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Geometric Average Versus Arithmetic Average
W. L. Silber
The Question Suppose you invest $435 in a zero coupon bond for one year and earn a return of 8%. You then reinvest the proceeds at 12% for a second year. How do you describe your average annual return over the two-year period?
A Simple Answer
One possibility is to calculate the simple arithmetic average of the annual returns, (8% + 12%)/2 = 10%. We suspect that it might be more complicated than that because we know that compounding is involved whenever the ‘proceeds of year 1 are reinvested in year 2’. So we check to make sure the arithmetic average tells the truth.
Brute Force Answer We know how to calculate the average annual return, assuming annual compounding, of an initial sum V 0 that grows to Vt over t years:
(1) Rann = ( Vt / V 0 ) 1/t^ - We can also calculate exactly what V 2 will be after 2 years in our case since we know that V 0 = 435. In particular,
(2) V 2 = 435 (1.08) (1.12) = 526. Therefore, we know that in this case,
(3) Rann = (526.176 / 435) ½^ -1 =. Thus, we see that the arithmetic mean is bigger than the true annual average return because we know that .0998 is correct since we calculated it from ‘first principles’, that is, we calculated it using the proper definition of annual returns.
A General Answer Let’s try to formulate an answer for calculating the average annual return over the two year period using the annual rates themselves, 8% and 12%, or more generally, r1 for year 1 and r2 for year 2.
We know the following is true (4) ( V 2 / V 0 ) = ( V 2 / V 1 ) ( V 1 / V 0 )
We also know that by the definition of annual return (V 1 / V 0 ) = 1+r1 and ( V 2 / V 1 ) = 1+r2. Hence, by substitution, expression (4) becomes:
(5) V 2 / V 0 = (1 +r2) (1 + r1) Given equation (1) above, this means (6) Rann = [(V 2 / V 1 ) (V 1 / V 0 )] 1/2^ - 1 (7) = [(1 + r1) (1 + r2)]1/2^ - 1 The last expression is called the geometric average of r1 and r2. Note that if you substitute .08 for r1 and .12 for r2, the calculation in (7) produces .0998, which is the correct answer. More generally, if r1 is the return in year 1, r2 is the return in year 2 and rt is the return in year t, then the correct formula for calculating the average annual
Is the Arithmetic Average Ever Correct?
The arithmetic average is the appropriate way to calculate the average return over more than one period if instead of reinvesting whatever you have at the end of each year, you start each year with the same amount of money, e.g. $100. Note that starting each year with $100 means that if you lose money at the end of a year you must add and if you make money you must withdraw. Here is the numerical example. Start with $100. If you make 100% you have $200. Withdraw $100. Reinvest only $100. If you lose 50% you have $50 at the end and you must put in $50 so you start the next year with $100. Add up the withdrawls ($100 -$50) and you accumulated $50 over 2 years. Thus you made $25 per annum on $100, or 25% per year, which is the arithmetic average of 100% and -50%. We prefer the geometric average because it tells us how an initial sum grows 'untouched by human hands'. But if you like to add and subtract at the end of each year to maintain the same dollar investment (you probably won't like the adding part), then the arithmetic mean tells the truth.
