SA(work)

ECOS1101 Undergraduate Programmes 2008/09Quantitative Methods

Z0905634

Summative Assignment


1.

To begin this assignment three stocks were selected from the S&P500 and the calendar monthly ultimo price for each stock for 60 consecutive months was collected. The stocks chosen are the ones that I have the most concern in and are: Apple, Starbucks and Ebay.

It is required to compute the log returns and provide a full statistical description of each return serious.

Logarithmic returns= Where P is a price and

t is a time period.

The advantage of logarithmic returns is that they take into consideration continuous compounding of returns.

In the table below prices and log returns for 60 consecutive months for 3 stocks are reflected. It is impossible to calculate the log return in the first month due to the absence of price in the previous month. Therefore there are 59 returns obtained.

For example in order to calculate return on 31.05.02 we need to use the formula mentioned above in such a way:

The return in this month is negative since the price has decreased.

Figure 1

Prices and logarithmic returns of stocks

Currency: $

MonthApple Starbucks Ebay

Price($) log return Price($) log return Price($) log return

30.04.02 12,135 11,41 13,275

31.05.02 11,65 -0,04079 12,14 0,06202 13,8025 0,03897

1


28.06.02 8,86 -0,27376 12,425 0,0232 15,405 0,10984

31.07.02 7,63 -0,14946 9,815 -0,2358 14,2725 -0,07636

30.08.02 7,375 -0,03399 10,05 0,02366 14,13 -0,01003

30.09.02 7,25 -0,01709 10,32 0,02651 13,2025 -0,06789

31.10.02 8,035 0,10281 11,92 0,14413 15,81 0,18024

29.11.02 7,75 -0,03611 10,87 -0,09221 17,23 0,08601

31.12.02 7,165 -0,07848 10,19 -0,0646 16,955 -0,01609

31.01.03 7,18 0,00209 11,36 0,10869 18,79 0,10276

28.02.03 7,505 0,04427 11,725 0,03162 19,605 0,04246

31.03.03 7,07 -0,05971 12,88 0,09395 21,3275 0,08421

30.04.03 7,11 0,00564 11,755 -0,0914 23,2275 0,08534

30.05.03 8,975 0,23294 12,335 0,04816 25,4125 0,0899

30.06.03 9,53 0,06 12,275 -0,00488 26 0,02286

31.07.03 10,54 0,10073 13,665 0,10727 26,825 0,03124

29.08.03 11,305 0,07007 14,195 0,03805 27,705 0,03228

30.09.03 10,36 -0,08729 14,4 0,01434 26,82 -0,03247

31.10.03 11,445 0,0996 15,8 0,09278 27,965 0,04181

28.11.03 10,455 -0,09047 16,085 0,01788 27,945 -0,00072

31.12.03 10,685 0,02176 16,58 0,03031 32,305 0,14498

30.01.04 11,28 0,05419 18,305 0,09898 33,465 0,03528

27.02.04 11,96 0,05854 18,7 0,02135 34,35 0,0261

31.03.04 13,52 0,1226 18,935 0,01249 34,64 0,00841

30.04.04 12,89 -0,04772 19,46 0,02735 40,015 0,14425

30.05.04 14,03 0,08475 20,3 0,04226 44,4 0,10399

30.06.04 16,27 0,14813 21,745 0,06876 45,975 0,03486

30.07.04 16,17 -0,00617 23,495 0,0774 39,165 -0,16031

31.08.04 17,245 0,06436 21,62 -0,08317 43,27 0,09968

30.09.04 19,375 0,11646 22,73 0,05007 45,97 0,06053

29.10.04 26,2 0,30178 26,44 0,15119 48,815 0,06005

30.11.04 33,525 0,24653 28,13 0,06196 56,15 0,13999

2


31.12.04 32,2 -0,04032 31,18 0,10294 58,17 0,03534

31.01.05 38,45 0,17739 27 -0,14394 40,75 -0,35591

28.02.05 44,86 0,15419 25,905 -0,0414 42,84 0,05002

31.03.05 41,67 -0,07377 25,83 -0,0029 37,26 -0,13955

29.04.05 36,06 -0,1446 24,76 -0,04231 31,71 -0,16129

31.05.05 39,76 0,09768 27,395 0,10113 38 0,18095

30.06.05 36,81 -0,07709 25,83 -0,05882 33,01 -0,14078

29.07.05 42,65 0,14726 26,275 0,01708 41,78 0,23561

31.08.05 46,89 0,09478 24,515 -0,06933 40,49 -0,03136

30.09.05 53,61 0,13393 25,05 0,02159 41,2 0,01738

31.10.05 57,59 0,07161 28,28 0,12128 39,61 -0,03936

30.11.05 67,82 0,16351 30,45 0,07393 44,81 0,12335

30.12.05 71,89 0,05828 30,01 -0,01456 43,22 -0,03613

31.01.06 75,51 0,04913 31,7 0,05479 43,1 -0,00278

28.02.06 68,49 -0,09758 36,32 0,13605 40,06 -0,07314

