Choosing Between Log and Simple Returns for FX Analysis

Summary

The question asks whether to use close-to-close percentage changes or logarithmic price differences for minute-level EUR/USD data, and whether the choice matters more when combining multiple currency pairs. The reply draws on a standard distinction: continuously compounded, or log, returns are often convenient when studying a return series through time, while simple returns are commonly used to compare assets across a cross-section. For short intervals, where returns are small, log returns approximate simple returns closely.

The answer also points out a portfolio calculation caveat. A portfolio's simple return can be computed as the weighted sum of its constituents' simple returns for the period, but the same weighted-sum formula does not generally apply to constituent log returns. The document gives conceptual guidance rather than an empirical comparison of FX data, and it does not quantify when the difference becomes material across a basket. Choice of return measure should therefore follow the analysis being performed, with portfolio aggregation handled consistently.

Key ideas

• For temporal analysis, the reply recommends log returns as a common representation.
• For cross-sectional comparisons, simple returns are commonly used.
• Over short intervals with small returns, simple and log returns are close.
• Portfolio simple returns aggregate by weights, whereas log returns do not generally follow the same weighted-sum rule.
• The discussion gives no FX-specific numerical test or threshold for when differences become material.

Tags

# Log Differences vs Percentage returns


# Log Differences vs Percentage returns












When working with a single TimeSeries of Foreign Exchange price data (EUR/USD : OHLC) on a minute by minute level, is it better to use the % difference of the close vs the lognormal difference of the close?

When scaling up to use a basket of fx pairs, do the differences between the two become more prevalent?

## Answer by skoestlmeier (score 1, accepted)

https://quant.stackexchange.com/a/41025

A practical hint is given by Campbell/Lo/MacKinlay p. 11f.:

> When returns are measured over short intervals of time, and are therefore close to zero, the continuously compounded return on a portfolio is close to the weighted average of the continuously compounded returns on the individual asset [...]. Nonetheless it is common to use simple returns when a cross-section of assets is being studied [...] and continuously compounded returns whe the temporal behavior of returns is the focus of interest.

So as you are analyzing the temporal behavior, it is recommended to use lognormal differences. You may also look at this detailed answer for the advantages of log returns. Nevertheless, you have to be aware, that the simple return of a portfolio $R_p$ of $N$ asset returns $R_{it}$ with weight $w_i$ is not the weighted average of log-returns, i.e. the convenient weighting calculation $R_p = \sum_{i=1}^N{w_i*R_{it}}$ does not hold.

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