Converting Log Returns to Simple Returns and Their Lower Bound
Summary
The document explains how to interpret a log return that is below negative one hundred percent. It gives the conversion from a log return to a simple return: exponentiate the log return, then subtract one. This reconciles an unusually large negative log return with the bounded loss on a long position whose value remains nonnegative.
A price change from ten dollars to one dollar illustrates the distinction: the simple return is a ninety percent loss, while the corresponding log return is about negative 2.30 in decimal units. The discussion also notes an important boundary: if an investment’s value becomes negative, the usual real-valued log return is undefined because the logarithm requires a positive argument. The examples clarify the mathematics, but the document does not discuss how to handle log returns in datasets with negative prices or other modeling conventions.
Key ideas
- A log return can take any real value, while the corresponding simple return cannot fall below a total loss for a nonnegative asset value.
- Convert a log return to a simple return by exponentiating it and subtracting one.
- A large negative log return can therefore correspond to a simple return between zero and a total loss.
- The real-valued logarithm is undefined when the ending value or price ratio is negative.
Tags
Full text
# How to interpret negative log return more than -100%?
# How to interpret negative log return more than -100%?
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
## Answer by Kiwiakos (score 4)
https://quant.stackexchange.com/a/22778
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 $+\infty$.
## Answer by sparkle (score 2)
https://quant.stackexchange.com/a/22779
The result is:
$ e^{(-230%)} - 1 = -89% $
## Answer by Chris Degnen (score 1)
https://quant.stackexchange.com/a/22783
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.
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