1. Arithmetic vs Continuously Compounded Returns
I want to demonstrate some of the ideas that we've just introduced on
Arithmetic Returns and Continuously Compounded Returns by walking through
some examples in Excel with you. So, can you please download the workbook
from the downloads section from this course called "Statistics for Finance
Template?" Once you've done that, press play to continue to watch this lesson,
and I'll try to give you some more insights into arithmetic and continuously
compounded returns. So, I've opened up the workbook called "Statistics for
Finance Template," and I've gone to the demo sheet, which is the second
sheet in that workbook. I want to think about and revisit what we've
just been discussing in terms of arithmetic returns and logarithmic returns.
So, the first scenario that we thought about was when an asset was
trading at 100 or had a price of 100 and then had a
closing price of 110. We can work out the arithmetic return by taking
the difference 110 minus 100 and dividing it by the opening price,
100. And we can see that we've got an arithmetic return of 10%.
Now, to get the logarithmic return, we take the natural log of those
two numbers. And to do that, we take the closing price and divide
it by the opening price, and close bracket, and then we get a
9.53% return. Now, we can check this by taking our opening price and
increasing it by an arithmetic return of 10%, we should get 110.
And also, by taking our opening price and increasing it by a log
return of 9.53%, and again we should get 110. So, let's do that
for the arithmetic return, first of all. Let's take our opening price and
increasing that by an arithmetic return of 10% means we multiply that by
1, plus the arithmetic return of 10%. And there we see we've got
110. So, how do we increase that by a log return of 9.53%? Well, we
use the exponential function. So, we take our opening price, and we multiply
that by the exponential of our log return, and we see that we
get back to 110. So, seems to work okay when we start with
a low price and we end up with a higher price,
but what about the scenario where we start with a high price and
we end with a lower price? In other words, if we start at
110 and we finish at 100, now what is the arithmetic return?
Well, again, we take the difference, and this time we divide it by
a starting price of 110. So, we see we have an arithmetic return
of 9.09%. But what about our logarithmic return? Well, we take our log
function, which is our closing price divided by our opening price.
And again, we get a negative return, but you can see that the
logarithmic return now is 9.53%. So, if we added the log return as
the asset increased from 100 to 110, and with the logarithmic return as
the asset decreased from 110 to 100, we'd end up with a net
return of zero, which is what we want. And again, we can do
the check. So, let's take our opening price of 110 and times that
by 1, plus the arithmetic return. We see, we go down to 100,
so that does work. And what about our logarithmic return? We'll take our
opening price of 100 and times that by the exponential of our logarithmic
return. And we see that when we do that, again, we get back
to 100.
2. Let's practice!