CBA #$2$

Load the data file, "Stocks.dat" into the data matrix, $x,$ where the columns are the stock

vectors. C D E$].$ Plot each column separetely in a single plot $($different color$)$ and

observe. Which stocks would you think that has the highest correlation?

Perform statistical normalization: zero mean $+$ unit variance

First perform the MATLAB comment:

$Z=$zscore$(x)$;

This command shifts and scales the column vectors to be zero$-$mean and unit$-$variance. Also

plot the columns of the $Z$ matrix and discuss what you see. Is each column really zero mean

and unit variance? Verify.

Compute and plot the joint pdf

Recall that we can use histograms to "approximate" the true pdf$.$ However, there is no standard

Matlab command to generate a $2$D histogram. Use the following code to compute the $2$D

histogram first and then to approximate the joint pdf$,$ say, between the $1^{st}$ and $2^{nd}$ stocks $($e$.$g$.,$

$x_1=$A and $x_2=B).$ Turn this into a MATLAB function so that you can use it to approximate all

other joint pdf$.$s $(1-3,.,4-5).$ This code uses Matlab's hist command to compute the $1-$D

histogram in an iterative way. The for loop segregates the stocks of $x_1$ by the $x_1-$value bins.

Then for each subset of $x_1$ stocks, it creates a $1-$D histogram of the corresponding $x_2$ stocks

and puts them into the appropriate bins in the $2-$D histogram. It requires you to set bi ns$,$ which

is the number of bins on each axis. You can use for instance: bins$=16.$

Next, as for the marginal pdf approximation, we need to normalize the histogram so that the

joint pdf sums up to one:

$pdf=n2d\frac{.}{{\displaystyle \underset{?}{\overset{?}{}}}({\displaystyle \underset{?}{\overset{?}{}}}(n2d))}$;

Finally, plot the pdf using one of the following options, so that the pdf is clearly visible.

Image Plot: $h=$imagesc$(x1,x2,pdf)$; colorbar;

Surface Plot: $h=$surf$(x1,x2,pdf)$; view $(20,30)$;

Mesh Plot: $h=$mesh$(x1,x2,pdf)$; view$(20,30)$;

Contour Plot: $h=$contour$(x1,x2,pdf)$; colorbar;

Do not forget to grid and label your plots. Insert and discuss two different joint pdf plots in

your report. Pick whichever two you think best show the features of your pdf$.$ Estimate the Covariance$/$Correlation from the joint pdf

As you have the approximated joint pdf$,$ now you can now compute the Covariance for any

pair of stocks using the well$-$known formula:

cov$(x,Y)={}_{xY}={\displaystyle \underset{(x)}{\overset{?}{}}},$yinSf$(x,y)(x-{}_x)(y-{}_Y)$

You can use the following code to implement this summation:

As you statistically normalize the data at the beginning, what do you say about Covariance

and Correlation? Repeat steps $2-3$ for other stock pairs $(1,3),$dots,$(4,5).$ Which pair of stocks is

giving you the highest and least correlation? Put all the cross$-$correlation results into a $55$

Cxy matrix where the $($i$,$j$)$th and $(j,i)$ th elements will both take the correlation value between

the $i^{th}$ and $j^{th}$ stock. The diagonal elements should be all $1($no need for calcuWHO WANTS TO BE A MILLIONAIRE?

Joint Distribution and Correlation

The weekly rates of return for five stocks listed on the New York Stock Exchange are given in

the file Stocks.dat. Call these stock column vectors: A$,$ B$,$ C$,$ D and E$.$ Let the data matrix be

C In this assignment, first you will approximate the joint distribution of the pair

of stocks. Then you will find the covariance and correlation between each pair of stocks using

the approximated pdf$\text{'}$s and directly over the data $($sample covariance$)$ too. The idea is to find

out which two stocks have the highest correlation and how to use this information for

investment.

Here are the main steps:

Load Stocks.DAT into MATLAB

Visualize $($plot$)$ each of the $5$ stocks. OUTPUT: Stock plots.

Perform statistical normalization $($zero mean, unit variance$)$ of each stock data vector

OUTPUT: Plots of all normalized stocks

Approximate the joint pdf for your data. OUTPUT: Two different plots of your joint pdf$.$

Compute the covariance matrix and the correlation coefficient by a$)$ from the joint

pdf$.$s that are estimated, b$)$ directly from the sample covariance matrix

OUTPUTS: the estimated correlation from the joint pdf$.$s and estimated correlation

matrix from pdf$.$s and the sample correlation matrix.

You should create ONLY one technical report $($in pdf$)$ containing: comments, discussions,

MATLAB script and all outputs $($plots$,$ etc.$)$ of your assignment.