Analysis of Static and Dynamic Investment Strategies

VSB — TECHNICAL UNIVERSITY OF OSTRAVA    FACULTY OF ECONOMICS 

Analysis of Static and Dynamic Investment Strategies

Analýza Statických a Dynamických Investičních Strategií 

Student:    Bc. Yuchun Zhou   

Supervisor of diploma thesis:       prof. Ing. Tomáš Tichý, Ph.D.       

Ostrava 2021   

VŠB - Technical University of Ostrava Faculty of Economics

Department of Finance

Diploma Thesis Assignment

Student:

Bc. Yuchun Zhou

Study Programme: N0412A050005 Finance

Title:

Analysis of Static and Dynamic Investment Strategies Analysis of Static and Dynamic Investment Strategies

The thesis language: English Description:

1. Introduction

2. Theoretical Foundations of Financial Investments 3. Description of Investment Strategies

4. Evaluation of Selected Strategies at Chinese Financial Market 5. Conclusion

Bibliography

List of Abbreviations

Declaration of Utilisation of Result from the Diploma Thesis List of Annexes

Annexes

References:

ANTONACCI, G. Dual Momentum Investing: An Innovative Strategy for Higher Returns with Lower Risk. 1st ed. New York: McGraw-Hill, 2014. 240p. ISBN 978-0071849449.

BODIE, Z., KANE, A., MARCUS, A.J. Investments and Portfolio Management, 9 th ed. New York: McGraw-Hill, 2010. 1056 p. ISBN 978-0071289146.

MALKIEL, B.G. A Random Walk Down Wall Street: The Time-Tested Strategy for Successful Investing. 12th ed. New York: W. W. Norton & Company, 2020. 432p. ISBN 978-

0393358384.

Extent and terms of a thesis are specified in directions for its elaboration that are opened to the public on the web sites of the faculty.

Supervisor: prof. Ing. Tomáš Tichý, Ph.D.

Date of issue: 20.11.2020 Date of submission: 23.04.2021

Ing. Petr Gurný, Ph.D. doc. Ing. Vojtěch Spáčil, CSc.

Head of Department Dean  

Declaration   

I hereby declare that I have elaborated the entire thesis including annexes myself. 

Ostrava dated ...      ...   

Student 's name and surname   

Content 

1. Introduction ... 7

2. Theoretical Foundations of Financial Investments ... 9

2.1 Portfolio Theory ... 9

2.1.1 Mean-Variance Model (MV)... 10

2.1.2 Capital Asset Pricing Model (CAPM) ... 11

2.1.3 Arbitrage Pricing Model (APM) ... 12

2.1.4 Option Pricing Model (BSM) ... 13

2.1.5 Efficient Markets Hypothesis (EMH) ... 14

2.2 Risk Measurement Theory ... 15

2.2.1 Safety-First Principle ... 15

2.2.2 Lower-Partial Moments (LPM) ... 16

2.2.3 Value at Risk (VaR) ... 16

2.2.4 Conditional Value at Risk (CVaR) ... 17

2.3 Return Measurement of Investment Portfolio... 19

2.4 Forecasting the Stock Volatility by GARCH Model ... 19

2.5 Forecasting the Stock Prices by Geometric Brownian Motion ... 21

3. Description of Investment Strategies ... 24

3.1 Static Investment Strategy ... 25

3.2 Dynamic Investment Strategy ... 27

4. Evaluation of Selected Strategies at Chinese Financial Market ... 30

4.1 Static Portfolio in Chinese Stock Market... 30

4.2 The Actual Performance of Static Investment ... 35

4.3 Dynamic Portfolio in Chinese Stock Market ... 38

4.4 The Actual Performance of Dynamic Investment ... 53

5. Conclusion ... 58

Bibliography ... 60

List of Abbreviations ... 61

Declaration of Utilisation of Result from the Diploma Thesis ... 62

List of Annexes ... 63

Annexes ... 64  

1. Introduction 

With the continuous development of the financial market, financial products, funds and other professional investment products have become important tools for investors to enter the capital market. These products are essentially an investment portfolio, so how to accurately evaluate the investment portfolio is an important subject faced by investors. However, in reality, investment is often a dynamic and constantly adjusted multi-stage process, and it is not reasonable to pay attention to the evaluation method of the final result only. Meanwhile, investment in different stages is correlated with each other. In order to make a reasonable evaluation of this kind of investment portfolio, it is necessary to comprehensively consider the performance of each stage and the relationship between stages. Therefore, it is of great significance to construct a reasonable dynamic portfolio efficiency evaluation model. 

The objective of this thesis is to form a practical guiding way of asset selection by using two distinct investment strategies: static investment strategy and dynamic investment strategy. Different investment groups have different expectations on the performance of asset portfolios due to their different investment capabilities and risk tolerance, so there are various types of optimal asset portfolios. The primary goal of the optimal portfolio set by this paper is to minimize the risk and volatility of the portfolio, to seek a stable rate of return, and then to choose a portfolio with the highest expected rate of return within an acceptable range of low risk as the optimal portfolio. 

Generally speaking, static investment strategy is based on the current point in time and conducts asset allocation according to the acquired information about asset volatility, expected return and so on. In a static investment strategy, once the asset allocation proportion is determined, no matter what happens to the market, the asset allocation proportion will not be changed or corrected before the maturity of the portfolio. Dynamic investment strategies are the opposite of this asset allocation approach. Under the dynamic investment strategy, the asset allocation weights will be regularly changed or revised in a specific interval and at a certain frequency according to the changes of market situation and asset information. In short, a static investment

strategy determines the optimal asset allocation at the time the decision is made.

However, dynamic investment strategy will periodically modify the portfolio based on market changes, so this investment strategy determines the dynamic optimal portfolio for each interval. 

The argument for static versus dynamic investment strategy is about application performance and efficiency. Theoretically speaking, dynamic investment strategy seems to be better than static investment strategy, but it cannot be proved without empirical analysis that the return rate of the portfolio under dynamic investment strategy is higher than the return performance of the portfolio under static investment strategy. Moreover, because dynamic investment strategies require asset allocation modifications and adjustments at defined intervals, it naturally takes more time, effort, and expense than static investment strategies. From the perspective of investment efficiency, the static investment strategy is likely to be higher than the dynamic investment strategy. 

Therefore, so as to make the detailed comparison of static investment strategy and dynamic strategy from the portfolio return and asset allocation efficiency and find out the more suitable method for asset allocation, the thesis will select 30 real stocks’ daily closed prices as a data source and evaluate the performances of both static and dynamic strategies. There are daily closed prices of these stocks from 2010 to 2020, and data from the previous 10 years will be used for asset allocation, data from the last year will be used to measure the performance of static and dynamic investments and to compare the two strategies.  

