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Calculation of discrete rate of return and arithmetic mean for each stock and the index

) The rate of return in each week for each stock and for the stock market index for the 27 weekly periods. Calculate the discrete rate of return for each week. Calculate the arithmetic mean return and the geometric mean return of each stock for the entire period. Use only the discrete returns for your calculations and for the calculations in the questions that follow. 

(b) The variance of returns for each stock and the index and the covariances of returns between each pair of stocks, the covariance between each stock and the stock market index, and the corresponding correlation coefficients.  

(c) Compare your results in (a) and (b) for each stock and the stock market index and comment on the risk return characteristics and performance of each of your stocks and the index. Illustrate with tables/charts as appropriate. Comment on the results, relating to what you have learnt in this course. Relate the risk return pattern and the performance of the market index and your stocks to relevant events that took place during this period. Draw on economic, political, industry and company related events that took place over this period that may have impacted on the performance of your stocks and the market index. Give bibliographic references to the sources of your information.  

a) Based on the discrete returns calculations in Part 1, compute the weekly rate of return and the variance of an equally weighted portfolio formed from the three stocks.  

(b). Examine and compare the pattern of the returns of your portfolio with those of the individual stocks, and the stock index. Compare the corresponding variances. Comment on your observations, relating to material learnt in this course.  

(a) Extract for each week, the yield of the 26-week Treasury bill (or equivalently the 90 day or 180-day bank accepted bill from the financial media (i.e. Federal Reserve Bank: http://www.federalreserve.gov/default.htm) over your sample period. (Remember sometimes reported yields are usually annualised figures. Convert the yields to weekly numbers. Use these as a proxy for the risk free rate).

(b) Estimate the Security Characteristic Line (SCL) for each of your stocks and the equal weighted portfolio, based on the ‘Market Model’, using excess returns (discrete returns less the risk free rate), using Excel regression analysis functions. Show your results graphically. From your results, compute the Beta and the Jensen’s Alpha of each stock and the portfolio.  

(c) Calculate the total risk (the return variance) of each stock and the portfolio. Partition the total risk to their respective systematic and unsystematic risk components.    

(d) Based on your observations and results in parts (b) and (c) above, comment on each of your stock's and portfolio's performance, and on their risk characteristics, comparing and contrasting the magnitude and the proportions of their systematic and unsystematic risk components. What further insights can you gain on the characteristics and behaviour of your stocks and portfolio compared to the analysis and observations you made in Part 1 (c) and Part 2 (b)?
 

Calculation of discrete rate of return and arithmetic mean for each stock and the index

The three stocks selected for this analysis are Adobe Systems Incorporated (ADBE), The Boeing Company (BA), and Paypal Holdings, Inc. (PYPL). The S&P500 index rates were included for comparison. For this analysis, historical weekly closing price data was extracted for each stock and for the index, for the past 28 weeks.

Using the weekly closing price data (Appendix A), the weekly rate of return for all three stocks and the index has been calculated in the table in Appendix B. The formula used for this calculation is:

Arithmetic and geometric means for these stocks and the index were calculated using the formulas:

These values are presented in the table below. 

S&P500

ADBE

BA

PYPL

Arithmetic mean

0.3470%

0.6699%

1.5764%

1.7721%

Geometric mean

0.3440%

0.6483%

1.5356%

1.7394%

Using the discrete rates of return calculated in Appendix A, as well as the arithmetic mean presented above in Part 1(a), the variance and standard deviationwere calculated for each of the stocks and for the index. The formulas that were used for the calculations was:

S&P500

ADBE

BA

PYPL

Variance

0.0000611

0.0004493

0.0008992

0.0006985

Standard deviation

0.0078186

0.0211971

0.0299860

0.0264296

From the standard deviation, and using the discrete rates of return and arithmetic mean calculated above, the covariance and correlation coefficient could be calculated using the following formulas:

The covariance matrix and correlation coefficient matrix have been included in Appendices C and D respectively. 

