research paper c - 1530 - 13-05-15

ECF 6102 - Quantitative Skills for Business

Tutorial / Research Paper CWednesday 13 May

15:30

ML14:114

Andrew Dash

What is Beta Coefficient?

• In finance evaluations the “beta coefficient” of a share is often considered a measure of the stocks volatility (or in financial terms

“risk”).

• Shares with beta coefficients greater than 1 generally bear greater risk, and hence more volatility, than the overall market whereas

• Shares with beta coefficients less than 1 generally are considered less risky (or less volatile) than the overall market.

• From previous studies it has been established that the “beta coefficient” has been normally distributed.

Research SituationImagine we take a random sample of 15 technology shares at the end

of 2014.

The mean and standard deviation of the beta coefficients for these data are calculated as: &

Other Solution Essentials

Previous studies have shown that the “beta coefficient” has been normally distributed.

Part ASet up the appropriate null and

alternate hypotheses to test whether the average technology

shares are riskier than the market as a whole.

One or Two tailed?• Keywords in the question are‘…riskier than the market as a

whole’.

• In other words, is the average beta coefficient of our sample greater than the beta coefficient of the market as a whole?

• Therefore, it is a one tailed test.

𝑛=15𝑋=1.23𝑠=0.37𝛼=0.10𝜇=1

The Golden Rule𝑋>𝜇=𝑈𝑝𝑝𝑒𝑟 𝑇𝑎𝑖𝑙 𝑇𝑒𝑠𝑡𝑋<𝜇=𝐿𝑜𝑤𝑒𝑟 𝑇𝑎𝑖𝑙𝑇𝑒𝑠𝑡

• As (1.23) is greater than (1), it is an upper tailed test.

The null & alternate hypotheses

• Setup the alternate hypothesis () first.

• The alternate hypothesis is setup to answer the question posed.

• Therefore:

𝑛=15𝑋=1.23𝑠=0.37𝛼=0.10𝜇=1

Part BUsing your sampling decision

tree, establish the appropriate test statistic and rejection region

for the test using alpha = 0.10.

Sampling Decision Tree

𝑛=15𝑋=1.23𝑠=0.37𝛼=0.10𝜇=1

(M Waring , personal communication, March 2015)

Critical Values of t

𝑛=15𝑋=1.23𝑠=0.37𝛼=0.10𝜇=1𝑡𝑐𝑣=+1.3450(M Waring , personal communication, March 2015)

Part CTest the hypothesis in Part A

The Decision Rule• Our hypothesis:

• Decision Rule: Reject if , otherwise do not reject.

Do not reject H0

0

Reject H0

tα=0.10

𝑡𝑐𝑣=+1.3450

𝑛=15𝑋=1.23𝑠=0.37𝛼=0.10𝜇=1𝑡𝑐𝑣=+1.3450

Image adapted from example provided by M Waring (personal communication, 30 March 2015)

Testing our hypothesis• Our test statistic:

Do not reject H0

0

Reject H0

t𝑡𝑐𝑣=+1.3450

𝑡𝑡𝑒𝑠𝑡=+2.4075

𝑡𝑡𝑒𝑠𝑡=+2.4075𝑛=15𝑋=1.23𝑠=0.37𝛼=0.10𝜇=1𝑡𝑐𝑣=+1.3450


Our Conclusion• Since , we reject as there is enough evidence to conclude

that the average technology shares are riskier than the market as a whole.

Null Hypothesis m= 1Level of Significance 0.1Sample Size 15Sample Mean 1.23Sample Standard Deviation 0.37

Standard Error of the Mean 0.0955Degrees of Freedom 14t Test Statistic 2.4075

Upper-Tail Test Calculations AreaUpper Critical Value 1.3450p -Value 0.0152

Reject the null hypothesis

Data

Intermediate Calculations

t Test for Hypothesis of the Mean

𝑡𝑡𝑒𝑠𝑡=+2.4075𝑛=15𝑋=1.23𝑠=0.37𝛼=0.10𝜇=1𝑡𝑐𝑣=+1.3450

Part DUse PhStat to determine the approximate

p-value associated with this test.

What does the p-value theoretically mean?

PhStat and the p-Value

• “The p-value is the probability of getting a test statistic equal to or more extreme than the sample result, given that the null hypothesis, is true. The p-value is also known as the observed level of significance” (Berenson, Levine, & Krehbiel, 2012).

𝑡𝑡𝑒𝑠𝑡=+2.4075𝑛=15𝑋=1.23𝑠=0.37𝛼=0.10𝜇=1𝑡𝑐𝑣=+1.3450

PhStat and the p-Value• The strength of the decision concerning H0 is

found by comparing the p-value to the alpha () level.

• If p-Value is low, must go…

𝑡𝑡𝑒𝑠𝑡=+2.4075𝑛=15𝑋=1.23𝑠=0.37𝛼=0.10𝜇=1𝑡𝑐𝑣=+1.3450





Data



Our Data• p-Value (0.0152) < (0.1).

• Therefore we reject .

PhStat and the p-Value• The p-Value is an important result because it measures the

amount of statistical evidence that supports the alternative hypothesis.

• A small p-Value indicates that there is ample evidence to support the alternative hypothesis.