In chapter 2, the theoretical foundations of investment will be described. Next to it, in chapter 3 there will be the more detailed static and dynamic investment strategies explained. The formula which would be applied in the following evaluation will also be in chapter 3. In chapter 4, I would exhibit the daily closed prices of those 30 stocks.

Therefore, the detailed evaluation of the static and dynamic investment strategies will be exhibited in chapter 4 step by step. After these procedures, it would be able to conduct the volatility and return comparison of the two strategies as well as determine the more ideal investment strategy in the last chapter. 

2. Theoretical Foundations of Financial Investments 

2.1 Portfolio Theory 

After the financial crisis in 2008, the developed economies in the world were basically in the era of zero or negative interest rate. In 2014 and 2015, the central bank of China cut interest rates and the reserve requirement for many times, and the benchmark interest rate fell to a record low. In the context of low global interest rates, other investment channels other than bank deposits are more popular with funds. For many investments with certain risks, “Don’t put all your eggs in one basket” illustrates the importance of controlling risks. Building investment portfolio has become the most effective way to spread risks. 

In recent decades, portfolio theory has been deeply studied by investment institutions and scholars all over the world, and widely used in investment activities. It is of great help to avoid risks, hedge and gain returns of assets, and also plays a crucial role in the stability and development of social economy. 

The concept of investment Portfolio was first proposed by Markowitz in 1952. He published the article “Portfolio Selection” and the book of the same name, laying the foundation for the development of modern investment Portfolio Theory. He used expectation to represent investment income and variance to represent risk to seek the optimal investment portfolio, so as to minimize the risk when the investment income is given, or to maximize the investment income when the risk is determined. Markowitz’s contribution is that he introduced the quantitative method into the field of financial investment and got rid of the previous state of completely relying on analysis and experience. Another contribution is that he constructed the framework of returns and risks of modern portfolio problems. Although some scholars later put forward the shortcomings of variance as a measure of risk, most of the research on portfolio are still in this framework to explore a better method of risk measurement. 

Portfolio Theory mainly revolves around two questions: how to reduce risk and how to increase returns. For the riskless assets, such as national debt, the value of national debt at every moment can be calculated by cash flow discount without the

influence of interest rate, politics and other factors. However, for risky assets, such as stocks and options, there is no precise value in the market, but only the trading price.

The trading price of risky assets often deviates greatly from its value. Therefore, finding an ideal pricing model is the content that institutions and scholars have been studying.

At present, the most widely used models in portfolio are mean-variance model, capital asset pricing model, arbitrage model and Black-Scholes option pricing model. 

2.1.1 Mean-Variance Model (M-V) 

Mean-variance model is a classic model in the portfolio. The key to Markowitz’s portfolio is how to balance the relationship between expected return and variance.

Firstly, Markowitz makes several assumptions about markets and investors: 

The first assumption is that the financial market is an efficient market in which the returns and risks of stocks are understood by every investor. The second assumption is that investors measure the return level by the expected value of the return rate, and they measure the risk by the variance of the return rate. Besides, the correlation between two assets in Markowitz is expressed as covariance. Expected rate of return and variance are also the parameters applied to measure portfolio performances in this study. The Markowitz also assumes that investor’s decision only depends on the expectation and variance of returns. What’s more, there is also the assumption of investors’ rationality.

Under Markowitz model, investors are rational and seek for higher returns on the same level of portfolio variances. 

Based on the above hypotheses, let N be the number of assets, 𝑟_𝑖 be the return rate of asset 𝑖 , Cov (𝑟_𝑖 , 𝑟_𝑗 ) be the covariance of asset 𝑖 and asset 𝑗 , 𝑟_𝑝 be the expected return rate, and 𝑥_𝑖 represent the proportion of investment in assets in 𝑖. The parameters of Markowitz are calculated by these formulas: 

Capital stock return: 𝑅_𝑖,𝑡 =^{𝑃𝑖,𝑡−𝑃𝑖,𝑡−1}

𝑃_{𝑖,𝑡−1} . (2.1)  Mean (expected) return: 𝐸(𝑅_𝑖) = ¹

𝑁∙ ∑^𝑁_𝑡=1𝑅_𝑖,𝑡. (2.2)  Sample variance of returns: 𝜎_𝑖² = ¹

𝑁−1∙ ∑^𝑁_𝑡=1[𝑅_𝑖,𝑡 – 𝐸(𝑅_𝑖)]². (2.3)  Sample standard deviation of returns:  

𝜎_𝑖 = √ ¹

𝑁−1∙ ∑^𝑁_𝑡=1[𝑅_𝑖,𝑡 – 𝐸(𝑅_𝑖)]² = √𝜎_𝑖² . (2.4)  Covariance between returns (sample): 

𝜎_𝑖𝑗 = ¹

𝑁−1∙ ∑^𝑁_𝑡=1[𝑅_𝑖,𝑡 – 𝐸(𝑅_𝑖)]∙ [𝑅_𝑗,𝑡 – 𝐸(𝑅_𝑗)] = ^𝑁

𝑁−1∙ 𝑐𝑜𝑣(𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛) . (2.5)  Thus, we have defined all the necessary parameters in static mean-variance model, and then the Markowitz model can be run by converging the results to satisfy some pre- defined constraints:  

Min Var (𝑟_𝑝)=¹

2∑^𝑛_{𝑖=1,𝑗=1}𝑥_𝑖𝑥_𝑗Cov (𝑟_𝑖, 𝑟_𝑗). (2.6)  Subject to { 𝑥_𝑖 ≥ 0

∑^𝑛_𝑖=1𝑥_𝑖 = 1. (2.7)  The above formulas (2.6) and (2.7) are constraints for obtaining point in the efficient frontier with the lowest risk. However, for constructing the efficient frontier it’s also needed to obtain point with the highest expected return. Hence, these formulas are for the calculation of the highest expected return in the efficient frontier: 

Max 𝑅̅_𝑝 =∑^𝑛_𝑖=1𝑥_𝑖𝑟̅_𝑖 . (2.8)  Subject to { 𝑥_𝑖 ≥ 0

∑^𝑛_𝑖=1𝑥_𝑖 = 1. (2.7)  In the model, if 𝑥_𝑖 is higher or equal to 0, since in Markowitz model the short selling is not allowed.  

2.1.2 Capital Asset Pricing Model (CAPM) 

On the basis of mean-variance model, Sharpe William put forward the single index-model in 1963, and in the following year, he put forward the Capital Asset Pricing Model (CAPM). Almost at the same time, Lintner. Capital markets are based on three ideal assumptions: (1) investors’ decisions are based on expected rates of return and variance; (2) All investors have the same expectations and are ideal mean-variance investors; (3) The market is a complete market with no friction.  