The arithmetic mean of return for ADBEover the 28-week period was 0.6699%.This return compared well with the index, being nearly double the S&P500 mean return of 0.3470%. However, compared to the other two stocks, BA and PYPL, ADBE had quite a low return. 

The volatility, or risk, of the stock is the calculated standard deviation, and for ADBE is equal to 0.0212. According to theory, a higher volatility will lead to a higher reward, or rate of return (Ghysels, Santa-Clara and Valkanov, 2005). As shown in parts 1(a) and 1(b), and graphed in Appendix E, ADBE has the lowest return and risk of the three stocks that were analysed, but a higher return and higher risk than the index. This stock has performed reasonably well for the last 28 weeks with a low return but also a low risk. 

The ADBE stock is fairly strongly correlated with the other two stocks and the index, meaning that all stocks move in the same direction. The correlation with the stocks BA and PYPL were 0.5780 and 0.6400 respectively.Similarly, the correlation of ADBE with the index was 0.6288. This is interesting as this stock has a correlation coefficient of around 0.6 for all stocks and index, even though they are from different industries. 

Calculation of variance and covariance of returns for each stock and the index

Over the last quarter, and therefore during the time that the data for this analysis was extracted from, the share price for ADBE has increased significantly. According to Zacks Equity Research (2017), this is in part due to strong demand in the market for the ADBE Creative Cloud software, as well as continued growth in demand for other ADBE services such as Photoshop and Illustrator. In fact, for programs like these two, ADBE has shifted to a “cloud-computing business model”, which is also boosting the stock price. Also, ADBE has recently enhanced its Adobe Experience Cloud, and though this project has now enabled companies to market to consumers within their own cars. A combination of these factors has caused a strong growth in the share price, which is expected to continue into 2018.  

The return for BA during the 28-week period was calculated as the arithmetic mean of the weekly returns, which was 1.5764%. This return is significantly higher than that for the index or for ADBE, and is closeto BA, which has the highest return. 

The volatility for this stock is its standard deviation. This was calculated above to be 0.02999. This value is the highest value for standard deviation across the three stocks. Unfortunately, BA has the highest volatility but does not have the highest return, so a rational investor would not be likely to choose to solely invest in this stock. 

As previously discussed, all stocks have a positive correlationwith one another, around the value of 0.6, despite the fact that they belong to different industries. Specifically, the correlation coefficient of BA with ADBE, PYPL and the S&P500 index are 0.5780, 0.5155 and 0.5463 respectively. The correlation coefficient values for BA with the other stocks and index are the lowest of any other stocks – while BA is still positively correlated with the other stocks, it is a weaker correlation than the other stocks have. 

During the past 28 weeks, the stock price for BA has grown somewhat, and particularly jumped to a record high just after the start of the new financial year, at the end of July. This was due to a surprise announcement from BA that they had recorded strong second-quarter profits beatingWall Street estimates, which was achieved by aggressively reducing payroll and development costs (Reuters, 2017). On the other hand, there was a small decline in stock prices immediately after the stock went ex-dividend, on 9 August 2017. However, BA is currently facing strong competition from Airbus, who has acquired a 50.1% stake in Bombardier, a plane manufacturer (Rich, 2017). According to analysts, BA will be in a dangerous and uncertain position unless it changes tack and chooses to pursue middle-of-the-market planes to stay competitive (Rich, 2017). This uncertainty has already caused a small decrease in stock price and will likely to continue falling gradually while the situation remains precarious.

Comparison of risk return characteristics and performance of stocks and the index

The return for PYPL is the arithmetic mean of discrete weekly returns, which was calculated in part 1(a) to be 1.7721%. This return is the highest of the stocks analysed, and is well above the index average return. 

