Now are going to take a look at our last stylized fact that's the fat tailed ness of asset return distributions
so stylized fact Number four suggests that the fat tailed ness of asset returns is a widely observed
phenomenon in markets.
I just should state before we go on that there actually is a caveat to say about this.
But while this is true for daily returns it gets to be less and less true as you increase the investment
horizon.
So as you as you increase the horizon from daily returns to weekly returns to monthly returns and to
ultimately to annual returns the fat tailed effect tends to diminish We'll come back to this point at
the end of the lecture.
It's certainly true for daily returns.
And that's what we're studying here.
So to investigate the fat tail this of the returns where the three assets that we've been studying we're
going to do two different things going to plot normal Q-Q plots for the return series for each of our
assets that we're been studying.
Second thing I'm going to do is I'm going to report the excess kurtosis for each of the assets.
So here we see the normal Q-Q plot for the S&P 500 returns the straight line that's plotted here.
This is based is based on a regression of these points and this represents the line the approximate
line that will be followed.
If this was a normal Q-Q plot of a sample from a normally from a normal distribution.
So we see the telltale signs here of fat tails as we see that are quantizes for S&P 500 returns deviating
away from that straight line on both the high and the low side.
We see clear clear evidence that the quintiles are too high on the big side and too low on the negative
side.
This is as we've noted already this is the telltale sign of fat tails.
So this is unequivocal evidence that S&P 500 returns have fat tailed distributions use the Q-Q plot
for pound USD exchange rate.
And once again deviations from the from the line at both ends are again unequivocal evidence that we
have fat tails for pound USD pound USD exchange rate returns.
Now for the soybean returns.
Same thing.
Deviations from the straight line on both ends indicated that soybean returns also have fat tails.
You might notice that on the on the positive side for soybean returns the deviation away from the straight
line is a little bit less than it was for the other two assets that we looked at and also it's a little
bit less on the low side.
Not sure what that really means when you could take the time to investigate that if one wanted to say
something significant about the soybean returns.
But no question that this still is an indicative of fat return of fat tails for the on the big side
as well.
So Q-Q plots normal Q-Q plots for all three of our assets show convincing evidence that we have fat
tailed distributions for all three of the return series I want to just report the excess kurtosis for
each of the three return series that we're looking at what's displayed in this table for the S&P 500
the returns had an excess kurtosis of ten point eight for the pound sterling USD exchange rate excess
kurtosis was eleven point six three soybeans excess kurtosis was 9.1 9.
All of those excess kurtosis values are consistent with fat tails.
Those are all very large values of the excess kurtosis.
Unless the the returns have exhibit fat tailed distributions.
So the evidence for fat tails in the return distributions is pretty clear from the evidence that we've
seen in this lecture.
Now I just want to come back to this little caveat about the return the horizon for the returns.
All of the return series that we've looked at in in this lecture in the last few lectures we've been
studying carrying out this empirical study.
These have all been daily returns.
It is it's empirically true that fat tailed defect diminishes as you increase the return horizon monthly
return data shows significantly less evidence of fat tails or not normal behavior.
What basically is happening as you increase the return horizon is that there's a central limit effect
kicking in.
And when you're taking a look at that longer return horizon this is equivalent to looking at the monthly
returns is equivalent to looking at the returns for about 22 days roughly speaking.
So it's the monthly return is just the sum of the daily returns for the 22 trading days in a month.
And that has an effect that essentially allows the central limit theorem to come into effect and normalize
the data.
Normalize the what you observe the result is that for longer return horizons the central limit theorem
is going to normalize your data and you're going to see much less fat paleness in the in the return
distribution.