Based on the above assumptions, the specific expression of CAPM model is as follows: 

E (𝑟_𝑖)= 𝑟_𝑓+𝛽_𝑖(E (𝑟_𝑚− 𝑟_𝑓). (2.9)  𝛽_𝑖 = ^{Cov (𝑟𝑖,𝑟𝑚)}

𝑉𝑎𝑟𝑖(𝑟_𝑚) =^𝜎𝑖𝑚

𝜎_𝑚² . (2.10) 

Where, E (𝑟_𝑖) represents the expected rate of return of asset 𝑖, E (𝑟_𝑚) represents the expected rate of return of market portfolio, 𝑟_𝑓 represents the rate of return of risk- free asset, 𝜎_𝑖𝑚= Cov (𝑟_𝑖, 𝑟_𝑚) represents the covariance of the rate of return of asset 𝑖 and the rate of return of market portfolio, and 𝜎_𝑚² represents the variance of market portfolio. 

The formula (2.3) in the model is also called the security market line, from which we can know that the risk of a single asset is composed of two parts. The first part is the return change of asset 𝑖 caused by the return change of market portfolio, namely 𝛽_𝑖, which can be considered as the system risk. The other part is residual risk, also known as non-systemic risk. According to the capital asset pricing model, the price of a single asset is only related to the system risk of the asset, while has nothing to do with the non-system risk. 

CAPM model is rather simple and practical, which is widely used in financial investment decision. The CAPM model can be used to find the assets in the market whose actual income is different from the expected income. Undervalued assets should be bought, and overvalued assets should be sold. However, the assumptions of CAPM model are all based on the ideal market and the ideal investor, which indicates that CAPM model is not exactly the case of reality financial market. Black, Jensen and Scholes made the famous BJS test, while Eugene Fama and Michaels made the FM test, which all verified the CAPM model. Among them, Roll Richard questioned the verification, which caused scholars to rethink the CAPM model. 

2.1.3 Arbitrage Pricing Model (APM) 

Based on Roll Richard’s doubts on CAPM model, scholars began to explore other portfolio models. In 1976, Ross Stephen proposed Arbitrage Pricing Theory (APT) in the expansion of Capital Asset Pricing Model. Arbitrage Pricing Model in the core assumption is that if you increase profits without adding risks, all investors will take this opportunity to do the same. In other words, arbitrage is the process which makes overvalued asset prices fall, makes an undervalued asset prices rise, eventually makes no arbitrage opportunity in the assets market. Here is the related formula: 

𝑅_𝑖 =𝑎_𝑖 +∑^𝑛_𝑗=1𝑏_𝑖𝑗𝐹_𝑗 +𝜀_𝑖 . (2.11)  Among them, 𝑅_𝑖 is the return rate of asset 𝑖, and 𝐹_𝑗 is the 𝑗^𝑡ℎ factor affecting the return rate of asset. 𝑏_𝑖𝑗 represents the sensitivity of the return rate of asset 𝑖 to factor 𝑗, and 𝜀_𝑖 is the random difference term. N = 1 represents a single factor model. For instance, Sharpe proposed a single exponential model in 1963. N = 2 represents the two-factor model, which is the more commonly used model. 

2.1.4 Option Pricing Model (BSM) 

In the 1970s, Black and Scholes used the capital asset pricing model to determine the relationship between the market’s expected return on European options and the expected return on stocks, which was not only related to stock prices but also related to time. Merton’s approach is different from that of Black and Scholes in that his approach involves the combination of options and stocks. Moreover, his approach is more universal and does not rely on the assumptions of capital asset pricing model. The pricing formula of BSM call option and put option is: 

C=𝑆₀N (𝑑₁)-K𝑒^−𝑟𝑇 N (𝑑₂). (2.12)  P= K𝑒^−𝑟𝑇 N (-𝑑₂)- 𝑆₀N (𝑑₁). (2.13)  In which, 

𝑑₁=^ln(

𝑆0

𝐾)+(𝑟+^𝜎2₂)𝑇

𝜎√𝑇 . (2.14)  𝑑₂ =^ln(

𝑆0

𝐾)+(𝑟−^𝜎2₂)𝑇

𝜎√𝑇 = 𝑑₁ - 𝜎√𝑇 . (2.15)  Function N (·) is the cumulative probability distribution function of the standard normal distribution, C and P are the prices of European call options and put options respectively, 𝑆₀ is the price of the stock at time 0, K is the strike price, 𝑟 is the risk- free interest rate, 𝜎 is the volatility of the stock price, and T is the term of the option.

There is no expected return rate of stocks in the BSM model. This result indicates that when pricing a European option that depends on stock price, the market can be assumed to be risk neutral, that is, the expected return rate of stocks is equal to the risk-free interest rate. BSM model has a great influence on option pricing and hedging. Therefore, it’s widely used in practical trading strategies. 

2.1.5 Efficient Markets Hypothesis (EMH) 

The Efficient Markets Hypothesis (EMH) was developed by Eugene Fama in 1970.

EMH indicates that stock prices reflect all available information. For example, a large premium may have to be paid if a company seeks a merger or acquisition. Because under the efficient market assumption, the announcement of a restructuring will cause a rise in share prices. EMH is a significant support for dynamic investment strategy, since under the efficient market, all valuable information is timely, accurately and fully reflected in the share price, including current and future value. Hence, the dynamic investment strategy appears to be reasonable as the stock prices indicate large amount of information. There are three main forms of efficient markets: Weak-Form Market Efficiency, Semi-Strong-Form Market Efficiency and Strong-Form Market Efficiency. 

Weak-Form Market Efficiency means that all the information in the stock market except the information in the stock market itself is reflected in the stock price, for example historical stock price, historical trading volume, price/earnings ratio, price-to- book ratio and financial indicators of listed companies. The assumption argues that technical analysis of stocks is of little use because basic information about these stocks is freely available to every investor. In this way, there is no problem of information asymmetry among all investors, and the investment decisions are based on the most basic information of the most popular stocks. 

Semi-Strong-Form Market Efficiency indicates that in addition to the information in the weakly efficient market, there is something relevant to the company’s growth prospects also reflected in its share price. For example, the production and operation situation, management situation and product market share of listed companies, profit forecast and other information. Therefore, there are also some public information, such as national macro-policy regulation, change of the listing system, the improvement of the delisting system and the integration with the international stock market fully reflected in the stock prices. 

Strong-Form Market Efficiency is that the stock price reflects all information related to the listed company, including the information known in advance by the

executives of the listed company. There is no insider trading in the strong efficient market, because there is no information asymmetry in the market, which is a favorable condition for investors. However, this assumption is ideal, including today’s developed securities markets in Europe and the United States do not exist strong efficient market.

What is not in doubt is that management has long traded on key information to make a profit before it has been published. In fact, most of the SEC’s activities are aimed at preventing insiders from taking advantage of their position. 

