Covariance and correlation of stock returns

We illustrate how mean, covariances and correlations can be computed using Matlab and how correlation can be visually inspected using scatter plots. We use returns of two commonly used stock market indices, NASDAQ-100 and SP-500.

ELEC-C5211 - Johdatus signaalien tilastolliseen mallintamiseen ja paattelyyn / Prof. Esa Ollila

Contents

Load the data

clear all; % read in the data if it is not already in the workspace
NDX100 = hist_stock_data('01012017','01012018','^NDX','frequency','d');
SP500 = hist_stock_data('01012017','01012018','^GSPC','frequency','d')

g1 = NDX100.AdjClose(2:end)./NDX100.AdjClose(1:(end-1)); % gross return
r1 = g1 - 1; % net return

g2 = SP500.AdjClose(2:end)./SP500.AdjClose(1:(end-1)); % gross return
r2 = g2 - 1; % net return

n = length(r1); % sample length
r = [r1 r2]; %  n x 2 matrix of returns

SP500 = 

  struct with fields:

        Date: {251×1 cell}
        Open: [251×1 double]
        High: [251×1 double]
         Low: [251×1 double]
       Close: [251×1 double]
    AdjClose: [251×1 double]
      Volume: [251×1 double]
      Ticker: '^GSPC'

Plot the data

xlist = cell(1,5);
for i=1:5
   xlist{i} = NDX100.Date{40*i};
end

figure(1); clf
subplot(2,1,1);
plot(2:(n+1),r1,'LineWidth',2.5)
title('NASDAQ-100 daily net returns');
xticks([40:40:220]); axis tight; grid on;
hold on;
plot([2 (n+1)],[0 0],'--','LineWidth',2.5)
xticklabels(xlist)
set(gca,'LineWidth',3,'FontSize',16);
ax = axis;

subplot(2,1,2);
plot(2:(n+1),r2,'LineWidth',2.5)
title('SP-500 daily net returns');
xticks([40:40:220]); axis tight; grid on;
hold on;
plot([2 (n+1)],[0 0],'--','LineWidth',2.5)
xticklabels(xlist)
set(gca,'LineWidth',3,'FontSize',16);
axis(ax);

What does the plot tell you about the risk (standard deviations) of these two market indices?

Compute the mean returns and covariance/correlation matrix of returns

mu = mean(r)  % estimated mean returns

Sigma = cov(r) % covariance matrix

Rho = corr(r)  % correlation matrix

risks = sqrt(diag(Sigma)) % standard deviations

mu =

    0.0011    0.0007


Sigma =

   1.0e-04 *

    0.4177    0.2162
    0.2162    0.1757


Rho =

    1.0000    0.7982
    0.7982    1.0000


risks =

    0.0065
    0.0042

Q: What can you infer about correlatedness of SP-500 and NASDAQ-100?

scatter plots: visual inspection of correlation between two variables

figure(2); clf
plot(r1,r2,'o')
set(gca,'LineWidth',3,'FontSize',16);
xlabel('$r_1$ (Nasdaq-100)','FontSize',22,'Interpreter','Latex')
ylabel('$r_2$ (SP 500)','FontSize',22,'Interpreter','Latex')
grid on

Q: Can you already infer the degree and type of correlation from this figure?

Published with MATLAB® R2018a