝑠𝑒𝑡 in the portfolio, and 𝑤 is the weight of the 𝑎𝑠𝑠𝑒𝑡. The
total number of assets is 𝑛. In this paper, 𝑛 is equal to five.
𝐸𝑅𝑤
𝐸𝑟
(2)
Where the ‘E’ and the ‘P’ is shorthand for ‘expected’ and ‘portfolio’ respectively. The 𝑅 is
the portfolio return and 𝑟 is the 𝑎𝑠𝑠𝑒𝑡 return.
𝜎
𝑤
𝑤𝐶𝑜𝑣𝑟,𝑟
(3)
Where 𝜎
is the variance and stands for the volatility of the portfolio, and 𝐶𝑜𝑣𝑟,𝑟
is the
covariance which is used to measure the overall error between the 𝑎𝑠𝑠𝑒𝑡 return and the 𝑎𝑠𝑠𝑒𝑡
return.
𝑆ℎ𝑎𝑟𝑝𝑒 𝑅𝑎𝑡𝑖𝑜 𝐸𝑅𝑅
𝜎 (4)
The 𝑅 means an interest rate without credit risk and market risk, and 𝜎 is the portfolio
standard deviation.
3.2 Capital Asset Pricing Model
The CAPM, first developed by Sharpe et al. in 1967, is widely recognized as a significant return
forecasting model in securities investment [3]. Although there are many new models after then, it is
still classic, and its central status is undoubtable in financial economics despite of some critiques [11].
It assumes that when the capital market reaches the equilibrium, the risk and return of each asset will
become balanced, which means the marginal price of risk will not change. In this case, the return
of an asset consists of the risk-free rate and the risk premium rate. The higher the risk is, the
correspondingly higher the value of the latter one will be. The well-known equation is showing below:
𝐸𝑟𝑅
𝛽
,𝐸𝑅𝑅
(5)
In this paper, 𝛽, is a coefficient with the interval range of 0.5 to 2.0 that measures the systematic
risk of 𝑎𝑠𝑠𝑒𝑡, and 𝐸𝑅 is the market expected return.
3.3 Fama-French Three-factor Model
A three-factor model was proposed by Fama and French, two renowned economists, for analyzing
stock returns and argued that the added two factors can compensate for the risk factors not captured
by market risk in CAPM. This model performs well in assessing portfolio when compared with
market returns [12]. Its equation is shown below:
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𝐸𝑟𝑅
𝛽
,𝐸𝑅𝑅
𝛽
,𝐸𝑆𝑀𝐵𝛽,𝐸𝐻𝑀𝐿 (6)
The 𝐸𝑆𝑀𝐵 is the expected returns from low market capitalization stocks less returns from high
market capitalization stocks, and the 𝐸𝐻𝑀𝐿 is the expected returns from high B/M stocks less
returns from low B/M stocks. Respectively, 𝛽, as well as 𝛽, are the risk factor coefficients.
4. Results
The five companies’ expected return can be calculated using the CAPM model and the FF3F model,
and the results are shown in Table 2.
Table 2. Expected returns under CAPM and FF3F Model (Training set)
AMZN BAC F NO
K
T
CAPM 1.010% 1.256% 1.077% 0.774% 0.732%
FF3F 0.792% 1.439% 1.321% 0.801% 0.790%
The expected returns under the two models are slightly different, but the highest one and the lowest
one are same, i. e. the ‘BAC’ and the ‘T’. The regression results of five companies’ beta coefficients
are shown in Table 3.
Table 3. Some details of five stocks under CAPM and FF3F Model (Training set)
AMZN BAC F NO
K
T
𝛽, (CAPM) 1.09 1.45 1.18 0.74 0.68
𝛽, (FF3F) 1.40 1.31 0.84 0.71 0.60
𝛽, (FF3F) -0.65 -0.22 0.70 0.03 0.15
𝛽, (FF3F) -0.93 1.01 1.05 0.13 0.26
It is shown in Table 3 that ‘T’ has the lowest market beta no matter in which model, while ‘BAC’
and ‘AMZN’ wins the highest market beta respectively in FF3F model and in CAPM. This paper
calculated the Sharpe ratios in two cases which is showing below in Table 4.
