Following is the table showing the standard deviation of equally weighted portfolio when firstly two, then five, ten, fifteen and twenty securities respectively are taken into account. This portfolio is made on the basis of the market returns for 132 months for each security.
No. of Securities in Portfolio
SD
2 Equally weighted securities
0.001339
5 Equally weighted securities
0.003857
10 Equally weighted securities
0.001999
15 Equally weighted securities
0.005476
20 Equally weighted securities
0.005106
Naïve Diversification takes into account the securities which are randomly selected from the number of securities (Naïve diversification of investment portfolio). For example, if we see in second part of question 2, there are five securities randomly selected from the number of securities and their mean and standard deviation are measured. Another case is observed when equally wighted portfolio of these securities are made, which is a Markowitz concept which says that risk increases as more and more securities are introduced, but his happens till 10-12 securities but when equally weighted portfolio of more than 10-12 securities are made then risk starts falling down and goes to a normal level (Portfolio Theory).
As we can see in the table of standard deviation above that standard deviation i.e. risk in case of the equally wighted portfolio when number of securities are 2 is least and as number of securities are introduced, the risk increases but this is happening till 5, the risk drops down when more than 5 securities are there in the portfolio. This suggests that increasing the number of securities in an equally weighted porfolio reduces the risk - retun ratio.
Looking at the plot of the standard deviation vs number of securities, we can say that there is some unexpected kind of result we have got from our analysis. According to the main concept, the standard deviation i.e. risk while increasing the number of securities in a portfolio should decrease after a considerable number of securities are added in the portfolio. This trend we can see till the number of securities in the portfolio are 10 but above 10 again the risk exposed by the portfolio increases and it finally decreases after 15. This suggests that our analysis shows a abnoral behavior as compared to the markowitz concept when number of securities in the portfolio are 10 to 15 but in case of less than 10 securities and more than 15 securities, our analysis absolutely matches with the markowitz concept.
Step by Step process
Take data of 132 months of two securities randomly selected from the entire data
Find avearge return of both the security
Give equal weight to average return of both the securities
Find out portfolio mean by giving equal weigh to each security in th portfolio
Now, with the help of portfolio mean and individual means, find out portfolio variance by assignning equal weights to both the securities
Find out standard deviation of the portfolio by sqaure rooting the portfolio variance
Repeat the same process for 5 securites, 10, 15 and 20 securities each
Compare the standard deviation in each case
Solution to Question 2(a)
Results of five securities chosen at random
AMVESCAP - TOT RETURN IND
BAE SYSTEMS - TOT RETURN IND
BP - TOT RETURN IND
BRITISH AIRWAYS - TOT RETURN IND
BRITISH SKY BCAST.GROUP - TOT RETURN IND
TOTAL RETURNS
Average Returns
0.0095
0.0122
0.0109
0.0016
0.0057
0.0399
Variance
0.0173
0.0099
0.0037
0.0167
0.0107
0.1285
Std deviation
0.1316
0.0993
0.0606
0.1294
0.1032
0.3584
Equally Weighted Portfolio of 5 companies
Mean of Portfolio
0.007973458
Variance of Portfolio
1.48774E-05
Standard Deviation
0.003857122
Comparison
Equally weighted Portfolio
5 Securities
Mean
0.007973458
0.0399
Variance
1.48774E-05
0.1285
Standard
0.003857122
0.3584
We can see tat from the above comparison that means return of five securities chosen at random is more than equally weighted portfolio of those five securities. It can also be seen that standard deviation that is the risk in case of five securities chosen at random is more than the equally weighted portfolio.
Now when we take the return -risk ratio of five securities chosen at random and the equally weighted portfolio then we can see that return-risk ratio of five securities is 0.111 where as Return- Risk ratio of equally weighted portfolio is 2.066. This means that equally weighted portfolio of five securities chosen at random is a better option.
Step by Step process
Take data of 132 months of five securities randomly selected from the entire data
Find avearge return, variance and standard deviation of all the five security as well as of the total of these securities
Give equal weight to average return of all five securities
Find out portfolio mean by giving equal weigh to each security in th portfolio
Now, with the help of portfolio mean and individual means, find out portfolio variance by assignning equal weights to both the securities
Find out standard deviation of the portfolio by sqaure rooting the portfolio variance
Compare the mean return and standard deviation of the total of five securities with the mean of the portfolio
Solution to Question 2(b)
Covariance
AMVESCAP - TOT RETURN IND
BAE SYSTEMS - TOT RETURN IND
BP - TOT RETURN IND
BRITISH AIRWAYS - TOT RETURN IND
BRITISH SKY BCAST.GROUP - TOT RETURN IND
AMVESCAP - TOT RETURN IND
0.017177
0.00512
0.001779
0.009711
0.004231
BAE SYSTEMS - TOT RETURN IND
0.00512
0.00978
0.00184
0.005049
0.000544
BP - TOT RETURN IND
0.001779
0.00184
0.003642
0.001694
0.000731
BRITISH AIRWAYS - TOT RETURN IND
0.009711
0.005049
0.001694
0.016611
0.004156
BRITISH SKY BCAST.GROUP - TOT RETURN IND
0.004231
0.000544
0.000731
0.004156
0.010577
Now, according to the given formula:
Average Variance
0.011557
Average Covariance
0.0051
Variance of the Portfolio
0.006391
SD of portfolio
0.079946
SD in Question 2(a)
0.003857
We can see that standard deviation here much more than the standard deviation of the portfolio of five securities which we have seen in the earlier question. Increase in the standard deviation is mainly due to involving co-variance in the list. Increase in the standard deviation means that this formula gives a value which lead to more risk being involved in the portfolio if covariance is taken into account. This approach is more realistic as compared to the earlier approach we used in Question 2(a).
Step by step Process
Find covariance of all the five securities with each other by using covar() function in MS- Excel
Find out the average of all the covariance values
Find out average of all the variance values (variance is covariance with self)
With the help of the above formula find out the portfolio variance and standard deviation
Compare this portfolio variance with one came in the above question
Solution to Question 2(c)
In this, FT ALL Share Index's data is the dependent variable and five securities taken in the portfolio are independent variable (Beta). With the approach of regression we could find BETA value of each security in the portfolio.
AMVESCAP - TOT RETURN IND
BAE SYSTEMS - TOT RETURN IND
BP - TOT RETURN IND
BRITISH AIRWAYS - TOT RETURN IND
BRITISH SKY BCAST.GROUP - TOT RETURN IND
Portfolio
S(XY)
0.5108
0.2897
0.1730
0.4412
0.1947
S(XX)
2.2846
1.3009
0.4845
2.2093
1.4068
Beta [S(XY)/S(XX)]
0.2236
0.2227
0.3571
0.1997
0.1384
0.246620889
BETA is considered to be a measure of systematic risk. This shows the market risk that company face. According to our observation in the above table, maximum risk is exposed by BP-TOT and the minimum systematic risk is exposed by British Sky BCast Group.
Talking about the equally weighted portfolio of the five securities we can see that if equally weighted portfolios of five securities are made then exposure of this port folio for systematic risk in the market becomes even lesser than the individual risk. But in our case, trading in portfolio is better in case of just one security
Step by Step process
Find out S(XX) of each security which is equal to variance of the security multiplied with 132
Find out S(XY) of each security, which is covariance of each security with FTSE all Share total multiplied by 132
Find out BETA for each security which is equal to S(XY)/S(XX)
Find out BETA for portfolio by taking the equal weights of their means
Compare BETA in individual case and BETA of portfolio
