portfolio optimization

PORTFOLIO OPTIMIZATION Return-Risk Analysis of Investment in Finance By:

Sabera Muna A Graduate Project in Applied Mathematics Presented to The Faculties of the Department of Mathematics Northeastern Illinois University December 2016



In Partial Fulfillment of the Requirements for the Degree Master of Science in Applied Mathematics

Abstract: The goal of this project is to create an interactive tool on Excel Spreadsheet for users, which will allow an investor to estimate the future return-risk characteristic of their investment portfolio. This work is based on the construction of a multi-asset investment model portfolio consisting of two, three, or five securities, first with the minimum risk analysis and then with the targeted returns. The mathematical method of Modern Portfolio Theory has been applied for minimization of risk with simultaneous maximization of return. Related factors, financial terminologies, mathematical formulation of the theory, and the solution of optimization process has been discussed to understand and support Modern Portfolio Theory. All functions on Excel are generated based on these formulas. The investor can use the built-in Excel template to update the data and stocks of their own choice as well as set their target return according to their level of risk tolerance.



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Acknowledgement: My special thanks to my project advisor Dr. Marina Polyashuk, Associate Professor, Northeastern Illinois University, who assisted me with her very sincere assistance, suggestion and cooperation for a long time. She went through the analysis of the optimization and considered her valuable times for me. I also would like to express my very special thanks to Dr. Marian Gidea, Associate Professor, Yeshiva University and Dr. Zbigniew Krysiak, Associate Professor, Warsaw School of Economics in Poland, who are my mentors of Financial Mathematics and whose high appreciation and encouragement drive me to pick up my graduate project on finance that is an extended work of my ex-project under Dr. Gidea. I am also sharing my gratefulness to Dr. Joseph Hibdon, Assistant Professor, Northeastern Illinois University, who directed me the successful finalization of this project. Finally, my ever respects, thanks and best gratitude to all the professors of Dept. of Mathematics, Northeastern Illinois University for their noble contribution for achieving me a vast knowledge in Applied Mathematics. Page 3





CONTENTS PAGES



1. Introduction Investigation and Structure 2. History Overview Concept Advancement in Work 3. Modern Portfolio Theory (MPT) Source & Extension 4. Risk and Return Definition Categories of Risk Mathematical Formulation 5. Expected Return Data Source Mathematical Format of Expected Return 6. Variance of Expected Return Mathematical Expression









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7. Standard Deviation as Risk Measure (Volatility) Mathematical Expression Examples of Risk in Finance 8. Weights Definition and Mathematical Formulation







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9. Return of a Portfolio Mathematical Derivation 10. Variance of Portfolio Consisting Two Securities Definition & Formulation 11. Risk and Return Trade-Off Definition and Formulation 12. Markowitz Portfolio Theory Introduction Theory and Extension 13. Feasible Set 14. Analysis of MPT for Portfolio of Two Securities Mathematical Structure of Variance









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15. Minimum Variance Portfolio (MVP) of Two Securities Solution of Variance Equation Diversified Weights as a Solution Multiple Condition for Correlation Coefficient 16. Analysis of MPT for Portfolio of Multi-Securities Expected Return and Risk of the Portfolio consisting n-Securities 17. Minimum Variance Portfolio (MVP) of Multi-Securities Objective Function and Constraint Solution for Minimum Variance Portfolio 18. Efficient Frontier Definition and Characteristic of Dominant Portfolios 19. Solving MPT for Targeted Expected Return Introduction Mathematical Equations and Solutions 20. Sharpe Ratio 21. Numerical Illustration Application One Application Two Application Three 22. Summary of The Excel Work Data Tables from Spreadsheet 23. Discussion Limitations of Modern Portfolio Theory Market and Theoretical Limitations



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24. Conclusion

















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25. Further work Idea on Capital Asset Pricing Model (CAPM)









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26. References

















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27. Links

















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28. Appendix



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1. INTRODUCTION In the modern financial market, investment is a major area that has drawn attention, expanded, and exercised in most. Varieties of methodologies as well as investment techniques have been researched in depth for over a decade and thereby numerous approaches have been established in order to maximize the profit and simultaneously to minimize the risk. Portfolio Optimization or Mean-Variance Optimization (MVO) is one out of these methods. Harry Markowitz is the Nobel laureate economist who is recognized as the founder of the Modern Portfolio Theory (MPT), he first published his article as ‘Portfolio Selection Theory’ in Journal of Finance (1952) and continued with more details in his book “Portfolio Selection: Efficient Diversifications” (1959). The technique of the Markowitz mean-variance optimization model was developed later and extended by his fellow William Sharpe (1964) which was extended to the Capital Asset Pricing Model (CAPM), the widely used updated method in the theory of financial asset price calculation. In 1990, Harry Markowitz together with William Sharpe received the Nobel Memorial Prize in Economics Science as a recognition for the importance of this valuable contribution in portfolio management theory (Elton & Gruber, 1997; Rubinstein, 2002). Most investigations related to modern portfolio theory (MPT) are with highly complex statistics-based mathematical modeling that are mainly focused on the theoretical assumptions. Since 1952, when this idea first came to light, the analysis focusing on Markowitz’s contribution in MPT includes many categorical factors and environmental parameters that leaded significant evolution with some advanced financial tools used and approached by the investors and financial professionals. But this project is purposely for the investors of comparatively longer investment horizon who are interested to invest in stock market or in volatile equities particularly. For easy assessment, some less complicated treatments have been introduced here, that are over simplified, less-comprehensive, and do not involve any serious speculative interface. Based on the key theoretical assumptions, there are some simplified modern computing techniques, mainly on Microsoft Excel Spreadsheet that demonstrate the optimizing goal of the portfolio. This financial tool can be used as a short-cut to calculate an estimated assumption for maximizing expected return and minimizing the risk. Here, the history of portfolio literature is reviewed and the concepts of Modern Portfolio Theory (MPT) is described including the definitions of risk and return, expected return, measures of risk and volatility, and diversification. Finally, the goal is set to establish an over-simplified short-cut method of using financial tools in Excel Spreadsheet for a rational investor, and accordingly an investment portfolio is generated Page 6





with the targeted expected return with the related minimum risk to which the investor is comfortable with. 2. HISTORY OVERVIEW The objective of the Modern Portfolio Theory established by Markowitz in his article ‘Portfolio Selection’ (Markowitz, 1952) was the description of the impact of asset allocation that is commonly also known as portfolio diversification by the number of securities within the portfolio and their covariance relationship. The concept based on Markowitz’s model, the economist James Tobin derived ‘Efficient Frontier’ and ‘Capital Asset Line’ as an extension of Markowitz’s Modern Portfolio Theory in his essay “Liquidity Preference as Behavior Toward Risk”, in Review of Economic Studies (Tobin, 1958). In his description, he suggested the market investors to preserve the same proportion of their assets as long as they expect identical outcomes, no matter what could be the level of risk tolerance is. So, the expected return would vary only in the relative proportion of different portfolios of stocks and bonds (Elton & Gruber, 1997). The work of Markowitz and Tobin was advanced by the individual modification by William Sharpe (1964), John Linter (1965) and Jan Mossin (1966). They distinctly derived the concept of Capital Asset Pricing Model (CAPM). CAPM derived better capital market equilibrium where the investors can value their securities as a function of systematic risk. Sharpe re-defined the ‘Efficient Frontier’ and ‘Capital Asset Line’ in his derivation of CAPM with significant advancement (Sharpe, 1964). Linter (1965) derived the CAPM from the perspective of a shares and stock issuing corporation. Finally, Mossin (1966) derived CAPM independently again with some specified quadratic utility function. Although the foundation of this portfolio optimization is established by Markowitz (1952), but also Sharpe, Linter and Mossin have various expansions and iterations in MPT which modernized this theory with simplification and investors’ friendliness (Mangram, 2013). 3. MODERN PORTFOLIO THEORY (MPT) Modern Portfolio Theory is comprised of Markowitz portfolio selection theory that was modified later by William Sharpe with Capital Asset Pricing Model (CAPM) and it allows investors to handle their portfolio in the pattern of construction by the selection of investment portfolio based on their own choice of risk tolerance, so as they can maximize the expected return by reducing the investment risk. There is the process of various mathematical formulation by which the risk component of portfolio can be measured; Page 7





some diversification of asset and/or number of assets is assessed as the tool of the optimization. This is known as asset allocation which is formulated as weighted collection of securities by the individual selection of investors. According to Modern Portfolio Theory (MPT), these contributions are extended to mean-variance analysis, where mean is the expected return of the portfolio and variance is the risk represented by the standard deviation of the return (Mangram, 2013; Rubinstein, 2002).

4. RISK AND RETURN In finance, risk is defined as the deviation, which shows how apart will the actual return be from the historical returns for a period of time. According to the Markowitz selection theory, the overall risk of an aggregate portfolio differs completely than the individual risk of the assets contained in that portfolio. There are two categories of risks: systematic risk; and unsystematic risk. Systematic risks are macro-level risks, that affect a big number of assets, such as the unexpected market crash due to general economic inflation, rate of interest, level of unemployment, rate of exchange of national gross product level are some factors of systemic risks. As an example, in this case we can say recessions of economy at the period of war, where the total finance market would be affected, with this result it is not possible to diversify away these risks. On the other hand, the unsystematic risks are micro-level risks that affect an individual asset and this doesn’t affect other securities or assets, i.e. the change of consumer policy may affect a certain security that is no way has an impact on overall stock market (Investopedia, n.d.). The unsystematic risks could be reduced by Modern Portfolio Theory (MPT) and/or with Capital Asset Pricing Model (CAPM) by a careful selection of weighted averages of assets by the mathematical process of diversification. (Sharpe 1964) If 𝑆(0) be the initial amount of money that was invested at time 0, and 𝑆(𝑇) be the future amount of money received after the certain time period 𝑇, then the difference 𝑆(𝑇) − 𝑆(0) as a fraction of the initial value represents the rate of return (simply referred as return) is, 𝑆 𝑇 − 𝑆(0) 𝑆(0) So, the relationship between the price and the return is 𝐾 𝑆 =

𝑆 𝑇 =𝑆 0 1+𝐾



Eq. 1

Eq. 2



(Capinski & Zastawniak, 2011, Springer) Page 8





5. EXPECTED RETURN To find out the predicted future returns that is the expected return for an investment, we need to examine the historical performance of that returns. Expected return is defined as “the average of a probability distribution of possible returns” (Investopedia, n.d.). As the future price or the return is ambiguous while buying the asset, so the return 𝐾 or the rate of return 𝐾(𝑆) are assumed as random variables with expected value 𝔼 𝐾 = 𝜇𝐾 = 𝜇 Here, 𝜇 is initiated as the expectation for several random returns. 𝔼 𝐾 =

𝔼(𝑆 𝑇 − 𝑆(0)) 𝑆(0)

𝔼(𝑆 𝑇 = 𝑆(0)(1 + 𝜇)

Eq. 3



If we define the probability space of the future value 𝑆(𝑇) by Ω that is a finite set considering some possible market scenarios 𝑠? , 𝑠A , … , 𝑠C that the future value S(T) could attain. And we define the set of possible future values as Ω = {𝑆EF (𝑇)} in ℝC . The probabilities of attaining these future values 𝑝J ′𝑠 gives the condition

C JL? 𝑝J

= 1. So, C

𝔼 𝑆 𝑇

𝑆EF 𝑇 𝑝J

= JL?

The return 𝐾 = 𝐾EF =

𝑆EF 𝑇 − 𝑆(0) 𝑆(0)



Eq. 4

The expected return, C

𝔼 𝐾 = (Capinski & Zastawniak, 2011, Springer)

𝐾EF 𝑝J

Eq. 5

JL?

