Simple Returns are defined as
Logarithmic Returns are defined as
Simple returns are additive accross securities. If are the weights of different securities within a portfolio, with linear returns , the linear return of the overall portfolio is
Logarithmic returns are additive over time. If the logarithmic returns of a security over time are , the logarithmic return over the whole period is:
If a security follows a Brownian motion, its logarithmic returns are normally distributed.
The Taylor series decomposition of log is as follows:
So the error of the log-return vs. the simple return is in for returns under 100%.
Typically, the error is less than 0.5% for returns below 10% in absolute value. However, it reaches several percentage points above 15% and more than 10% for returns above 50% or under -40%. The following 2 charts illustrates the extent of the error for small and large values.
The mean of the log return can be approximated as follows:
The 2nd, 3rd and 4th moment are also affected.
Whenever the terminal wealth is assessed or returns can be higher than a few percentage points, log returns can lead to significant estimation errors.