2.2 Risk Measurement Theory 

After the Mean - Variance portfolio model is proposed, variance, as a measure of risk has a major drawback. Variance only considers the deviation of the return rate, without distinguishing whether the return is positive deviation or lower deviation, both of which are regarded as risks. However, in the actual situation, the return of the investment with positive deviation exceeding the expectation is the return pursued by investors. Hence, positive deviation should not be punished as risks. The risk that investors really care about is the lower deviation of the return lower than the expectation.

In 1959, Markowitz also found the inadequacy of variance risk measurement and proposed to replace it with semi-variance, but there were still some defects.

Subsequently, a variety of lower bias risk measures appeared, and the best representative ones were LPM, VaR and CVaR. 

2.2.1 Safety-First Principle 

The Safety-First Principle (SFP) was first proposed by Roy in 1952. When studying the downside risk, he proposed that investors should consider the probability of catastrophic loss or investment return less than the minimum required return, that is, the Safety-First Principle. He believed that the lower the probability, the better. Let P (x, y) represent the expected return of the investment portfolio and A represent the minimum return level. The expression of the safety priority principle SFP is as follows: 

SFP= min{𝑷(𝑃(𝑥, 𝑦) ≤ 𝑎)} . (2.16)  The above formula indicates that SFP is the minimum probability that P (x, y) is less than 𝑎. SFP is a downward-skewed risk measurement study. The basics, while very

simple, still have very important practical implications. 

2.2.2 Lower-Partial Moments (LPM) 

The lower-partial moments free investors from the bondage of relying solely on quadratic functions such as variance or semi-deviation as the objective function and covers almost all the degree of risk preference. LPM takes into account the situation where revenue is below target, and the formula is as follows: 

𝐿𝑃𝑀_𝑛(h, R) =∫ (ℎ − 𝑅)_−∞^{ℎ 𝑛}𝑑F(R). (2.17)  Wherein, F(R) represents the cumulative distribution function of investment return 𝑅, ℎ represents the target return rate, and 𝑛 represents the degree of risk preference of investors. For the discrete case, let a total of 𝑇 values of rate of return 𝑅 be observed, and expressed by 𝑅₁, 𝑅₂,···, 𝑅_𝑇, then there will be: 

𝐿𝑃𝑀_𝑛(h, R) = ¹

𝑇−1∑^𝑇_𝑖=1[max (0, ℎ − 𝑅_𝑖)^𝑛]. (2.18)  The lower-partial moments provide a wider risk measure than semi-deviation and semi-variogram. The probability distribution of investment return 𝑅 in the lower offset can be arbitrary. Some common risk measures are special cases of the lower-partial moments, such as loss probability 𝐿𝑃𝑀₀ (h) and expected loss 𝐿𝑃𝑀₁ (h). When calculating the lower offset, how to set a reasonable benchmark target profit rate h and risk preference 𝑛 is also a problem worthy of further study. 

2.2.3 Value at Risk (VaR) 

Value-at-risk (VaR) is also a kind of downside risk measurement, which is not limited to general risk measurement, but also emphasizes on catastrophic risk management. In the 1990s, several shocking bankruptcies took place in the world, which made people pay more attention to the management of financial risk. VaR refers to the expected maximum loss of a portfolio within a certain confidence level 𝛼 and holding period 𝑡 under normal market conditions. The confidence level 𝛼 and holding period 𝑡 are important factors in the model, and their values are subjective to a certain extent. The value of 𝛼 in international investment institutions is 95% and 99%, and the value of 𝑡 is 1 and 10 days. Suppose P (𝑥, 𝜔) represents the expected return of the

investment group, 𝑥 = (𝑥₁ , 𝑥₂ ,···, 𝑥 nn) represents the investment portfolio, 𝑥_𝑖 represents the proportion of investment in asset 𝑖, 𝜔 represents uncertainty and obeys the probability distribution of density 𝑓(𝜔). In the case of a given confidence level 𝛼, VaR meets the following requirements: 

1- 𝛼 =∫_−∞^𝑉𝑎𝑅𝑓(𝜔)𝑑𝜔 . (2.19)  𝑷(𝑃(𝑥, 𝜔) ≤ −𝑉𝑎𝑅) =1- 𝛼 . (2.20) 

Figure 2.1 Value at Risk 

Source: ZMESKAL.Z. DLUHOSOVA. D. and Tomas. TICHY. Financial Models  Although the probability of VaR loss is small, if it happens, it will have a catastrophic impact on the portfolio, so it needs to be managed. 

2.2.4 Conditional Value at Risk (CVaR) 

Although VaR is widely used, VaR does not satisfy subadditivity. In addition, VaR only provides the minimum possible loss within the confidence interval α, and it does not consider the loss in the probability long tail, which may occur when the confidence interval α is the same VaR, but the tail information is very different, as shown in figure 2.2. 

Figure 2.2 Tail Information of VaR 

Source: ZMESKAL.Z. DLUHOSOVA. D. and Tomas. TICHY. Financial Models  In order to take into account that the tail information of VaR and improve the deficiency of VaR, Rockafellar and Uryasev proposed conditional value at risk (CVaR), which represents the conditional period expectation of loss when the loss exceeds VaR.

Here we use the P (𝑥 , 𝜔 ) to present the investment loss function, other marks are constant as them in VaR, set threshold limit probability of alpha 𝛹 (𝑥, 𝛼) , in the confidence interval 𝛽 risk value is: 

𝑉𝑎𝑅_𝛽= min{𝛼 ∈ 𝑅: 𝛹 (𝑥, 𝛼) ≥ 𝛽}. (2.21)  The corresponding conditional value at risk is:  

𝐶𝑉𝑎𝑅_𝛽 = ¹

1−𝛽∫P (𝑥,𝜔)≥𝑉𝑎𝑅_𝛽P (𝑥, 𝜔) 𝑓(𝜔)𝑑𝜔 . (2.22)  Figure 2.3 The Relationship Between VaR and CVaR 

Source: ZMESKAL.Z. DLUHOSOVA. D. and Tomas. TICHY. Financial Models 

2.3 Return Measurement of Investment Portfolio 

In general, the return on a stock is made up of two components: dividends and capital gains from price changes. Dividends are very volatile and unpredictable, depending on the business and profitability of the company issuing the stock. In this paper, the stock return rate excludes the influence of dividends and only considers the price changes. Hence, the stock rate of return is defined as: 

𝑅_𝑖 =(𝑃₁ -𝑃₀ )/ 𝑃₀ . (2.23)  The return rate of an individual asset is calculated by the expected value of its return rate time series, while the return rate of a portfolio asset is obtained by weighting the return rates of various assets in the portfolio, that is: 

𝑅̅_𝑝=∑^𝑛_𝑖=1𝑥_𝑖𝑟̅_𝑖. (2.24)  Among them: 𝑅̅_𝑝 is the yields for portfolio, 𝑥_𝑖 is the weight of all the assets, 𝑟̅_𝑖 is the yield of all assets. 

