I'm doing some analysis on log returns and I notice that returns can exceed 100%. For example, if a security's close price \$1 today and \$10 yesterday, the log return is $ln(1) - ln(10) = -230\%$! Under arithmetic computation, returns for a long position cannot exceed 100% (i.e. the initial investment). So how would i interpret a log return of -230%?
Thanks,
Alex
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Large? ?
The relationship between normal and log returns is $$(normal return) = exp(log return)-1$$ Therefore log-returns can be from $-\infty$ to $+\infty$ while normal ones can only be between $-1$ and $+\infty$.
The result is:
$ e^{(-230%)} - 1 = -89% $
If the value is $1 today and was $10 yesterday
return = today/yesterday - 1 = 1/10 - 1 = -0.9 = -90%
log return = ln(1 - 0.9) = -2.302585
check : A = P e^rt = $10 * e^-2.302585 = $1 (i.e. today's value)
Since investments can end up in the red it's interesting to consider a return that exceeds -100%, for instance if today's value is -$1
return = today/yesterday - 1 = -1/10 - 1 = -1.1 = -110%
Now the log return method breaks down since the natural log cannot be taken for a negative number (unless imaginary numbers are used)
log return = ln(1 - 1.1) = -2.30259 + 3.14159 i
The curve for ln(x) is only in the positive domain of x.