6. VARIANCE OF EXPECTED RETURN Variance 𝜎 A is the mathematical expectation of the average squared deviations from the mean. To compute this, the probability weighted average of squared deviations from the expected value needs to be calculated. Variance is the tool to calculate the volatility, and volatility is the risk of the investment. So, Page 9





while buying a security in a portfolio, it is possible to measure the risk in advance by an investor with this statistic process. The variance of the return 𝐾 is: 𝑉𝑎𝑟 𝐾 = 𝔼 𝐾 − 𝔼 𝐾

A

= 𝔼 𝐾 A − 𝔼 𝐾 A

𝑉𝑎𝑟 𝐾 = 𝔼 𝐾 A − 𝜇 A

Eq. 6

(Amu & Millegard, 2009; Capinski & Zastawniak, 2011, Springer) 7. STANDARD DEVIATION AS RISK MEASURE Standard deviation of the portfolio expected return is the risk of that investment portfolio. Graphically, standard deviation is a measurement of the diffusion of a set of data points from the mean of the data set. The more scattered apart the data points are from the mean, the higher the deviation is. Standard deviation is computed as the square root of the variance. In finance, standard deviation is known as the risk or volatility and is applied on the annual return of that investment to estimate the volatility. It is a statistical measurement to estimate the historical volatility too. For example, a volatile stock i.e. “Maximus” stock is having a high standard deviation while a stable “Blue Chip” stock will have a lower standard deviation. A larger dispersal tells us how much the return of the investment is deviating from the normal expected returns (Marling & Emanuelson, 2012). Let us consider an example for a portfolio risk: If there are two investments: One is in a risk free asset such as in a zero-coupon bond where the return is 7% expectedly and second investment is in a portfolio of expected returns of 10% and 11% where the estimated risk measured are 5% and 20% respectively depending on the market scenario; then it is obvious that the risk is much smaller in the zero-coupon bond than the investment in the portfolio unless there is a market catastrophe. However, probability is an another important factor that should be taken care of while estimating the risk. If we consider a first portfolio investment consists of two securities with expected returns 20% and 5% with probabilities 0.69 and 0.08 respectively and another second portfolio consists similarly of two

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investments with same returns but with probability 0.5 for each; then the first one is considered as less risky rather than the second. Graphically, we consider the uncertainty as the dispersal of the attained points from the reference points. Reference points are nothing but the values of expected returns. This dispersal is measured by the standard deviation that is considered as risk in finance. This concept depends on two properties of risk: (1) distance between the possible values and expected values; (2) the probabilities of achieving these values. Mathematically, the measurement of risk as standard deviation is: 𝜎Q = 𝑉𝑎𝑟 (𝐾)

Eq. 7



for the return 𝐾 (Investopedia: Risk Measure, n.d.). 8. WEIGHTS Portfolio weights are the percentage compositions of the assets in that portfolio. Weights are for describing the allocation of funds between the securities as a convenient alternative to specifying portfolios in terms of the number of shares of each security. The weights are defined by 𝑤? =

𝑥? 𝑆? (0) 𝑥A 𝑆A (0) , 𝑤A = , 𝑉(0) 𝑉(0)

…, 𝑤C =

𝑥C 𝑆C (0) 𝑉(0)

Eq. 8



If 𝑥J denote the numbers of shares of kind 𝑖 = 1, 2, … , 𝑛 in the portfolio, then 𝑤J is the percentage of the initial value of the portfolio invested in security 𝑖. Weights always add up to 100%. 𝑤? + 𝑤A + ⋯ + 𝑤C =

𝑥? 𝑆? 0 + 𝑥A 𝑆A 0 + ⋯ + 𝑥C 𝑆C (0) 𝑉(0) = = 1 𝑉(0) 𝑉(0)

Eq. 9

(Capinski & Zastawniak, 2011, Springer)

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9. RETURN OF A PORTFOLIO If a portfolio consists of 𝑛 securities, and the numbers of shares for each are 𝑥? , 𝑥A , … , 𝑥C , then the initial value of that portfolio is 𝑉 0 = 𝑥? 𝑆? 0 + 𝑥A 𝑆A 0 + ⋯ + 𝑥C 𝑆C 0



Eq. 10

And the final value at time 𝑇 is 𝑉 𝑇 = 𝑥? 𝑆? 𝑇 + 𝑥A 𝑆A 𝑇 + ⋯ + 𝑥C 𝑆C 𝑇 𝑉 𝑇 = 𝑥? 𝑆? 0 1 + 𝐾? + 𝑥A 𝑆A 0 1 + 𝐾A + ⋯ + 𝑥C 𝑆C 0 1 + 𝐾C [𝑏𝑦 𝐸𝑞. 9] = 𝑉 0 𝑤? 1 + 𝐾? + 𝑤A 1 + 𝐾A + ⋯ 𝑤C 1 + 𝐾C 𝑏𝑦 𝐸𝑞. 8 𝑎𝑛𝑑 𝐸𝑞. 10; 𝑠𝑖𝑛𝑐𝑒 𝑎𝑠 𝑤J =

𝑥J 𝑆J 0 , 𝑉 0

𝑠𝑜 𝑡ℎ𝑎𝑡 𝑥J 𝑆J 0 = 𝑤J 𝑉(0)

C

𝑤J + 𝑤? 𝐾? + 𝑤A 𝐾A + ⋯ + 𝑤C 𝐾C

𝑉 𝑇 =𝑉 0 JL?

𝑉 𝑇 = 𝑉 0 1 + 𝑤? 𝐾? + 𝑤A 𝐾A + ⋯ + 𝑤C 𝐾C [𝑏𝑦 𝐸𝑞. 9] As a result, the expected return of a portfolio is 𝐾d =

𝑉 𝑇 −𝑉 0 [𝑓𝑟𝑜𝑚 𝐸𝑞. 4] 𝑉 0 =

𝑉 𝑇 − 1 𝑉 0

𝐾d = 1 + 𝑤? 𝐾? + 𝑤A 𝐾A + ⋯ + 𝑤C 𝐾C − 1 And finally, we see that the return of a portfolio 𝐾d is the weighted average of the individual returns of the assets contained in that portfolio 𝐾d = 𝑤? 𝐾? + 𝑤A 𝐾A + ⋯ + 𝑤C 𝐾C

Eq. 11

And the expected return of the portfolio is 𝔼(𝐾d ) = 𝑤? 𝔼(𝐾? ) + 𝑤A 𝔼(𝐾A ) + ⋯ + 𝑤C 𝔼(𝐾C )

Eq. 12

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C

𝑂𝑅 𝜇 = 𝔼 𝐾d =

𝜇J 𝑤J

Eq. 13

JL?

(Capinski & Zastawniak, 2011, Springer)

10. VARIANCE OF A PORTFOLIO CONSISTING TWO SECURITIES The variance of the expected return of a portfolio consisting two securities is the function of the covariance between each of the assets contained in that portfolio. Covariance is the measurement of the degree of the direction in which two risky assets will move in cycle. A positive covariance means that the returns of the both assets move together and negative covariance means that it goes inverse that is in opposite direction. Covariance is closely related to correlation, where the difference between the two is considered as factors while calculating standard deviation (Thorpe, 2011). As an example, the variance of the portfolio consisting two securities with returns 𝐾? and 𝐾A is given by: 𝑉𝑎𝑟 𝐾d = 𝑤?A 𝑉𝑎𝑟 𝐾? + 𝑤AA 𝑉𝑎𝑟 𝐾A + 2𝑤? 𝑤A 𝐶𝑜𝑣 𝐾? , 𝐾A



Eq. 14



Proof: Substituting 𝐾d = 𝑤? 𝐾? + 𝑤A 𝐾A and collecting the terms with 𝑤?A , 𝑤AA and 𝑤? 𝑤A , we compute 𝑉𝑎𝑟 𝐾d = 𝔼 𝐾dA − 𝔼 𝐾d A = 𝑤?A 𝔼 𝐾?A − 𝔼 𝐾?

A

+ 𝑤AA 𝔼 𝐾AA − 𝔼 𝐾A

A

+ 2𝑤? 𝑤A 𝔼 𝐾? 𝐾A − 𝔼 𝐾? 𝔼 𝐾A

= 𝑤?A 𝑉𝑎𝑟 𝐾? + 𝑤AA 𝑉𝑎𝑟 𝐾A + 2𝑤? 𝑤A 𝐶𝑜𝑣 𝐾? , 𝐾A . We introduce the following notations: 𝜇d = 𝐸 𝐾d , 𝜎d = 𝑉𝑎𝑟 𝐾d , 𝜇? = 𝐸 𝐾? , 𝜎? = 𝑉𝑎𝑟 𝐾? , 𝜇A = 𝐸 𝐾A , 𝜎A = 𝑉𝑎𝑟 𝐾A 𝑐?A = 𝐶𝑜𝑣 𝐾? , 𝐾A .

Eq. 13 and Eq. 14 can be written for the portfolio of two securities as, 𝜇d = 𝑤? 𝜇? + 𝑤A 𝜇A 𝑂𝑅 𝜇d = 𝑤?

𝑤A

𝜇? 𝜇A

𝜎dA = 𝑤?A 𝜎?A + 𝑤AA 𝜎AA + 2𝑤? 𝑤A 𝑐?A

Eq. 15



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𝑂𝑅 𝜎dA = 𝑤?

𝑤A

𝜎? 𝜎? 𝜎? 𝜎A

𝜎? 𝜎A 𝜎A 𝜎A

𝑤? 𝑤A

(Yahaya, 2012) We will also use the correlation coefficient 𝜌?A =

𝑐?A 𝜎? 𝜎A

If the correlation coefficient 𝜌?A is undefined, then 𝜎? 𝜎A = 0. And this means that at least one of the asset is risk free. Modern Portfolio Theory (MPT) states that the portfolio variance could be reduced by choosing asset classes with lower variance, i.e. the combination of stocks and bonds. This is the process of the best diversification to reduce the risk (Capinski & Zastawniak, 2011, Springer). 11. RISK AND RETURN TRADE-OFF The idea of ‘Risk and Return Trade-Off’ is founded on Markowitz’s basic principle that is the riskier the invested portfolio is, the better the potential expected return. In fact, investors will continue with a risky asset only if a sufficiently high enough expected return is to offset them for assuming the risk. From historical analysis it has been observed that making the riskier investments is the only way for investors to earn a higher return (Thorp, 2011). In Markowitz’ portfolio selection theory, risk is identical with volatility, greater portfolio volatility indicates the greater risk. Volatility is the amount of risk that is uncertainly relates the size of changes in the value of a security. This volatility can be measured by a numerous portfolio tools, such as (i) the calculation of expected return, (ii) the variance of the expected return, (iii) the standard deviation of the expected return, (iv) the covariance of security portfolio, and (v) the correlation between different investments (Mangram, 2013). 12. MARKOWITZ PORTFOLIO THEORY Markowitz portfolio selection is an analyzing method for diversifying to control portfolio risk based on the mean and variance of the returns of the assets in that portfolio. So, it is a process of justifying how good Page 14





a portfolio is. In general, investors are risk-averse who like to take small risk with a high expected returns. Markowitz theory is the foundation of this process to retain this goal. Let us consider, there are 𝑛 assets in a portfolio and 𝐾J , 𝜇J and 𝜎JA are the corresponding returns, means and variances of the 𝑖𝑡ℎ asset respectively. Let 𝜎J,{ be the covariance between the returns 𝐾J and 𝐾{ . Suppose that the relative weights of the assets invested in that portfolio are 𝑤J and 𝑤{ respectively. If 𝐾d is the total return of the entire portfolio, then C

𝜇J 𝑤J

𝜇d = 𝔼 𝐾d = JL? C

𝜎dA

C

= 𝑉𝑎𝑟 𝐾d =

C

𝜎J,{ 𝑤J 𝑤{ 𝑤ℎ𝑒𝑟𝑒 JL? {L?

𝑤J = 1 JL?

𝑤J ≥ 0,1,2, … , 𝑛 (Markowitz, 1952, collected from Marling & Emanuelson, 2012) The aim of this model is to figure out some new weights 𝑤J , so that an investor can allot his investment fund to different assets 𝑖 to minimize the risk (𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒, 𝜎dA ) of the entire portfolio’s expected return and also at the same time to ensure that the portfolio’s total expected return will attain a specified target, say 𝜇. The mathematical formulation of Markowitz Portfolio selection, for a required level of expected return 𝜇 is summarized as: C

𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒 𝜎dA

C

𝜎J,{ 𝑤J 𝑤{

= 𝑉𝑎𝑟 𝐾𝐾d = JL? {L? C

𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜 𝜇 = 𝔼 𝐾 =

𝜇J 𝑤J = 𝜇 JL?

C

𝑎𝑛𝑑

𝑤J = 1 JL?

