### Blog Contributors

Ryan Nauman
VP, Product and Market Strategist
Stephen Berei
VP, Client Services & Implementation
Jeremy Poulin
Senior Client Consultant

# Arithmetic vs. Geometric Mean Returns

Mar 12, 2013

At Zephyr we occasionally get questions regarding returns in StyleADVISOR vs. AllocationADVISOR.  Specifically, clients notice that the annual returns do not match.  This is by design, as StyleADVISOR uses geometrically-compounded returns and AllocationADVISOR uses arithmetically averaged returns.  StyleADVISOR is used primarily for historical, ex-post analysis of what a manager or index’s performance has been.  AllocationADVISOR, on the other hand, is used for establishing a strategic plan going forward.  For those different roles, different calculation methodologies are recommended.

This obviously leads to the question, “Well, WHY is geometric-compounding preferred for historical analysis and arithmetic-averaging for forward-looking projections?”  To be honest, I’ve always struggled to articulate why this convention is standard, at least in a way that anyone ever found helpful.  However, I’ve been studying for the CIPM exam and came across a few lines that explain the different approaches in a much more concise manner than I’ve ever been able to muster.  I’ll simply quote Carl Bacon, David Carino, and Arin Stancil and thank them for articulating what I could not:

“Arithmetic and geometric mean returns are useful for different purposes.  If we need to estimate an expected or average return over a one-period time horizon of stated length, a historical arithmetic mean return is an appropriate estimator.  For statistical analysis, when a model is fitted with parameters estimated from data, the arithmetic mean normally is the appropriate estimator.  The geometric mean is affected by volatility, whereas the arithmetic mean is not.  In contrast, for summarizing the effect on wealth of observed investment returns, the geometric mean return is the only return to use: It represents the compound rate of growth of an investment- that is, the rate of return that if earned each period on the initial investment would match the actual cumulative return achieved.

Geometric mean returns are useful for accurately summarizing the ending wealth that has been produced by an investment…There is a one-to-one relationship between the geometric mean return and the relative value.  In contrast, ending wealth cannot be recovered from the arithmetic mean return.  In general, therefore, if the intent is to accurately summarize and report past returns, as is the case in performance evaluation, the geometric mean return must be used.  If the intent is to forecast future returns or infer parameters of statistical models, the arithmetic mean return is more useful.”

“Performance Evaluation: Rate-of-Return Measurement”, CIPM Principles, vol.1, Reading 4, section 3.2.3.