In terms of risk, the standard deviation for PYPL was calculated as 0.0264. This is a comparatively high volatility, as it was the second highest risk term for the stocks behind BA. Overall PYPL has performed quite well over the last 28 weeks, having the highest return and, while reasonably high, not the highest volatility. 

Again, this stock is positively correlated with the other stocks, with correlation coefficients of 0.6400, 0.5155 and 0.7588 with ADBE, BA and the index, respectively. Most notably, PYPL and the index have the highest correlation coefficient of any other pairs of stocks, meaning that PYPL and the index most closely mirror one another. 

Similar to the other two stocks, ADBE and BA, PYPL has experienced an increase in stock price over the past 28 weeks. For PYPL, this was due to a large increase in customer growth rates, up to 88% (Business Wire, 2017). PYPL has also been able to introduce more features to its product, increase its partnerships (for example, with Skype), and closed two acquisitions during this time period (Business Wire, 2017). All of these factors would have contributed to the growth of the stock price during the time period, and have left investors feeling hopeful for the future.

Portfolio return can be calculated as the total weighted average of returns of each rate of return and it is a financial return experienced by an investor’s portfolio. (investopedia 2017).  For this assignment, we will assume that the three stocks (ADBE, BA, PYPL) we chose from the market had equal weighted, which is 1/3 for each stock in the portfolio. With the equal weighted for each stock, the formula used to calculate the portfolio weekly return shown below:

Portfolio variance can be calculated as the total actual returns of a set of securities compensate a portfolio due to the irregularly changes and the different weighted average of variance. (investopedia 2017). Covariance and correlation play an important role as a key to lower down the risk such as unsystematic risk in the portfolio. The lower portfolio variance shows that the lower correlation. The formula used to calculate the portfolio variance shown below:

The three selected stocks are used to calculate the average return of the portfolio: the average return of the portfolio is 1.34% (appendix G). Based on the observation, the portfolio return is higher than ADBE and S&P 500, which was 0.67% and 0.35% respectively. But, the returns for BA and PYPL were higher compared to the portfolio return.  According to the appendix A, the closing date on 18 September 2017, the share price for ADBE had a significant dropped. Therefore, this might be one of the factors that will reduce the discrete weekly portfolio rate. According to the appendix B, on 24 July 2017, the share price for BA increased by 13.7% and on 24 April 2017, the share price for PYPL increased by 8.92%. According to part 1,the reason that the rise of BA share might be due to the achievement on strong second-quarter profits beating Wall Street estimate by their reducing payroll and development costs. (Reuters, 2017). The increase of share price for PYPL because of the increase in customer growth and new product introduced to the market.   

Calculation and comparison of portfolio returns and variances

The portfolio variance is 0.032% (appendix G). The variance for the three selected stocks calculated at part 1 have a higher value than the portfolio variance. The variance for S&P 500 is much lower than the portfolio variance because there are 500 different shares, which we only have 3 shares. The lower variance shows that the portfolio is well diversified and it will reduce the unsystematic risk. Based on the observation, since all three selected stocks have a higher variance than the portfolio, it can conclude that most likely the risk is systematic risk. 

The yield of 26-week Treasury bill has been identified and the amount has been converted into the weekly numbers through dividing the yield number by 360 and multiplying it by 7 days. The yield has been converted into weekly numbers to identify the risk free rate of the economy and it has been evaluated that the average risk free rate of USA of 3 months is 2.12%. The risk free rate explains the interest of an investor in the absolutely risk free security or investment in a particular time period (Burger, 2012).