𝑤ℎ𝑒𝑟𝑒 𝑤J ≥ 0 The condition 𝑤J ≥ 0,1,2, … , 𝑛 is stating that only the long positions for selling is allowed, if short positions are allowed, then this condition should be omitted (Sharpe, 1964; collected from Engels & Marnix, 2004). Page 15





13. FEASIBLE SET For different choices of 𝑤J s’, there are different combinations of 𝜇 and 𝜎 A . The set of all possible combinations of (𝜇, 𝜎 A ) is called the feasible or attainable set. Each portfolio can be represented by a point with coordinated 𝜇d and 𝜎d in the 𝜎, 𝜇 plane (Marling & Emanuelson, 2012). Here, we attempt to discuss the possible scenarios that might arise for portfolios of two securities and so as for portfolio consists of multi-securities:

14. ANALYSIS OF MPT FOR PORTFOLIO OF TWO SECURITIES

According to the Markowitz’s portfolio selection process, we know the coordinates of the feasible set for the portfolio of two securities consist of, Expected return, 𝜇d = 𝑤? 𝜇? + 𝑤A 𝜇A and Variance,

𝜎dA = 𝑤?A 𝜎?A + 𝑤AA 𝜎AA + 2𝑤? 𝑤A 𝑐?A

where 𝑤? , 𝑤A ∈ ℝ and 𝑤? + 𝑤A = 1 Now, if we parameterize the portfolios in the feasible set by one of the weights, such as, let 𝑡 = 𝑤? as a parameter. So, 1 − 𝑡 = 𝑤A , and the above expression can be written as 𝜇d = 𝑡𝜇? + 1 − 𝑡 𝜇A

Eq. 16

𝜎dA = 𝑡 A 𝜎?A + (1 − 𝑡)A 𝜎AA + 2𝑡 1 − 𝑡 𝑐?A

Eq. 17

where 𝑡 ∈ ℝ

(Capinski & Zastawniak, 2011, Springer) 15. MINIMUM VARIANCE PORTFOLIO (MVP) FOR TWO SECURITIES

To find the portfolio with the smallest variance (that is the smallest risk) among all the feasible portfolios, we have to find the value of 𝑡 for which 𝜎dA attains a minimum. We differentiate 𝜎dA with respect to 𝑡 and equate the derivative to zero. 𝑑(𝜎dA ) = 2𝑡 𝜎?A + 𝜎AA − 2𝑐?A − 2 𝜎AA − 𝑐?A = 0 𝑑𝑡

Page 16





𝑡… =

𝜎AA − 𝑐?A 𝜎?A + 𝜎AA − 2𝑐?A

as long as the denominator is non-zero. This is definite because the correlation coefficient that −1 < 𝜌?A < 1 and/or 𝜎? ≠ 𝜎A . And 𝑡… = 𝑤? , 1 − 𝑡… = 𝑤A are confirming the new diversified weights for the minimum variance portfolio.

Condition 1: For 𝜌?A =

ˆ‰Š ‹‰ ‹Š

< 1, we find, 𝑐?A < 𝜎? 𝜎A −2𝑐?A > −2𝜎? 𝜎A

𝜎?A + 𝜎AA − 2𝑐?A > 𝜎?A + 𝜎AA − 2𝜎? 𝜎A = 𝜎? − 𝜎A

A

≥ 0

If 𝜎? ≠ 𝜎A , then 𝜎?A + 𝜎AA − 2𝑐?A > 𝜎?A + 𝜎AA − 2𝜎? 𝜎A = 𝜎? − 𝜎A

A

> 0

For the second derivative, 𝑑 A (𝜎dA ) = 2 𝜎?A + 𝜎AA − 2𝑐?A > 0 𝑑𝑡 A As both first and second derivative are positive, so we accomplish here that 𝜎dA attains its minimum at 𝑡… . The expression for 𝜇•Ž and 𝜎•AŽ follow by substituting 𝑡… for 𝑡 in 𝐸𝑞. 16 and 𝐸𝑞. 17 Variance for minimum Risk: 𝜎•AŽ =

A 𝜎?A 𝜎AA − 𝑐?A 𝜎?A + 𝜎AA − 2𝑐?A

and the related expected return of that portfolio: 𝜇 •Ž =

𝜇? 𝜎AA + 𝜇A 𝜎?A − (𝜇? + 𝜇A )𝑐?A 𝜎?A + 𝜎AA − 2𝑐?A

Condition 2: If 𝜌?A = 1 and 𝜎? = 𝜎A , then 𝑐?A = 𝜎? 𝜎A , and for any 𝑡 ∈ ℝ 𝜎dA = 𝑡 A 𝜎?A + (1 − 𝑡)A 𝜎AA + 2𝑡 1 − 𝑡 𝑐?A = (𝑡𝜎? + (1 − 𝑡)𝜎A )A = 𝜎?A = 𝜎AA

𝝁 (𝝈𝟐 , 𝝁𝟐)



𝝁𝟎 (𝝈𝟏 , 𝝁𝟏)



𝝈

𝝈𝟎



Figure: Hyperbola representing the feasible portfolio on the 𝜎, 𝜇 plane

Page 17



Condition 3: If 𝜌?A = 1 and 𝜎? ≠ 𝜎A , then for 𝑐?A = 𝜎? 𝜎A equation (15) will take the form, 𝜎dA = 𝑤?A 𝜎?A + 𝑤AA 𝜎AA + 2𝑤? 𝑤A 𝜎? 𝜎A = (𝑤? 𝜎? + 𝑤A 𝜎A )A and 𝜎dA = 0 if and only if 𝑤? 𝜎? + 𝑤A 𝜎A = 0. And as 𝑤? + 𝑤A = 1, we get 𝑤? = −

𝜎A 𝜎? 𝑎𝑛𝑑 𝑤A = 𝜎? − 𝜎A 𝜎? − 𝜎A

Eq. 18



Either weight is negative means that short selling is required. Condition 4: If 𝜌?A = −1, then for 𝑐?A = − 𝜎? 𝜎A , eqn. (15) becomes, 𝜎dA = 𝑤?A 𝜎?A + 𝑤AA 𝜎AA + 2𝑤? 𝑤A 𝑐?A = (𝑤? 𝜎? − 𝑤A 𝜎A )A and 𝜎dA = 0 if and only if 𝑤? 𝜎? − 𝑤A 𝜎A = 0. And also as 𝑤? + 𝑤A = 1, so 𝑤? =

𝜎A 𝜎? 𝑎𝑛𝑑 𝑤A = 𝜎? + 𝜎A 𝜎? + 𝜎A

Eq. 19



No short selling is required, since both 𝑤? and 𝑤A are positive. We know that 𝜎? + 𝜎A > 0, because both securities are risky. (Capinski & Zastawniak, 2011, Springer)





𝝁

(𝜎A , 𝜇A )

𝝁

(𝜎A , 𝜇A )



(𝜎? , 𝜇? ) (𝜎? , 𝜇? )

𝝈



𝝈

Figure: Typical portfolio lines with 𝜌?A = −1 𝑎𝑛𝑑 1. The bold segments correspond to portfolios without short selling.

(Capinski & Zastawniak, 2011, Springer) Page 18





16. ANALYSIS OF MPT FOR PORTFOLIO OF MULTI-SECURITIES For investment in the portfolio of two risky assets, we describe the portfolio return 𝜇d and risk 𝜎d in (𝜎d , 𝜇d )-plane that all possible portfolios lie on a curve that is one side of a parabola. But, for the investment in the portfolios of three or more risky assets, describing the general shape of (𝜎d , 𝜇d )-space is more complicated, and it crucially depends on covariance 𝜎J{ . Therefore, after moving past the portfolio with two securities, it is necessary to use matrix operations for determination of the optimal asset weights in the portfolio (FTSE Russell; 2015). Risk and Expected return of portfolio consisting 𝒏 securities: From equation (8), we know that the weights of a portfolio consisting n securities are: 𝑤J =

𝑥J 𝑆J (0) , 𝑖 = 1, … , 𝑛 𝑉(0)

Let, a 1×𝑛 row-matrix 𝒘 is described as, 𝒘 = 𝑤? ,

𝑤A ,

…

𝑤C



Eq. 20



As the sum of the weights equals to 1, so, in the matrix form 𝒘𝒖ž = 1

Eq. 21



where 𝒖 is a 1×𝑛 unit row-matrix with all 𝑛 entries equal to 1, and 𝒖ž is a 𝑛×1 column-matrix, the transpose of 𝒖 . 𝒖 = 1

1

…

1

Eq. 22



In this case, for the portfolio of 𝑛- securities, all the attainable the feasible sets consist of portfolios with weights 𝒘 satisfying the equation (21) and are called the feasible or attainable portfolios. Let us arrange the expected returns 𝜇J = 𝔼(𝐾J ), for the individual returns of the securities 𝐾? , 𝐾A , … , 𝐾C , for 𝑖 = 1, 2, … , 𝑛 in a 1×𝑛 row-matrix, as 𝒎 = 𝜇?

𝜇A

…

𝜇C

Let, the covariance between the returns is denoted by 𝑐J{ = 𝐶𝑜𝑣(𝐾J , 𝐾{ ) and these are arranged as entries of 𝑛 × 𝑛 covariance matrix as, Page 19





𝑐?? 𝑐A? 𝑪 = … 𝑐C?

𝑐?A 𝑐AA … 𝑐CA

… … … …

𝑐?C 𝑐AC … 𝑐CC

Here, the diagonal elements of 𝑪, that are 𝑐?? , 𝑐AA , 𝑐¡¡ , … , 𝑐CC are the variances of returns 𝐾? , 𝐾A , … , 𝐾C 𝑐JJ = 𝜎J = 𝑉𝑎𝑟(𝐾J ) As the covariance matrix 𝑪 is symmetric and non-negative definite. So, 𝑪 ≠ 0, that indicates that the inverse of 𝑪 exists. Eq. 11 for the expected return of a portfolio consisting 𝑛 securities is: 𝑲d = 𝑤? 𝑘? + 𝑤A 𝑘A + ⋯ + 𝑤C 𝑘C From Eq. 12 and Eq.13, we have 𝔼(𝐾d ) = 𝑤? 𝔼(𝐾? ) + 𝑤A 𝔼(𝐾A ) + ⋯ + 𝑤C 𝔼(𝐾C ) and 𝜇 = 𝔼(𝐾d ) =

C JL? 𝜇J 𝑤J , and by the row matrices for weights 𝑤J ′𝑠 and expected returns 𝜇J ′𝑠 , we do

have now, C

𝜇d = 𝔼 𝐾d = 𝔼

C

𝑤J 𝜇J = 𝒘𝒎ž

𝑤J 𝐾J = JL?

JL?

𝜇d = 𝒘𝒎ž

Eq. 23

With the same matrices, the variance is C

𝜎dA

𝑤J 𝐾J

= 𝑉𝑎𝑟 𝐾d = 𝑉𝑎𝑟 JL?

= 𝐶𝑜𝑣

C JL? 𝑤J 𝐾J

C JL? 𝑤{ 𝐾{

=

C JL? 𝑤J 𝑤{ 𝑐J{

𝜎dA = 𝒘𝑪𝒘ž

Eq. 24





(Capinski & Zastawniak, 2011, Springer) 17. MINIMUM VARIANCE PORTFOLIO (MVP) OF MULTI-SECURITIES

As the portfolio with the smallest variance is known as the minimum variance portfolio (MVP), to find this MVP portfolio we need to minimize the variance 𝜎dA = 𝒘𝑪𝒘ž over all weights 𝒘 . Because the weights must add up to 1 , and this leads to the constrained minimum problem: min 𝒘𝑪𝒘ž where the minimum is taken over all vectors 𝒘 ∈ ℝ that satisfies the condition 𝒘𝒖ž = 1 Page 20





To compute this constrained minimum, we use the method of Lagrange’s Multiplier. In order to find the minimum of 𝒘𝑪𝒘ž subject to the constraint 𝒘𝒖ž = 1, with the Lagrange’s Multiplier 𝜆, let, 𝐹 𝑤, 𝜆 = 𝒘𝑪𝒘ž − 𝜆(𝒘𝒖ž − 1) Taking the partial derivative with respect to 𝑤 and 𝜆, 𝛿𝐹 = 2𝒘𝑪 − 𝜆𝒖 = 0 𝛿𝑤 𝛿𝐹 and = − 𝒘𝒖ž + 1 = 0 𝛿 𝜆 𝛿𝐹 𝜆 Hence from , 𝒘 = 𝒖𝑪§? 𝛿𝑤 2 𝜆 𝛿𝐹 𝒖𝑪§? 𝒖ž = 𝒘𝒖ž = 1 𝑏𝑦 2 𝛿 𝜆 𝜆 𝒖𝑪§? 𝒖ž = 1 2 As 𝒖𝑪§? 𝒖ž is a numerical quantity and also as by the property of covariance matrix, 𝒖𝑪§? 𝒖ž ≠ 0 , so the above equation takes the pattern, 𝜆 1 = 2 𝒖𝑪§? 𝒖ž ¨

¨

A

A

By substituting the value of in 𝒘 = 𝒖𝑪§? , we can compute the expression for 𝒘 to obtain the formula for the diversified weights of the portfolio 𝒘©dª as 𝒘©dª =

𝒖𝑪§? 𝒖𝑪§? 𝒖ž

Eq. 25

(Capinski & Zastawniak, 2011, Springer) 18. EFFICIENT FRONTIER A portfolio is considered to be efficient if it is not possible to obtain a higher return without increasing the standard deviation (McNair, 2003). It is the set of all dominant portfolios. Dominant portfolios are either with the following characteristics: (1) If a portfolio has higher expected return than other portfolios of equal level of risks; or (2) If a portfolio has lower level of risk than other portfolios with equal level of expected returns; or (3) If a portfolio has higher expected return with lower level of risk. Page 21





Efficient frontier represents the relationship between the total expected return of a portfolio and the volatility or risk of that portfolio. The knowledge of the efficient frontier allows an investor to choose a diversified dominant portfolio according to his/her level of risk tolerance. (Yahaya, 2012) The efficient frontier represents a parabolic pattern on a risk-return plane after solving the quadratic optimization problem by dropping the option of short selling that would cause non-negativity for the constraints. This curve starts with the point of global MVP (Minimum variance portfolio).

