Arithmetic and Geometric Return Assumptions and the Variance Approximation

Summary

The document raises a question about converting annualized arithmetic and geometric return assumptions shown in a capital market assumptions publication. It cites the familiar approximation that geometric return is approximately arithmetic return minus half the variance of periodic returns, and asks whether the gap from the displayed figures could arise because the relation is approximate.

No answer, calculation, or source methodology is included in the document. It therefore identifies a useful interpretation issue but does not establish how the published figures were computed. A direct comparison can depend on the return frequency, the variance estimate, compounding horizon, and whether the arithmetic and geometric figures use matching inputs. The approximation is not an exact conversion in general; without the underlying return assumptions and calculation conventions, the document cannot determine whether the observed discrepancy is expected or signals a different translation method.

Key ideas

• The cited relation approximates geometric return using arithmetic return and return variance.
• The document asks whether the approximation explains a discrepancy between published return measures.
• It provides no calculations or answer for the particular capital market assumptions.
• Comparisons require consistent return frequency, horizon, variance input, and compounding conventions.

Tags

# Interpreting capital market assumptions (geometric versus arithmetic returns)


# Interpreting capital market assumptions (geometric versus arithmetic returns)












I am going through capital market assumptions from Amundi(Link, Amundi). However, I am having trouble converting the displayed average annualised geometric return to the displayed average annualised arithmetic return. I am aware of the following approximation for converting the returns:

\begin{equation} r_G \approx r_A - 0.5V(r_t). \end{equation}

However, applying this formula I can not generally convert the "Average Annualised Arithmetic" return (10y) provided to the "Average Annualised Geometric" (10y) provided.

Am I doing something wrong or is this error in translation due to formula being an approximation?

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