2.4 Forecasting the Stock Volatility by GARCH Model 

In the previous chapters, there was methodology for evaluating the risk and expected return according to the historical information of selected stocks. However, in the stock market, future estimations of stock volatility are attached great significance when investors are determining the investment portfolios. Therefore, this thesis is supposed to consider the future estimations as well in order to construct good portfolios in the process of dynamically adjust the stock composition.  

As for the dynamic investment strategy, the GARCH (Generalized Autoregressive Conditional Heteroskedastic) model will be applied for obtaining the estimated stock volatilities. Traditional econometrics has an assumption about time series variables: it is assumed that the fluctuation amplitude (variance) of time series variables is fixed.

This is not true in reality. For example, the variance of stock is not constant because it has long been known that stock returns fluctuate over time. This makes the traditional time series analysis not effective for practical problems. The ARCH model (Autoregressive conditional heteroskedasticity model) solves the problem caused by the traditional econometrics’ assumption about time series variables (constant variance).

If the variance is expressed by the ARMA model, then the ARCH model is transformed into the GARCH model. The GARCH (Generalized Autoregressive Conditional Heteroskedastic) model of volatility is introduced by Engle (1982) and Bollerslev (1986). Moreover, the GARCH model is specifically designed to capture the volatility clustering of returns. The forecasts which are made from the GARCH model is not equal to the current estimations. Instead, the evaluated volatility can be higher or lower than the average volatility over short-term. However, as the forecast time horizon increases, estimated volatility would converge to the long-term values. To understand a GARCH model, it’s necessary to clarify the distinctions between unconditional variance and the conditional variance of a time series of returns. The unconditional variance is just the variance of the unconditional returns of distribution, which is constant over the entire data period considered. It can be thought of as the long-term average variance over the whole period. The conditional variance, on the other hand, will change at any point in time because it depends on the history of returns up to this point. That is, we account for the dynamic properties of returns by regarding their distribution at any point in time as being conditional on all the information up to this point.  

The strength of GARCH is that it provides short-term and medium-term volatility forecasts, which are based on a proper econometric model. There are a number of kinds of GARCH models, such as symmetric normal GARCH, normal GJR-GARCH, normal E-GARCH, symmetric Student t GARCH and so on. Among these methods, symmetric normal GARCH is relatively simple to estimate and much superior to any moving average models. 

The symmetric normal GARCH model was developed by Bollerslev (1986), which is the plain vanilla version of a GARCH model. The calculation formula of GARCH (1:1) model for one period is: 

𝑦_𝑡+1 =𝑦̂_𝑡+1 +𝜀_𝑡+1 . (2.25)  𝜎_𝑡+1,𝑡² =𝜔 +𝛼 ⋅ 𝜀_𝑡² + 𝛽 ⋅ 𝜎_{𝑡,𝑡−1}² . (2.26)  In which, 𝜔>0, 𝛼>0,𝛽>0, and meanwhile 𝛼 + 𝛽 <1. In the formula (2.25), 𝑦_𝑡+1 is the observed value of the forecasted financial asset. 𝑦̂_𝑡+1 is the estimated value of

the forecasted financial asset at time t. 𝜀_𝑡+1 is the forecast error, which is the difference between the forecasted value and observed value.  

In formula (2.26), 𝜀_𝑡² is the observed variance, while 𝜎_{𝑡,𝑡−1}² defines the forecasted value of the variance. These are estimated parameters: [𝜔, 𝛼, 𝛽, 𝑎, 𝑏]=𝜃⃗.

For calculation, the expected variance of the forecasted financial asset can be obtained by applying the maximum likelihood method (MLM). 

2.5 Forecasting the Stock Prices by Geometric Brownian Motion 

Geometric Brownian motion is also known as exponential Brownian motion. An exponential Brownian motion is a process where the logarithm of a random variable follows Brownian motion in continuous time. Geometric Brownian motion is used in financial mathematics to simulate stock prices in the Black-Scholes pricing model.

Therefore, it has also been the most commonly used model to describe stock prices.

This is also one of the reasons why this model is chosen in this paper to predict the future price of the target stock. 

The Geometric Brownian Motion model is chosen to predict price parameters in dynamic investments because it has several advantages: 

The expectation of Geometric Brownian Motion is independent of the price of the random process, which is consistent with our expectation of the real market. What’s more, Geometric Brownian Motion processes only consider positive stock prices, as the case in real stock market that only positive prices count in the reality. Besides, the process of Geometric Brownian Motion has big similarity with the volatility of the price trajectory we observe in the stock market. Last but not the least, Geometric Brownian Motion process is relatively easy to simulate. It’s necessary to readjust the optimal assets allocation at the beginning of each under dynamic investment. Hence, the simplicity of Geometric Brownian Motion would give the advantage to save investor’s time and energy. 

To use GBM successfully, we first need to simulate a set of random numbers as a set of parameters in the GBM process. And in the process of simulating the target financial assets in GBM, we need to apply some Monte Carlo theories. In order to derive

the formula and process of GBM, a concept, namely Wiener Process, must be introduced here. Wiener Process is the basic element of GMB, and Wiener Process is based on 2 assumptions: One of the assumptions is that Wiener Process follows the Markov process. Future prices in Wiener Process are influenced by the current prices, while are not infected by the historical prices. The second assumption of Wiener Process is that the increments are independent in time. Wiener Process is defined by the formula: 

𝑧̃_𝑡-𝑧₀= 𝑑𝑧=𝑧̃ ⋅ √𝑑𝑡. (2.27)  In this formula (2.27), 𝑧̃ stands for a random variable from the standard normal distribution. Then we could consider several more evolution steps, it would be able to derive the formula: 

𝑧̃_𝑡-𝑧₀=∑^𝑛_𝑖=1𝑧̃ ⋅ √𝑑𝑡. (2.28)  Next, it would be able to introduce a kind of general stochastic process. The Arithmetic Brownian Motion can be defined as: 

𝑑𝑥 = 𝜇 ⋅ 𝑑𝑡 + 𝜎 ⋅ 𝑑𝑧. (2.29)  In this formula (2.29), each variable is independent from each other, and the parameters are constant. Geometric Brownian Motion is similar with Arithmetic Brownian Motion, and can be defined as: 

𝑑𝑥 = 𝜇 ⋅ 𝑥 ⋅ 𝑑𝑡 + 𝜎 ⋅ 𝑥 ⋅ 𝑑𝑧 . (2.30)  After integration, we can deform and deduce this formula, and get another form of it, which will make it easier for us to understand this formula：

