# Arithmetic vs. Geometric Mean

#### rwwoods

##### Dryer sheet wannabe
In reviewing the paper "A Gentle Introduction to the Calculus of Retirement Income" by Milevsky (http://www.ifid.ca/pdf_workingpaper...duction to the calculus of retirement income"), something is not clear to me. Two of the input variables for his equation are the mean and standard deviation of the investment. I understand that these are arithmetic rather than geometric values. If I know the geometric values, I understand that I can compute the approximate arithmetic mean by adding half of the variance (square of the standard deviation). For example, IFA reports that over the last 50 years, their Index Portfolio 75 has an annualized (geometric) return of 12.54% and a standard deviation of 12.11%. Adding in half of the variance gives an arithmetic mean of 13.27%. If I am considering this particular investment, should I use 13.27% for the mean and 12.11% for the standard deviation when using Milevsly's equation?

I believe the answer is given on page 6 of 6:

"Finally, assume that your client’s investment portfolio is allocated to a mix of balanced mutual funds (or ETFs, SMAs, etc) and you estimate that this portfolio will earn {μ = 0.075 }, which is an arithmetic average of 7.5% in any given year, with a standard deviation or volatility of {σ = 0.18}, which is 18% per year, both in inflationadjusted terms. These numbers are consistent with the well-known and widely-cited numbers by Ibbotson Associates (2004). In some sense, the precise magnitude of these numbers is less important than the general awareness that they can fluctuate and hence are yet another source of retirement risk."