Converting Between Simple Returns and Log Returns

Summary

The document explains how to convert a series of simple, or arithmetic, returns into continuously compounded log returns when the underlying price series is unavailable. The conversion uses each return itself, so past prices are not required. It also gives the reverse transformation from log returns back to simple returns.

The method is an algebraic relationship between the two return conventions, rather than an estimation technique. The answer offers no empirical evidence because the conversion follows directly from the definitions of the return measures. Its practical limit is that the simple return must be greater than negative one for the logarithm to be defined. The discussion does not address aggregation across periods, annualization, or the distinct ways arithmetic and log returns behave when summed over time.

Key ideas

• A simple return can be converted to a log return using the natural logarithm of one plus the return.
• A log return can be converted to a simple return by exponentiating it and subtracting one.
• The conversion requires no underlying price series.
• The log conversion is defined only when the simple return is greater than negative one.

Tags

# Convert arithmetic returns to log returns


# Convert arithmetic returns to log returns












I have a series of arithmetic returns and I need log returns. I do not have the underlying prices. How do I convert?

All the posts I have found explain why using one versus the other is appropriate but how do I get from one to the other without the underlying data?

Thanks

## Answer by amdopt (score 11, accepted)

https://quant.stackexchange.com/a/47538

Transmuting one to the other is pretty straightforward without the underlying sequence of prices.

To go from log to simple:

$R = exp(r) - 1$

To go from simple to log:

$r = log(R+1)$

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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