𝑑𝑥

𝑥= 𝜇 ⋅ 𝑑𝑡 + 𝜎 ⋅ 𝑑𝑧. (2.31)  From this formula, we can clearly see that it is a good fit to express the income generated by the price change of a financial asset. We need a few letters to represent the key parameters. Hence, we get 𝛼 to stand for average return of an asset; 𝜎 to stand for the standard deviation of this asset. Then average return and standard deviation can be defined as: 

E (𝑑𝑥)= 𝜇 ⋅ 𝑑𝑡. (2.32)  E (𝑥_𝑇)= 𝑥₀+𝑥₀⋅ 𝜇 ⋅ 𝑇. (2.33) 

Var (𝑑𝑥)=𝜎²⋅ 𝑑𝑡. (2.34)  Var (𝑥_𝑇)= 𝑥₀+𝑥₀⋅ 𝜎² ⋅ 𝑇. (2.35)  If we apply the formula (2.30) into a case of an asset with continuous returns, there will be more formulas which would be meaningful in this study. Assume the returns of asset 𝑆 are calculated continuously. We could write down the formula of returns derived by asset 𝑆: 

𝑆_𝑇 = 𝑆 ⋅ 𝑒^𝜇 . (2.36)  And just by bending this formula we can get an expression for the continuous rate of return 𝜇=ln^𝑆𝑇

𝑆. (2.37) As the price return follows the formula (2.30), and the price 𝑆 is lognormally distributed. Hence, we can derive the following formulas：

𝑆_𝑇= 𝑆 ⋅exp⋅(𝛼 ⋅ 𝑇 + 𝜎 ⋅ 𝑑𝑧). (2.38)  E (𝑆_𝑇)= 𝑆 ⋅exp⋅(𝜇 ⋅ 𝑇). (2.39)  Var (𝑆_𝑇)=𝑆²⋅exp⋅(2⋅ 𝛼 ⋅ 𝑇)⋅[exp⋅(𝜎² ⋅ 𝑇)-1]. (2.40)  The 𝑦-quantile for the lognormal distribution can be defined as: 

𝑆_𝑇^𝑦= 𝑆 ⋅ exp⋅ [𝛼 ⋅ 𝑇 +Φ⁻¹ (𝑦 )⋅ 𝜎 ⋅ √𝑇 ]. (2.41)  In this formula (2.41), Φ⁻¹(𝑦) is the 𝑦-quantile for the lognormal distribution, and it can be calculated in Excel by the formula NORMSINV (𝑦 ). The 𝑦 -quantile specifies the boundary within which the estimated prices of the asset are supposed to evolve randomly.  

3. Description of Investment Strategies 

The portfolio strategy in the stock market refers to the allocation of investment weight to a variety of stocks in order to achieve the goal of maximizing the return or minimizing the risk with a certain return. Portfolio strategy is mainly divided into static portfolio strategy and dynamic portfolio strategy, the difference between the two is mainly the static or dynamic stock asset allocation, the similarity is based on the Markowitz mean-variance model as the core framework. The use of portfolio strategy can play a role in investment diversification to increase returns and disperse risks.

Portfolio strategy has been developed in foreign countries for more than 60 years and has gradually matured. However, Chinese financial market started late, and the development of portfolio strategy is lagging behind that of foreign countries. However, in recent years, with the rapid development of stock market and securities investment fund industry in China, the application of portfolio strategy in Chinses stock market is more and more urgent.  

In order to further compare the specific performance of static investment and dynamic investment in the actual financial market, this study selected 30 stocks of Shenzhen Stock Exchange for calculation and research. These 30 stocks come from 10 popular investment sectors in China: food and beverage (A), medicine and biology (B), real estate (C), leisure services (D), commercial trading (E), household appliances (F), clothing (G), car (H), electronic industry (I), information technology (J). I selected 3 stocks from each of these 10 popular investment sectors to simulate how the normal investor narrowed down his investment decision when faced with countless stocks in the market. From now on, these stocks will be represented by A1, A2, A3…J1, J2, J3. 

In China, over 87% investors hold their stocks less than one year, the majority of these investors who make short-term investments will sell the stocks in months. In this thesis, I assume there is no dividend for simplicity. There is a lot of randomness in the short-term stock investment. So as the get the research more practical to the real situation in China stock market, I will set the investment period as 1 year and under the dynamic investment situation, the portfolio will be readjusted at the end of each month.

In other words, the portfolio will be reconstructed 12 times. For drawing the final conclusion, I will compare the annual return of the portfolios as well as the annual volatility of the portfolios. 

Referring to the basic assumptions of mean-variance model, the basic assumptions of modeling in this chapter are as follows: All the 30 stocks can be traded, but short selling factor is not taken into account; Investors are rational investors who avoid risks and expect to gain returns; Investors mainly consider risks and returns when making investment decisions; In accordance with the hypothesis of Markowitz’s mean-variance model, the expected value of return time series is used to represent investment return, and the volatility of return rate, namely variance or standard deviation, is used to measure risk. 

3.1 Static Investment Strategy 

The static portfolio model is assumed as a single period, that is to say, investors take the investment period in a certain period of time, and only consider the return and risk situation during this period. In this thesis, the input data is from 2010 to 2020, which is 11 years. It’s assumed that the investor was at the beginning of 2020 and the investing period is 1 year. In other words, the investor will consider the historical data from 2010 to 2019 and make a portfolio, which will be applied to the year 2020. Once the static investment is decided on the historical data, there would be no investment adjustments for the whole year 2020. For measuring the actual performance of the static investment, the assets allocation weights will be applied to the market prices of the year 2020. Such an investment and measurement method simplified the complex stock market. 

Here in this chapter, A1, A2, A3…J1, J2, J3 are 30 stocks from 10 different industries in Shenzhen Exchange as the assets to composite the portfolio. Assuming that investor’s total wealth will be invested in these 30 stocks, then 1≤N≤30. Where, N is number of assets which are included in the portfolio. 

