Why Negative Returns Can Have Negative Log Returns

Summary

The document clears up a common confusion between negative returns and taking the logarithm of a negative number. A log return is calculated as the logarithm of the ending price minus the logarithm of the starting price, or equivalently the logarithm of their ratio. When both prices are positive, that ratio can be below one, producing a negative log return without taking the logarithm of a negative input.

An example compares a price decline from one period to the next using an ordinary percentage return and a natural-log return; the two values are close for the small decline shown. The key limitation is that the price levels used in the logarithms must be positive. Negative price observations therefore create a genuine problem, even though negative returns do not. The explanation is brief and does not address broader statistical assumptions or cases where returns are large.

Key ideas

• A negative return does not imply that a logarithm is being taken of a negative number.
• For positive prices, log return is the difference between the logarithms of ending and starting prices.
• A price ratio below one produces a negative log return while remaining valid for the logarithm.
• Negative price levels, unlike negative returns, prevent the usual log-price calculation.

Tags

# in time series analysis or finance people use log return for inference but returns can take negative value


# in time series analysis or finance people use log return for inference but returns can take negative value












in time series analysis or finance people use log return for inference but returns can take negative value. but log cant take negative values. so why we use it when log is not defined on most of values

## Answer by AKdemy (score 3)

https://quant.stackexchange.com/a/65816

It is not negative returns that are an issue.

Assume the following prices in two periods: $P_o = 100$ ; $P_1 = 99$ Standard percent calc: $$\frac{P_1}{P_0}-1 = -0.01$$ which is -1% . Using the natural logarithm you get $$ln(99)-ln(100) = -0.01005$$ which is essentially identical.

Negative prices are an issue - but these are not observed for many economic variables like stock prices etc. For a general discussion about logs, you can have a look at this. It also discusses this calculation here in more detail.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.