• A large p-Value indicates that there is little evidence to support the alternative hypothesis.

𝑡𝑡𝑒𝑠𝑡=+2.4075𝑛=15𝑋=1.23𝑠=0.37𝛼=0.10𝜇=1𝑡𝑐𝑣=+1.3450

Part EIf we had tested this same

hypothesis using the Z distribution at alpha = 0.10 would you have

drawn the same conclusion?

Finding the value of &

𝑡𝑡𝑒𝑠𝑡=+2.4075𝑛=15𝑋=1.23𝑠=0.37𝛼=0.10𝜇=1𝑡𝑐𝑣=+1.3450

𝑍 𝑐𝑣=+1.28

𝑍𝑡𝑒𝑠𝑡=𝑋−𝜇𝑠𝑋

= 1.23−10.09553

=+2.4075


Testing our hypothesis• Our decision rule:Reject if , otherwise do

not reject

Do not reject H0

0

Reject H0

t𝑍 𝑐𝑣=+1.28

𝑍𝑡𝑒𝑠𝑡=+2.4075

𝑍𝑡𝑒𝑠𝑡=+2.4075𝑛=15𝑋=1.23𝑠=0.37𝛼=0.10𝜇=1𝑍 𝑐𝑣=+1.28


Our Conclusion• Since , therefore we reject as there is enough evidence to

conclude that the average technology shares are riskier than the market as a whole.

𝑍𝑡𝑒𝑠𝑡=+2.4075𝑛=15𝑋=1.23𝑠=0.37𝛼=0.10𝜇=1𝑍 𝑐𝑣=+1.28

Null Hypothesis m= 1Level of Significance 0.1Population Standard Deviation 0.37Sample Size 15Sample Mean 1.23

Standard Error of the Mean 0.0955Z Test Statistic 2.4075

Upper-Tail TestUpper Critical Value 1.2816p -Value 0.0080


Data


Z Test of Hypothesis for the Mean

Part FDevelop a 95% confidence

interval for the “beta coefficient”.

Solution Essentials

𝑛=15𝑥=1.23𝑠=0.37(1−𝛼 )=95 %


Critical Values of t

Therefore, = 2.1448

𝑛=15𝑥=1.23𝑠=0.37(1−𝛼 )=95 %𝑡 0.025; 14=±2.1448


• The formula for the confidence interval estimation of the mean is:

The Calculations.5000 .5000

0.95(1-)

.4750 .4750

/2= 0.025 /2=0.025

s x -2.1448 +2.1448 s x

t-2.1448 0 2.1448

𝑛=15𝑥=1.23𝑠=0.37(1−𝛼 )=95 %𝑡 0.025; 14=±2.1448


Therefore, we can be 95% confident that the mean beta coefficient lies between 1.0251 and 1.4349

The Calculations

1.23−2.1448 ( 0.37

√15 )≤𝜇≤1.23+2.1448( 0.37

√15 )

𝑋− 𝑡𝛼 /2( 𝑠√𝑛 )≤𝜇≤𝑋+𝑡𝛼/2( 𝑠

√𝑛 )

1 .0251≤𝜇≤1.4349

1.23−0.2049≤𝜇 ≤1.23+0.2049

𝑛=15𝑥=1.23𝑠=0.37(1−𝛼 )=95 %𝑡 0.025; 14=±2.1448

Confidence Interval Estimate for the Mean

DataSample Standard Deviation 0.37Sample Mean 1.23Sample Size 15Confidence Level 95%

Standard Error of the Mean 0.095533589Degrees of Freedom 14t Value 2.1448Interval Half Width 0.2049

Interval Lower Limit 1.0251Interval Upper Limit 1.4349


Confidence Interval

Just in case…

1 .0251≤𝜇≤1.4349

𝑛=15𝑥=1.23𝑠=0.37(1−𝛼 )=95 %𝑡 0.025; 14=±2.1448

Part GUse PhStat software to conduct a

test to determine if the variance of the shares beta value differs from

0.15 at alpha = 0.05. Show this printout

Chi-Square Test of Variance

DataNull Hypothesis s^2= 0.15Level of Significance 0.05Sample Size 15Sample Standard Deviation 0.37

Degrees of Freedom 14Half Area 0.025Chi-Square Statistic 12.7773

Two-Tail TestLower Critical Value 5.6287Upper Critical Value 26.1189p -Value 0.4559

Do not reject the null hypothesis


Chi-Square Test of Variance

Part HUse PhStat output to confirm your

answer to a) – d) above.

The Confirmation

Since , we reject as there is enough evidence to conclude that the average technology shares are riskier than the market as a whole.





Data



𝑡𝑡𝑒𝑠𝑡=+2.4075𝑛=15𝑋=1.23𝑠=0.37𝛼=0.10𝜇=1𝑡𝑐𝑣=+1.3450

References• Mark L. Berenson, T. C. K. D. M. L. (2012). Basic

Business Statistics: Concepts and Applications (E. Svendsen Ed. Vol. 12): Pearson Education, Inc.

research paper c - 1530 - 13-05-15

Documents

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t test statistic

alternate hypothesis

average beta coefficient

upper tailed test

reject h0 t

appropriate test statistic

beta coefficients greater