Calculating Log Excess Returns from Index Levels and Risk-Free Rates

Summary

The document asks how to calculate monthly log excess returns from a total return index, a price index, and the Fama–French risk-free rate. It proposes calculating gross return from the current total return index relative to the prior price index, then subtracting the log risk-free rate. The central issue is that a quoted risk-free rate is ordinarily a simple return, so taking its logarithm directly can fail when the rate is zero or negative. To express it as a log return, convert it using the logarithm of one plus the simple rate, with rates aligned to the same period and units as the asset return.

The question also raises a possible index-construction mismatch: combining a total return index with a prior price index may not represent a consistent gross return unless those series are defined on compatible bases. The document contains no worked data or answer confirming the proposed index formula, so that detail requires checking against the data definitions before implementation.

Key ideas

• A simple risk-free return is converted to a log return with the logarithm of one plus the rate.
• Subtract the log risk-free return from the asset log return to obtain log excess return.
• Align the risk-free rate period and units with the return interval.
• Check that the total return and price index levels are compatible before forming a ratio.

Tags

# Log excess returns calculation


# Log excess returns calculation












I need to calculate log excess returns. I'm given market level monthly total return index, price index and a risk-free rate (from Fama/French 3/5/etc. factors). I'm not sure whether I'm calculating the log excess returns correctly.

$log\ excess\ return = log\ gross\ return - log\ risk\ free\ rate$

$log\ gross\ return_{t} = log(total\ return\ index_t/price\ index_{t-1}) $ (Campbell-Shiller, 1988)

I assume that FF monthly risk-free rates are not log-transformed, but since there are zeros in RF column, I'm getting `Inf` values for $log\ risk\ free\ rate$.

Should I then calculate gross return on levels, take the difference and then apply log-transformation?

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