Table 4. Sharpe ratios under CAPM and FF3F Model (Training set)
AMZN BAC F NO
K
T
CAPM 8.84% 11.90% 8.91% 5.12% 8.33%
FF3F 6.23% 14.11% 11.60% 5.39% 9.38%
The highest Sharpe ratio and the lowest one under two cases are same, i. e. BAC and NOK. Under
these two models, the portfolio can be constructed with the objectives of maximum Sharpe Ratio,
minimum variance, and equal weights. The results are shown in the following tables.
Table 5. Weights of tangency portfolios under CAPM and FF3F Model (Training set)
AMZN BAC F NO
K
T
CAPM 27.03% 30.34% 11.77% 1.94% 28.93%
FF3F 3.13% 53.77% 19.32% 4.23% 19.56%
Table 5 illustrates the maximum Sharpe ratio portfolios, which is also called tangency portfolios.
This kind of portfolios is the adoption of the mean-variance efficient frontier. After comparing 4
portfolios, Theron and Vuuren [14] concluded that the TG portfolio outperforms others at most of the
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time, with positive cumulative return and relatively low long-term volatility. TG portfolios in this
paper prefer to give ‘BAC’ a high weight.
Table 6. Minimum variance portfolios under CAPM and FF3F Model (Training set)
AMZN BAC F NO
K
T
CAPM 26.10% 0.00% 4.50% 5.81% 63.59%
FF3F 26.10% 0.00% 4.50% 5.81% 63.59%
Table 6 presents the weights of minimum variance portfolios. This strategy is of great concern
over the past several years as well, especially when a financial crisis happens, and risk management
is needed urgently [15]. Bednarek and Patel [16] came out the result that the MVP portfolio tends to
long low beta stocks and short high beta stocks, which did happen in this paper. Both models seem
to choose BAC as the greatest weight stock, and it is possible because the Sharpe ratio of FF3F Model
is higher than that of CAPM, the weights increase when change model from CAPM to FF3F Model.
When it comes to minimum variance portfolio, the reliable stock is T for the two cases.
The data used to construct the portfolio above are from the training set. When the weights are taken
into the test set, the returns are shown in Table 7.
Table 7. Yearly returns of the test set.
return
tangency portfolio (CAPM) 27.95%
tangency portfolio (FF3F) 48.97%
minimum variance portfolio (CAPM) 4.92%
minimum variance portfolio (FF3F) 4.92%
equal weights portfolio 45.28%
The return of SPDR S&P 500EFT Trust in the same year is approximately 28.82%, which means
the only two cases where the asset portfolio outperforms it are maximum Sharpe ratio (48.97%) and
equal weights (45.28%). It can be observed from Figure 1 and Figure 2 as well.
Figure 1. Comparison between 4 portfolios under CAPM model
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Figure 2. Comparison between 4 portfolios under the FF3F model
5. Conclusion
In summary, Existing asset portfolio studies tend to focus on the whole market or a specific popular
sector, while this paper provides five portfolios (maximum Sharpe ratio and minimum variance for
CAPM model, maximum Sharpe ratio and minimum variance for FF3F model, and equal weights)
consisting of five stocks in different sectors for diversification purposes. In this paper, the expected
return of each stock in a specific period and the covariance matrix are calculated based on the data
from Yahoo Finance. After applying the mean-variance analysis to optimize the portfolio and
construct the tangency portfolio and the minimum volatility portfolio, the cumulative portfolio returns
for 2021 are available using the test set data.
The comparison with SPY shows that only the Sharpe ratio maximizing portfolio and the equally
weighted portfolio under the FF3F model outperforms the stock index, with the former having the
highest returns. ‘T’ has a share of over 60% in the minimum variance assets allocation, indicating
that it is a worthwhile long position for risk-averse investors. In conclusion, the asset portfolio
constructed by applying the FF3F model is superior, with a maximum Sharpe ratio portfolio providing
the highest returns and a moderate risk. The minimum variance strategy, while reducing risk, also
significantly reduces expected returns.
References
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