Case Study 3
According to the capital Asset pricing model (CAPM), the risk associated with a capital asset is proportional to the slope obtaining by regressing the asset’s past returns with the corresponding returns of the average portfolio called the market portfolio. (The return of the market portfolio represents the return earned by the average investor. It is a weighted average of the returns from all the assets in the market). The larger the slope of an asset, the larger is the risk associated with that asset. A of 1.00 represents average risk. The return from an electronics firm’s stock and the corresponding returns for the market portfolio for the past 15 years are given below. Market Return (%)Stock’s Return (%)

1.Carry out the regression and find the for the stock. What is the regression equation?

2.Does the value of the slope indicate that the stock has above average risk? (For the purpose of this case assume that the risk is average if the slope is in the range , below average if it is less than 0.9 and above average if it is more than 1.1).

3.Give a 95% confidence interval for this . Can we say the risk is above average with 95% confidence?

4.If the market portfolio return for the current year is 10%, what is the stock’s return predicted by the regression equation? Give a 95% confidence interval for this prediction.

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Case Study 1.
A company supplies pins in bulk to a customer. The company uses an automatic lathe to produce the pins. Due to many causes- vibration, temperature, wear and tear and the like-the length of the pins made by the machines are normally distributed with a mean of 1.012 inches and a standard deviation of...

...CAPMCAPM provides a framework for measuring the systematic risk of an individual security and relate it to the systematic risk of a well-diversified portfolio. The risk of individual securities is measured by β (beta). Thus, the equation for security market line (SML) is:
E(Rj) = Rf + [E(Rm) – Rf] βj
(Equation 1)
Where E(Rj) is the expected return on security j, Rf the risk-free rate of interest, Rm the expected return on the market portfolio and βj the undiversifiable risk of security j. βj can be measured as follows:
βj = Cov (Rj, Rm)
Var (Rm)
= σj σm Cor jm
σ2 m
= σj Cor jm
σm
(Equation 2)
In terms of Equation 2, the undiversifiable (systematic) risk (βj) of a security is the product of its standard deviation (σj) and its correlation with the market portfolio divided by the market portfolio’s standard deviation. It can be noted that if a security is perfectly positively correlated with the market portfolio, then CML totally coincides with SML.
Equation 1 shows that the expected rate of return on a security is equal to a risk-free rate plus the risk-premium. The risk-premium equals to the difference between the expected market return and the risk-free rate multiplied by the security’s beta. The risk premium varies directly with systematic risk measured by beta.
The figure above illustrates the security market line. For a given amount of systematic risk (β), SML shows the...

...REGRESSION
1. Prediction Equation
2. Sample Slope
SSx= ∑ x2- (∑ x)2/n
SSxy= ∑ xy- ∑ x*∑ y/n
3. Sample Y Intercept
4. Coeff. Of Determination
5. Std. Error of Estimate
6. Standard Error of 0 and
1
7. Test Statistic
8. Confidence Interval of 0 and 1
9. Confidence interval for mean value of Y given x
10. Prediction interval for a randomly chosen value of Y given x
11. Coeff. of Correlation
12. Adjusted R2
13. Variance Inflation Factor
14. Beta Weights
15. Partial F Test
SSER - sum of squares of error of reduced model SSEF - sum of squares of error of full model
r – no. of variables dropped from full model.
16. Outliers
Measure | Potential Outliers |
Standardized residual, Studentized residual | > 3 (3 sigma level) |
Mahalanobis distance | > Critical chi-square value with df = number of explanatory variables(Outliers in independent variable) |
Cook’s distance | > 1 implies potential outlier |
Leverage values | > 2(k+1)/n, then the point is influential (k is the number of independent variables and n is the sample size) |
SDFBeta | > 2/n |
SDFFit | |
17. Mahalanobis Distance
Mi = (Xi – X)2/ Sx
18. Cook’s Distance
Di =
∑j (Yj – Yj(i))2/k x MSE
19. Durbin Watson Test
Durbin Watson value close to 2 implies no auto-correlation
Durbin Watson value close to 0 implies positive auto-correlation...

...of 1000 flights and proportions of three routes in the sample. He divides them into different sub-groups such as satisfaction, refreshments and departure time and then selects proportionally to highlight specific subgroup within the population. The reasons why Mr Kwok used this sampling method are that the cost per observation in the survey may be reduced and it also enables to increase the accuracy at a given cost.
TABLE 1: Data Summaries of Three Routes
Route 1
Route 2
Route 3
Normal(88.532,5.07943)
Normal(97.1033,5.04488)
Normal(107.15,5.15367)
Summary Statistics
Mean
88.532
Std Dev
5.0794269
Std Err Mean
0.2271589
Upper 95% Mean
88.978306
Lower 95% Mean
88.085694
N
500
Sum
44266
Summary Statistics
Mean
97.103333
Std Dev
5.0448811
Std Err Mean
0.2912663
Upper 95% Mean
97.676525
Lower 95% Mean
96.530142
N
300
Sum
29131
Summary Statistics
Mean
107.15
Std Dev
5.1536687
Std Err Mean
0.3644194
Upper 95% Mean
107.86862
Lower 95% Mean
106.43138
N
200
Sum
21430
From the table above, the total number of passengers for route 1 is 44,266, route 2 is 29,131 and route 3 is 21,430 and the total numbers of passengers for 3 routes are 94,827.
Although route 1 has the highest number of passengers and flights but it has the lowest means of passengers among the 3 routes. From...