31.03.06 62,72 -0,08801 37,63 0,03543 39 -0,02682

28.04.06 70,39 0,11537 37,27 -0,00961 34,41 -0,12521

31.05.06 59,77 -0,16355 35,65 -0,04444 32,81 -0,04761

30.06.06 57,27 -0,04273 37,76 0,0575 29,29 -0,11349

31.07.06 67,96 0,17114 34,23 -0,09815 24,07 -0,19628

31.08.06 67,85 -0,00162 31,01 -0,09879 27,82 0,14479

29.09.06 76,98 0,12625 34,05 0,09352 28,36 0,01922

31.10.06 81,08 0,05189 37,75 0,10316 32,13 0,12481

30.11.06 91,66 0,12265 35,295 -0,06724 32,34 0,00651

29.12.06 84,84 -0,07732 35,42 0,00354 30,07 -0,07278

31.01.07 85,73 0,01044 34,94 -0,01364 32,39 0,07432

28.02.07 84,61 -0,01315 30,9 -0,12288 32,06 -0,01024

30.03.07 92,91 0,09358 31,36 0,01478 33,15 0,03343

3


To understand features of the returns obtained, statistical description of return series should be computed.

Figure 2

Statistical description of Apple return series

mean 0,034500847

median 0,05189

Q1 -0,04176

Q3 0,10909

Interquartile Range 0,15085

Midhinge 0,033665

Sample Varience 0,01202383

Sample Standard Deviation 0,109653225

Max 0,30178

Min -0,27376

Range 0,57554

Count 59

Sum 2,03555

Figure 3

Statistical description of Starbucks return series

mean 0,017136102

median 0,02366

Q1 -0,02798

Q3 0,071345


Midhinge 0,0216825



Min -0,2358

Max 0,15119

Range 0,38699

4


Count 59

Sum 1,01103

Figure 4

Statistical description of Ebay return series

median 0,03124

Q1 -0,0343

Q3 0,085675


Midhinge 0,0256875



Min -0,35591

Max 0,23561

Range 0,59152

Count 59

Sum 0,91517

2.

Descriptive statistics is a tool, which can be used to check whether continuous variables follow a normal distribution. Therefore in order to test the normality of each return series characteristics of the data will be compared with theoretical properties of the normal distribution.

Let’s start with Apple return series.

1. The mean of 0,0345 is slightly lower than the median of 0,0518, which means that the distribution is negative , or left skewed. Distortion to the left is cause by some extremely small values.

5


2.Figure 5

-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8

Lower Whisker

Lower Hinge Median

Upper Hinge

Upper Whisker

-0,16355 -0,04176 0,05189 0,10909 0,30178

The boxplot graphically shows the skewness of the distribution to the left with 1 lower outlier at -0,27376.

3. The interquartile range of 0,151 is approximately 1,38( ) standard deviations, which is close to normal distribution where the IQR is 1,33 standard deviations.

4. The range of 0,58 is equal to 5,25 standars deviations. In a normal distribution, the range is nearly 6 standard deviations. The difference is insignifficant and it is caused because values are situated closer to the mean.

5. 66,1%(39 values) of the returns are within 1 standard deviation of the mean. This is approximately the same as in the normal distribution, where there are 68,26%.

6. 86,4%(51 values) of the returns are within 1,28 standard deviations of the mean. In the normal distribution it is 80%.

According to the characteristics of the data examined, returns are slightly left-skewed and its values are closer to the mean than expected in the normal distribution. In general data characteristics do not differ signifficantly from the theoretical properties of the normal distribution.

6


Another way for evaluating whether data is normally distributed is to construct a normal probability plot. One common approach is called the quantile-quantile plot.1 First of all each ordered value have to be transformed to a Z value. In our case there are 59 values(n=59), Z value for a smallest value

corresponds to a cumulative area of for this

cumulative area Z value is -2,128(see Appendix1or use formula NORMSINV in the Xcel). Figure 6 illustrates the complete set of Z values.