According to the return measurement from Markowitz mean-variance model introduced in chapter 2, the expected return of each single asset will be calculated at first. It can be obtained by the following formula: 

𝑅_𝑖,𝑡 = ^{𝑃𝑖,𝑡−𝑃𝑖,𝑡−1}

𝑃_{𝑖,𝑡−1} . (2.1)  𝐸(𝑅_𝑖) = ¹

𝑁∙ ∑^𝑁_𝑡=1𝑅_𝑖,𝑡. (2.2)  After getting the expected return of each single the portfolio can be obtained: 

𝑅̅_𝑝=∑^𝑛_𝑖=1𝑥_𝑖𝑟̅_𝑖. (2.24)  Afterwards, it would be able to calculate the risk parameters which will be needed to form a model. The risk is measured by variance and standard deviation of returns in this research, and they are calculated by: 

𝜎_𝑖² = ¹

𝑁−1∙ ∑^𝑁_𝑡=1[𝑅_𝑖,𝑡 – 𝐸(𝑅_𝑖)]². (2.3)  𝜎_𝑖 = √ ¹

𝑁−1∙ ∑^𝑁_𝑡=1[𝑅_𝑖,𝑡 – 𝐸(𝑅_𝑖)]² = √𝜎_𝑖² . (2.4)  Since the stocks have relationship with each other and each individual in the financial market is related in a way, the covariance of these stocks should also be taken into account. Hence, it’s necessary to calculate the covariance to complete the Markowitz model. Here the formula of covariance is obtained from the formula expressed in chapter 2, covariance between returns (sample): 

𝜎_𝑖𝑗 = ¹

𝑁−1∙ ∑^𝑁_𝑡=1[𝑅_𝑖,𝑡 – 𝐸(𝑅_𝑖)]∙ [𝑅_𝑗,𝑡 – 𝐸(𝑅_𝑗)] = ^𝑁

𝑁−1∙ 𝑐𝑜𝑣(𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛) . (2.5)  Hence, we got all the parameters which are necessary for the calculation of Markowitz model. Therefore, it’s able to apply these formulas in Excel. According to them, the expected return of the portfolio can be expressed as: 

E (R_p) = ∑ x_{i i}∙ E(R_i)= x⃗⃗^T∙ E(R⃗⃗⃗). (3.1)  Variance and standard deviation of the portfolio can be expressed as: 

𝜎_𝑝²=∑ ∑ 𝑥_{𝑖 𝑗 𝑖} ∙ 𝜎_𝑖𝑗∙ 𝑥_𝑗 = x⃗⃗^T∙ C ∙ x⃗⃗. (3.2)  𝜎_𝑝 = √∑ ∑ 𝑥_{𝑖 𝑗} 𝑖∙ 𝜎_𝑖𝑗 ∙ 𝑥_𝑗 = √x⃗⃗^T∙ C ∙ x⃗⃗ . (3.3)  As the investment horizon is crucial for a portfolio investment, in this thesis the investment horizon is settled as 1 year. The daily closed prices of stocks are selected as the input data. Under the static investment strategy, the Markowitz model will be applied to form the optimal portfolio. At the beginning of the investment, weights of each asset will be determined according to the formulas mentioned above. By applying

the Markowitz model, the investment portfolio will be constructed at the beginning of the investment. Once settled, the portfolio composition will not be adjusted in the following investment period, that is the whole year 2020. 

For evaluating the performance of static investment strategy, the asset allocation obtained from Markowitz model will be applied in the market price of 2020. There will be real return exhibited. 

3.2 Dynamic Investment Strategy 

As the measurement of investment return and risk both require the time series of stock return rate, the establishment of dynamic M-V model requires the selection of the length of the historical stock closing price data, and the length of the input data is crucial, and even affects the success of the entire investment strategy. Since the stock price always changes with time, the longer the input data length of the M-V combination model is not the better. It is necessary to select an appropriate range value, and the length of the input data should be selected according to the actual situation.  

As is settled in the previous part, the investment period of the portfolio is 1 year.

Since it’s dynamic investment strategy, the portfolio will be adjusted monthly. In other words, the portfolio will be readjusted 12 times after being settled by applying the Markowitz mean-variance model. It’s known that stock prices are always of high volatility and there is currently no reliable method to capture the stock piece changes.

Hence, this chapter will draw on the GARCH model to forecast the future volatility of stocks. As for the stock expected return, the Geometric Brownian Motion will be applied for the estimation.  

For applying GARCH model, it’s mentioned in chapter 2 that this study takes the MLM. Hence, the objective function is: 

L (𝜔, 𝛼, 𝛽)=∑ 𝑧_{𝑡 𝑡}→max. (3.3)  The constraints in GARCH model are already mentioned in chapter 2, which are:

𝜔>0, 𝛼>0,𝛽>0, and meanwhile 𝛼 + 𝛽 <1. For the calculation of estimated variance by the observed variance, the formula (2.26): 𝜎_𝑡+1,𝑡² =𝜔+𝛼 ⋅ 𝜀_𝑡²+ 𝛽 ⋅ 𝜎_{𝑡,𝑡−1}² . Besides, the formula for calculating the parameter which stands for the likelihood, we apply: 

𝑧_𝑡=-ln𝜎_{𝑡,𝑡−1}² - ^𝑅𝑡2

𝜎_{𝑡,𝑡−1}² . (3.4)   Since the investor is assumed to be in the beginning of year 2020, and he is seeking to invest in the 30 selected stocks in the following year, the asset allocation of the first month in 2020 will be determined according to the previous 10 years (2010-2019). To begin with, I get the initial optimal portfolio by Markowitz mean-variance model. At the end of each year, I will readjust the weight of each stock according to the GARCH model and the GBM process. There are 30 stocks in this study, hence, the process of estimating the future variance by GARCH model will be repeated 30 times for each month in 2020. From February 2020, this paper will use the stock price of the previous month for the data to calculate the variance estimated by the GARCH model. Because last month’s data is relatively new market information, it is more meaningful in terms of data effectiveness than the historical data of the past 10 years. 

As for the estimated returns of the target stocks, the GBM process will be applied for calculation. GBM is relatively easy to calculate and can be simulated simply by the key stock parameters: average return, stock variance, time interval and initial stock price. As the study of investment practice, the feasibility is an important part of the study, if I seek to explore practical guiding significance for investors for asset allocation, then the asset allocation process requires investment presents the characteristics of a good feasibility. Otherwise, the operation difficulty or cost of too much energy will gives no meaning to ordinary investors. Therefore, the GBM process in this paper is carried out as follows: 

The first step is to construct a random evolution of the stock price. First and foremost, it’s necessary to determine time interval, ∆𝑡. ∆𝑡 indicates how much time the researchers want between the two predicted periods. To simulate the number of trading days in a year without rest days, this paper assumes that there are 250 trading days in a year. For instance, assume the study predicts prices with an interval of 1 day,

∆𝑡 will be 0.004, for 1/250=0.004. And for other days of time interval, the same rule will apply. Then, it would be able to simulate a random evolution of the stock price if we apply the following formula: 

𝑆_𝑡=𝑆_𝑡−1⋅exp⋅(𝛼 ⋅ ∆𝑡+𝜎 ⋅ 𝑧̃ ⋅ √∆𝑡). (3.5)  Mean value of the stock price will be: 

E (𝑆_𝑇)= 𝑆₀⋅exp⋅(𝜇 ⋅ ∆𝑡 ⋅ 𝑛)= 𝑆₀⋅exp⋅(𝜇 ⋅ 𝑇). (3.6)  Besides, variance of the asset’s price and the stock price quantile for the lognormal probability distribution are here as follows. These two formulas only substitute the parameter t on the basis of formulas (2.40) to (2.41).

