Portfolio theory and portfolio analysis are important instruments in determining the appropriate approach towards building a portfolio of stocks and other investments. In this case, the analysis looks at three types of blue chip stock (IBM, Microsoft, Apple) and uses a series of statistical tools to determine the return and risk for each of these during October 2013. It then attempts several combinations of stock to produce the best results.
¶ … Download stock prices companies - 22 daily closing prices company. You database obtain prices e.g. datastream, yahoo finance . Use prices September 2013 onwards. 2.Present a table companies, dates, prices returns ( %).
The 22 daily closing prices for IBM, Microsoft and Apple reflect the period from October 1 to October 30. In order to calculate the price return for each stock, the previous day closing price has been subtracted from the present day closing price and the result was divided by the present day value. The result has been multiplied by 100, to reflect a percentage price return. The formula below resumes this calculation:
R% = ((Rt-Rt-1)/Rt) * 100
The table below shows the closing prices for the three selected companies (IBM, Microsoft and Apple), as well as the percentage price returns for each of these companies, for the interval October 1 -- October 30.
IBM
Microsoft
Apple
Close Price
Price Return
Close Price
Price Return
Close Price
Price
(a) Before discussing the results for each stock in part and before drawing a conclusion in terms of which stock dominates the other, it is useful to make some general observations about the three types of stock. IBM, Microsoft and Apple are all considered blue chips, namely stocks with low risk, which yield a regular rate of return through time. In the case of the period in analysis here (October 1-October 30), this is reflected through the fact that the daily fluctuations are generally quite small, usually less than $3. The only stock that has a significant fluctuation in a day is IBM, whose closing price drops on October 17 with almost $12.
In terms of average returns, this translates in the fact that both Microsoft and Apple had moderate average returns in the analyzed period: 0.26% for Microsoft and 0.34% for Apple. IBM had a negative return (-0.18%), primarily explained with the one-day drop that has been mentioned previously (a company-related event triggered that most likely, since the other two companies gained on that day).
In terms of variance, a measure of risk, the low values for all the three stocks reflect the blue chip quality. IBM has the highest risk of the three, probably because of some of the price fluctuations (the one-day drop and a significant gain on October 29). Apple has the highest average return and the lowest variance of the three stocks that are analyzed. Given the theory discussed in class, this makes Apple the dominant investment. Since Apple dominates both Microsoft and IBM, it is thus not dominated by any of the other stock. This means that Apple is the mean-variance efficient investment.
4. Variance-covariance matrix
IBM
Microsoft
Apple
IBM
0.000362242
0.000300194
0.00021607
Microsoft
0.000300194
0.000248773
0.00017906
Apple
0.000216069
0.000179059
0.00012888
Correlations
IBM
Microsoft
Apple
IBM
1
-0.037804343
-0.13020998
Microsoft
-0.344617385
1
-0.17470316
Apple
-0.130209981
-0.174703161
1
5. (a) The companies with the highest correlation coefficient are IBM and Apple. Let's consider XIBM the weight for IBM stock and XAPPLE the weight for Apple stock. Assuming the correlation coefficient is zero, the two equations for the return and standard deviation are
Rp = -0.18* XIBM + 0.34*XAPPLE
p = (0.0192* XIBM2 + 0.0112* XAPPLE2)1/2
Combinations
Weight IBM
Weight Apple
Return (%)
Standard deviation (%)
1
1
0
-0.001789
0.019032664
2
0.8
0.2
-0.000749
0.015394489
3
0.75
0.25
-0.00049
0.014553911
4
0.6
0.4
0.0002899
0.012289349
5
0.5
0.5
0.0008096
0.011080645
6
0.4
0.6
0.0013292
0.010215465
7
0.25
0.75
0.0021087
0.009753738
8
0.2
0.8
0.0023685
0.009847496
9
0
1
0.0034077
0.011352553
(b) The companies with the lowest correlation coefficients are Microsoft and IBM.
Combinations
Weight IBM
Weight Microsoft
Return (%)
Standard deviation (%)
1
1
0
-0.001789
0.019032664
2
0.8
0.2
-0.000916
0.01554947
3
0.75
0.25
-0.000697
0.014809106
4
0.6
0.4
-4.25E-05
0.013046492
5
0.5
0.5
0.0003941
0.012359365
6
0.4
0.6
0.0008306
0.012145664
7
0.25
0.75
0.0014854
0.012750494
8
0.2
0.8
0.0017037
0.013179703
9
0
1
0.0025768
0.015772548
6. For the IBM-Apple combination, the Minimum Variance Portfolio (MVP) is for the N= 7 combination, namely for a standard deviation of 0.009754. The MEF is the line that results from joining N=7, N=8 and N=9.
For the IBM-Microsoft combination, the Minimum Variance Portfolio (MVP) is for the N= 6 combination, namely for a standard deviation of 0.012146. The MEF is the line that results from joining N=6, N=7, N=8 and N=9.
7. (a) The two curves are surprisingly similar, the main difference being the MVP, which occurs for the 6th combination (40% IBM, 60% Microsoft) in the case of the IBM-Microsoft portfolio and for the 7th combination (0.25% IBM, 0.75% Apple). So, in both cases, the efficient portfolios occur for a rather small number of combinations: for the IBM-Apple portfolio, for only 4 combinations out of 9, while for the IBM-Microsoft portfolio, for only 5 combinations out of 9, less than 50% of the total combinations (and this only if we count the MVP as reflecting an efficient portfolio).
In case of the MEF, the IBM-Apple MEF looks almost like a straight line, with little curvature and a high gradient. The IBM-Microsoft MEF is significantly more curved.
All the portfolios that are on the two MEFs represent dominating portfolios, according to the mean-variance efficiency criterion. Any of these would represent a viable investor choice, but the choice would be determined by the investors' risk preference. For a particular level of risk, each portfolio/combination would provide a return, and the relationship between the two is one of direct proportionality. This means that the highest the level of risk, the highest the likely return. If the investor is highly risk averse (he does not embrace risk), he will chose one of the lower combinations (N=7 for the IBM-Apple or N=6 for IBM-Microsoft). The investor's risk preference is thus fundamental because it eventually drives the choice itself.
The optimal portfolio will thus be formed at the intersection of an indifference curve representing a level of acceptable risk for the investor and the MEF. This intersection maximizes the utility of the investment, showing the highest return at a certain risk level.
(b) The modern portfolio theory generally recommends diversification as a way of reducing risk, even if the returns may be smaller. In this case, I would also choose one of the combinations that reflect a portfolio rather than a single stock option (excluding combination 9). Other than the mentioned explanation regarding risk diversification, this can also be argued from a historical perspective: in October, IBM had a bad month, in which it suffered a significant one day drop. However, this is not traditionally the case: IBM is a blue chip, with low volatility and, as a consequence, a safe investment. A combination with Apple or Microsoft could be an optimal combination for a normal level of risk.
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