Log Returns Use the Logarithm of the Price Ratio

Summary

The document distinguishes daily log returns from the logarithm of a simple percentage return. For consecutive closing prices, the standard log-return definition takes the natural logarithm of the current price divided by the previous price. This expresses the change as a log price ratio.

The alternative expression first calculates the relative price change and then takes its logarithm. The accepted answer explains why that is not the usual definition: the relative return can be negative, and the natural logarithm is not defined for negative values. The exchange gives a concise definitional clarification, but does not discuss other return conventions, compounding properties, or practical applications. It also does not explore edge cases such as zero prices, which would make the price ratio unsuitable for taking a logarithm.

Key ideas

• A daily log return is the natural logarithm of the current price divided by the prior price.
• Taking the logarithm of a simple relative return is a different calculation.
• A negative simple return cannot be passed directly to the real natural logarithm.
• The explanation addresses the definition but not broader uses or edge cases.

Tags

# Definition of log return of an asset


# Definition of log return of an asset












What is the general usage of the term daily log returns $Y_t$ of an asset? (1) or (2)? $$(1) \text{ } Y_t = log (\frac{p_t}{p_{t-1}})$$ OR $$(2) \text{ } Y_t = log (\frac{p_t-p_{t-1}}{p_{t-1}})$$ for $p_t$ being the close price of day $t$.

## Answer by Sanjay (score 1, accepted)

https://quant.stackexchange.com/a/36686

The first one is a correct way. The second one is simply taking log to the relative returns which can be negative and natural log is not defined for negative values...

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