Figure 6

Ordered Values and Corresponding Probabilities and Z Values for a Sample of n=59

Ordered Value Probability Z value

1 0,016666667 -2,128045234

2 0,033333333 -1,833914636

3 0,05 -1,644853627

4 0,066666667 -1,501085946

5 0,083333333 -1,382994127

6 0,1 -1,281551566

7 0,116666667 -1,191816172

8 0,133333333 -1,110771617

9 0,15 -1,036433389

10 0,166666667 -0,967421566

11 0,183333333 -0,902734792

12 0,2 -0,841621234

13 0,216666667 -0,783500375

14 0,233333333 -0,727913291

15 0,25 -0,67448975

16 0,266666667 -0,622925723

17 0,283333333 -0,572967548

1 Pr. A.Darnell, Dr. M. Lau, Quantitative Methods For Finance, second edition, 2009, Pearson Education Limited, Great Britain;

7


18 0,3 -0,524400513

19 0,316666667 -0,477040428

20 0,333333333 -0,430727299

21 0,35 -0,385320466

22 0,366666667 -0,340694827

23 0,383333333 -0,296737838

24 0,4 -0,253347103

25 0,416666667 -0,210428394

26 0,433333333 -0,167894005

27 0,45 -0,125661347

28 0,466666667 -0,083651734

29 0,483333333 -0,041789298

30 0,5 -1,39146E-16

31 0,516666667 0,041789298

32 0,533333333 0,083651734

33 0,55 0,125661347

34 0,566666667 0,167894005

35 0,583333333 0,210428394

36 0,6 0,253347103

37 0,616666667 0,296737838

38 0,633333333 0,340694827

39 0,65 0,385320466

40 0,666666667 0,430727299

41 0,683333333 0,477040428

42 0,7 0,524400513

43 0,716666667 0,572967548

44 0,733333333 0,622925723

45 0,75 0,67448975

46 0,766666667 0,727913291

47 0,783333333 0,783500375

8


48 0,8 0,841621234

49 0,816666667 0,902734792

50 0,833333333 0,967421566

51 0,85 1,036433389

52 0,866666667 1,110771617

53 0,883333333 1,191816172

54 0,9 1,281551566

55 0,916666667 1,382994127

56 0,933333333 1,501085946

57 0,95 1,644853627

58 0,966666667 1,833914636

59 0,983333333 2,128045234

To create a normal probability plot, Z values should be plotted on X axis and corresponding cumulative probabilities on the Y axis. If the data are normally distributed the points should fall along a straight line.

Figure7

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

R² = 0.982875547598428

Normal Probability Plot of Apple Returns

Z value

Ap

ple

Ret

urn

9


Figure 7 shows that Apple returns approximate a straight line despite its lower outlier. Therefore it is reasonable to conclude that returns are almost normally

distributed. (see Appendix 2) = 98.3% which is close to 1 and therefore justifies that returns approximate normal distribution.

Now, let’s have a look how Starbucks’ returns are distributed.

1. The mean of 0,017 roughly equals to the median of 0,02. This means that data is normally distributed or might be slightly left-skewed.

2. Figure 8

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

Lower Whisker

Lower Hinge Median

Upper Hinge

Upper Whisker

-0,14394 -0,02798 0,02366 0,071345 0,15119

The boxplot looks to be symmetrical, with 1 lower outlier -0,2358.

3. The IQR of 0,099 is approximately 1,27 standard deviations.

4. The range of 0,39 is equal to 4,95 standars deviations.

5. 64,4% of the returns are within 1 standard deviation of the mean.

6. 81,4% of the returns are within 1,28 standard deviations of the mean.

According to these statements it can be assumed that Starbucks' returns are normally distributed. Even though there is lower outlier, the boxplot appears to be symmetrical. The percentage of data that lies within 1 and 1,25 is approximately the same as in normal distribution.

10


Figure 9

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2R² = 0.958626765755029

Normal Probability Plot of Starbucks Returns

Z value

Star

bu

cks

Ret

urn

From Figure 9 it can be seen that points plot along an approximately straight line with only one lower outlier. This proves the fact that data follows normal

distribution. is close to 1 though it is less than in Apple returns.

Lets have a look at Ebay log returns now and find out whether they can be approximated by the normal distribution.

1. The mean of 0,016 is lower than the median of 0,031. Therefore the data is slightly skewed to the left.

Figure10

11


2.-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

The boxplot shows the left-skewness of the data which is caused by extremely small numbers such as the lower outlier of -0,36.

