Investment Principles: Portfolio of Three Stocks

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  1. Data Sourcing
  2. Trend Lines and descriptive statistics
  3. Equally-weighted portfolio risk and return
  4. Regression: Beta estimation
  5. Implementing CAPM
  6. The optimal portfolio
  7. Works Cited
  8. Appendices

Data Sourcing

Data for five assets will be collected for the analysis. The five assets are KapStone Paper and Packaging Corporation (KS), Baxter Inc. (BAX), Bank of America Inc. (BAC), S&P 500 index, and the US 13-week Treasury bill. The S&P 500 index represents the market index while the US 13-week Treasury bill represents the risk free asset. Further, monthly data will collected for the period between December 1995 and December 2005. Finally, the data are collected from the Yahoo Finance Website (Yahoo Inc. 1a; Yahoo Inc. 1b; Yahoo Inc. 1c; Yahoo Inc. 1d; Yahoo Inc. 1e). The table presented in appendix 1 shows the monthly values of the five items.

Trend Lines and descriptive statistics

The time series plots for the monthly returns of the three companies, market index, and the risk free rate of return are presented in appendix B. In the graph, it can be noted that the monthly returns of the five securities are oscillating around zero. This shows that the monthly returns are stationary. A stationary data set is consistent with the random walk hypothesis and the efficient market hypothesis. The two theories state that the past trends cannot be used to predict future trends. Thus, the monthly prices reported in the future are independent of past performance. Since trend does not appear in the values of monthly return, it implies that the mean, variance, correlations and other variables are constant over the period of analysis. The concept of stationary data is an important aspect in statistics because it ensures that the results of the analysis are meaningful.

A summary of the descriptive statistics is presented in appendix B. The mean monthly return for KapStone Paper and Packaging Corporation was 3.92%, while the standard deviation was 15.66%. The minimum return was -46.28%, while the maximum return was 62.91%. In the case of Baxter Corporation, the mean return was 1.14%, while the standard deviation was 8.36%. The minimum return was -34.34%, while the maximum return was 27.86%. In the case of Bank of America Inc., the average monthly return was 1.40%, while the standard deviation was 7.6%. The minimum return was -27.9%, while the maximum return was 17.32%. Therefore, out of the three stocks, KapStone Paper and Packaging Corporation had the highest value of the mean and standard deviation. The S&P 500 index had a mean of 0.70% and a standard deviation of 4.50%. The maximum monthly return was -14.58%, while the maximum return was 9.672%. Finally, the 13-weeks US Treasury bill had a mean of 0.30% and a standard deviation of 0.14%. The maximum and the minimum monthly returns were 0.06% and 0.52% respectively.

The X-Y plot for the standard deviation and mean is presented in appendix B below. In the graph, it is evident that a positive relationship exists between the risk (standard deviation) and return (monthly return). This implies that return increases with an increase in risk. This is consistent with the theory which outlines that risky assets offer high return in order to compensate investors for pursuing such investments.

The correlation coefficients of the three securities and the S&P 500 index are presented in appendix B. There is a strong positive correlation between the monthly return for Bank of America and S&P 500 index (52.66%). This shows that the movement in stock prices of this stock will tend to follow the direction of the market. Thus, if the S&P 500 index increases, then the stock prices of Bank of America are also likely to increase. This is followed by the coefficient between monthly returns for Baxter Corporation and S&P 500 index (35.35%), and KapStone Paper and Packaging Corporation (9.9%). Therefore, the monthly returns for Bank of America are most likely to move in the same direction as the market while KapStone Paper and Packaging Corporation is the least likely to follow market trends. Also, it can be noted that there is positive correlation between the three securities.

Equally-weighted portfolio risk and return

In this section, a portfolio of the three stocks will be created. It will be assumed that the stocks have equal weight that is 100/3%. The results of the calculations are presented in appendix C. The results show that the equally-weighted portfolio mean is 2.15%, while the standard deviation is 6.78%. The Sharpe ratio for this portfolio is 0.2726.

