Log return is used for statistical evaluation such MSPE and out-of-sample R-square. Simple return is used for calculating economic value such as CER gain and Sharpe ratio. This is because Log return and simple return have the additivity property for, respectively, time-series and cross-section perspectives. In addition, stock return is always assumed to follow a Log Normal Distribution, so that Log return is used for statistical evaluation. Can you provide more reasons? Besides, Is it OK that simple return is used for statistical evaluation?
It must be remembered that capital market is a long run activity. Huge investments are made in millions of Dollars and returned as expressed in small percentages. For comparability of the time series data the log returned is preferred in the data analysis.
log values effectively captures the compounding effect.
the compounding of returns of stocks can be easily understand by assuming the normal distribution.
Natural log value is the more appropriate measure for computing the change in variable which follows normal distribution.
I think that in most cases, no cosideration is made with this regardand researchers are choosing arbitrarily, because difference is very small.
Another reason is that the log transformation is usually employed to (approximately) validate the constant variance assumption of the statistical model. While for calculating the economic value no such assumption is needed.
Dear Yaojie:
In quantitative finance is assumed that the evolution of asset prices follows a "geometric brownian motion", the equation is in the form:
St=St-1·exp(drift·dt+sigma·sqrt(dt)·dW)
where drift is the expected return of the asset adjusted by variance per unit of time; sigma is the standard deviation of the return of the asset adjusted by the square root of unit of time and multiplied by dW, a realization of a normal distribution N(0,1)
If we want to know the return of the asset we need to clear the equation obtaining the following one:
ln(St/St-1) = ln (exp(drift·dt+sigma·sqrt(dt)·dW)), then we obtain:
ln(St/St-1) = (drift·dt+sigma·sqrt(dt)·dW)
Then, ln(St/St-1) is the expected return of the risky asset per unit of time
Best regards,
Mariano
This web link (youtube video) might be of help!
https://www.youtube.com/watch?v=PtoUlt3V0CI
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