1. Measuring Returns
When examining the past performance of a security, it's the return and not
the price that's really important. So being able to calculate a return from
price is critical. There are actually two ways to go about calculating the
return of a security over time, but only one will give you the
quick answer. The first way is using what's called arithmetic returns.
The arithmetic return assumes the price change of the security has happened
in one step at the end of the period.
To calculate the arthritic return, you can take the difference between the
process at end of the period and the price at the start of
the period, and then divide this difference by the price at the start
of the period. For example, if the starting price is $100 and the
ending price is $110, the difference is $10.
Dividing this by the starting price of $100 gives an arthritic return of
10%. Arithmetic returns work well in the numbers, in this case,
the start and end price are not related to each other.
In math described this as being independent from each other.
This isn't the case with security prices though.
The price tomorrow is related to the price today,
so the prices of what we call dependent. We can demonstrate the issue
with arithmetic returns by calculating the return when this security falls
from $110 back to $100. The arithmetic return is 9.1%. Adding the returns
over two periods gives a total return of 0.9%, but this clearly isn't
correct. The total return should be 0%. Rather than moving in one big
step at the end of a period, security prices move in lots of
small steps. Let's go back to our example, and assume again, a security
starts at $100 and increases to $110, but travels this journey in a
nice smooth manner. Every small step forward in time, the price increases
by just a little bit more than the previous step, because the starting
price at the start of each small step was a little bit higher
than the starting price from the previous step.
This means that the price of the security is growing exponentially,
which we can draw with this nice smooth curved line.
To calculate the return, we use the inverse of the exponential function,
which is called the natural log function. If we take the natural log
of a $110 provided by $100, we get 9.53%. Additionally, when we reverse
this and take the natural log of $100 divided by $110,
we get 9.53%. The total return is now equal to zero.
2. Let's practice!