Security characteristic line is a line of regression which plots the performance of a portfolio or a specific security against the market portfolio or market index at each time. The intercept point on SCL is Alpha and the slope is the beta of the security. The security characteristics line of each stock is as follows:

On the basis of Excel regression analysis, the beta and alpha of each stock has been calculated. Beta explains about the total volatility in the stock price of a company in a specific time period. Whereas, alpha explains about the total return of company less than the total return of market index (Carlen, 2010). Beta and alpha of each stock explains that the higher the alpha of the company, higher the performance of the company would be. And the lower the beta of the company, the better the position and the profitability level of the security would be. The alpha and beta of each stock is as follows: 

Alpha

Beta

Variance

ADBE

0.008694346

1.308958081

0.0433%

BA

0.01423954

1.10997813

0.0866%

PYPL

0.035273641

2.188445989

0.0673%

The above table express that the alpha, beta and variances of each stock. Alpha is considered as the active return of a security. Beta is calculated to analyse the volatility in the stock price of a security and variances explains that how much difference is among the observations. On the basis of above table, it has been evaluated that the alpha of PYPL is higher than other securities whereas if the beta calculations are done, than the BA’s stock is holding lesser beta. The variance calculations explain that the stock of PYPL is best option for the purpose of investment as the associated risk (beta) of the company is average and the active return (alpha) of the security is highest (Low, 2009). On the other hand, the variance calculations of the company also explain about the less difference among the observations. The calculations of alpha, beta and variance of each stock have been given in the appendix.

Estimation of Security Characteristic Line and computation of beta and Jensen's Alpha

Total risk of a business explains about the total systematic risk and unsystematic risk of a portfolio. Every investment has risk which is carried by the assets and liabilities of a security (systematic risk) and any risk which is unique in nature (unsystematic risk). Investors are required to evaluate the total risk of an investment while making the decision about the investment. The total risk of each stock is as follows: 

Systematic Risk

Unsystematic Risk

ADBE's Value

0.0107%

0.0326%

BA's value

0.0079%

0.0787%

PYPL's value

0.0297%

0.0376%

The systematic risk of BA is higher and the unsystematic risk of ADBE is higher and the total risk of ADBE is higher. It explains that the highest risk of ASBE stock is highest. The calculations of systematic risk, unsystematic risk, portfolio calculations have been given in the appendix.

According to the evaluation on the each stock, it has been found that each stock is holding different returns and thus it becomes important to evaluate each stock and measure that which stock would offer the highest return to the return and in which stock, highest risk is associated. For evaluating the each stock, associated total risk, systematic risk, unsystematic risk, alpha, beta and the return has been calculated and it has been compared with the market index. So that a better conclusion could be made and a better recommendation could be given about the investment in a particular security or the portfolio. 


The risk and return of each portfolio has been calculated to identify the best security. On the basis of calculations, it has been found that the risk of ABDE, BA and PYPL is 0.0433%, 0.0866% and 0.0673% whereas the risk of S&P 500 (market index) 0.0059%. The figures explain that the associated risk of ADBE’s stock is lowest among all the securities. Though, it is higher than the market index. It explains that the ADBE stock is way better but the risk level of ADBE is also higher and it is risky security (Waemustafa & Sukri, 2016). 

Further, the return of all the three securities and market index explains that the market return of ADBE is 0.67%, BA is 1.576% and PYPL is 1.772%. Whereas the market index returns is 0.347%.  It explains that the return of PYPL is higher and ADBE is lowest. The market return of PYPL is quite higher than the market index as well. The calculations of risk and return calculations have been given in the appendix. 

Table 5: Risk return characteristics for stocks & index

Return (Y)

Risk (X)

S&P500

0.347%

0.0059%

ADBE

0.670%

0.0433%

BA

1.576%

0.0866%

PYPL

1.772%

0.0673%

 On the basis of risk and return factors, it has been identified that the PYPL is the best option for the purpose of investment as the risk of the company is average in the market and the return of the security is highest. The return is higher than the portfolio as well.

