Download Introduction - Google Groups

QUANTITATIVE
METHODS
CHRISTOPHER PACK GROUP
2011
COURSEWORK JULY
Introduction
This report makes investment recommendations on shares from the FTSE100 to investors with
the primary aim of achieving superior performance in the form of higher return on their
investments. For the purpose of this report, investments have been limited to ten selected shares
listed on the FTSE100 to enable individual investors to invest either separately or as a portfolio
investment.
The shares have been chosen from different sectors i.e. banking, mining, oil & gas, retail and
telecom based on their market capitalisation to provide diversification to the analysis. The ten
shares selected are Anglo American, BHP Billiton, BP, BT, HSBC, Marks & Spencer, Rio Tinto,
Sainsbury’s, Standard Chartered and Tesco.
The share prices have been taken at the close of business on the last day of the month and have
been studied on a monthly basis from December 2002 to May 2011. The data used for the
analysis is from publicly available sources.1
1. Returns
We began our analysis by calculating the returns of the FTSE100 and each of the ten shares
separately.2 .
2.
i.
Scatterplot
We then prepared a scatterplot of the returns of each of the ten shares against the returns of the
FTSE100 to analyse the correlation between the market returns and the individual share returns,
which is summarised in table1below.
The table shows that the returns of Anglo American, BHP Billiton and Standard Chartered have a high
correlation to the FTSE100 returns, whereas the returns of Marks & Spencer have a lower correlation to the
FTSE100 returns.
1
2
. Appendix A provides a glossary of terms
Please refer to Appendix 1 to see the results
Page 1 of 12
Table 1: Correlation of returns
Shares
Return Anglo American vs. FTSE100
Return BHP Billiton vs. FTSE100
Return BP vs. FTSE100
Return BT vs. FTSE100
Return HSBC vs. FTSE100
Return Marks & Spencer vs. FTSE 100
Return Rio Tinto vs. FTSE100
Return Sainsbury’s vs. FTSE100
Return Standard Chartered vs. FTSE100
Return Tesco vs. FTSE100
Correlation coefficient
0.712
0.683
0.621
0.519
0.586
0.377
0.547
0.530
0.682
0.559
From a visual assessment the scatterplots indicate a positive relationship between the returns of
the shares and the FTSE100 returns.3
ii.
Time Series Graphs
The time series graph is used to visualise the relationship between the returns of the shares and
the FTSE100 returns over our period of analysis. This time series graph shows that the individual
share returns have generally tracked the market returns.4
iii.
Mean Returns
As the next step, we compared the mean returns of the ten shares to the mean returns of the
FTSE100. The table 2 shows that the mean returns of the shares are positive and six of them have
outperformed FTSE100 returns.
.3 Please refer to Appendix 2(i)
.4 Please refer to Appendix 2(ii)
Page 2 of 12
Table2: Summary analysis of the mean returns
Shares
FTSE 100
Anglo American
BHP Billiton
BP
BT
HSBC
Marks & Spencer’s
Rio Tinto
Sainsbury’s
Standard Chartered
Tesco
iv.
Mean Returns
0.005
0.015
0.024
0.004
0.004
0.001
0.006
0.019
0.004
0.012
0.009
The highest mean
return was achieved by
BHP Billiton at 0.024
against the lowest for
HSBC at 0.001.
Standard Deviation
When looking at the standard deviation of the FTSE100, it is interesting to note that none of the
shares outperformed the market that had a standard deviation of only 0.041. The three largest
standard deviations were all shared amongst the leading mining and mineral resource companies,
Rio Tinto, Anglo American and BHP Billiton and the smallest standard deviations by the retailing
giants, Tesco and Sainsbury’s (an apt reflection of the difference in volatility of the commodities
and retail markets over this period).
Table3: Summary analysis of the standard deviations
Shares
FTSE 100
Anglo American
BHP Billiton
BP
BT
HSBC
Marks & Spencer’s
Rio Tinto
Sainsbury’s
Standard Chartered
Tesco
Standard Deviation
0.041
0.100
0.092
0.074
0.081
0.065
0.084
0.112
0.064
0.085
0.059
Against this, the shares
with the lowest
standard deviation were
Tesco at 0.059 and the
worst performing share
being Rio Tinto with
the highest standard
deviation of 0.112.