Var (𝑆_𝑇)=𝑆₀²⋅exp⋅(2⋅ 𝛼 ⋅ ∆𝑡 ⋅ 𝑛)⋅[exp⋅(𝜎²⋅ ∆𝑡 ⋅ 𝑛)-1]. (3.7)  The 𝑦-quantile for the lognormal distribution is: 

𝑆_𝑇^𝑦= 𝑆₀⋅exp⋅[𝛼 ⋅ ∆𝑡 ⋅ 𝑛+Φ⁻¹(𝑦)⋅ 𝜎 ⋅ √∆𝑡 ⋅ 𝑛]. (3.8)  In the same way that GARCH is used to estimate the risk of a target stock over the next 12 months, GBM requires 30 calculations, each of which estimates the price of the stock over the next 12 investment periods. Instead of directly obtaining the return rate of each stock in the next 12 months through GBM, we first estimate the price trend of the 30 target stocks in the next 12 investment periods through GBM, and then obtain the estimated return rate through this formula based on the estimated stock prices: 

𝑅_𝑖,𝑡 = ^{𝑃𝑖,𝑡−𝑃𝑖,𝑡−1}

𝑃_{𝑖,𝑡−1} . (2.1)  By applying the GARCH model and the GBM process, we will know the estimated value of expected returns and volatilities in the following one period. Therefore, we apply Markowitz mean-variance model once again to get a new portfolio for next investment period. At the end of the investment periods, there will be 12 portfolios in total. For evaluating the performance of dynamic investment strategy, the average return and average variance will be calculated. 

4. Evaluation of Selected Strategies at Chinese Financial Market   

4.1 Static Portfolio in Chinese Stock Market 

The static portfolio model is assumed as a single period, that is to say, investors take the investment period in a certain period of time, and only consider the return and risk situation during this period. Such a static idea simplifies the complex stock market.

In the thesis, Markowitz model will be applied to construct the static investment portfolio. The input data is from Shenzhen Stock Exchange, and there are 30 stocks which can be invested in. The assets are from 10 different industries which are attached great importance in Chinese stock market, and they are food and beverage, medicine and biology, real estate, leisure services, commercial trade, household appliances, clothing, car, electronic and information technology. 

I collected the daily closing prices of these stocks from January 2010 to December 2019. There are daily closing prices of 10 years. Stock is one of the riskiest assets in the investment market with extremely high volatility. Generally speaking, the holding period of stocks are from days to decades. Hence, it’s reasonable to set the time span over which benefits are measured as one month. Therefore, the daily data will be selected. Since monthly accounting units are needed to measure the risk and expected return of a stock, I will first calculate the monthly average of the daily closing price of each stock to get some data samples. That gives each stock 120 observations from 2010 to 2019. In reality, companies pay their dividends in different terms and the stocks dividends are directly influenced by the profits gained by the invested companies. Thus, in this thesis, I don’t take the stock dividends into account, the stock returns are only calculated from the price changes. 

According to formula 𝑅_𝑖,𝑡 =^{𝑃𝑖,𝑡−𝑃𝑖,𝑡−1}

𝑃_{𝑖,𝑡−1} (2.1), 𝐸(𝑅_𝑖) = ¹

𝑁∙ ∑^𝑁_𝑡=1𝑅_𝑖,𝑡 (2.2) and 𝜎_𝑖 = √ ¹

𝑁−1∙ ∑^𝑁_𝑡=1[𝑅_𝑖,𝑡 – 𝐸(𝑅_𝑖)]² (2.4), it’s able to get the monthly returns and standard deviation of these stocks. The monthly return is settled to measure the stock rate of return, and the standard deviation is settled to measure the volatility of stocks. The two indexes mentioned above to measure the stock performances are shown in the table 4.1

below. 

Table 4.1 Monthly Returns and Volatilities of Selected Assets 

  𝑹_𝒊  𝝈_𝒊    𝑹_𝒊  𝝈_𝒊    𝑹_𝒊  𝝈_𝒊 

A1  0.47%  7.76%  D2  0.40%  10.56%  G3  -0.55%  11.36% 

A2  1.00%  8.21%  D3  0.45%  9.36%  H1  -0.06%  11.04% 

A3  -0.20%  13.20%  E1  0.58%  11.50%  H2  0.13%  9.32% 

B1  0.29%  9.37%  E2  -0.18%  8.07%  H3  0.18%  10.28% 

B2  -0.36%  13.51%  E3  0.84%  12.28%  I1  0.15%  12.31% 

B3  1.22%  10.86%  F1  -0.12%  10.06%  I2  2.00%  13.22% 

C1  0.06%  13.74%  F2  0.53%  11.17%  I3  0.47%  11.14% 

C2  0.16%  10.37%  F3  0.53%  13.09%  J1  0.78%  13.46% 

C3  1.30%  9.07%  G1  -0.54%  11.38%  J2  1.55%  13.09% 

D1  -0.63%  8.09%  G2  0.19%  13.11%  J3  1.36%  11.92% 

As it’s shown obviously in the figure, the stocks have rather high volatility, and all of them have volatility from 7.76% to 13.74%. However, their average returns are not satisfying at all, most of them have monthly returns at around 0.15%, some stocks even have negative return, and the lowest monthly return is -0.55%. Hence, it’s an evidence of prolonging the investment period as much as possible so as to gain profits from the stock investment. It also argues the necessity of constructing portfolios to diversify the risks from stock market. 

Since the mean returns and standard deviations are all available, now I still need covariance matrix to conduct the optimal portfolio constriction by applying Markowitz model. The covariance matrix can be calculated from standard deviation and correlation matrix by applying the formula (2.5): 𝜎_𝑖𝑗 = ¹

𝑁−1∙ ∑^𝑁_𝑡=1[𝑅_𝑖,𝑡 – 𝐸(𝑅_𝑖)]∙ [𝑅_𝑗,𝑡 – 𝐸(𝑅_𝑗)] =

𝑁

𝑁−1∙ 𝑐𝑜𝑣(𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛).  

Therefore, it’s now able to construct the optimal assets allocation by applying Markowitz model. In chapter 2, there are the constraints and objectives: 

Min Var (𝑟_𝑝)=¹

2∑^𝑛_{𝑖=1,𝑗=1}𝑥_𝑖𝑥_𝑗Cov (𝑟_𝑖, 𝑟_𝑗). (2.6)  Subject to { 𝑥_𝑖 ≥ 0

∑^𝑛_𝑖=1𝑥_𝑖 = 1. (2.7)  Max 𝑅̅_𝑝 =∑^𝑛_𝑖=1𝑥_𝑖𝑟̅_𝑖 . (2.8)  The Markowitz model is conducted by using SOLVER under the minimum