Investment Principles: Portfolio of Three Stocks Report

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Table of Contents

Data Sourcing
Trend Lines and descriptive statistics
Equally-weighted portfolio risk and return
Regression: Beta estimation
Implementing CAPM
The optimal portfolio
Works Cited
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. 13 Week Treasury Bill (^˄IRX). 2016a. Web.

Yahoo Inc. Bank of America Corporation (BAC). 2016b. Web.

Yahoo Inc. Baxter International Inc. (BAX). 2016c. Web.

Yahoo Inc. KapStone Paper and Packaging Corporation (KS).2016d. Web.

Yahoo Inc.S&P 500 (^˄GSPC).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

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

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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