VIX vs S&P 500 Expected Returns Range

In this post, I give out a ready to use figure or plot that one could keep handy to relate VIX level to the Expected S&P 500 returns Range over the next 30 days. I also show how the plot is derived. DOUBLE CLICK THE PLOT FOR A BIGGER IMAGE

As you can see from the Above figure, one can easily figure out the expected range of S&P 500 Returns with a given VIX Level for a given Probability. For example, Currently the VIX level is 30%. So looking at the above plot, for 90% Probability,30% VIX,on the X-Axis and following it vertically and across, we can see that over the next 30 days, the expected range for S&P 500 Returns is +/- 14.24% .

In general, The linear relationship is as follows:

Expected Range with 50% Probability = 0.1947 * VIX

Expected Range with 68% Probability = 0.2871 * VIX

Expected Range with 75% Probability = 0.3321* VIX

Expected Range with 90% Probability = 0.4748* VIX

Expected Range with 95% Probability = 0.5658 * VIX

Expected Range with 99% Probability = 0.7436 * VIX

Now I show how I derived the Above Plot

% probabilities
p = [0.50 0.68 0.75 0.90 0.95 0.99]
% Properly adjust those probabilities to be passed into norminv program
p2 = 0.50 + p./2;

% Before we proceed we should make an important assumption that
% The rate of return on the S&P 500 over the next 30 days is normally
% distributed

% USe the norminv command to get the number of standard deviations
% that a number drawn from unit normal distribution will be

nstd = norminv(p2,0,1);

% since VIX is is an annualized standard deviation, We divide the values
% Obtainded above by sqrt(12)

nstd = nstd ./ sqrt(12);

% Now the Linear Relationship between Level of VIX and S&P 500 return range
% would be: ExpRangeSP500 = nstd .* VIX
% Now let us plot using various values of VIX

VIX = 0:10:100;
ExpRangeSP500 = nstd' * VIX;

%%PLOT
plot(VIX,ExpRangeSP500,'-*')
set(gca,'YTick',[0:5:max(ExpRangeSP500(:))])
set(gca,'YtickLabel',cellstr(strcat(num2str([0:5:max(ExpRangeSP500(:))]'),'%')))
set(gca,'XTickLabel',cellstr(strcat(num2str(VIX'),'%')))
title('VIX vs S&P 500 returns Expected Range over next 30 days')
xlabel('VIX')
ylabel('S&P 500 returns Expected Range over next 30 days')
plottitles = strcat(cellstr(num2str(p' .*100)),'%');
legend(plottitles,'Location','best')
text(VIX(9)*ones(6,1),ExpRangeSP500(:,9),plottitles,'FontSize',14,'FontWeight','bold')
grid on
axis tight