(Author’s note: This year I have volunteered to coordinate the Society of Actuaries (SOA) Investment Section’s sessions at the SOA annual meeting, which will be held this October in Chicago. For more information, visit the SOA annual meeting website).
At this year’s Society of Actuaries’ annual meeting, one of the more anticipated discussions is likely to be one involving Richard Joss and Alex Kane, discussing “Is the Arithmetic Mean of Past Returns the Best Estimate for the Future?”
The issue at hand has to do with several attributes of the arithmetic average return that call its appropriateness as the “expected” return into question. This discussion will pertain to choosing expected return inputs for the purpose of modeling possible outcomes, such as what you would input into an optimizer or a Monte Carlo simulation. This discussion will have nothing to do with whether future asset market returns will be like past ones.
Prof. Kane is co-author of the ubiquitous finance textbook Investments along with Zvi Bodie and Alan Marcus. The Ninth Edition suggests using Arithmetic Mean as the expected return (pp. 130-132). Richard Joss’ book Six Myths in Modern Finance that Every Investor Needs to Know devotes an entire chapter to this topic, referring to it as Myth #2.
If you think this is a trivial topic of interest only to actuaries and finance professors, consider how you decide what to put into your optimizer and Monte Carlo simulations. Your decision makes an enormous difference.
A convention most finance students learn is how to convert arithmetic returns to geometric. The convention resembles the first few elements in a Taylor series expansion:
E[ geometric return ] = E[ arithmetic return ] – 
where 
is the estimated variance of arithmetic returns.
While this convention is handy, its accuracy varies. But understanding the relationship is crucial for making informed allocation decisions, especially when the result of your decisions will be compounded over time.
Don’t fool yourself. Make sure you, your analysts, and your advisors comprehend how and how not to use arithmetic returns when modeling compounded results.