title: “ECOM 2001 Term Project: (GSK, C, BP)”
“`{r setup, include=FALSE}
knitr::opts_chunk$set(echo = TRUE, message= FALSE, warning = FALSE, tidy.opts = list(width.cutoff = 90), tidy = TRUE )
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rm(list=ls(all=TRUE)) # this code clears all output from your Environment
# Include comments in your coding to explain what you are doing.
# You can delete unnecessary comments/hints that I have provided.
# Replace your name, student ID and your three assigned stocks in the YAML.
# Knit frequently to ensure your coding is working and explanations are formatted in the text as you intend.
# packages
library(tidyquant) # for importing stock data
library(tidyverse) # for working with data
library(broom) # for tidying output from various statistical procedures
library(knitr) # for tables
library(kableExtra) # for improving the appearance of tables
library(moments)
library(e1071)
library(ggpubr)
library(onewaytests)
library(corrplot)
# Add any additional packages that you use to this code chunk
# Import the Data (2 points)
“`{r data}
## 1) Import your assigned stocks
## Use the package tidyquant. You may need to install this package first.
## Replace Stock1, Stock2, Stock3 with your assigned stock names (in quotation marks), uncomment the code, and Run
Stocks <-c("GSK", "C","BP") %>%
tq_get(get = “stock.prices”, from = “2000-01-01”)%>%
select(symbol, date, adjusted)
## This is your data set for this project (rename yourDataName to something more descriptive)
## output the first 6 rows of your data frame:
head(Stocks, n = 6 )%>%
kable(caption = “First 6 row of data”)
# The Analysis
## Plot prices over time (3 points)
Plot the prices of each asset over time separately. Succinctly describe in words the evolution of each asset over time. (limit: 100 words for each time series).
“`{r timeSeriesPrices}
ggplot(Stocks, aes(x=date, y=adjusted))+geom_point()+facet_wrap(~symbol)
fig.cap= “Price of Each Asset Over Time”
## Don’t forget to add fig.cap= “Your caption” to the code chunk header.
## facet_wrap() may be useful
## Calculate returns and plot returns over time (4 points)
Calculate the daily percentage returns of each asset using the following formula:
r_t = 100*\ln\Big(\frac{P_t}{P_{t-1}}\Big)
Where $P_t$ is the asset price at time $t$. Then plot the **returns** for each asset over time.
“`{r timeSeriesReturns}
## Hint: you need to add a column to your data frame (yourDataName).
## You can use the mutate() function
## Don’t forget to group_by()
## The lag() function can be used to find the price in the previous date
## Histogram of returns (4 points)
Create a histogram for each of the returns series (explain how you determined the number of bins to use).
## Summary table of returns (4 points)
Report the descriptive statistics **in a single table** which includes the mean, median, variance, standard deviation, skewness and kurtosis for each series. What conclusions can you draw from these descriptive statistics?
“`{r summaryTable}
## Your summary table here. Be sure to format the table appropriately.
## Are average returns significantly different from zero? (5 points)
Under the assumption that the returns of each asset are drawn from an independently and identically distributed normal distribution, are the expected returns of each asset statistically different from zero at the 1\% level of significance? Provide details for **all 5 steps to conduct a hypothesis test**, including **the equation for the test statistic**. Calculate and report all the relevant values for your conclusion and be sure to provide an interpretation of the results.
“`{r sigTests}
## Hint: you can extract specific values from t.test objects using the $
## Eg. using t.test(x,y)$statistic will extract the value of the test statistic.
## Consult the help file for the other values generated by the t.test() function.
## The relevant values are: the t-test method, the estimated mean , the test statistic, whether the test is one or two tailed, the degrees of freedom, and the p-value. (You might wish to present this in a table)\
## Test statistic exceed critical value, we reject the null
## Are average returns different from each other? (6 points)
Assume the returns of each asset are independent from each other. With this assumption, are the mean returns statistically different from each other at the 1\% level of significance? Provide **details for all 5 steps to conduct each of the hypothesis tests** using what your have learned in the unit. Calculate and report all the relevant values for your conclusion and be sure to provide and interpretation of the results. (*Hint: You need to discuss the equality of variances to determine which type of test to use.*)
“`{r multiSigTests}
## Decide on which test is appropriate for testing differences in mean returns
## Hint: Include the results of your supporting test for the differences in variances (include all 5 hypothesis step tests and the equation for the test statistics, and a clear interpretation of the result).