Regression: Beta estimation

Regression analysis will be used to estimate the value of beta for each stock. The excess return of each stock will be regressed on the excess return of the market. All the regression results are presented in appendix D below. For KapStone Paper and Packaging Corporation, the estimated value of beta is 0.3559. The p-value of the coefficient is 0.382. This shows that the estimated value of beta is not statistically significant. The coefficient of determination is 0.9812%. It is low and this implies that the explanatory variable does not adequately explain the variations in the dependent variable. The F-test also shows that the overall regression line is not statistically significant. Therefore, the beta of this company is not valid and alternative values of beta such as industry beta can be used for future analysis. In the case of Baxter Corporation, the estimated value of beta is 0.656. The p-value of the coefficient is 06.94E-05. This shows that the estimated value of beta is statistically significant.

The R-square is 12.5%. It is low and this implies that the data does not fit the model well. The F-test also shows that the overall regression line is statistically significant. Therefore, the estimated value of beta is statistically significant and can be used for further analysis. Finally, for Bank of America, the coefficient of S&P 500 index is 0.8887. It represents the value of beta. The estimated value of the intercept (0.0073) represents the alpha (Verbeek 138). The p-value of the coefficient is 5.427E-10. This shows that the estimated value of beta is statistically significant. The R-square is 27.74%. The value is low and this implies that the data does not fit the model well. The explanatory variable only explains 27.74% of the variations in the dependent variables. The F-test also shows that the overall regression line is statistically significant. Therefore, the estimated value of beta is statistically significant and can be used for further analysis. Based on the measure of the goodness of fit, the estimated regression lines for the three securities are weak. This indicates that there are several other factors that affect the value of beta of a stock (Verbeek 138).

Implementing CAPM

The results of the required rate of return are presented in appendix E. Using the CAPM model, the required rate of return for KapStone Paper and Packaging Corporation is 0.44%. Further, the estimated required rate of return for Baxter Corporation is 0.56%, while for Bank of America is 0.66%. A comparison of the required rate of return using the CAPM model and the mean monthly return shows that the later overstates the required rate of return. The results of mean monthly return of all the three stocks are higher than those generated using CAPM model. Further, it can also be observed that the raw mean of the three stocks are not consistent with the beta. Beta is a measure of risk. From a theoretical point of view, it is expected that as the value of beta increases, then the return on the stock should also increase. In the results, KapStone Paper and Packaging Corporation has the lowest value of beta yet it has the highest value of mean as compared to the other two securities. This shows a lack of consistency.

The optimal portfolio

A summary of the results for the minimum variance optimal portfolio is presented in appendix F. In the results, the optimal weights for KapStone Paper and Packaging Corporation, Baxter Corporation and Bank of America are 12.67%, 39.01%, and 48.32% respectively. The estimated portfolio return is 1.61%, while the portfolio standard deviation is 5.84%. The Sharpe ratio for this portfolio is 0.2248. When iterations are carried out, the weights of stock that maximizes the Sharpe ratio are 37.32%, 22.72% and 39.96% for KapStone Paper and Packaging Corporation, Baxter Corporation and Bank of America respectively. The mean for the portfolio is 2.28%, while the standard deviation is 7.18%. The Sharpe ratio is 0.2758.

The results of part 11 above differ from those of the equally-weighted portfolio. The minimum variance portfolio yields a lower mean and standard deviation than the equally weighted portfolio. On the other hand, the maximum Sharpe ratio portfolio yields higher return and standard deviation than both the equally distributed portfolio and the minimum variance portfolio.

Works Cited

Verbeek, Marno. A Guide to Modern Econometrics, England: John Wiley & Sons Ltd., 2008. Print.

Yahoo Inc. 2016a. Web.

Yahoo Inc. 2016b. Web.

Yahoo Inc. 2016c. Web.

Yahoo Inc. 2016d. Web.

Yahoo Inc.2016e. Web.

Appendices

Appendix A

Part a – Data sourcing

Data of stock prices, S&P index and Treasury bill.