Page 3 of 12
v.
Median Returns
Table 4: Summary analysis of the median returns
Shares
FTSE 100
Anglo American
BHP Billiton
BP
BT
HSBC
Marks & Spencer’s
Rio Tinto
Sainsbury’s
Standard Chartered
Tesco
Median
0.010
0.008
0.026
0.007
0.006
0.000
-0.001
0.009
0.012
0.006
0.011
The median returns show
that BHP Billiton
significantly outperforms
the market at 0.026,
whereas M&S
underperforms at -0.001.
5
The graph below summarises the position.
.5 Please refer to Appendix 2(iii, iv, v).
Page 4 of 12
vi.
Correlation
6
. On the whole correlations are strong.
Correlations go as high as 0.712 with Anglo American and 0.683 with both BHP Billiton and Standard
Chartered. However, Marks & Spencer’s monthly returns have a relatively weak correlation with the
FTSE100 returns at only 0.377. An explanation of this could be the higher market capitalisation of the Anglo
American, BHP and Standard Chartered shares and their inevitable correlation with the market they
represent.
vii.
Confidence Interval
The 95% confidence interval for the share returns is constructed using the mean returns and the
standard deviations. Appendix 2(vii) shows the 95% confidence interval for the range of our
share returns and the market returns. The table of the confidence interval suggests that there is a
probability of 95% that the mean return of the FTSE100 in any given month will be between 7.63% and 8.62%.This table also shows that the lower the standard deviation is, the tighter the
95% confidence interval will be, which implies less fluctuations in the mean returns during the
period of sampling of the returns. The highest volatility in this 95% confidence interval is again
Rio Tinto, as expected, given its high standard deviation, which ranges between -20.6% and
24.4%. After FTSE100, Tesco’s monthly mean return has provided least volatility with a
relatively small variation from -11% to 13% around the 95% confidence interval.
viii.
Hypothesis Test if mean returns are different amongst the stocks and market
We adopted a null hypothesis, that the hypothesized mean returns of each of our shares are equal
to the FTSE100 returns. The alternative hypothesis is therefore that the mean returns of our shares
are not equal to the FTSE100 returns. This represents a two-tailed hypothesis test.7 To summarise
the table we can say that in the case of BHP Billiton we can confidently reject the null hypothesis.
A very high t-value t=2.69 and a very low p-value p=0.0082 provides a lot of evidence in
favour of the alternative hypothesis. As the p-value of BHP Billiton is 0.01, the null hypothesis
can be rejected at the 10% and the 5% significance level but not at the 1% level. For the
remaining shares the hypothesis strongly suggests in favour of the null hypothesis at 10%, 5%
and 1% significance levels.
In addition we conducted another test to compare the mean returns of the combined index of
shares with the mean market returns. As seen in Appendix 2(viii) (2), we can reject the null
hypothesis at 10% significance level however, not at 5% and 1% levels.
.6 Appendix 2(vi) examines the interdependence between the share returns and market returns
7
Please refer to Appendix 2(viii) and 2(viii)2.
Page 5 of 12
ix.
Scatterplot of mean returns versus standard deviations
Table 5: Summary analysis of the mean returns and the standard deviations:8
Returns
Mean
Return FTSE 100
0.00499 0.04063
0.01520 0.10045
Return Anglo - American
Return
BHP Billiton
Return
BP
Return
BT
Return HSBC
Return
Marks & Spencer
Return
Rio Tinto
Return Sainsbury
Return Standard Chartered
Return Tesco
Std. Dev.
0.02396 0.09185
0.00381 0.07446
0.00359 0.08051
0.00133 0.06487
0.00587 0.08402
0.11247
0.01896
0.00421 0.06400
0.01193 0.08533
0.00940 0.05949
Data labels are for
the mean returns
(x-axis)
Positive linear relationship of
0.682
While FTSE100 has a low mean return of 0.005, it also has low standard deviation of 0.040. Alternatively BHP
Billiton has the highest return of 0.024 with a relatively high standard deviation of 0.092. Rio Tinto follows the
same pattern, with a high return of 0.018 and an even higher standard deviation of 0.11, followed by Anglo
American that has a mean return of 0.015 and standard deviation of 0.100
8
The scatterplot in Appendix 2(ix) plots the mean returns against the standard deviations of the shares and the FTSE100.