...Model commonly known as CAPM defines the relationship between risk and the return for individual securities. CAPM was first published by William Sharpe in 1964. CAPM extended “Harry Markowitz’s portfolio theory” to include the notions of specific and systematic risk. CAPM is a very useful tool that has enabled financial analysts or the independent investors to evaluate the risk of a specific investment while at the same time setting a specific rate of return with respect to the amount of the risk of a portfolio or an individual investment. The CAPM method takes into consideration the factor of time and does not get wrapped up over by the systematic risk factors, which are rarely controlled. In this research paper, I will look at the implications of CAPM in the light of the recent development. I will start by attempting to explain and discuss the various assumptions of the CAPM. Secondly, I will discuss the main theories and moreover, the whole debate that is surrounding this area more specifically through the various critics of the CAPM assumptions.
When Sharpe (1964) and Lintner (1965) proposed CAPM, it was majorly seen as the leading tool in measuring and determining whether an investment will yield negative or positive return. The model attempts to expound the relationship between expected reward/return and the investment risk of very risky...

...Linear Regression deals with the numerical measures to express the relationship between two variables. Relationships between variables can either be strong or weak or even direct or inverse. A few examples may be the amount McDonald’s spends on advertising per month and the amount of total sales in a month. Additionally the amount of study time one puts toward this statistics in comparison to the grades they receive may be analyzed using theregression method. The formal definition of Regression Analysis is the equation that allows one to estimate the value of one variable based on the value of another.
Key objectives in performing a regression analysis include estimating the dependent variable Y based on a selected value of the independent variable X. To explain, Nike could possibly measurer how much they spend on celebrity endorsements and the affect it has on sales in a month. When measuring, the amount spent celebrity endorsements would be the independent X variable. Without the X variable, Y would be impossible to estimate. The general from of the regression equation is Y hat "=a + bX" where Y hat is the estimated value of the estimated value of the Y variable for a selected X value. a represents the Y-Intercept, therefore, it is the estimated value of Y when X=0. Furthermore, b is the slope of the line or the average change in Y hat for each change of one unit in the independent...

...Trajico, Maria Liticia D.
BSEd III-A2
REFLECTION
The first thing that puffs in my mind when I heard the word STATISTIC is that it was a very hard subject because it is another branch of mathematics that will make my head or brain bleed of thinking of how I will handle it. I have learned that statistic is a branch of mathematics concerned with the study of information that is expressed in numbers, for example information about the number of times something happens. As I examined on what the statement says, the phrase “number of times something happens” really caught my attention because my subconscious says “here we go again the non-stop solving, analyzing of problems” and I was right. This course of basic statistic has provided me with the analytical skills to crunch numerical data and to make inference from it. At first I thought that I will be alright all along with this subject but it seems that just some part of it maybe it is because I don’t pay much of my attention to it but I have learned many things. I have learned my lesson.
During our every session in this subject before having our midterm examination I really had hard and bad times in coping up with this subject. When we have our very first quiz I thought that I would fail it but it did not happen but after that, my next quizzes I have taken I failed. I was always feeling down when in every quiz I failed because even though I don’t...

...Regression Analysis: A Complete Example
This section works out an example that includes all the topics we have discussed so far in this chapter.
A complete example of regression analysis.
PhotoDisc, Inc./Getty Images
A random sample of eight drivers insured with a company and having similar auto insurance policies was selected. The following table lists their driving experiences (in years) and monthly auto insurance premiums.
Driving Experience (years) Monthly Auto Insurance Premium
5 2 12 9 15 6 25 16
$64 87 50 71 44 56 42 60
a. Does the insurance premium depend on the driving experience or does the driving experience depend on the insurance premium? Do you expect a positive or a negative relationship between these two variables? b. Compute SSxx, SSyy, and SSxy. c. Find the least squares regression line by choosing appropriate dependent and independent variables based on your answer in part a. d. Interpret the meaning of the values of a and b calculated in part c. e. Plot the scatter diagram and the regression line. f. Calculate r and r2 and explain what they mean. g. Predict the monthly auto insurance premium for a driver with 10 years of driving experience. h. Compute the standard deviation of errors. i. Construct a 90% confidence interval for B. j. Test at the 5% significance level whether B is negative. k. Using α = .05, test whether ρ is different from zero.
Solution a. Based on theory and intuition, we...

...compliments the regular mathematics and therefore both are tested in primary schools. Mathematics is the written application of operation. It teaches students to think clearly, reason well and strategize effectively. Mental Mathematics is the ability to utilise mathematical skills to solve problems mentally. The marks scored by pupils generate statistics which are used by teachers to analyse a student’s performance and development of theories to explain the differences in performance.
The Standard 3 class is where the transition from junior to senior level occurs where teachers expect the transference of concrete to abstract thinking would have occurred.
A common theory by many primary school teachers is ‘Students perform better in Mathematics than Mental math. Mental math is something that has to be developed and involves critical thinking. Mental math requires quick thinking and the student must solve the problem in their minds whereas in regular mathematics, the problem can be solved visually. Therefore, teachers should take these factors into consideration while testing and marking students in these areas.’
In this study, the statistics of 30 students of a standard 3 class of San Fernando Boys’ Government School will be analysed to determine the truth of this theory.
DATA COLLECTION METHODS
Mathematics and mental mathematics marks of term 1 of the class of 2013 were obtained from a Standard 3 teacher of San Fernando Boys’...