3. The IQR of 0,12 is about 1,15 standard deviations. 4. The range of 0,59 is equal to 5,65 standard deviations.

5. 71,2% of the returns are within 1 standard deviation of the mean.

6. 83,1% of the returns are within 1,28 standard deviations of the mean.

According to results, the data approximates normal distribution, however it is slightly left-skewed and values are situated closer to the mean than expected.

Figure 11

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

R² = 0.950381696246258

Normal Probability Plot of Ebay Returns

Z value

Eb

ay R

etu

rn

12


The graph is almost a straight line and is slightly less than 1. Therefore the

returns for this stock follow normal distribution however not as good as

returns on Apple and Starbucks.

944 words

3.

In order to construct 95% confidence interval estimates of the population mean return ( unknown) the following formula should be used:

(see Appendix3)

Therefore:

= 0,0064 0,0636

= -0,0032 0,037

= -0,0118 0,0428

Figure12

0,0064 0,0636 With 95% confidence it can be concluded that the mean amount of all Apple returns is between $0,0064 and

$0,0636.

-0,0032 0,037 With 95% confidence it can be concluded that the mean amount of all

Starbucks returns is between $-0,0032and $0,037.

13


-0,0118 0,0428 With 95% confidence it can be concluded that the mean amount of all Ebay returns is between $ -0,0118 and

$0,0428.

Due to the fact that there are no negative mean amounts, it is reasonable to conclude that Apple returns are the most advantageous.

The higher the return the higher is capital gain(see Appendix4) for stockholder, therefore by checking the mean amount of returns it is possible to evaluate how profitable the stock is. Hypothesis test ca be used to estimate the amount of the mean. In given case null hypothesis test will be used to check whether mean amount of returns is 0.

So the null hypothesis is that mean=0. (H0 : =$0), according toμ

alternative hypothesis, the mean ≠0, respectively (H0 : μ≠ $0).

n=59 ; n-sample size

=0,05 -level of significance (0,05 is 5% which exceed 95% confidenceα α interval).

Since population standard deviation( ) σ is unknown, t distribution and t STAT test are used. T-test can be used only uder assumption that the log returns population(of every stock) is normally distrubuted and the sampling distribution of the mean follows normal distribution with 59-1=58 degrres of freedom. The areas in rejection regions of the t-distribution left and right tails are both 0,025. According to t-table (see Appendix 3) the critical values are ± 2,0017.

⇒ Reject H0 if t STAT <

−tα /2=-2 , 0017

or H0 if

t STAT > tα /2=+2, 0017

If not, fail to reject H0 .

14


Figure 13

t-Test for the Hypothesis of the Mean for Apple Returns

Null Hypothesis μ= 0

Level of Significance 0,05

Sample Size 59

Sample Mean 0,0345


Intermediate Calculations

Standard Error of the Mean 0,0143

Degrees of Freedom 58

t-Test Statistic 2,4168

Two-Tail Test

Lower Critical Value -2,0017

Upper Critical Value 2,0017

p-value 0,0188

Reject the null hypothesis ⇒ μ¹ 0

Figure 14

t-Test for the Hypothesis of the Mean for Starbucks Returns



Sample Size 59

Sample Mean 0,0171

15







Two-Tail Test



p-value 0,0973

Do not reject the null hypothesis ⇒ μ=0

Figure 15

t-Test for the Hypothesis of the Mean for Ebay Returns



Sample Size 59

Sample Mean 0,0155






Two-Tail Test


16



p-value 0,2602

Do not reject the null hypothesis ⇒ μ=0

17


Appendix

Appendix1:

The Cumulative Standardized Normal Distribution

(Entry represents area under the cumulative standartized normal distribution from -∞ to Z )

