Geometric Mean Return

The Ashgrove Balanced strategy returned +18% in year one, -10% in year two and +12% in year three. Its marketing sheet advertises an average annual return of 6.67%. What figure should an adviser present instead, and why?

1. A.About 18.94%, the cumulative three-year return, since that is what the account actually gained18.94% is the correct cumulative figure, but the question asks for an ANNUAL rate. Presenting a three-year cumulative number as an annual return would be misleading.
2. B.About 6.67%, the arithmetic mean, which is the standard measure of realized annual performanceThe arithmetic mean overstates realized results whenever returns vary, because it ignores that the loss year applied to a larger base than the recovery.
3. C.About 5.95%, the geometric mean, because only compounding reflects what an investor actually earnedCorrect. 1.18944^(1/3) - 1 = 5.95%, the rate that grows the starting balance to the ending balance.
4. D.About 6.31%, the average of the arithmetic and geometric means, which balances the two conventionsThere is no such convention. Splitting the difference between two defined statistics produces a number that describes nothing.

Why: The 6.67% is the arithmetic mean, (18 - 10 + 12) / 3, which is not the return an investor actually earned. Compounding the three years gives 1.18 x 0.90 x 1.12 = 1.18944, a cumulative gain of 18.94%. The geometric mean, or compound annual growth rate, is 1.18944 raised to the one-third power minus 1, which is about 5.95%. The geometric mean is what turns the starting value into the ending value, and it is always at or below the arithmetic mean whenever returns vary.

A portfolio returns +10% in year one and -1% in year two. Its cumulative two-year return, properly computed by GEOMETRIC LINKING, is:

1. A.9.0%Wrong-but-tempting. Adding percentages ignores compounding on the changed base.
2. B.8.9%Correct. (1.10)(0.99) - 1.
3. C.11.0%Wrong. No valid arithmetic yields 11 from these inputs.
4. D.4.45% per year guaranteedWrong. Halving the sum mislabels an average as a guarantee.

Why: Cumulative performance multiplies sub-period growth factors: 1.10 x 0.99 = 1.089, an 8.9% total - simple addition overstates results whenever returns vary. Citation: geometric return linking. Takeaway: multiply the factors - never add the percentages.