Calculation of total risk and partitioning into systematic and unsystematic risk components

The systematic risk and unsystematic risk of the securities have also been calculated and it has been found that the systematic risk of ADBE, BA and PYPL is 0.0107%, 0.0079% and 0.0297%. It explains that the systematic risk of BA is lowest and it explains that the risk of BA is lowest. Further, the unsystematic risk of ADBE, BA and PYPL is 0.0326%, 0.0787% and 0.0376%. It explains that the unsystematic risk of ADBE is lowest and it explains that the BA is required to maintain the risk (Rego, Billett  & Morgan, 2009).  The calculations of systematic and unsystematic risk have been given in the appendix.


Further, the portfolio performance of all the three stocks has been identified and it has been found that the portfolio’s risk and return both are lower which is 0.033% and 1.034%. The risk and return of the portfolio is lower than the PYPL’s stock performance. It explains that the investors should not invest into the portfolio of the company as the investor will not be able to achieve the expected return from the portfolio (Li et al, 2008). The calculations of portfolio of all the three stocks have been given in the appendix.

On the basis of the above study on the risk factor, return factors, systematic risk, unsystematic risk, and beta. alpha etc of each security, it has been concluded that the investor should take the concern of PYPL stock as this stock would offer the highest return to the investors in average associated risk as well as the unsystematic risk of stock is lower which could be controlled by the company through managing and diversifying some activities. It explains that the investors should definitely invest into PYPL security to enhance the performance and the return of the security.

References

“Adobe Systems Incorporated (ADBE): Historical Data.” 2017. Yahoo Finance. Accessed 11 October 2017.

“Boeing shares make biggest gain since 2009 on second-quarter profit.” 2017. Reuters. Accessed 20 October 2017. https://www.reuters.com/article/us-boeing-results/boeing-shares-make-biggest-gain-since-2009-on-second-quarter-profit-idUSKBN1AB1L3

“Here’s Why Adobe (ADBE) Should Feature In Your Portfolio.” 2017. NASDAQ. Accessed 20 October 2017. https://www.nasdaq.com/article/heres-why-adobe-adbe-should-feature-in-your-portfolio-cm854644

“PayPal Holdings, Inc. (PYPL): Historical Data.” 2017.Yahoo Finance. Accessed 11 October 2017.

“Paypal Reports Third Quarter 2017 Results.” 2017. Business Wire. Accessed 20 October 2017. https://www.businesswire.com/news/home/20171019006420/en

“The Boeing Company (BA): Historical Data.” 2017.Yahoo Finance. Accessed 11 October 2017.

Bodie, Zvi, Michael E. Drew, Anup Basu, Alex Kane and Alan J. Marcus, 2013. Principles of Investments. Australia: McGraw Hill Education (Australia) Pty Ltd.

Burger, M. A. (2012). Accounting measurement and beta risk measures (Doctoral dissertation, David Eccles School of Business, University of Utah).

Carlen, W. B. (2010). Density Functional Calculation of X-Ray Absorption Spectra within the Core Hole Approximation: An Implementation in NWChem.

Ghysels, Eric, Pedro Santa-Clara, and RossenValkanov, 2005. There is a risk return trade-off after all.Journal of Financial Economics. 76(3): 509-548.

Li, X., Miffre, J., Brooks, C., & O’Sullivan, N. (2008). Momentum profits and time-varying unsystematic risk. Journal of Banking & Finance, 32(4), 541-558.

Low, A. (2009). Managerial risk-taking behavior and equity-based compensation. Journal of Financial Economics, 92(3), 470-490.

Rego, L. L., Billett, M. T., & Morgan, N. A. (2009). Consumer-based brand equity and firm risk. Journal of Marketing, 73(6), 47-60.

Rich, Gillian, 2017. “This Is How Boeing Could Get Squeezed By Airbus, Bombardier.” Investor’s Business Daily. Accessed 20 October 2017. https://www.investors.com/news/this-is-how-boeing-could-get-squeezed-by-airbus-bombardier/?data-src=A00220&yptr=yahoo

Waemustafa, W., & Sukri, S. (2016). Systematic and unsystematic risk determinants of liquidity risk between Islamic and conventional banks.

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