Statistics of Business and Finance

This text provides a detailed explanation of hypothesis testing and analysis in business and finance, covering topics such as normality test, t-test, F-test, CAPM model, and more. It includes examples and step-by-step procedures for each test, along with interpretations of the results. The text also discusses the risk and return characteristics of two stocks, Boeing and General Dynamics, and provides insights into their performance based on sample statistics.

STATISTICS OF BUSINESS AND FINANCE
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(1) Returns
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Test for normality of variable has been performed through Jarque- Berra Test. The respective test
statistics is known as JB stat and the relevant formula to find the JB stat is highlighted below.
JB stat = N
6 {(S2 + ( K −3 )2
4 ) }
Where,
N=Number of observations
S=Valueof skew
K=Valueof kurtosis
Step 1: Null and alternative hypotheses
H0: Returns of variable is normally distributed.
H1: Returns of variable is not normally distributed.
Step 2: JB stat
The JB stat has been computed in excel with the help of description statistics and the final table
represents the value of JB stat.
Step 3: Compare with applicable critical value
It can be concluded from JB stat table that for Boeing Returns the JB stat value is significantly
lesser than the critical value which represents that null hypothesis would not be taken for
rejection. Thus, Boeing returns are considered to be normally distributed. It can be concluded
from JB stat table that for General Dynamics Returns the JB stat value is significantly greater
than the critical value which represents that null hypothesis would not be taken for rejection and
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alternative hypothesis would be taken for acceptance. Thus, General Dynamics Returns cannot
be considered to be normally distributed.
(2) Average General Dynamics Returns differs from 2.8% (hypothesis claim)
Step 1: Null and alternative hypotheses
H0: Average General Dynamics Returns is not considered to be different than 2.8%
H1: Average General Dynamics Returns is considered to be different than 2.8%
Step 2: t statistics (z stat would not be used because the population standard deviation is not
given)
Step 3: Degree of freedom
Total number of data points = 60
Degree of freedom = Total number of data points-1 = 59
Step 4: The p value
The p value would be taken here would be two tailed value because the hypothesis test is a two
tailed test.
Inputs: t stat =-3.0279, df =59
The p value = 0.003658
Step 5: Assumption for significance level
Let the significance level = 5%
Step 6: Compare p value with significance level
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It can be concluded from above that p value is significantly lower than significance level which
represents that null hypothesis would be rejected and alternative hypothesis would be accepted.
Thus, average General Dynamics Returns is considered to be different than 2.8%
(3) Risk for Boeing Returns and General Dynamics Returns is different. (hypothesis claim)
Step 1: Null and alternative hypotheses
H0: Risk for Boeing Returns and General Dynamics Returns is not different.
H1: Risk for Boeing Returns and General Dynamics Returns is different.
Step 2: F stat
F stat = 1.5592
Step 3: The p value
The p value would be taken here would be two tailed value because the hypothesis test is a two
tailed test.
The p value = 2 times of one tailed value = (2*0.0453) = 0.09067
Step 4: Assumption for significance level
Let the significance level = 5%
Step 5: Compare p value with significance level
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It can be concluded from above that p value is significantly higher than significance level which
represents that null hypothesis would not be rejected and alternative hypothesis would be not
accepted. Thus, risk for Boeing Returns and General Dynamics Returns is not different.
(4) Returns for Boeing Returns and General Dynamics Returns are different. (Hypothesis claim)
Step 1: Null and alternative hypotheses
H0: Returns for Boeing Returns and General Dynamics Returns are not different.
H1: Returns for Boeing Returns and General Dynamics Returns are different.
Step 2: two sample t stat
t stat = 0.0011
Step 3: The p value
The p value would be taken here would be two tailed value because the hypothesis test is a two
tailed test.
The p value = 0.9991
Step 4: Assumption for significance level
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Let the significance level = 5%
Step 5: Compare p value with significance level
It can be concluded from above that p value is significantly higher than significance level which
represents that null hypothesis would not be rejected and alternative hypothesis would be not
accepted. Thus, returns for Boeing Returns and General Dynamics Returns are not different.
The population risk and return of the two stocks i.e. Boeing and General Dynamics (GD) do not
tend to differ in any significant manner as is apparent from the hypothesis test deployed above.
In such circumstance, the only potential option is to rely on the sample statistics with regards to
return and risk. Boeing stock has higher sample returns but the same is achieved at higher risk.
The preferred stock choice would be GD owing to the higher return per unit risk in the sample
selected.
(5) The excess return computation has been performed by deduction of risk free rate (indicated
by treasury yields). The CAPM model computation has been henceforth performed on the basis
of linear regression analysis. The excess market return (S&P index used as proxy) serves as the
predictor variable for estimation of the excess stock return (General Dynamics).
The requisite equation would be captured as stated below.
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GD stock excess returns = -0.08 + 0.836*Excess returns on market
(c) R2 = 0.3808
(d) The GD stock beta had a 95% confidence interval in the range (0.556,1.116).
(6) Hypothesis Testing
H0: β = 0 indicating stock beta is insignificant.
H1: β ≠ 0 indicating stock beta is significant.
For hypothesis testing using confidence interval, the key decision rule is that the underlying null
hypothesis can be rejected only when the value hypothesised does not fall in the confidence
interval. The value hypothesised is 1 which is present in the confidence interval. Hence, no
rejection of null hypothesis can happen.
(7) Test for normality of variable has been performed through Jarque- Berra Test. The respective
test statistics is known as JB stat and the relevant formula to find the JB stat is highlighted
below.
JB stat = N
6 {(S2 + ( K −3 )2
4 ) }
Step 1: Null and alternative hypotheses
H0: Returns of variable is normally distributed.
H1: Returns of variable is not normally distributed.
Step 2: JB stat
The JB stat has been computed in excel with the help of description statistics and has come out
as 23.81
Step 3: It is evident that the above computed JB statistic is higher than the critical value and
hence rejection of null hypothesis is caused.
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PART B: Interpretation
(1) In line with the inferential testing, it would be appropriate to conclude that normality is
observed for stock monthly returns of Boeing stock while non-normality is observed for stock
monthly returns of General Dynamic stock.
In relation to stock, two pivotal characteristics are relates risk (represented by standard deviation)
and returns. This is in line with the theoretical underpinning whereby for an investment with
higher risk, a higher return in expected. For the sample period stock returns, Boeing tend to have
a higher average returns than GD but the associated risk is also higher for Boeing.
(2) For the null hypothesis, the relevant distribution would be student T considering the fact that
the population standard deviation is unknown. Further, the relevant hypothesis testing indicates
that average returns on GD stock are significantly deviant from 2.8%.
(3) The population variances for the monthly stocks returns of Boeing and GD are not different
and can be assumed to be same.
(4) The population returns for the monthly stocks returns of Boeing and GD are not different and
can be assumed to be same. GD emerges as the preferable stock on back of sample
characteristics.
(5) (b) The GD stock as per CAPM is 0.836. This is indicative of the fact that as there is a
change in the excess returns on market index by 1%, there is corresponding 0.836% change in
the excess returns on GD stock.
(c) The R2 value can be interpreted as the given predictor variation i.e. excess returns on market
provides explanation for a paltry 28.08% of the changes in excess returns on GD stock and the
remaining variation is not explained by the given model.
(d) The GD stock beta would lie between 0.556 and 0.116 with a 95% chance.
(6) The beta of the GD stock does not vary significantly from 1 based on the confidence interval
related hypothesis testing and hence it is correct to assume this as a neutral stock.
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(7) The normality of residuals is not satisfied for the CAPM model as the relevant hypothesis test
indicates non-normal distribution.
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