Quantitative Methods
Homework 2
Due on Monday, NOVEMBER 18th, 2024 at 5pm. (10% of the final grade)
Round up to SIXth decimal places.
PartI: Use the monthly return of S&P sample in EXCEL. [20 points in total]
1. (10 points) Use Z-approach, construct the confidence interval of the sample mean
a. if confidence level is 90%
b. if confidence level is 95%
c. if confidence level is 99%
d. Briefly discuss the difference in the size of the three intervals.
2. (2 points) Build a test and determine whether or not to reject the null that
S&P has a monthly return no more than 0.3% given the level of significance is 0.05 and 0.01.
3. (2 points) Build a test and determine whether or not to reject the null that the
monthly return of S&P has a standard deviation is smaller than 5% given the level of significance is 0.05 and 0.01. Assume the monthly returns follow a normal distribution and the observations are independent.
Part II: Use the four-asset sample in EXCEL [30 points in total]
. Assume normally distributed and independent populations.
. level of significance=0.05
1. (5 points) Build a test and state any two stocks can be considered as
(linearly) correlated.
2. (5 points) Build a test and determine whether or not to reject the null
hypothesis that Amazon(AMZN) and Apple(AAPL) are equally risky.
3. (5 points) Based on the conclusion of question 2, build a test and determine
whether the difference between the stock returns of Amazon(AMZN) and Apple(AAPL) is statistically significant.
4. (5 points) Build a test and determine whether or not to reject the null
hypothesis that Amazon(AMZN) is not as risky as Microsoft(MSFT).
5. (5 points) Based on the conclusion of question 4, build a test and determine
whether or not to reject the null hypothesis that the stock returns of
Amazon(AMZN) is greater than or equal to the returns of Microsoft(MSFT).
6. (5 points) Build a paired test regarding the null in question 5, is the result consistent?
Part III: [14 points] A stock is priced at $1.00 and follows a one-period binomial process with an up move that equals $1.03 and a down move that equals $0.98. If one hundred Bernoulli trials are conducted, and the mean terminal stock price is $1.01.
a. [2 points] Calculate the probability of an up move (p).
b. [3 points] Calculate the variance of the stock price using the probability calculated in Part a.
c. [3 points] Describe the distribution of this sample mean.
d. [6 points] Construct a 90% of confidence interval of the sample mean (use the more conservative approach).
Part IV: Linear Regression [36 points in total]
An analyst estimates a simple regression to investigate the effect of the debt ratio on a company’s short interest ratio. She calculates the short interest ratio (the ratio of short interest to average daily share volume, expressed in days) for 50 companies as of the end of 2016 and compares this ratio with the companies’ debt ratio (the ratio of total liabilities to total assets, expressed in decimal form). The statistics are given as
Given the level of significance is 0.05
a. (5pts) What is the correlation between the debt ratio and the short interest
ratio? Test the null hypothesis that the correlation is equal to zero against the alternative hypothesis that it is not equal to zero.
b. (5pts) Based on the previous study, the coefficient of the debt ratio is mostly smaller than -3. Build an appropriate test and determine whether or not to reject the null hypothesis.
c. (5pts) State the null and alternative hypothesis of the F-statistics in ANOVA. Without the F-table, briefly explain whether or not to reject the null hypothesis.
d. (6pts) A debt ratio above 40% is considered to be critical. Compute the 95% prediction interval for short interest ratio if the debt ratio equals 40%.