## Hint: http://www.sthda.com/english/wiki/one-way-anova-test-in-r
## So this section has (at least) 2 significance tests.
## Test for equality of variance
## Value (1538.8) exceeds critical value, so we reject the null
## Variances are not equal to each other
## One way ANOVA not assuming equal variances
onewaytests::aov.test(returns ~ symbol, data = Stocks)
## Correlations (2 points)
Calculate and present the correlation matrix of the returns. Discuss the direction and strength of the correlations.
“`{r correlations}
## Include a formatted correlation matrix here
## Hint: http://www.sthda.com/english/wiki/correlation-matrix-a-quick-start-guide-to-analyze-format-and-visualize-a-correlation-matrix-using-r-software
## Testing the significance of correlations (2 points)
Is the assumption of independence of stock returns realistic? Provide evidence (the hypothesis test including **all 5 steps of the hypothesis test** and **the equation for the test statistic**) and a rationale to support your conclusion.
“`{r correlationTest}
## Report the results of tests for statistical significance of the correlations here.
## Hint: http://www.sthda.com/english/wiki/correlation-matrix-a-quick-start-guide-to-analyze-format-and-visualize-a-correlation-matrix-using-r-software
## Advising an investor (12 points)
Suppose that an investor has asked you to assist them in choosing **two** of these three stocks to include in their portfolio. The portfolio is defined by
r = w_1r_1 + w_2r_2
Where $r_1$ and $r_2$ represent the returns from the first and second stock, respectively, and $w_1$ and $w_2$ represent the proportion of the investment placed in each stock. The entire investment is allocated between the two stocks, so $w+1+w_2=1$.
The investor favours the combination of stocks that provides the highest return, but dislikes risk. Thus the investor’s happiness is a function of the portfolio, $r$:
h(r) = \mathbb E(r) – \mathbb Var(r)
Where $\mathbb E(r)$ is the expected return of the portfolio, and $\mathbb Var(r)$ is the variance of the portfolio.^[Note that $\mathbb E(r) = w_1 E(r_1) + w_2 \mathbb E(r_2)$, and $\mathbb Var(r) = w_1^2\mathbb Var(r_1) + w_2^2 \mathbb Var(r_2) + 2w_1w_2 \mathbb Cov (r_1,r_2)$]
Given your values for $\mathbb E(r_1)$, $\mathbb E(r_2)$, $\mathbb Var(r_1)$, $\mathbb Var(r_2)$ and $\mathbb Cov(r_1, r_2)$ which portfolio would you recommend to the investor? What is the expected return to this portfolio?
Provide evidence to support your answer, including all the steps undertaken to arrive at the result. (*Hint: review your notes from tutorial 6 on portfolio optimisation. A complete answer will include the optimal weights for each possible portfolio (pair of stocks) and the expected return for each of these portfolios.)
“`{r optimalPortfolio}
# You can use this section to create a table of your results.
## The impact of financial events on returns (6 points)
Two significant financial events have occurred in recent history. On September 15, 2008 declared bankruptcy and a Global Financial Crisis started. On March 11, 2020 the WHO declared COVID-19 a pandemic. Use linear regression to determine if
a. Any of the stocks in your data exhibit positive returns over time.
b. Either of the two events had a significant impact on returns.
Report the regression output for each stock and interpret the results to address these two questions. How would you interpret this information in the context of your chosen portfolio?
“`{r regression}
## Add a column to your returns data set.
## This is a factor variable with three levels:
## ‘ ‘ for the date 2008-09-15,
## ‘Pandemic’ for the date 2020-03-11, and
## ‘BAU’ (Business as usual) for all other dates.
## Then run a regression analysis to determine whether returns to each stock are increasing over time and if the events had and statistically significant impact on the returns of each stock.
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