z .00 .01 .02 .03 .04 .05 .06 .07 .08 .09

-3.4 .0003 .0003 .0003 .0003 .0003 .0003 .0003 .0003 .0003 .0002

-3.3 .0005 .0005 .0005 .0004 .0004 .0004 .0004 .0004 .0004 .0003

-3.2 .0007 .0007 .0006 .0006 .0006 .0006 .0006 .0005 .0005 .0005

-3.1 .0010 .0009 .0009 .0009 .0008 .0008 .0008 .0008 .0007 .0007

-3 .0013 .0013 .0013 .0012 .0012 .0011 .0011 .0011 .0010 .0010

-2.9 .0019 .0018 .0018 .0017 .0016 .0016 .0015 .0015 .0014 .0014

-2.8 .0026 .0025 .0024 .0023 .0023 .0022 .0021 .0021 .0020 .0019

-2.7 .0035 .0034 .0033 .0032 .0031 .0030 .0029 .0028 .0027 .0026

-2.6 .0047 .0045 .0044 .0043 .0041 .0040 .0039 .0038 .0037 .0036

-2.5 .0062 .0060 .0059 .0057 .0055 .0054 .0052 .0051 .0049 .0048

-2.4 .0082 .0080 .0078 .0075 .0073 .0071 .0069 .0068 .0066 .0064

-2.3 .0107 .0104 .0102 .0099 .0096 .0094 .0091 .0089 .0087 .0084

-2.2 .0139 .0136 .0132 .0129 .0125 .0122 .0119 .0116 .0113 .0110

-2.1 .0179 .0174 .0170 .0166 .0162 .0158 .0154 .0150 .0146 .0143

-2 .0228 .0222 .0217 .0212 .0207 .0202 .0197 .0192 .0188 .0183

-1.9 .0287 .0281 .0274 .0268 .0262 .0256 .0250 .0244 .0239 .0233

-1.8 .0359 .0351 .0344 .0336 .0329 .0322 .0314 .0307 .0301 .0294

-1.7 .0446 .0436 .0427 .0418 .0409 .0401 .0392 .0384 .0375 .0367

-1.6 .0548 .0537 .0526 .0516 .0505 .0495 .0485 .0475 .0465 .0455

-1.5 .0668 .0655 .0643 .0630 .0618 .0606 .0594 .0582 .0571 .0559

-1.4 .0808 .0793 .0778 .0764 .0749 .0735 .0721 .0708 .0694 .0681

-1.3 .0968 .0951 .0934 .0918 .0901 .0885 .0869 .0853 .0838 .0823

-1.2 .1151 .1131 .1112 .1093 .1075 .1056 .1038 .1020 .1003 .0985

18


-1.1 .1357 .1335 .1314 .1292 .1271 .1251 .1230 .1210 .1190 .1170

-1 .1587 .1562 .1539 .1515 .1492 .1469 .1446 .1423 .1401 .1379

-0.9 .1841 .1814 .1788 .1762 .1736 .1711 .1685 .1660 .1635 .1611

-0.8 .2119 .2090 .2061 .2033 .2005 .1977 .1949 .1922 .1894 .1867

-0.7 .2420 .2389 .2358 .2327 .2296 .2266 .2236 .2206 .2177 .2148

-0.6 .2743 .2709 .2676 .2643 .2611 .2578 .2546 .2514 .2483 .2451

-0.5 .3085 .3050 .3015 .2s981 .2946 .2912 .2877 .2843 .2810 .2776

-0.4 .3446 .3409 .3372 .3336 .3300 .3264 .3228 .3192 .3156 .3121

-0.3 .3821 .3783 .3745 .3707 .3669 .3632 .3594 .3557 .3520 .3483

-0.2 .4207 .4168 .4129 .4090 .4052 .4013 .3974 .3936 .3897 .3859

-0.1 .4602 .4562 .4522 .4483 .4443 .4404 .4364 .4325 .4286 .4247

0.0 .5000 .4960 .4920 .4880 .4840 .4801 .4761 .4721 .4681 .4641

0.1 .5000 .5040 .5080 .5120

.5160

.5199 .5239

.5279

.5319 .5359

0.2 .5398 .5438 .5478 .5517 .5557 .5596 .5636 .5675 .5714 .5753

0.3 .5793 .5832 .5871 .5910 .5948 .5987 .6026 .6064 .6103 .6141

0.4 .6179 .6217 .6255 .6293 .6331 .6368 .6406 .6443 .6480 .6517

0.5 .6554 .6591 .6628 .6664 .6700 .6736 .6772 .6808 .6844 .6879

0.6 .6915 .6950 .6985 .7019 .7054 .7088 .7123 .7157 .7190 .7224

0.7 .7257 .7291 .7324 .7357 .7389 .7422 .7454 .7486 .7517 .7549

0.8 .7580 .7611 .7642 .7673 .7704 .7734 .7764 .7794 .7823 .7852

0.9 .7881 .7910 .7939 .7967 .7995 .8023 .8051 .8078 .8106 .8133

1 .8159 .8186 .8212 .8238 .8264 .8289 .8315 .8340 .8365 .8389

1.1 .8413 .8438 .8461 .8485 .8508 .8531 .8554 .8577 .8599 .8621

1.2 .8643 .8665 .8686 .8708 .8729 .8749 .8770 .8790 .8810 .8830

1.3 .8849 .8869 .8888 .8907 .8925 .8944 .8962 .8980 .8997 .9015

1.4 .9032 .9049 .9066 .9082 .9099 .9115 .9131 .9147 .9162 .9177

