Why Log Returns Add Across Periods in Commodity Research

Summary

The document addresses why some commodity research annualizes monthly returns by multiplying by twelve instead of using a compounded-return calculation. The explanation is that the reported quantities may be logarithmic returns even when authors omit that qualifier. A log return is the change in the natural logarithm of price between two dates; over consecutive periods, these changes add exactly to the total log return across the full interval.

This additive property makes summing monthly log returns—and scaling a representative monthly log return by twelve—consistent with an annualized log-return convention. It does not mean that ordinary percentage returns compound by simple addition: converting log returns into cumulative simple returns requires exponentiation. The answer is brief and gives no empirical evidence about how common the convention is across commodity studies, so readers should check each paper’s definitions before comparing annualized figures.

Key ideas

• Log returns measure changes in the logarithm of price.
• Consecutive log returns add to the log return over the combined interval.
• Multiplying a monthly log return by twelve follows an annualized log-return convention.
• Ordinary percentage returns compound differently, so definitions matter when comparing research.

Tags

# Why do academics use simple interest rather than compounding when calculating annualised returns?


# Why do academics use simple interest rather than compounding when calculating annualised returns?












I've seen than in commodities research, returns seem to be treated as simple rather than compounding. For example, the formula for annualised return would be $r_Y = r_M \times 12$ rather than $r_Y = r_M^{12}$ where $r_Y$ and $r_M$ are the annualised and monthly returns. However, most would agree that the formula for futures/forward prices is

$$ S_t = S_0 e^{rt} $$

which implies compounding (I'm ignoring storage costs, convenience yield etc). So why would the annualised return not be calculated as the monthly return to the power of 12 but is instead multiplied by 12?

## Answer by Richard Hardy (score 1)

https://quant.stackexchange.com/a/82095

In commodities research, it is not uncommon to work with logarithmic returns. Sometimes the term logarithmic is omitted and the reader is assumed to figure that out by themselves. Then summation makes sense, as the cumulative logarithmic return is the sum of period-by-period logarithmic returns: \begin{aligned} r_{t\rightarrow (t+h)} &:= \ln(P_{t+h})-\ln(P_t) \\ &= [\ln(P_{t+h})-\ln(P_{t+h-1})] + \dots + [\ln(P_{t+1})-\ln(P_t)] \\ &= r_{(t+h-1)\rightarrow (t+h)} + \dots + r_{t\rightarrow (t+1)}. \end{aligned}

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