Date KapStone Date BAX BAC S&P IRX
11/1/2013 25.89679 12/1/2005 17.01027 37.05899 1264.67 3.877
10/1/2013 25.2552 11/1/2005 16.71759 37.06706 1249.48 3.857
9/3/2013 20.80298 10/3/2005 16.45503 34.95281 1207.01 3.805
8/1/2013 20.41414 9/1/2005 17.16092 33.64227 1228.81 3.47
7/1/2013 21.41054 8/1/2005 17.35892 34.38544 1220.33 3.43
6/3/2013 19.52952 7/1/2005 16.90267 34.43674 1234.18 3.332
5/1/2013 14.10033 6/1/2005 15.96865 36.02431 1191.33 3.06
4/1/2013 14.37738 5/2/2005 15.88257 36.22966 1191.5 2.882
3/1/2013 13.51221 4/1/2005 15.96865 35.2285 1156.85 2.837
2/1/2013 12.94839 3/1/2005 14.62574 34.49327 1180.59 2.722
1/2/2013 11.66522 2/1/2005 15.34885 36.13902 1203.6 2.69
12/3/2012 10.78547 1/3/2005 14.53104 35.92211 1181.27 2.422
11/1/2012 9.674984 12/1/2004 14.86677 36.40241 1211.92 2.182
10/1/2012 9.697053 11/1/2004 13.37663 35.49603 1173.82 2.182
9/4/2012 9.882431 10/1/2004 13.00048 34.36065 1130.2 1.867
8/1/2012 8.840782 9/1/2004 13.59218 33.24061 1114.58 1.674
7/2/2012 7.419548 8/2/2004 12.9075 34.16119 1104.24 1.568
6/1/2012 6.995826 7/1/2004 12.70885 32.28149 1101.72 1.412
5/1/2012 6.814861 6/1/2004 14.58539 32.13339 1140.84 1.303
4/2/2012 7.971269 5/3/2004 13.28787 31.26346 1120.68 1.052
3/1/2012 8.695127 4/1/2004 13.37663 30.27062 1107.3 0.952
2/1/2012 8.871678 3/1/2004 13.05542 30.4549 1126.21 0.923
1/3/2012 7.706442 2/2/2004 12.30734 30.50966 1144.94 0.927
12/1/2011 6.947274 1/2/2004 12.32002 30.33834 1131.13 0.897
11/1/2011 7.318031 12/1/2003 12.89904 29.95474 1111.92 0.907
10/3/2011 7.238583 11/3/2003 11.52515 27.79922 1058.2 0.908
9/1/2011 6.130727 10/1/2003 11.01145 27.90978 1050.71 0.932
8/1/2011 6.633896 9/2/2003 12.03885 28.76111 995.97 0.926
7/1/2011 6.881068 8/1/2003 11.64115 28.91289 1008.01 0.96
6/1/2011 7.313617 7/1/2003 11.43816 30.12413 990.31 0.927
5/2/2011 7.260652 6/2/2003 10.77117 28.83262 974.5 0.838
4/1/2011 7.671132 5/1/2003 10.49775 26.84165 963.59 1.085
3/1/2011 7.578444 4/1/2003 9.528342 26.78739 916.92 1.101
2/1/2011 7.569616 3/3/2003 7.7221 24.17919 848.18 1.09
1/3/2011 7.516651 2/3/2003 11.76129 24.81182 841.15 1.175
12/1/2010 6.753068 1/2/2003 11.67429 25.10208 855.7 1.148
11/1/2010 6.49707 12/2/2002 11.59972 24.93007 879.82 1.182
10/1/2010 5.649626 11/1/2002 13.00626 24.88091 936.31 1.202
9/1/2010 5.358317 10/1/2002 10.17245 24.7815 885.76 1.417
8/2/2010 5.027284 9/3/2002 12.4208 22.65128 815.28 1.527
7/1/2010 5.049353 8/1/2002 14.75452 24.65852 916.07 1.645
6/1/2010 4.91694 7/1/2002 16.22632 23.39886 911.62 1.665
5/3/2010 4.877216 6/3/2002 18.07216 24.75705 989.82 1.659
4/1/2010 5.693763 5/1/2002 21.83295 26.45958 1067.14 1.705
3/1/2010 5.239145 4/1/2002 23.13399 25.29733 1076.92 1.73
2/1/2010 4.043014 3/1/2002 24.19921 23.74067 1147.39 1.74
1/4/2010 4.078323 2/1/2002 22.55665 22.32014 1106.73 1.72
12/1/2009 4.338736 1/2/2002 22.69895 21.79163 1130.2 1.72
11/2/2009 3.129363 12/3/2001 21.80449 21.76397 1148.08 1.67
10/1/2009 3.063157 11/1/2001 20.9023 21.01692 1139.45 1.73
9/1/2009 3.592809 10/1/2001 19.44316 20.19857 1059.78 2.01
8/3/2009 3.03226 9/4/2001 22.1283 19.99655 1040.94 2.3
7/1/2009 2.22454 8/1/2001 20.74151 20.86754 1133.58 3.28