1.5 .9192 .9207 .9222 .9236 .9251 .9265 .9279 .9292 .9306 .9319

1.6 .9332 .9345 .9357 .9370 .9382 .9394 .9406 .9418 .9429 .9441

1.7 .9452 .9463 .9474 .9484 .9495 .9505 .9515 .9525 .9535 .9545

1.8 .9554 .9564 .9573 .9582 .9591 .9599 .9608 .9616 .9625 .9633

1.9 .9641 .9649 .9656 .9664 .9671 .9678 .9686 .9693 .9699 .9706

19


2 .9713 .9719 .9726 .9732 .9738 .9744 .9750 .9756 .9761 .9767

2.1 .9772 .9778 .9783 .9788 .9793 .9798 .9803 .9808 .9812 .9817

2.2 .9821 .9826 .9830 .9834 .9838 .9842 .9846 .9850 .9854 .9857

2.3 .9861 .9864 .9868 .9871 .9875 .9878 .9881 .9884 .9887 .9890

2.4 .9893 .9896 .9898 .9901 .9904 .9906 .9909 .9911 .9913 .9916

2.5 .9918 .9920 .9922 .9925 .9927 .9929 .9931 .9932 .9934 .9936

2.6 .9938 .9940 .9941 .9943 .9945 .9946 .9948 .9949 .9951 .9952

2.7 .9953 .9955 .9956 .9957 .9959 .9960 .9961 .9962 .9963 .9964

2.8 .9965 .9966 .9967 .9968 .9969 .9970 .9971 .9972 .9973 .9974

2.9 .9974 .9975 .9976 .9977 .9977 .9978 .9979 .9979 .9980 .9981

3 .9981 .9982 .9982 .9983 .9984 .9984 .9985 .9985 .9986 .9986

3.1 .9987 .9987 .9987 .9988 .9988 .9989 .9989 .9989 .9990 .9990

3.2 .9990 .9991 .9991 .9991 .9992 .9992 .9992 .9992 .9993 .9993

3.3 .9993 .9993 .9994 .9994 .9994 .9994 .9994 .9995 .9995 .9995

3.4 .9995 .9995 .9995 .9996 .9996 .9996 .9996 .9996 .9996 .9997

3.5 .9997 .9997 .9997 .9997 .9997 .9997 .9997 .9997 .9997 .9998

Source: file:///Users/macuser/Desktop/Z%20Table.webarchive

Appendix 2:

“The coefficient of determination, r 2 (0 < r 2 < 1) represents the proportion of the variance of one variable that is predictable from the other variable. It represents the percent of the data that is the closest to the line of best fit.”

Source: http://mathbits.com/Mathbits/TISection/Statistics2/correlation.htm

Appendix 3:

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http://mathbits.com/Mathbits/TISection/Statistics2/correlation.htm


“Where is a critical value corresponding to an upper-tail probability of from the t distribution with n-1 degrees of freedom.” (A.Darnell p.652)

Degrees of Feedom

Critical values of t for 58 degrees of freedom

Cumulative Probabilities

0,75 0,90 0,95 0,975 0,99 0,995

0,25 0,10 0,05 0,025 0,01 0,005

58 0,6787 1,2963 1,6716 2,0017 2,3924 2,6633

Appendix 4:

“Capital Gains is a profit that results from investments into a capital asset. It is the difference between a higher selling price and a lower purchase price, resulting in a financial gain for the seller.”

Source: Sullivan, arthur; Steven M. Sheffrin (2003). Economics: Principles in action. Upper Saddle River, New Jersey 07458: Pearson Prentice Hall. pp. 268, 508

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Bibliography

1. Lecture notes;

2. Pr. A.Darnell, Dr. M. Lau, Quantitative Methods For Finance, second edition, 2009, Pearson Education Limited, Great Britain;

3. Steven M. Sheffrin (2003). Economics: Principles in action. Upper Saddle River, New Jersey 07458: Pearson Prentice Hall. pp. 268, 508

4. file:///Users/macuser/Desktop/Z%20Table.webarchive

5. http://mathbits.com/Mathbits/TISection/Statistics2/correlation.htm

6.

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http://mathbits.com/Mathbits/TISection/Statistics2/correlation.htm

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