Stock returns, arithmetic and geometric averages

◈ 8 cards

Total return with the dividend, annualised over a fractional year; and why −40 % then +60 % is not +10 % — the geometric mean is what your money actually grew at.

Total return on a share

You buy Ossington Brewing at $62.00 * *, co l l ec t * *$ 1.40 in dividends over the holding period, and sell at $68.50 after 18 months. The holding-period return (HPR) counts both the price change and the income:

$HPR = \frac{P _{1} - P _{0} + D}{P _{0}} = \frac{68.50 - 62.00 + 1.40}{62.00}$

=(68.5-62+1.40)/62 = 0.127419 — 12.74 % over the eighteen months. Leave out the dividend and you understate the return by 1.40 ÷ 62 = 2.26 points; the paper always includes it.

Annualising

An 18-month return is not comparable with a one-year bond yield until it is expressed per year. Compound, do not divide:

$r_{annual} = (1 + HPR)^{1/ years} - 1 = 1.12741 9^{1/1.5} - 1$

=(1+0.127419)^(1/1.5)-1 = 0.083237 — 8.32 % a year. Dividing 12.74 % by 1.5 gives 8.49 %, which is wrong for the same reason that simple interest is wrong for a compound instrument: the first year's return earned a return in the half-year that followed.

Two averages — and why they differ

Lakehead Robotics returns −40 % in year 1 and +60 % in year 2. The arithmetic mean is =AVERAGE(-0.40,0.60) = 10 % a year. But watch the money: 100 becomes =100*0.6 = 60, then =60*1.6 = 96. Two years, and you are down 4 %. The geometric mean is the constant rate that turns 100 into 96 in two years:

$r_{g} = (\prod (1 + r_{i}))^{1/ n} - 1 = (0.60 \times 1.60)^{1/2} - 1$

=(0.6*1.6)^(1/2)-1 = −0.020204 — −2.02 % a year. That is what the money actually grew at. The arithmetic mean is the average of the rates; the geometric mean is the rate of the wealth. Volatility drives them apart: the geometric mean is never greater than the arithmetic, and the two are equal only when every year's return is identical.

A gentler series — 8 %, −5 %, 12 % — has arithmetic mean =AVERAGE(0.08,-0.05,0.12) = 5.00 % and geometric mean =(1.08*0.95*1.12)^(1/3)-1 = 4.74 %: closer, because the swings are smaller. Excel has =GEOMEAN(1+range)-1 for the same thing, or =(PRODUCT(1+range))^(1/n)-1 entered as an array.

Which average when

Use the geometric mean to report what a fund or a share did — the compound annual growth rate of the wealth. Use the arithmetic mean as the best forecast of a single future year's return from a sample of past years (it is the unbiased estimate of the expected value, Lesson 13.2). A fund that advertises the arithmetic average of a volatile history is advertising a number nobody earned.

Recompute

A 20 % return earned over 30 months: =(1.20)^(1/2.5)-1 = 7.57 % a year. Three fresh years, 15 %, −10 %, 20 %: arithmetic 8.33 %, geometric =(1.15*0.90*1.20)^(1/3)-1 = 7.49 %. Cold: total return includes the dividend; annualise by the root, not by division; geometric ≤ arithmetic.

Where the two averages part company. The arithmetic mean averages the rates and says +10 %; the wealth index says 100 became 96, and the geometric mean reports that: −2.02 % a year.