Figure: Efficient Frontier



19. SOLVING MPT FOR TARGETED EXPECTED RETURN According to the original frame work of Markowitz’s theory, we assume that the investors always choose portfolios that maximize the expected return of the portfolio subject to a certain level of risk, or, equivalently to minimize risk subject to a targeted expected return. Therefore, we can focus only on the set of efficient portfolios for the simplification of this asset allocation problem. These portfolios lie on the boundary of the set of attainable portfolios and above the global minimum variance portfolio. According to Markowitz, the investors wish to find the portfolios that will have the best expected return-risk tradeoff. Markowitz categorized these efficient portfolios in two parallel ways. He stated, first, investors are either interested for the portfolio that maximizes the expected return for a given level of risk or second, for the portfolio that minimizes the risk subject to a target expected return.

Page 22





In practice of finding the efficient portfolios of risky assets, the second minimization optimal problem is solved most often for computational conveniences. And also for in practical finance field, investors are more likely to pre-set their target expected returns rather than pre-planning the target risk level (Engels, 2004). If 𝜇 is a pre-set target of expected return of a minimum variance portfolio, then the family of portfolios 𝑉, parameterized by 𝜇 ∈ ℝ, such that 𝜇« = 𝜇 and 𝜎dA ≤ 𝜎dA- for each portfolio 𝑉 ® with 𝜇𝑉 ® = 𝜇 is called the minimum variance line (MVL). To compute the portfolio with the minimum variance line for any 𝜇 ∈ ℝ, we are to solve the constrained minimum portfolio min 𝒘𝑪𝒘ž 𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜 𝒘𝒎ž = 𝜇 𝑎𝑛𝑑 𝒘𝒖ž = 1 where the minimum is calculated over the all vectors 𝒘 ∈ ℝC By introducing Lagrange’s Multipliers 𝜆? , 𝜆A , the minimization function is assumed as, 𝐺 𝒘, 𝜆? . 𝜆A = 𝒘𝑪𝒘ž − 𝜆? 𝒘𝒎ž − 1 − 𝜆A (𝒘𝒖ž − 1) The gradient of 𝐺 with respect to 𝒘 and the partial derivatives with respect to 𝜆? , 𝜆A give the necessary condition for a minimum,



𝛿𝐺 = 2𝒘𝑪 − 𝜆? 𝒎 − 𝜆A 𝒖 = 𝟎 𝛿𝒘 𝛿𝐺 = 𝒘𝒎 ž − 𝜇 = 0 𝛿𝜆? 𝛿𝐺 = 𝒘𝒖 ž − 1 = 0 𝛿𝜆A

Eq. 26 Eq. 27 Eq. 28

Equation 26 is a system of 𝑛 equations, 𝟎 representing a vector of n-zeros. As the covariance 𝑪 is nonsingular and 𝑪 ≠ 0, so 𝑪§? exists, Eq. 26 can be written as 2𝒘 = 𝜆? 𝒎𝑪§? + 𝜆A 𝒖𝑪§?



1 𝒘 = 𝜆? 𝒎𝑪§? + 𝜆A 𝒖𝑪§? 2

Eq. 29



Eq. 30

Multiplying the above Eq. 19 by 𝒎ž on the right and then by 𝒖ž on the right, Eq. 27 and Eq. 28 give the following system of linear equations for the Lagrange’s Multipliers: 𝜆? 𝒎𝑪§? 𝒎ž + 𝜆A 𝒖𝑪§? 𝒎ž = 2𝜇 Page 23





𝜆? 𝒎𝑪§? 𝒖ž + 𝜆A 𝒖𝑪§? 𝒖ž = 2 Let the coefficient matrix of the above system is denoted by §? ž 𝑀 = 𝒎𝑪 §?𝒎ž 𝒎𝑪 𝒖

𝒖𝑪§? 𝒎ž 𝒖𝑪§? 𝒖ž

As the vectors 𝒖, 𝒎 are linearly independent, then 𝑀 §? exists. The solution of the system of linear equations for 𝜆? and 𝜆A are: 𝜆? 𝜇 = 2𝑀 §? 1 𝜆A



Eq. 31

Inserting these values of 𝜆? and 𝜆A into equation 30, we get the form, 𝒘 = 𝜇𝑀 §? 𝒎𝑪§? + 𝑀 §? 𝒖𝑪§? Or, 𝒘 = 𝜇𝒂 + 𝒃 for some vectors 𝒂, 𝒃 ∈ ℝ, 𝑤ℎ𝑒𝑟𝑒 𝒂 = 𝑀 §? 𝒎𝑪§? 𝑎𝑛𝑑 𝒃 = 𝑀 §? 𝒖𝑪§? . The important here is that the vectors 𝒂, 𝒃 are the same for each portfolio on the minimum variance line. (Capinski & Zastawniak, 2011, Springer). 20. SHARPE RATIO 𝑆ℎ𝑎𝑟𝑝𝑒 𝑅𝑎𝑡𝑖𝑜 =

𝑀𝑒𝑎𝑛 𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝐷𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛

Sharpe Ratio is the ratio of the total portfolio expected return to the total portfolio risk. The higher the Sharpe Ratio is; the better the efficiency of the portfolio is estimated. Sharpe Ratio is the component to estimate the change in a portfolio’s overall return-risk feature when a new asset or asset class is added or is changed. It was developed by Nobel laureate William Sharpe (1964). 21. NUMERICAL ILLUSTRATION The goal of this project is to establish an interacting and built-in excel spreadsheet for end-user investors to verify the risk and expected return of the portfolio that they will be going to invest for different assets. Using Markowitz portfolio selection theory, diversification of assets or asset allocation is the process to create the minimum variance portfolio that have been discussed so far. Here, an excel spreadsheet has been created applying all of the above described Modern Portfolio Optimization theory to calculate the Page 24





diversified weights and thereby to figure out what number of stocks are needed to be purchased for the minimum variance portfolio, and then is for set targeted return according to the tolerable level of risk. Three different approaches have been categorized in Excel work. In all cases, the sample data is collected from US NASDAQ Stock market and they are for the past one year’s adjusted daily closing prices since October 21st, 2015 to October 20th, 2016. In Application 1, a portfolio has been created consisting of two securities and the method of asset allocation is applied for determining the weights of the assets for the minimum variance portfolio that is to obtain the smallest risk and maximum return. In application 2, one more asset is added in the same portfolio for the purpose of diversification of allocated assets to lower the prior risk and as well as to increase the expected return of the portfolio. In Application 3, expected return is increased as a multiplicative of the previous return and so as the risk has been measured and also the diversified weights have been generated for the minimum variance portfolio. In all cases, we observe the “Sharpe Ratio”, that is the ratio of the portfolio expected return and risk. The larger this quantity is, the better the selection is associated with higher return and higher risk. APPLICATION 1

Portfolio consists of two stocks Microsoft Inc. (MSFT) and People’s United Financial Inc. (PBCT). Historical data of daily closing prices from NASDAQ (National Association of Securities Dealers Automated Quotations System, the largest US stock exchange market) for one year since October 21st 2015 to October 20th 2016 are used to calculate the allocated weights for the minimum variance portfolio (MVP) that ensures the maximum portfolio return with the minimum risk. We consider, Stock 1--- MSFT; Stock 2 ---- PBCT as of October 20th, 2016: Initial Price, 𝑆? (0) = $ 57.25 Initial Price, 𝑆A (0) = $ 16.09 Let, the total investment for this portfolio, 𝑉 0 = $ 20,000.00 Page 25





Let, the random number of shares purchased for MSFT and PBCT are respectively 𝑥? = 200, 𝑥A = 531 So, the weights for them are respectively 𝑤? =

𝑥? 𝑆? (0) 200×57.25 = = 57.25% 𝑉(0) 20000

𝑤A =

𝑥A 𝑆A 0 531×16.09 = = 42.72% 𝑉 0 20000

We verify that 𝑤? + 𝑤A = 57.25% + 42.72% ≈ 100% Calculation of the average rate of change of the closing prices per day for both stocks MSFT and PBCT, that are the sample means of them respectively are 𝜇? = 0.10%, 𝜇A = 0.03% And the sample variances respectively are 𝜎?A = 0.02% 𝑎𝑛𝑑 𝜎?A = 0.02% Calculated standard deviations are 𝜎? = 𝑠𝑞𝑟𝑡 𝜎?A = 1.56%, 𝜎A = 𝑠𝑞𝑟𝑡 𝜎?A = 1.42% Covariance 𝐶𝑜𝑣. 𝐾? , 𝐾A = 0.000005 But, to calculate the diversified weights for minimum variance portfolio, we calculate the correlation coefficient 𝜌?A = 0.0232 < 1 So, the new allocated weights found are, 𝑤? =

𝜎AA − 𝑐?A = 45%, 𝑤A = 1 − 𝑤? = 55% 𝜎?A + 𝜎AA − 2𝑐?A

The calculated expected return and risk of the minimum variance portfolio respectively are 𝜇d = 𝑤? 𝜇? + 𝑤A 𝜇A = 𝜎d = 𝑠𝑞𝑟𝑡 Sharpe Ratio =

….…º ?.…º

𝜇? 𝜎AA + 𝜇A 𝜎?A − (𝜇? + 𝜇A )𝑐?A = 0.06% 𝜎?A + 𝜎AA − 2𝑐?A A 𝜎?A 𝜎AA − 𝑐?A = 1.06% 𝜎?A + 𝜎AA − 2𝑐?A

= 5.57%

APPLICATION 2 Here, one new stock Yahoo Inc. (YHOO) is added in the existing portfolio with initial price 𝑆¡ (0) = $ 42.38



Page 26



Now, to create again the minimum variance portfolio to maximize the expected return with lower the risk than before, the calculations are as follows: 𝐶𝑜𝑣. 𝐾? , 𝐾? 𝐶𝑜𝑣. 𝐾? , 𝐾A 𝐶𝑜𝑣. 𝐾? , 𝐾¡ 𝐶𝑜𝑣. 𝐾A , 𝐾A 𝐶𝑜𝑣. 𝐾A , 𝐾¡ 𝐶𝑜𝑣. 𝐾¡ , 𝐾¡

= 0.00024403 = 0.00000516 = 0.00014022 = 0.00020124 = −0.00000291 = 0.00037844

Then the calculated new diversified weights are 𝒖𝑪§? 𝒘©dª = §? ž = 𝑤? 𝑤A 𝒖𝑪 𝒖

𝑤¡ = 33%

51%

16%

So that the number of shares have to be purchased: 𝑥J = 𝑤J

𝑆J (0) , 𝑉(0)

where V(0) is the initial money invested for the portfolio to be determined by the individual investors. The related expected returns of the assets MSFT, PBCT and YHOO of the portfolio 𝜇? , 𝜇A , 𝜇¡ are: 𝜇?

𝜇A

𝜇¡ = 0.10%

0.03%

0.14%

The calculated portfolio expected return and portfolio risk for the minimum variance portfolio are: 𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑅𝑒𝑡𝑢𝑟𝑛 𝜇d = 𝒘𝒎ž = 0.10%

0.03%

0.14% 33%

51%

16%

ž

= 0.07%

𝑅𝑖𝑠𝑘 𝜎d = 𝑠𝑞𝑟𝑡 𝒘𝑪𝒘ž = 1.02% Sharpe Ratio =

….…» ?.…A

= 6.69%

APPLICATION 3 The portfolio expected return is increased by 50% higher than the expected return obtained for the in Application 2 for the same combination of three stocks, that is the new expected return is 𝜇dA = 1.5 × 𝜇d = 1.5 × 0.07% = 0.10% To compute the weight with the portfolio with the minimum related variance, we solve the constrained minimum problem min 𝑊𝐶𝑊 ž To calculate the values of Lagrange’s multipliers 𝜆? 𝑎𝑛𝑑 𝜆A , the matrix M is created as: §? ž 𝑴 = 𝒎𝑪 §?𝒎ž 𝒎𝑪 𝒖

𝒖𝑪§? 𝒎ž = 0.006766005 6.538577569 𝒖𝑪§? 𝒖ž

6.538577569 9553.943538

Then the computed 𝜆? 𝑎𝑛𝑑 𝜆A are:

Page 27





𝜆? 𝜇 0.2987147 = 2𝑀 §? = 1 𝜆A 0.0000049 Diversified new weights are: 𝑤?

