Logarithmic Returns

For a price time series p(t), discrete with a time increment dt, a logarithmic return variable is

x(t|dt) = log(p(t)/p(t-dt))

where dt is the time increment separating adjacent points in the time series.

This variable has several advantages. It is additive: the return of the entire series is the sum of the returns comprising the series:

x(t_n|t_n-t₁) = x(t₂|dt) + … + x(t_n|dt),

dt = (t_n-t₁)/(n-1)

Non-negativity of the price is “built in” — especially useful when simulating artificial time series.

When used in the correlation analysis, logarithmic returns (as do ordinary returns p(t)/p(t-dt)) eliminate one trivial source of non-stationarity of the correlation functions which is the possible long time-scale trend in the time dependence of the price. Long-term absolute level of the price is almost irrelevant to a forex trader, what matters is relative movements.

Finally, the moments of the logarithmic returns may converge better than they would for the ordinary returns — although, notably, Mandelbrot postulated that the variance of this variable would be infinite.