6/1/2009 2.070058 7/2/2001 20.01797 21.58687 1211.23 3.44
5/1/2009 1.875852 6/1/2001 20.29935 20.36875 1224.38 3.56
4/1/2009 1.257924 5/1/2001 19.84915 20.10409 1255.82 3.54
3/2/2009 1.085787 4/2/2001 18.31966 18.8228 1249.46 3.83
2/2/2009 0.666479 3/1/2001 18.9206 18.40264 1160.33 4.18
1/2/2009 0.790065 2/1/2001 18.50859 17.04135 1239.94 4.72
12/1/2008 1.050477 1/2/2001 17.66244 17.89308 1366.01 4.84
11/3/2008 1.9553 12/1/2000 17.74937 15.25167 1320.28 5.73
10/1/2008 2.114196 11/1/2000 17.158 13.27768 1314.95 6.01
9/2/2008 2.802744 10/2/2000 16.2908 15.75799 1429.4 6.15
8/1/2008 3.3015 9/1/2000 15.82005 17.1719 1436.51 6.03
7/1/2008 3.376534 8/1/2000 16.50451 17.56636 1517.68 6.11
6/2/2008 2.943985 7/3/2000 15.41123 15.37993 1430.83 6.02
5/1/2008 3.147018 6/1/2000 13.937 13.95962 1454.6 5.7
4/1/2008 3.036674 5/1/2000 13.18131 17.99735 1420.6 5.49
3/3/2008 2.921916 4/3/2000 12.90876 15.76269 1452.43 5.65
2/1/2008 2.935157 3/1/2000 11.88785 16.86849 1498.58 5.72
1/2/2008 2.846882 2/1/2000 10.3352 14.63679 1366.42 5.64
12/3/2007 3.089639 1/3/2000 12.11304 15.41237 1394.46 5.53
11/1/2007 3.041088 12/1/1999 11.91156 15.96921 1469.25 5.17
10/1/2007 3.089639 11/1/1999 12.75436 18.48988 1388.91 5.15
9/4/2007 3.169087 10/1/1999 12.27062 20.32851 1362.93 4.96
8/1/2007 3.03226 9/1/1999 11.37392 17.56809 1282.71 4.71
7/2/2007 3.217638 8/2/1999 12.5996 18.94436 1320.41 4.85
6/1/2007 3.350052 7/1/1999 12.9049 20.784 1328.72 4.62
5/1/2007 3.398603 6/1/1999 11.39013 22.95634 1372.71 4.67
4/2/2007 2.877778 5/3/1999 12.07798 20.11177 1301.84 4.51
3/1/2007 2.913088 4/1/1999 11.78568 22.24925 1335.18 4.43
3/1/1999 12.3469 21.95777 1286.37 4.36
2/1/1999 13.11223 20.17121 1238.33 4.55
1/4/1999 13.21704 20.65377 1279.64 4.36
12/1/1998 11.98267 18.56909 1229.23 4.35
11/2/1998 11.78801 19.99336 1163.63 4.42
10/1/1998 11.11574 17.63556 1098.67 4.21
9/1/1998 11.06937 16.40874 1017.01 4.25
8/3/1998 9.802196 17.51876 957.28 4.76
7/1/1998 11.02459 24.29776 1120.67 4.95
6/1/1998 9.929048 23.36469 1133.84 4.97
5/1/1998 10.49614 22.88683 1090.82 4.89
4/1/1998 10.17495 23.0384 1111.75 4.85
3/2/1998 10.11759 22.11004 1101.75 4.99
2/2/1998 10.33932 20.65041 1049.34 5.18
1/2/1998 10.16814 18.05027 980.28 5.04
12/1/1997 9.209526 18.33289 970.43 5.2
11/3/1997 9.19348 17.99525 955.4 5.06
10/1/1997 8.421682 17.92035 914.62 5.06
9/2/1997 9.48858 18.53829 947.28 4.97
8/1/1997 9.608794 17.80107 899.47 5.09
7/1/1997 10.43305 21.2086 954.31 5.1
6/2/1997 9.450718 19.23484 885.14 5.05
5/1/1997 9.479287 17.44371 848.28 4.82
4/1/1997 8.603239 17.88814 801.34 5.1
3/3/1997 7.749654 16.44375 757.12 5.17
2/3/1997 8.213742 17.68279 790.82 5.07
1/2/1997 8.213742 15.93293 786.16 5.01
12/2/1996 7.320944 14.40411 740.74 5.05
11/1/1996 7.537189 15.16905 757.02 4.99
10/1/1996 7.382011 13.7967 705.27 5.01
9/3/1996 7.608128 12.71711 687.33 4.91
8/1/1996 7.27211 12.3749 651.99 5.14
7/1/1996 6.764282 12.5021 639.95 5.18
6/3/1996 7.678374 12.01147 670.63 5.03
5/1/1996 7.144031 11.71198 669.12 5.04
4/1/1996 7.144031 11.53152 654.17 5.01
3/1/1996 7.305477 11.56761 645.5 5
2/1/1996 7.338727 10.64725 640.43 4.88
1/2/1996 7.298624 10.00981 636.02 4.91
12/1/1995 6.71714 9.973994 615.93 4.96