𝑤A

1 𝑤¡ = 𝒘 = 𝜆? 𝒎𝑪§? + 𝜆A 𝒖𝑪§? = 36.71% 2

20.26%

43.03%

Now, for the new portfolio expected return 0.10%, the risk (𝜎d = 𝑠𝑞𝑟𝑡 𝒘𝑪𝒘ž ) is 1.25%, but the Sharpe Ratio =

….?… ?.A¾

= 8.23% that is much higher than before.

22. SUMMARY OF THE EXCEL WORK

The spread-sheet, as a part of this project can be used by any investor just by changing the historical dataset by the updated closing prices of any portfolio that consists of two stocks or more. By application one, investors can estimate the expected return and related risk as the best optimal portfolio that could be achieved with the highest Sharpe Ratio. By application two, an investor can add one more suitable stock based on one-year historical data to upgrade the Sharpe Ratio that will offer a better option. And by application three, the investor can target a certain increase in expected return for the portfolio and justify the related risk. 23. DISCUSSION

Real time practice in financial investment market, investors sometimes stay apart from this idea of MPT. The reasons observed for the contradiction of MPT with the real life investment are mainly for the followings. Investor’s rationality: The fundamental assumption for MPT is that the investors are always rational that is they seek to maximize the return and simultaneously minimize the risk. Though rarely but this idea sometimes goes reverse in real time market practice, some investors go for ‘herd behavior’ that is they go for market rumor for ‘hot’ sectors, no matter the level of risk is. In reality, financial markets are frequently associated with hypothetical excesses (Morein, n.d.). Page 28





Higher Risk-Return: The assumption that the investors will go for higher risk only if they are counterweighed by higher expected return frequently contradicts the market scenario by the contrary action of the investors. For instance, to reduce the overall portfolio expected return, sometimes the investors add some super risky asset/s without any evident increase or impact in expected returns (McClure, 2010). There are also some theoretical limitations of MPT that have been noticed while analyzing. Some of the underlying assumptions that are the key component in this process have few contradictions with real time finance market. Unlimited Capital: One key contradiction for MPT is that it is assumed that the investors are having unlimited financial balance or borrowing capacity, especially for investing risk-free assets. Asset allocation is the key feature of portfolio optimization, but if short selling is not allowed, then to allocate the weights, only option for the investors is to add newer asset/s which demands more investments. But in real world market, every investor has financial limit or credit limit (Morein, n.d.). Efficiency of the Market: Markowitz’s theoretical contributions to MPT are built upon the assumption that the market is perfectly efficient (Markowitz, 1952). But this contradicts as the real time market is not in steady condition and also vulnerable for various market impulses such as environmental, personal, strategic, or social investment phenomenon. The risk and expectations are derived from historical closing prices for certain period of time, but it could fail to summarize newer unusual circumstances that was not existing during that time period; upcoming prices might have some dimensional phase change due to several market impulses (Mangrum, 2013). No Tax or Transaction Cost: In financial investment, there are always tax and transaction fees, these may have some effects on optimization selection process, but MPT is not assuming these as parameters. Overall, it is undoubtedly reasonable to support MPT as the most efficient optimized model. Only by overwhelming the above limited limitations that may occur occasionally, Markowitz’s asset allocation process is the most efficient method for risk-return optimization. Page 29





24. CONCLUSION In real time finance market, with my observation I have categorized the investors into two sections. One portion of the investors are generally interested to invest their excess balance of earnings in some prospective business for long position of future. This categorized investor is generally interested to invest only in stock exchange rather than investing in multi-class assets. The second categorized investor is the principally investing business professionals who are with higher credit limits and are capable to diversify their portfolio at any time by any dimensional additional investment. The goal of this project is attempted to create a template in excel spreadsheet as a short cut estimation tool for the first categorized investor who are dealing their portfolio only with the risky assets such as in stock market. That is, in this project I tried to go for over simplified process of MPT, and I limit the assets only with risky stock market. Ignoring any unfortunate market condition and avoiding related tax and transaction cost, this worksheet is expected to be considered helpful and a quick measuring tool to estimate risk-return ratio. And obviously this is generous for risky assets such as stock market, American Call Options etc. where short selling is not allowed. 25. FURTHER WORK Further work could be continued to Capital Asset Pricing Model (CAPM) that is to include some risk-free asset/s in the existing portfolio, and thereby to optimize the portfolio more with higher Sharpe ratio. There are more privileges of financial credit opportunities for investment in risk-free assets than that of risky assets. And also for the cases of negative weights that arise after diversification, but short selling is not allowed that is no chance of deducting that asset completely from the existing portfolio, a better approach is to add some risk-free assets to eliminate the negativity of the diversified weights. MPT has been developed with the advanced work of Markowitz with his fellow Sharpe. The extension of this process is CAPM and this have mentioned prior as the development of Markowitz Selection process to Modern Portfolio Theory (MPT). Page 30





26. REFERENCES

Amu, Farhad and Millegard, Marcus (2009): ‘Markowitz Portfolio Theory’. Retrieved from http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.145.90&rep=rep1&type=pdf

Barnes, Randal J. (2006): ‘Matrix Differentiation’. Retrieved from http://www.atmos.washington.edu/~dennis/MatrixCalculus.pdf

Capinski, Marek and Zastawniak, Tomasz (2003,2011): ’Mathematics for Finance’, Second Edition, Springer Undergraduate Mathematics Series ISSN 1615-2085; ISBN 978-0-85729-081-6

Elton, Edwin J. and Gruber, Martin J. (1997): ‘Modern Portfolio Theory, 1950 to Date’. Retrieved from http://pages.stern.nyu.edu/~eelton/papers/97-dec.pdf

Engels, Marnix (2004): ‘Portfolio Optimization: Beyond Markowitz’. Retrieved from https://www.math.leidenuniv.nl/scripties/Engels.pdf

FTSE Russell (2015): ‘Low Volatility or Minimum Variance’. Retrieved from http://www.ftse.com/products/downloads/Low-Vol-Whitepaper.pdf

Investopedia (n.d.): ‘Expected Return’. Retrieved from http://www.investopedia.com/terms/e/expectedreturn.asp Investopedia (n.d.): ‘Risk Measure’. Retrieved from http://www.investopedia.com/terms/r/riskmeasures.asp Kazan, Halim and Uludag, Kultigin 1914: ‘Credit Portfolio Selection According to the Sectors in Risky Environment: Markowitz Practice’. Retrieved from: http://www.aessweb.com/pdf-files/aefr-2014-4(9)-1208-1219.pdf Linter, John (1965): ‘Security Prices, Risk, and Maximal Gains from Diversification’. Retrieved from http://efinance.org.cn/cn/fm/Security%20Prices,%20Risk,%20and%20Maximal%20Gains%20From%20Di versification.pdf Page 31





Lummer, Scott L. and Riepe, Mark W. (1994): ‘The Role of asset Allocation in Portfolio Management’. Retrieved from https://corporate.morningstar.com/ib/documents/MethodologyDocuments/IBBAssociates/RoleAssetAll ocation.pdf Mangram, Myles E. (2013): ‘A Simplified Perspective of Markowitz Portfolio Theory’. Retrieved from ftp://ftp.repec.org/opt/ReDIF/RePEc/ibf/gjbres/gjbr-v7n1-2013/GJBR-V7N1-2013-6.pdf Markowitz, Harry (1952): ‘Portfolio Selection’. Retrieved from http://www.math.ust.hk/~maykwok/courses/ma362/07F/markowitz_JF.pdf

Marling, Hannes and Emanuelson, Sara (2012): ‘The Markowitz Portfolio Theory’. Retrieved from http://www.smallake.kr/wp-content/uploads/2016/04/HannesMarling_SaraEmanuelsson_MPT.pdf

McClure, Ben (2010): ‘Modern Portfolio Theory: Why It’s Still Hip’. Retrieved from http://www.investopedia.com/articles/06/mpt.asp

McNair, Alexander (2003): ‘Using Microsoft Excel to Build Efficient Frontier Via the Mean Variance Optimization Method’. Retrieved from: http://ms.mcmaster.ca/~grasselli/john.pdf Megginson, William L. (1996): ‘A Historical Overview of Research in Finance’. Retrieved from https://www.google.com/#q=A+historical+overview+of+research+in+finance+by+Megginson

Morien, Travis (n.d.): ‘MPT Criticism’. Retrieved from http://www.travismorien.com/FAQ/portfolios/mptcriticism.htm

Mossin, Jan (1966): ‘Equilibrium in a Capital Asset Market’. Retrieved from http://efinance.org.cn/cn/fm/Equilibrium%20in%20a%20Capital%20Asset%20Market.pdf Rubinstein, Mark (2002): ‘Markowitz’s Portfolio Selection: A Fifty-Year Retrospective’. Retrieved from http://www.efalken.com/pdfs/rubinsteinMarkowitz.pdf Page 32





Sharpe, William F. (1964): ‘Capital Asset Prices: A Theory of Market Equilibrium Under Condition of Risk’. Retrieved from https://psc.ky.gov/pscecf/2012-00221/[email protected]/10252012f/sharpe_-_CAPM.pdf Thorp, Wayne A. (2011): ‘Mean Variance Optimization: Multi Asset Portfolio’. Retrieved from http://www.aaii.com/computerized-investing/article/mean-variance-optimization-multi-asset-portfolio

Tobin, J. (1958): ‘Liquidity Preference as Behavior Towards Risk’. Retrieved from http://web.uconn.edu/ahking/Tobin58.pdf Yahaya, Abubakar (2012): ‘On Numerical Solution for Optimal Allocation of Investment Funds in Portfolio Selection Problem’. Retrieved from https://www.researchgate.net/profile/Abubakar_Yahaya2/publication/280069155_On_Numerical_Solut ion_for_Optimal_Allocation_of_Investment_funds_in_Portfolio_Selection_Problem/links/55a65ab208a ebe1d24699e4a.pdf 27. LINKS The web-links used multiple times of NASDAQ US market to retrieve the historical data in this project are: http://www.advfn.com/nasdaq/nasdaq.asp http://finance.yahoo.com/q/hp?a=&b=&c=&d=9&e=20&f=2012&g=d&s=PBCT&ql=1 Page 33





28. APPENDIX Attachment from the Excel Spread Sheet (1) Adjusted Daily closing prices of Microsoft Inc. (MSFT), People’s United Financial Inc. (PBCT), Yahoo Inc. (YHOO), Apple Inc. (AAPL), Netflix, Inc. (NFLX) from Oct. 20, 2016 to past one year Oct. 21, 2015 Date 10/20/16 10/19/16 10/18/16 10/17/16 10/14/16 10/13/16 10/12/16 10/11/16 10/10/16 10/7/16 10/6/16 10/5/16 10/4/16 10/3/16 9/30/16 9/29/16 9/28/16 9/27/16 9/26/16 9/23/16 9/22/16 9/21/16 9/20/16 9/19/16 9/16/16 9/15/16 9/14/16 9/13/16 9/12/16 9/9/16 9/8/16 9/7/16 9/6/16 9/2/16 9/1/16 8/31/16 8/30/16 8/29/16 8/26/16 8/25/16 8/24/16 8/23/16 8/22/16 8/19/16 8/18/16 8/17/16 8/16/16 8/15/16 8/12/16 8/11/16 8/10/16 8/9/16 8/8/16 8/5/16 8/4/16 8/3/16 8/2/16

MSFT 57.25 57.53 57.66 57.22 57.42 56.92 57.11 57.19 58.04 57.80 57.74 57.64 57.24 57.42 57.60 57.40 58.03 57.95 56.90 57.43 57.82 57.76 56.81 56.93 57.25 57.19 56.26 56.53 57.05 56.21 57.43 57.66 57.61 57.67 57.59 57.46 57.89 58.10 58.03 58.17 57.95 57.89 57.67 57.62 57.60 57.56 57.44 57.76 57.58 57.94 57.66 57.84 57.70 57.60 57.03 56.62 56.23

Adjusted Daily Closing Prices PBCT YHOO 16.09 42.38 15.93 42.73 15.81 41.68 15.64 41.79 15.44 41.44 15.51 41.62 15.41 42.36 15.74 42.68 15.78 43.92 15.89 43.22 15.83 43.68 15.87 43.71 15.88 43.18 15.76 43.13 15.75 43.10 15.82 42.57 15.62 43.69 15.84 43.37 15.67 42.29 15.53 42.80 15.81 44.15 15.85 44.14 15.72 42.79 15.68 43.19 15.71 43.67 15.71 43.99 15.93 43.46 15.86 43.04 16.00 43.46 16.16 42.92 16.08 44.36 16.27 44.35 16.28 44.71 16.12 43.28 16.33 42.93 16.18 42.75 16.25 42.58 16.19 42.26 15.97 42.27 15.87 42.03 15.81 41.91 15.70 42.60 15.70 42.52 15.69 43.02 15.61 42.90 15.52 42.70 15.50 42.49 15.43 42.67 15.51 42.94 15.39 41.27 15.45 39.93 15.40 39.24 15.50 39.24 15.44 38.99 15.48 38.92 15.08 38.39 15.07 38.57

