Wednesday, September 5, 2007

Ignore Average Annual Return Rates: Geometric Mean V. Arithmetic Mean

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Protect yourself from slick marketing: This article explains the importance of the geometric mean and how to calculate it to read and report returns on investment (ROI) accurately.

Misleading "Average Annual Returns"

The average annual rate of return tends to overestimate your gains because it is an arithmetic mean (an average based on additive units) which is inappropriate for multiplicative products such as compounded interest.

Using the arithmetic "average annual rate of return" for stock performance is like trying to describe how tall you have grown in ounces or asking, "How many inches do you weigh?"

Simple hypothetical of a $100 lump-sum buy-and-hold:

Year - Investment - Return
0 ............ $100 ......... -
1 .............. $50 ...... (50%)
2 ............ $100 ...... 100%

  • Average annual rate of return: (100 + (-50))/2 = 25%

You started with $100 and you ended with $100 but your $0 gain shows +25% average annual return.

Of course, you actually have 0% gain on your initial value after 2 years.

Use the Geometric Mean Instead

Ignore the arithmetic mean (average annual rate of return) and instead calculate the more helpful geometric mean (annualized rate) to find the factor that, if repeated, would result in your current/desired balance; for n years, the nth-root of the products of the rates-expressed-as-positive-growth-factors. For our example above that halved (*0.50) and then doubled (*2.00) in 2 years:

  • The square-root of (0.50 * 2.00) = a factor of 1.00
So $100 * 1.00 = $100. The factor of 1.00 is equivalent to 0% interest, since you can multiply by 1 forever and still have the same number with which you started (in our example, 0% per year for 2 years). A factor of 1.12 would equal 12% growth (per time period). Note that 3 years would require the cubed-root, etc.

A shortcut is the nth-root of the last-year's-balance-divided-by-the-first-year's-balance. For our example:
  • The square-root of (100/100) = a factor of 1.00

The shortcut shows that you can ignore all the "paper profits" ups and downs (unrealized gains and losses) of your stocks or home equity and concentrate on the end points of initial investment v. final cash-out (assuming no intervening hard cash inputs/withdrawals). The geometric mean simulates a consistent year-after-year interest rate so you can compare a volatile stock to something with steady progress such as a 5-year Certificate of Deposit (CD).

Use Geometric Standard Deviation

Ignore arithmetic standard deviation and use geometric standard deviation. That calculation is a bit more complicated (involving logs) but at least know how to read it. Unlike the arithmetic version which is reported as a quantity (e.g. 5% mean with standard deviation of 3% indicates a range of 2-8%), geometric standard deviation is reported as a factor (e.g. 1.05 mean with standard deviation of 1.03 indicates a range of 1.02-1.08, and the nth standard deviation is the nth power of the geometric standard deviation).

Always use the right tool for the job and do not let Wall Street or Madison Avenue tell you otherwise.

Geometric Mean Calculator for Annualized Returns on Investment (ROI)


Tom said...

great insight..

Jason said...

I found your article on the carnival of personal finance, and I enjoyed it allot. In fact, I am picking it as my favorite on my blog,

J at IHB and HFF said...

Thank you both and please visit again.