Page 6 of 12
3
The scatterplot above indicates that, based on the performance during the sampling
period, the FTSE100 represents a conservative investment approach.
Using the graph above and based on our analysis in the sampling period, we recommend
investing in BHP Billiton as there is a better balance between higher returns and an
acceptable level of standard deviation or volatility.
4
To regress the equation: α+β Market, we have used α as the co-efficient of the
constant and β as the co-efficient of the FTSE100 to run a simple regression for each
of the shares.9
Table 6: Summary of the overall regression
Adjusted RSquare
Anglo American
BHP Billiton
BP
BT
HSBC
M&S
Rio Tinto
0.9295
0.5536
0.6106
0.6851
0.4616
0.7871
0.7310
0.8640
0.3064
0.3728
0.4693
0.2131
0.6195
0.5343
0.8626
0.2994
0.3665
0.4639
0.2052
0.6157
0.5296
St error of
estimate
281.62
524.52
62.79
44.02
118.64
84.05
905.58
Sainsbury’s
Standard Chartered
Tesco
0.7721
0.8448
0.8007
0.5962
0.7137
0.6411
0.5921
0.7109
0.6375
52.68
193.57
46.09
Summary
correlation
R-square
ANOVA
table-p value
< 0.0001
< 0.0001
< 0.0001
< 0.0001
< 0.0001
< 0.0001
< 0.0001
< 0.0001
< 0.0001
< 0.0001
It can be concluded from the table above that most of the share prices have a high
correlation with the FTSE100. The highest correlation is observed in Anglo
American’s share price (0.9295) while HSBC and BHP shares have the lowest
correlations. Similarly, for the majority of regression equations, it can be said that
there is a considerably high value for the R square, which indicates that a high
percentage of the change in the share price can be explained by the change in the
explanatory variable FTSE100 unlike shares of HSBC, BHP and BP which have a
very low R square.
The standard error of estimate helps to define the confidence interval for the
dependent variable i.e. share price. Thus a lower standard error of estimate is
desirable. It is observed that the regression equations provide varying standard
error of estimates, where Tesco has the least standard error of estimate and Rio
Tinto is clearly an outlier. Finally, a p-value less than 0.0001 in the ANOVA table
indicates an overall good fit for the equation.
.9 The table below summarises the position in Appendix 4
Page 7 of 12
Table 7: Summary of FTSE regression
Regression
Table
Anglo
American
BHP Billiton
BP
BT
HSBC
M&S
Rio Tinto
Sainsbury’s
Standard
Chartered
Tesco
Coefficient
Standard
error
t-Value
p-Value
0.8676
0.0346
25.0773
0.4262
0.0592
0.0506
0.0755
0.1311
1.1857
0.0782
0.0644
0.0644
0.0054
0.0146
0.0103
0.1113
0.0065
6.6137
7.6715
9.3565
5.1780
12.6966
10.6578
12.0891
< 0.0001
< 0.0001
< 0.0001
< 0.0001
< 0.0001
< 0.0001
< 0.0001
< 0.0001
0.3736
0.0238
15.7112
0.0753
0.0057
13.2984
Confidence Interval 95%
Lower
Upper
0.7990
0.9363
0.2983
0.0439
0.0399
0.0465
0.1106
0.9649
0.0654
0.5540
0.0745
0.0613
0.1044
0.1516
1.4064
0.0911
< 0.0001
0.3264
0.4208
< 0.0001
0.0641
0.0865
The Table above summarises the coefficient for the FTSE100 index from the
output of the regression. It can be concluded that the FTSE100’s coefficient is
positive for all the regressions and it has the p-value less than 0.001. Similarly the
t-value for all the equations is further than 3 standard errors on the t-distribution
curve. Thus we can confidently say that for all the regressions, we can reject the
Null Hypothesis at 99% significance level (that the coefficient of the FTSE100
index is 0, Ho: µ=0).