AAPL 117.06 117.12 117.47 117.55 117.63 116.98 117.34 116.30 116.05 114.06 113.89 113.05 113.00 112.52 113.05 112.18 113.95 113.09 112.88 112.71 114.62 113.55 113.57 113.58 114.92 115.57 111.77 107.95 105.44 103.13 105.52 108.36 107.70 107.73 106.73 106.10 106.00 106.82 106.94 107.57 108.03 108.85 108.51 109.36 109.08 109.22 109.38 109.48 108.18 107.93 108.00 108.81 108.37 107.48 105.87 105.22 103.92

NFLX 123.35 121.87 118.79 99.80 101.47 100.23 99.50 100.59 103.33 104.82 105.07 106.28 102.34 102.63 98.55 96.67 97.48 97.07 94.56 95.94 95.83 94.88 98.25 98.06 99.48 97.34 97.01 96.09 99.05 96.50 99.66 99.15 100.09 97.38 97.38 97.45 97.45 97.30 97.58 97.32 95.18 95.94 95.26 95.87 96.16 96.37 95.12 95.31 96.59 95.89 93.93 93.99 95.11 97.03 93.44 93.10 93.56

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8/1/16 7/29/16 7/28/16 7/27/16 7/26/16 7/25/16 7/22/16 7/21/16 7/20/16 7/19/16 7/18/16 7/15/16 7/14/16 7/13/16 7/12/16 7/11/16 7/8/16 7/7/16 7/6/16 7/5/16 7/1/16 6/30/16 6/29/16 6/28/16 6/27/16 6/24/16 6/23/16 6/22/16 6/21/16 6/20/16 6/17/16 6/16/16 6/15/16 6/14/16 6/13/16 6/10/16 6/9/16 6/8/16 6/7/16 6/6/16 6/3/16 6/2/16 6/1/16 5/31/16 5/27/16 5/26/16 5/25/16 5/24/16 5/23/16 5/20/16 5/19/16 5/18/16 5/17/16 5/16/16 5/13/16 5/12/16 5/11/16 5/10/16 5/9/16 5/6/16 5/5/16 5/4/16 5/3/16 5/2/16 4/29/16 4/28/16 4/27/16 4/26/16 4/25/16 4/22/16 4/21/16

56.23 56.33 55.86 55.84 56.41 56.38 56.22 55.45 55.56 52.76 53.63 53.37 53.41 53.18 52.88 52.26 51.98 51.06 51.06 50.85 50.84 50.85 50.23 49.13 48.13 49.52 51.59 50.67 50.87 49.76 49.82 50.08 49.38 49.52 49.83 51.16 51.30 51.72 51.78 51.81 51.47 52.15 52.52 52.67 52.00 51.57 51.80 51.27 49.72 50.31 50.01 50.50 50.20 51.15 50.41 50.84 50.38 50.35 49.41 49.73 49.29 49.22 49.13 49.95 49.22 49.25 50.27 50.77 51.43 51.10 55.05

14.84 14.98 15.16 15.15 15.08 15.16 15.21 15.25 15.00 15.12 15.22 15.15 15.30 15.29 15.05 15.07 14.81 14.66 14.49 14.38 14.23 14.49 14.50 14.28 13.97 13.69 14.91 15.73 15.26 15.26 15.15 15.00 14.97 15.11 15.00 15.31 15.52 15.56 15.71 15.59 15.71 15.57 15.82 15.79 15.70 15.67 15.49 15.66 15.46 15.23 15.25 15.10 15.21 14.82 15.04 14.87 15.11 15.07 15.04 14.96 14.87 14.83 14.89 15.13 15.38 15.33 15.36 15.63 15.80 15.69 15.74

38.80 38.19 38.52 38.66 38.76 38.32 39.38 38.85 38.90 38.17 37.95 37.72 37.96 37.64 37.89 37.96 37.74 37.52 37.51 37.50 37.99 37.56 36.86 36.04 35.22 36.24 37.78 37.36 37.40 37.29 36.94 37.39 37.32 37.40 36.47 36.83 37.35 36.97 36.73 37.07 36.60 37.15 36.65 37.94 37.82 36.76 35.59 37.53 36.66 36.50 37.02 37.24 37.27 37.48 36.48 37.03 37.37 37.44 37.18 37.23 36.94 36.00 36.01 36.53 36.60 36.59 36.95 37.11 37.23 37.48 37.67

105.48 103.65 103.78 102.40 96.15 96.82 98.13 98.89 99.42 99.33 99.29 98.25 98.26 96.35 96.90 96.46 96.16 95.42 95.02 94.48 95.37 95.08 93.89 93.09 91.54 92.90 95.58 95.04 95.39 94.59 94.82 97.02 96.62 96.93 96.82 98.30 99.11 98.41 98.50 98.10 97.39 97.19 97.93 99.32 99.81 99.87 99.08 97.37 95.91 94.71 93.69 94.05 92.99 93.37 90.03 89.85 92.01 92.92 92.29 92.22 92.74 93.12 94.09 92.57 92.67 93.75 96.70 103.16 103.88 104.47 104.76

94.37 91.25 91.65 92.04 91.41 87.66 85.89 85.99 87.91 85.84 98.81 98.39 98.02 96.43 95.97 94.67 97.06 95.10 94.60 97.91 96.67 91.48 91.06 87.97 85.33 88.44 91.66 90.01 90.99 93.80 94.45 95.44 94.29 94.12 93.85 93.75 97.09 97.86 99.89 100.74 99.59 101.25 101.51 102.57 103.30 102.81 100.20 97.89 94.89 92.49 89.55 90.50 88.63 89.12 87.88 87.74 90.02 92.89 90.54 90.84 89.37 90.79 91.54 93.11 90.03 90.28 91.04 92.43 93.56 95.90 94.98

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4/20/16 4/19/16 4/18/16 4/15/16 4/14/16 4/13/16 4/12/16 4/11/16 4/8/16 4/7/16 4/6/16 4/5/16 4/4/16 4/1/16 3/31/16 3/30/16 3/29/16 3/28/16 3/24/16 3/23/16 3/22/16 3/21/16 3/18/16 3/17/16 3/16/16 3/15/16 3/14/16 3/11/16 3/10/16 3/9/16 3/8/16 3/7/16 3/4/16 3/3/16 3/2/16 3/1/16 2/29/16 2/26/16 2/25/16 2/24/16 2/23/16 2/22/16 2/19/16 2/18/16 2/17/16 2/16/16 2/12/16 2/11/16 2/10/16 2/9/16 2/8/16 2/5/16 2/4/16 2/3/16 2/2/16 2/1/16 1/29/16 1/28/16 1/27/16 1/26/16 1/25/16 1/22/16 1/21/16 1/20/16 1/19/16 1/15/16 1/14/16 1/13/16 1/12/16 1/11/16 1/8/16

54.86 55.65 55.72 54.92 54.63 54.63 53.93 53.60 53.71 53.75 54.40 53.85 54.70 54.84 54.51 54.33 53.99 52.84 53.50 53.26 53.36 53.15 52.79 53.94 53.64 52.89 52.47 52.37 51.37 52.15 50.97 50.36 51.35 51.66 52.26 51.89 50.21 50.63 51.42 50.69 50.51 51.96 51.14 51.51 51.73 50.42 49.48 48.69 48.71 48.29 48.42 49.15 50.95 51.11 51.93 53.61 53.98 51.01 50.19 51.12 50.75 51.24 49.46 49.77 49.54 49.96 52.04 50.60 51.72 51.25 51.28

15.53 16.12 16.00 15.84 15.72 15.78 15.68 15.31 15.13 15.05 14.98 15.38 15.30 15.70 15.78 15.59 15.74 15.58 15.51 15.52 15.62 15.82 15.78 15.67 15.60 15.50 15.54 15.56 15.60 15.38 15.08 15.15 15.29 15.27 15.20 14.96 14.80 14.30 14.59 14.46 14.25 14.38 14.54 14.35 14.19 14.36 14.51 14.05 13.55 13.89 14.23 14.05 13.94 14.02 13.86 13.77 14.02 14.06 13.74 13.66 13.64 13.19 13.48 13.85 13.96 13.98 14.00 14.32 14.01 14.52 14.49

37.84 36.33 36.52 36.51 37.17 37.31 36.66 36.48 36.07 36.17 36.66 36.41 37.02 36.48 36.81 36.56 36.32 35.23 34.86 34.80 35.41 35.47 35.17 34.28 34.01 33.26 33.58 33.81 32.82 33.51 32.93 33.96 33.86 32.88 32.91 32.80 31.79 31.37 31.36 30.95 30.67 31.17 30.04 29.42 29.37 29.28 27.04 26.76 27.10 26.82 27.05 27.97 29.15 27.68 29.06 29.57 29.51 28.75 29.69 29.98 29.78 29.75 29.31 28.78 29.74 29.14 30.32 29.44 30.69 30.17 30.63

105.91 105.69 106.25 108.60 110.82 110.76 109.18 107.78 107.42 107.30 109.69 108.56 109.85 108.74 107.75 108.31 106.45 103.99 104.46 104.92 105.50 104.70 104.71 104.59 104.76 103.39 101.35 101.09 100.02 99.97 99.88 100.71 101.83 100.34 99.60 99.38 95.59 95.80 95.66 95.00 93.61 95.77 94.94 95.16 97.00 95.54 92.92 92.63 93.19 93.91 93.93 92.95 95.50 94.74 92.90 94.82 95.71 92.51 91.86 98.32 97.78 99.72 94.69 95.17 95.04 95.50 97.85 95.76 98.29 96.88 95.34

96.77 94.34 108.40 111.51 110.42 109.65 106.98 102.68 103.81 104.45 104.83 104.94 104.35 105.70 102.23 102.19 104.13 101.21 98.36 99.59 99.84 101.06 101.12 99.72 99.35 97.86 98.13 97.66 97.36 98.00 96.23 95.49 101.58 97.93 97.61 98.30 93.41 94.79 94.53 91.61 89.12 91.93 89.23 90.49 94.76 89.05 87.40 86.35 88.45 86.13 83.32 82.79 89.71 90.74 91.49 94.09 91.84 94.41 91.15 97.83 99.12 100.72 102.35 107.74 107.89 104.04 107.06 106.56 116.58 114.97 111.39

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1/7/16 1/6/16 1/5/16 1/4/16 12/31/15 12/30/15 12/29/15 12/28/15 12/24/15 12/23/15 12/22/15 12/21/15 12/18/15 12/17/15 12/16/15 12/15/15 12/14/15 12/11/15 12/10/15 12/9/15 12/8/15 12/7/15 12/4/15 12/3/15 12/2/15 12/1/15 11/30/15 11/27/15 11/25/15 11/24/15 11/23/15 11/20/15 11/19/15 11/18/15 11/17/15 11/16/15 11/13/15 11/12/15 11/11/15 11/10/15 11/9/15 11/6/15 11/5/15 11/4/15 11/3/15 11/2/15 10/30/15 10/29/15 10/28/15 10/27/15 10/26/15 10/23/15 10/22/15 10/21/15

51.12 52.96 53.94 53.70 54.36 55.18 55.41 54.82 54.55 54.70 54.24 53.73 53.04 54.58 55.00 54.09 54.03 52.97 54.16 53.87 54.67 54.69 54.78 53.11 54.10 54.11 53.26 52.84 52.61 53.16 53.10 53.10 52.85 52.77 51.90 52.33 51.43 51.90 52.22 52.08 52.71 53.45 52.93 52.95 52.70 51.82 51.23 51.94 52.54 52.26 52.80 51.46 46.75 45.94

14.45 14.74 15.09 15.24 15.25 15.61 15.81 15.97 15.80 15.81 15.78 15.64 15.49 15.37 15.72 15.98 15.78 15.46 15.46 15.69 15.64 15.70 15.94 16.26 15.92 16.10 16.29 16.19 16.15 16.13 16.09 16.03 16.05 15.88 16.07 16.01 15.99 15.75 15.96 16.17 16.22 16.16 16.23 15.88 15.68 15.68 15.74 15.42 15.88 16.24 15.82 15.77 15.73 15.46

30.16 32.16 32.20 31.40 33.26 33.37 34.04 33.60 34.11 34.45 34.19 32.97 32.95 33.23 33.78 33.03 32.59 32.91 34.63 34.40 34.85 34.68 34.91 34.34 35.65 33.71 33.81 32.94 33.16 32.96 33.36 33.11 32.63 32.98 32.86 32.95 32.19 33.23 33.38 33.99 33.68 34.20 35.12 35.07 34.72 35.27 35.62 35.05 35.19 34.30 33.40 33.17 31.67 31.12

