library(nimble)From the course website, download the historical stock prices of Walmart for the last 252 (or so) trading days, i.e., for a one-year period from our course website. Do the same for IBM.
(5 points) Draw the time-plot of the evolution of the closing stock prices (not the adjusted) for both of the stocks on the same coordinate system. You do not need to put the calendar days on the horizontal axis, but you do need to label your axes and give your time-plot a title indicating the dates. Make sure that you plot the two trajectories in different colors indicating in the text which color corresponds to which company. Include the legend in your plot.
The simple daily return of the stock over a day indexed by \(t\) is defined as \[ \frac{\text{price at end of day $t$}-\text{price at end of day $(t-1)$}}{\text{price at end of day $(t-1)$}} \] (5 points) Construct the vector of simple daily returns over the last year for both stocks. With \(R\) denoting the daily simple return, the daily volatility of the stock is defined as the standard deviation of \(R\). Assuming independent, identically distributed daily returns, what is the daily volatility point estimate for the daily volatility of both stocks?
(10 points) If you wanted to study the relationship between the returns of the two stocks, which plot would you create? What (if anything) can you say after looking at the plot?
(5 points) If you wanted to provide one single, unitless quantity which provided you a measure of association between the returns of Walmart and IBM, which value would you report?
(5 points) You create a portfolio in which half your wealth is maintained in the Walmart stock and the remaining half of your wealth is kept in the IBM stock. What is your estimate of the daily volatility of this portfolio?
(25 points) You create a portfolio in which the
weight \(w\) of your wealth is
maintained in the Walmart stock and the remainder of your
wealth is kept in the IBM stock. Create a function in
R which calculates the daily volatility of this portfolio
as it depends on \(w\)? Plot the
portfolio’s volatility as a function of \(w\) for \(w\) ranging from \(-2\) to \(2\). Yes, it’s perfectly acceptable
to have negative weights; if you want to know why, come ask in office
hours. In the same plot, add two horizontal lines at the values
of the daily volatilities of Walmart and IBM. Now,
review the optimal two-stock portfolio work we did in the first week of
classes. Add the horizontal line corresponding to the optimal portfolio
whose Walmart weight is \(\hat w=\hat
\alpha\) given by Hastie and Tibshirani. Add the vertical line
corresponding to the optimal \(\hat
w\).
From the course website, download the historical prices for the NASDAQ Composite index for the maximum time period and at the monthly frequency available on our course website.
(5 points) Use the above definitions of returns, changing the period length to a month (rather than a day). Assuming, as usual independent, identically distributed monthly returns, what is your point estimate for the mean monthly return of NASDAQ?
(5 points) Assuming, as usual independent, identically distributed monthly returns, what is your point estimate for the monthly volatility of NASDAQ?
(20 points) You know that there is uncertainty in point estimation due to sampling variability. So, you want to provide a confidence interval.
For more about confidence intervals in general, please watch:
A follow up video for the mean parameter is
Let \(n\) stand for the number of observations. Let \(\hat\mu\) denote our point estimate for the mean parameter. Let \(s\) denote the sample standard deviation. At a confidence level \(C\), the structure of a confidence interval is
\[ (\hat \mu - \frac{s}{\sqrt{n}} z^*, \hat \mu + \frac{s}{\sqrt{n}} z^*) \] where \(z^*\) is a critical value of the standard normal distribution such that \(\mathbb{P}[-z^* < Z < z^*]=C\) with \(Z \sim N(0,1)\).
You should input ?qnorm into the console in
R to learn more about the different functions which have to
do with the normal distribution in R.
Provide confidence intervals for the monthly mean return at the confidence level \(C=0.95\).
(5 points) There is a “shortcut” for constructing
confidence intervals in this case which uses the t.test
function. You should input ?t.test into the console in
R to learn more about it. Create the \(95\%-\)confidence interval using this
command and compare to your result from the “pedestrian” implementation
above.
(10 points) Finally, create a bootstrap \(2SE\) confidence interval and a \(95\%\) percentile interval for the monthly mean return. Please, set your seed to \(1\) for comparison. Plot the histogram of the bootstrap point estimates for \(N=10^5\) draws. Indicate your confidence intervals on the histogram. Compare your confidence intervals to what you obtained above without resampling.