5
To regress the equation: α+β1 Market+β2 variable, we evaluated
various10variables that would produce better regression equations. We
considered the following options as the other variable:
Market Data: Log of the FTSE100, index of 10 selected shares, mean returns of
all the shares, cumulative returns till date and multiple lags of the share price. All
these were rejected as they did not produce a strong regression, which was
observed in low correlations and low R-squares. Similarly the p-values for the
coefficients did not give enough confidence in rejecting the null hypothesis that
their coefficient is 0 (Ho: β2 =0). For example, the logarithmic equation for Rio
Tinto, which was based on three variables, has given almost the same output with
the lag1 of the market and lag1 of the share price.10
Economic Indicators: Macroeconomic indicators for the UK’s National Economy
such as Consumer Price Index as well as commodity prices such as Brent Crude
Oil prices were considered. However, it was observed that the share prices were
10. Log(Rio Tinto)=0,2569+-0,8378Log(FTSE100)+1,9844Log(Index10Shares)
Page 8 of 12
poorly correlated to these variables. This was evident from the correlation test,
their scatter plot and a visual comparison of the time series charts.
The table below shows that the regression indicates that the lag of the FTSE100
and the lag of the share prices have a high correlation with the share price. The
highest correlation is observed in BHP Billiton’s share price (0.9807) followed
closely by others. A considerably high value for the R-square as well as the
adjusted R-square for all the shares indicates that high percentage of change in the
share price is explained by the explanatory variables. BT has the lowest standard
error of estimate followed by Tesco, Marks & Spencer and BP. Rio Tinto is still an
outlier. The p-value in the ANOVA table is less than 0.0001 which indicates that
there is an overall good fit for the equation.
Table 8: Summary of regression with lag 11
Share Lag with
market lag
ANOVA
Summary
Correlation R
Square
0.9544
0.9108
Adjusted
R Square
0.9090
Std Err of
Est.
227.2900
p Value
0.9807
0.9618
0.9610
122.9425
< 0.0001
0.8880
0.9758
0.9460
0.9719
0.7885
0.9521
0.8950
0.9445
0.7842
0.9511
0.8928
0.9434
36.2191
13.3471
43.4600
32.3310
< 0.0001
< 0.0001
< 0.0001
< 0.0001
0.9631
0.9276
0.9261
357.6189
< 0.0001
Rio Tinto (Log)
0.9632
0.9277
0.9262
0.129
< 0.0001
Sainsbury
0.9602
0.9220
0.9204
23.2002
< 0.0001
Standard Chartered
Tesco
0.9558
0.9135
0.9117
105.6128
< 0.0001
0.9655
0.9322
0.9308
19.6430
< 0.0001
Anglo – American
BHP Billiton
BP
BT
HSBC
Marks& Spencer
Rio Tinto
< 0.0001
11. Refer to Appendix 5(a) for lag and 5(b) for log
11
Below is the regression equation we adopted:
Share price = α + β1 Market (lag1) + β2 share price (lag1)
. 11Lag 1 is used as the other variable in 5.
Page 9 of 12
6
Investment recommendation
The suggestions made here are primarily based on the information derived from sections 2(iii),
2(iv) and 5 and on standard error of estimation in the regression in section 5. For this analysis,
standard error of estimate is calculated as a percentage of the share price. The data is interpreted
as below:
High standard error of estimate as a percentage of share price indicates a wide confidence interval
and thus poor predictability. Hence such shares should not be considered for investment based on
this analysis.12
High Standard deviation indicates wider range of returns or in other words, high risk. Thus shares
with high risk but low returns should not be considered for investment based on this analysis.
Shares with higher risk and medium to higher returns can be suitable for an aggressive investor.
Similarly shares with low risk and low returns are recommended for a risk-averse (conservative)
investor. Lastly, low risk, high return shares are recommended across the board.
Table 9 below depicts the suggestions per share:
Share
Anglo - American
BHP Billiton
BP
BT
HSBC
Marks& Spencer
Rio Tinto
Sainsbury
Low
X
Predictability
High
High Risk
Low Risk
Medium
Low
Medium to
Low
to High
Returns
High
Returns
Rerurns
Returns
X
X
X
X
X
X
X
Standard
Chartered
Tesco
Not
recommended
12
X
X
Aggressive
Conservative
Best pick
Please refer to appendix 6.
Page 10 of 12
In summary, this analysis found Tesco to be the overall best investment option. Sainsbury’s is
better suited for risk-averse investor while BHP and Standard Chartered are better suited for
aggressive investors. Despite the predictability and low risk, HSBC does not seem to be attractive
due to its very low monthly mean return (0.00133%) that is significantly below the mean return
of market.
Page 11 of 12