94.84 99.01 100.99 103.59 103.50 105.52 106.92 105.03 106.22 106.79 105.43 105.53 104.25 107.16 109.48 108.64 110.60 111.29 114.23 113.68 116.25 116.30 117.04 113.27 114.33 115.38 116.32 115.84 116.05 116.89 115.78 117.30 116.79 115.33 111.79 112.27 110.46 113.78 114.17 114.81 118.55 119.03 118.90 119.45 120.00 118.64 117.00 118.01 116.77 112.15 112.87 116.59 113.08 111.38

114.56 117.68 107.66 109.96 114.38 116.71 119.12 117.11 117.33 118.16 116.24 116.63 118.02 122.51 122.64 118.60 120.67 118.91 122.91 124.20 126.98 125.36 130.93 126.81 128.93 125.37 123.33 125.44 124.16 123.31 125.03 123.84 120.22 120.63 117.10 111.35 103.65 108.92 112.86 112.70 109.86 114.06 113.50 114.05 109.74 107.64 108.38 105.12 105.80 103.07 103.04 100.04 97.32 97.96

(2) Expected Returns: MSFT K1 -0.004867009 -0.002254613 0.007689601 -0.003483055 0.008784259 -0.003326965 -0.001398811

PBCT K2 0.010043942 0.007590133 0.010869565 0.012953368 -0.004513217 0.006489293 -0.020965693

Expected Returns YHOO K3 -0.008190943 0.025191939 -0.002632233 0.008445994 -0.004324844 -0.017469357 -0.007497634

AAPL K4 -0.000512338 -0.002979467 -0.000680578 -0.000680048 0.005556454 -0.003067948 0.00894233

NFLX K5 0.012144047 0.025928125 0.190280535 -0.016458047 0.012371525 0.007336714 -0.010836028

Page 37





-0.014645107 0.004152284 0.001039089 0.001734958 0.006988068 -0.003134727 -0.003125 0.003484251 -0.010856402 0.001380466 0.018453409 -0.009228591 -0.006745071 0.001038816 0.016722355 -0.002107834 -0.00558952 0.001049152 0.016530413 -0.004776243 -0.009114812 0.01494396 -0.02124327 -0.0039889 0.000867888 -0.00104035 0.001389095 0.002262461 -0.007427881 -0.003614441 0.001206255 -0.002406722 0.003796324 0.001036483 0.003814826 0.000867737 0.00034724 0.000694875 0.002089171 -0.005540184 0.003106661 -0.006174972 0.004825911 -0.0030928 0.002411284 0.00172537 0.009932037 0.007372275 0.006892889 0 -0.001764262 0.008361523 0.000355933 -0.010042277 0.0005288 0.002828355 0.013799291 -0.001967454 0.053117333 -0.016123032 0.004841684 -0.000744339 0.00429833 0.005638023 0.011789281 0.005544958 0.017905751 0 0.004104041 0.000195425

-0.002534854 -0.006922593 0.003790272 -0.002520479 -0.000629723 0.007614213 0.000634921 -0.004424779 0.012804097 -0.013888889 0.010848756 0.00901481 -0.01771031 -0.002523659 0.00826972 0.00255102 -0.001909612 0 -0.013810421 0.004413619 -0.00875 -0.00990099 0.004975124 -0.011677935 -0.000614312 0.009925558 -0.012859706 0.009270705 -0.004307692 0.003705929 0.013775892 0.006301197 0.003795066 0.007006369 0 0.000637349 0.00512492 0.005798969 0.001290323 0.004536617 -0.005157963 0.007797271 -0.003883495 0.003246753 -0.006451613 0.00388601 -0.002583979 0.026525199 0.00066357 0.015498652 -0.009345794 -0.011873351 0.000660066 0.00464191 -0.005218513 -0.003251 -0.002594027 0.016479887 -0.007848247 -0.00649771 0.004569212 -0.00969626 0.000646862 0.015768753 -0.001312333 0.017356501 0.010114573 0.012286728 0.007565353 0.010423844

-0.028233107 0.016196136 -0.010531113 -0.000686319 0.012274178 0.001159263 0.000696125 0.012450035 -0.025635134 0.007378372 0.025537904 -0.011915841 -0.030577643 0.00022662 0.03154938 -0.009261357 -0.010991505 -0.007274471 0.012195191 0.009758318 -0.009664013 0.012581571 -0.032461744 0.000225547 -0.008051913 0.033040666 0.008152784 0.004210526 0.003992438 0.007572267 -0.000236622 0.005710231 0.002863255 -0.016197137 0.001881421 -0.011622501 0.002797156 0.004683864 0.004942316 -0.004218327 -0.006287867 0.040465205 0.033558728 0.017584046 0 0.0064119 0.001798664 0.013805653 -0.004666865 -0.005927809 0.015972768 -0.008567004 -0.003621314 -0.002579928 0.011482203 -0.026917242 0.013642292 -0.00128545 0.019125073 0.005797022 0.006097561 -0.006322392 0.008501594 -0.006598047 -0.001844046 0.005829279 0.005863593 0.000266649 0.000266613 -0.012898183

0.002154244 0.017447002 0.00149266 0.007430305 0.000442504 0.004265935 -0.004688244 0.00775542 -0.015533103 0.007604572 0.001860374 0.001508278 -0.016663793 0.009423161 -0.000176076 -8.80613E-05 -0.011660251 -0.005624314 0.033998417 0.035386754 0.02380496 0.022398963 -0.022649735 -0.02620897 0.006128171 -0.00027853 0.009369437 0.005937842 0.000943377 -0.007676465 -0.001122143 -0.005856633 -0.004258067 -0.007533294 0.003133315 -0.007772485 0.002566914 -0.001281807 -0.001462754 -0.000913464 0.012017036 0.002316316 -0.000648148 -0.00744415 0.004060118 0.00828061 0.015207329 0.006177542 0.012538259 -0.01480433 0.017656693 -0.001245902 0.013501694 0.064963266 -0.00688307 -0.013379355 -0.007744109 -0.005302107 0.00090113 0.000400686 0.010629717 -0.000101244 0.019820357 -0.005645611 0.004536963 0.003103056 0.007713124 0.004291889 0.005684815 -0.009385767

-0.026517042 -0.014214825 -0.002379366 -0.011385011 0.038499151 -0.002825694 0.041400242 0.019447657 -0.008309448 0.004223787 0.026544015 -0.014384031 0.001147866 0.010012701 -0.034300285 0.00193761 -0.014274276 0.021984868 0.003401649 0.00957442 -0.029883967 0.026424902 -0.031707845 0.005143742 -0.009391488 0.027829114 0 -0.000718317 0 0.001541562 -0.00286943 0.002671619 0.022483715 -0.007921638 0.007138358 -0.006362793 -0.003015817 -0.002179091 0.013141295 -0.001993442 -0.013251869 0.0073 0.020866592 -0.000638345 -0.01177587 -0.019787674 0.038420344 0.00365203 -0.004916631 -0.008583289 0.034191814 -0.004364452 -0.004237277 0.006891992 0.042778917 0.020607813 -0.001162914 -0.021840586 0.024114726 -0.131262041 0.004268716 0.00377476 0.016488614 0.004793154 0.013731943 -0.024623944 0.020609885 0.005285412 -0.033806617 0.012827206

Page 38





-0.000195387 0.012465319 0.022249221 0.020854821 -0.028095558 -0.040069305 0.018042704 -0.003906951 0.022368671 -0.001196921 -0.005159723 0.014087359 -0.002809616 -0.006182632 -0.026029546 -0.002712111 -0.008070744 -0.001151586 -0.000575539 0.006564973 -0.013147845 -0.007000923 -0.00283021 0.012996941 0.008286769 -0.004412886 0.010273286 0.031181327 -0.011655482 0.005961829 -0.009643792 0.005939462 -0.018651648 0.014682843 -0.008347828 0.009010765 0.000587983 0.018973458 -0.006350466 0.009010825 0.001403645 0.001807949 -0.016399949 0.014838615 -0.000601261 -0.020416112 -0.009720071 -0.012857454 0.006373156 -0.071710283 0.003417852 -0.014186904 -0.001239812 0.014555209 0.005238459 0.000180705 0.012808724 0.006260375 -0.002021272 -0.0007345 -0.011973874 0.010263897 -0.015695458 -0.002519337 0.006156092 0.003269744 0.006214603 0.021852769 -0.012359304 0.004446882

-0.017747397 -0.000682161 0.01523549 0.021939118 0.020953742 -0.082228114 -0.052168415 0.03110822 0 0.007180172 0.009887879 0.001981534 -0.009162281 0.007251102 -0.020025759 -0.014012769 -0.00254129 -0.009439971 0.00760943 -0.007551964 0.008888931 -0.015624993 0.001878486 0.005667524 0.001892708 0.01213289 -0.011363672 0.013435699 0.014935127 -0.001297079 0.009823191 -0.007152161 0.026017382 -0.014464199 0.011303197 -0.015706768 0.002624665 0.001972348 0.005287495 0.005984061 0.00266666 -0.003984054 -0.015686301 -0.016077163 0.003225831 -0.001931708 -0.017710393 -0.01063824 0.006857741 -0.003107446 0.013862662 -0.037014532 0.007950966 0.009882646 0.00746738 -0.003719738 0.006238217 0.024281247 0.012289696 0.005201595 0.004572172 -0.026081435 0.005115058 -0.025545056 -0.004959736 0.012554883 -0.009944067 0.010678366 0.004416401 -0.000630489

0.011448376 0.018990775 0.022752497 0.023282225 -0.028145721 -0.04076223 0.011241916 -0.001069545 0.002949879 0.009474878 -0.012035304 0.001875643 -0.002139091 0.025500438 -0.009774667 -0.01392225 0.010278523 0.006534195 -0.009171837 0.012841585 -0.014804952 0.013642564 -0.034000976 0.003172898 0.028835747 0.032874347 -0.051691954 0.02373156 0.004383562 -0.014046461 -0.00590768 -0.000804883 -0.005602988 0.027412281 -0.014852795 -0.009098207 -0.001869658 0.00699298 -0.001343003 0.007850596 0.026111083 -0.000277645 -0.014234903 -0.001912541 0.000273244 -0.009742923 -0.004311506 -0.00322318 -0.006670224 -0.005043749 -0.004492653 0.041563389 -0.005202574 0.000273952 -0.017756256 -0.003752426 0.017730524 0.004934211 0.011366787 -0.002764667 -0.013366121 0.006866246 -0.01647758 0.014802632 -0.008964982 0.006838074 0.006607957 0.03093954 0.010613855 0.001724195

0.003033478 0.012711839 0.008654828 0.016840454 -0.014561035 -0.028095709 0.0057561 -0.003753526 0.008517406 -0.002412706 -0.022757574 0.004220755 -0.0032834 0.001232829 -0.015076456 -0.008228792 0.007176051 -0.000908774 0.004055574 0.007250812 0.002046629 -0.007515713 -0.01401965 -0.004882891 -0.000597593 0.00793014 0.017568962 0.015244233 0.012707398 0.010828077 -0.003807114 0.011445066 -0.004154232 0.037118878 0.001992482 -0.023456988 -0.009740913 0.006789497 0.000754952 -0.005576971 -0.004058999 -0.010401323 0.016445972 -0.001066766 -0.011494292 -0.030566326 -0.062577853 -0.006947118 -0.005677509 -0.002736633 -0.01082792 0.002057744 -0.005303312 -0.021574835 -0.020071363 0.0005355 0.014487497 0.013025179 0.003313026 0.00110561 -0.021809643 0.010472645 -0.011789104 0.010273707 0.009175154 -0.005202629 0.017459112 0.023671436 -0.004542409 -0.00433429

0.056733656 0.004612399 0.035125577 0.030938696 -0.035165083 -0.035129848 0.018331318 -0.010770371 -0.029957408 -0.006881885 -0.010373061 0.012196426 0.001806184 0.002876985 0.001066645 -0.034401031 -0.007868434 -0.020322335 -0.008437552 0.011547365 -0.016395101 -0.002561344 -0.010334386 -0.007066825 0.004766122 0.026047915 0.023597896 0.031615555 0.025948763 0.032830764 -0.010497204 0.021098985 -0.005498272 0.014110219 0.001595612 -0.025327695 -0.030896781 0.025955356 -0.003302455 0.016448394 -0.015640467 -0.00819314 -0.016861776 0.034210841 -0.002769163 -0.008348001 -0.015038397 -0.01207779 -0.024400458 0.009686239 -0.018497407 0.025757909 -0.12970485 -0.027889875 0.009871437 0.007022307 0.024957926 0.041877707 -0.010885252 -0.006127324 -0.003624964 -0.001048218 0.005654087 -0.012771987 0.03394301 0.000391438 -0.01863051 0.028850885 0.028975173 -0.012350588