Appendix B

Part b – Trend lines and descriptive statistics

Time series plots
Time series plots.

Descriptive statistics.

KS BAX BAC S&P IRX
Mean 0.039151054 0.011366569 0.013947 0.007031 0.003002
Standard Error 0.017503671 0.007629677 0.006938 0.004112 0.000132
Median 0.017506298 0.016875139 0.012985 0.00922 0.003609
Mode #N/A 0 #N/A #N/A 0.00421
Standard Deviation 0.1565575 0.0835789 0.075999 0.045047 0.001448
Sample Variance 0.024510279 0.006985437 0.005776 0.002029 2.1E-06
Kurtosis 3.596351876 2.712374433 1.662344 0.377647 -1.45692
Skewness 0.798813067 -0.5346251 -0.62809 -0.52181 -0.34173
Range 1.091893156 0.622007876 0.452185 0.242517 0.004479
Minimum -0.462754053 -0.3434308 -0.279 -0.1458 0.000699
Maximum 0.629139103 0.278577066 0.173188 0.09672 0.005178
Sum 3.132084294 1.3639883 1.673644 0.843683 0.36028
Count 80 120 120 120 120
Largest(1) 0.629139103 0.278577066 0.173188 0.09672 0.005178
Smallest(1) -0.462754 -0.3434308 -0.279 -0.1458 0.000699
XY scatter plots
XY scatter plots.

Correlation matrix.

KS BAX BAC S&P
KS 1 0.005079 0.048907 0.099058
BAX 0.005079 1 0.221558 0.353456
BAC 0.048907 0.221558 1 0.526647
S&P 0.099058 0.353456 0.526647 1

Appendix C

Part c – Equally weighted portfolio risk and return

Equally weighted portfolio.