Page 39





-0.00184943 0.003898985 0.006917159 -0.021405019 0.005703791 0.014181724 0.007899224 0.001884279 0.019596558 -0.014950824 0.023039675 0.012149757 -0.019219679 -0.006112684 -0.011331504 0.007036893 0.033411963 -0.008187109 -0.015355077 0.014408074 0.003516988 -0.027920235 0.016017009 -0.007089458 -0.004387625 0.026032449 0.018946908 0.01630111 -0.000402347 0.008725646 -0.002631071 -0.014952136 -0.03538462 -0.003067488 -0.015849057 -0.031255695 -0.006897826 0.058202044 0.016399844 -0.018209633 0.007337269 -0.009562059 0.03585581 -0.006103583 0.004549038 -0.008433043 -0.039917119 0.028466332 -0.021599085 0.00917781 -0.000573343 0.003066978 -0.034782637 -0.018165308 0.004562045 -0.012256667 -0.014739864 -0.004243991 0.01072383 0.005029676 -0.002687224 0.008491439 0.009483798 0.01293184 -0.028186706 -0.007660803 0.016847843 0.001088157 0.019977779 -0.021892526

-0.006265642 -0.012987067 0.002479931 0.006866367 0.005018854 0.006313109 -0.002518908 -0.001257806 -0.002509427 0.013994874 0.020116842 -0.004521961 -0.00959696 0.001281238 0.005151354 0.015696504 0.010575019 0.035592088 -0.020120722 0.008795624 0.015109849 -0.009523734 -0.010767163 0.012951584 0.01172411 -0.012261592 -0.010114596 0.032729846 0.036823016 -0.024647871 -0.023383783 0.012534903 0.007719242 -0.00558266 0.011291391 0.007107368 -0.018143733 -0.002783595 0.023504247 0.005586622 0.001417493 0.034457418 -0.021520727 -0.027215706 -0.007617649 -0.001383183 -0.001381201 -0.022282286 0.02208425 -0.035286326 0.002001382 0.002675557 -0.019672128 -0.023062097 -0.009517798 -0.000634174 -0.023529357 -0.012232476 -0.010290511 0.011015923 -0.000611678 0.001838279 0.00865263 0.00998754 0.007547221 -0.022140306 -0.016333953 0.012867702 0.020637941 0

-0.0172268 -0.001691598 0.008530083 0.025962632 0.007938871 0.02254961 -0.0095296 -0.006802691 0.030164564 -0.02059081 0.017613058 -0.030329771 0.002953278 0.029805352 -0.000911547 0.003353689 0.031770933 0.013388587 0.000318878 0.013247172 0.009129475 -0.016041065 0.037616477 0.021074133 0.001702383 0.00307377 0.082840234 0.010463416 -0.012546125 0.01043997 -0.008502736 -0.032892386 -0.040480309 0.053106936 -0.047487923 -0.017247244 0.002033209 0.026434783 -0.031660524 -0.009673082 0.006715883 0.001008437 0.015011976 0.018415496 -0.032279724 0.020590289 -0.038918239 0.029891269 -0.040729878 0.017235698 -0.015017924 0.015583521 -0.062189055 -0.001242267 0.025477739 -0.055922974 -0.003296404 -0.019682784 0.013095328 -0.014951715 -0.009869376 0.007604621 0.037003275 0.00060698 -0.008426091 -0.016281794 0.022706631 0.013501043 -0.009723488 -0.049667945

-0.005528525 0.007647981 -9.43638E-05 0.001134166 -0.00160421 0.013291258 0.020093695 0.002542487 0.010773982 0.000494416 0.000890861 -0.008245842 -0.011066877 0.014876863 0.007444171 0.002188417 0.039714508 -0.002270164 0.001550249 0.006867884 0.014890652 -0.022605227 0.008746312 -0.002285493 -0.01895638 0.015314605 0.028194507 0.003094995 -0.006046463 -0.007579757 -0.000210548 0.010529728 -0.026708097 0.008035088 0.019792499 -0.020221889 -0.00934864 0.034541398 0.007171895 -0.065706572 0.005530936 -0.019522738 0.053167136 -0.005062483 0.001344884 -0.004838802 -0.024015269 0.021870802 -0.025710286 0.014513354 0.016192241 0.005287731 -0.042204564 -0.019569686 -0.025059327 0.000854995 -0.019194907 -0.013058656 0.017974148 -0.011200587 -0.005340223 0.01286951 -0.000931699 0.012260712 -0.027069226 -0.021196274 0.007692987 -0.017692083 -0.006184809 -0.025738125

-0.002504007 -0.012072056 -0.000593404 0.01403933 0.003724238 0.015225802 -0.002751411 0.004812543 0.003081378 -0.006530602 0.018393401 0.007749555 -0.059952785 0.037271541 0.003278342 -0.007019349 0.052349843 -0.014558466 0.002750471 0.031874227 0.027939833 -0.030566703 0.030258847 -0.013924136 -0.045061249 0.064121267 0.01887873 0.012159861 -0.023742217 0.026936028 0.03372536 0.006401727 -0.077137421 -0.011351102 -0.008197617 -0.027633097 0.02449913 -0.027221776 0.035765243 -0.068281712 -0.013014538 -0.015885603 -0.015925716 -0.050027846 -0.001390314 0.037004978 -0.028208454 0.004692192 -0.085949595 0.014003662 0.032139349 -0.027671081 -0.026512593 0.093070738 -0.020916652 -0.038643103 -0.019964031 -0.020231732 0.017163368 -0.001875062 -0.00702439 0.016517602 -0.0033439 -0.011777665 -0.03665011 -0.001059989 0.03406409 -0.017154223 0.014801059 -0.032544137

Page 40





0.00527467 -0.01451875 -0.00035837 -0.001788556 0.031549797 -0.01829376 -0.000181137 0.016007417 0.007787826 0.004470139 -0.0103226 0.001107233 0 0.00463478 0.001671326 0.016613125 -0.008238161 0.017600304 -0.00900224 -0.006151008 0.00261639 -0.012001499 -0.01383828 0.009930064 -0.000367663 0.004616808 0.017092416 0.011398206 -0.013493266 -0.011485724 0.005401395 -0.010322597 0.026101774 0.100770355 0.01758472

-0.014787466 0.003090202 -0.003694542 -0.01516067 -0.01961954 0.021250787 -0.010810822 -0.011869495 0.00597021 0.002993918 0.00119909 0.002403883 0.003618842 -0.001204869 0.010346943 -0.011432075 0.003623275 0.001209115 0.015346803 -0.01332518 -0.013150015 -0.002979768 0.004189188 -0.004764875 0.021911223 0.012947035 0 -0.003685592 0.020689697 -0.029214859 -0.022023764 0.026769225 0.003034031 0.002433067 0.017956695

0.006686017 -0.012912368 0.004901903 -0.00658837 0.016598719 -0.036746197 0.057549779 -0.002957764 0.026411719 -0.00663453 0.006067992 -0.011990467 0.007550589 0.014710389 -0.010612462 0.003651826 -0.002731411 0.02360988 -0.031297051 -0.004493739 -0.017946483 0.009204335 -0.015204707 -0.026195844 0.001425691 0.010080616 -0.015593961 -0.009825913 0.016262483 -0.003978403 0.025947523 0.026946016 0.006934097 0.047363372 0.017673489

0.004756921 -0.022075613 -0.000422692 -0.006300931 0.033246543 -0.009287942 -0.009033552 -0.00811502 0.00415929 -0.001863944 -0.00715006 0.009596586 -0.012992488 0.004377885 0.012703532 0.031665043 -0.00429145 0.016378879 -0.029208474 -0.00335888 -0.005652102 -0.031516981 -0.004047561 0.001157781 -0.004609836 -0.004650404 0.011470535 0.014058578 -0.008545577 0.010564284 0.041204656 -0.006332376 -0.031911344 0.030995694 0.015295332

-0.010386417 -0.021893258 0.012922798 -0.042541757 0.032489512 -0.016442993 0.028395868 0.016540995 -0.016820791 0.010309262 0.006893245 -0.013756707 0.009609198 0.03011142 -0.00339879 0.030145167 0.051638977 0.074288431 -0.048384099 -0.034910535 0.001419734 0.025851047 -0.036822699 0.004933903 -0.004822472 0.039274695 0.019509467 -0.00682781 0.031012119 -0.006427221 0.026486883 0.000291139 0.029988004 0.027949044 -0.006533269





(3) Application 1, MVP for two assets MSFT and PBCT:

APPLICATION 1 Average Returns MSFT µ 1 PBCT µ2 0.10% 0.03% Corr. Coeff. (K1, K2) 0.023284 Cov (K1, K2) 0.000005 Weight w1 45% Weight w2 55% SUM 100% Investment 20000.00 Int. Price MSFT 57.25 Int. Price PBCT 16.09 No. of MSFT (x1) 157.490852 No. of PBCT (x2) 682.638206 Portfolio Exp. Return 0.06% Portfolio Risk 1.06% Sharpe Ratio 5.57%

Page 41





(4) Application 2, MVP for three assets MSFT, PBCT, and YHOO:



APPLICATION 2 MINIMUM VARIANCE PORTFOLIO (MVP) Unit Matrix U

1

1

1















Cov. (K1, K1)

0.00024403

Cov. (K1, K2)

0.00000516



MSFT

PBCT

YHOO

Cov. (K1, K3)

0.00014022

MSFT

0.000244026

5.15993E-06

0.000140223

Covariance Matrix

Cov. (K2, K2)

0.00020124

PBCT

5.15993E-06

0.000201243

-2.90752E-06

Cov. (K2, K3)

-0.00000291

YHOO

0.000140223

-2.90752E-06

0.000378441

Cov. (K3, K3)

0.00037844



























Weights for MVP w1

w2

w3

SUM of Weights





33%

51%

16%

100%



















µ1

µ2

µ3





m

0.10%

0.03%

0.14%

















Portfolio Expected Return

0.07%

Sharpe Ratio



Portfolio Risk



1.02%











6.69%







(5) Efficient Frontier for the Portfolio of three Assets MSFT, PBCT, and YHOO: Page 42





(6) Application 2, MVP for multi-assets MSFT, PBCT, YHOO, AAPL and NFLX:

APPLICATION 2 U

1

1

1

1

1















µ1

µ2

µ3

µ4

µ5

m

0.00099433

0.000260891

0.001414603

0.000321682



0.001346636













C

MSFT

PBCT

YHOO

AAPL

NFLX



MSFT

0.000243058

5.13946E-06

0.000139667

0.000122707

0.000154668



PBCT

5.13946E-06

0.000200444

-2.89599E-06

-1.1682E-05

-1.40614E-05



YHOO

0.000139667

-2.89599E-06

0.000376939

0.0001185

0.000177779



AAPL

0.000122707

-1.1682E-05

0.0001185

0.000248204

0.000144043



NFLX

0.000154668

-1.40614E-05

0.000177779

0.000144043

0.000866143















W

w1

w2

w3

w4

w5



17.28%





Portfolio Exp. Rtn.



Portfolio Risk







46.64%

24.58%

Sum of W 2.17%

100.00%









0.05%









0.95%



























Sharpe Ratio

9.33%



5.60%



(7) Application 3, targeted return for portfolio of three assets MSFT, PBCT and YHOO:

APPLICATION 3









0.10%













0.006766005

6.538577569





6.538577569

9553.943538











0.10%

1













λ1

0.29871473







λ2

4.9017E-06















Matrix W

New w1

New w2

New w3

Sum



36.71%

20.26%

43.03%

100.00%







New Expected Return Matrix M Matrix µ,1





Portfolio Expected Return 0.10%

Portfolio Risk







1.25%













8.23%





Sharpe Ratio

(8) Assumption for some investment in 3 assets:

INVESTMENT

100000

No. of stocks MSFT (x1)

641

No. of stock PBCT (x2)

1259

No. of stocks YHOO (x3)

1015

Page 43





(9) Application 3, targeted return for portfolio of Multi-Assets, MSFT, PBCT, YHOO, AAPL and NFLX:

APPLICATION 3 New Exp. Rtn.

0.08%







1











M

0.007740694 5.86954861













5.86954861







10996.60234



















λ1

0.115839346









λ2

0.000120044









w1

w2

New Weights

27.87%

w3 37.82%

w4 23.19%

w5

5.16%

5.95%

Portfolio Exp. Rtn. 0.08% Sharpe Ratio

Portfolio Risk



0.95%

8.40%



(10) Efficient Frontier for Portfolio of 5 Assets MSFT, PBCT, YHOO, AAPL and NFLX:







Page 44