KS BAX BAC
Weights 0.3333 0.3333 0.3333
0.3333 0.002723364 7.38456E-06 6.46559E-05
0.3333 7.38456E-06 0.00077616 0.000156368
0.3333 6.46559E-05 0.000156368 0.000641755
1.0000 0.0028 0.0009 0.0009
0.0392 0.0114 0.0139
Variance 0.0046
St. Dev 0.067809262
R * weight 0.013050351 0.003788856 0.004649011
Mean 0.021488218 0.00300233
Risk free rate of return 0.00300233
Mean – rf 0.018485888
Sharpe ratio 0.272615978

Appendix D

Part D: Regression results

KapStone Paper and Packaging Corporation.

SUMMARY OUTPUT
Regression Statistics
Multiple R 0.099058329
R Square 0.0098125
Adjusted R Square -0.0028828
Standard Error 0.1567839
Observations 80
ANOVA
Df SS MS F Significance F
Regression 1 0.0190001 0.0190001 0.7729638 0.8391856
Residual 78 1.9173118 0.0245809
Total 79 1.9363120
Coefficients Standard Error t Stat P-value Lower 95%
Intercept 0.03696223 0.0175705 2.1036492 0.0386318 0.0019819
S&P 0.3559284 0.4048394 0.8791836 0.3820022 -0.4500452

Baxter Corporation.

SUMMARY OUTPUT
Regression Statistics
Multiple R 0.353570164
R Square 0.125011861
Adjusted R Square 0.117659019
Standard Error 0.078186424
Observations 121
ANOVA
df SS MS F Significance F
Regression 1 0.103934257 0.103934 17.00184 6.94432E-05
Residual 119 0.727460912 0.006113
Total 120 0.831395169
Coefficients Standard Error t Stat P-value Lower 95%
Intercept 0.005580005 0.007132462 0.782339 0.435569 -0.008542983
S&P 0.655991205 0.159092611 4.123329 6.94E-05 0.340971946

Bank of America.

SUMMARY OUTPUT
Regression Statistics
Multiple R 0.5267579
R Square 0.2774739
Adjusted R Square 0.2714022
Standard Error 0.0646091
Observations 121
ANOVA
Df SS MS F Significance F
Regression 1 0.1907669 0.1907669 45.699935 5.42664E-10
Residual 119 0.4967461 0.0041743
Total 120 0.6875130
Coefficients Standard Error t Stat P-value Lower 95%
Intercept 0.0072731 0.0058938 1.2340212 0.2196265 -0.0043973
S&P 0.888730 0.1314656 6.7601727 5.42664E-10 0.6284155

Appendix E

Part E – CAPM

Summary of calculations.

Return on stock = risk free rate of return + β * (expected return on market – risk free rate of return)
Risk free rate beta expected return on market risk free rate Required rate of return
KS 0.003002 0.355928 0.007031 0.003002 0.004436
BAX 0.003002 0.655991 0.007031 0.003002 0.005645
BAC 0.003002 0.888731 0.007031 0.003002 0.006582

Appendix F

Part F – Optimal portfolio results

Minimum variance.

Portfolio KS BAX BAC
Weight 0.126735 0.3901 0.483165
0.126735441 0.000394 3.29E-06 3.56E-05
0.390099662 3.29E-06 0.001063 0.000265
0.483164897 3.56E-05 0.000265 0.001348
1 0.000433 0.001332 0.001649
Variance 0.003413
St. Dev 0.058424
R * weight 0.004962 0.004434 0.006739
Mean 0.016135
Mean – Rf 0.013132
Sharpe ratio 0.224775

Maximum Sharpe ratio.

Portfolio KS BAX BAC
Weight 0.3732 0.2272 0.3996
0.373241753 0.003414512 5.63568E-06 8.67825E-05
0.227189906 5.63568E-06 0.000360555 0.000127753
0.399568341 8.67825E-05 0.000127753 0.000922134
1.0000 0.0035 0.0005 0.0011
Variance 0.0051
St. Dev 0.071676659
R * weight 0.014612808 0.00258237 0.005572792
Mean 0.02276797
Mean – Rf 0.01976564
Sharpe ratio 0.275761179
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