14.17 The Geometric Average Return: Multi-year Returns
Generally, we quote return in annual terms. In order to calculate the return for multiple years, we must arrive at a reasonable, average annual return figure.
Suppose you observe the following three (historical or projected) annual returns:
.10 .25 .35
The simple, arithmetic average return would be:
[.10 + .25 + .35] ÷ 3 = .2333
For the average to be valid in terms of the time value of money, its future value should equal the product of the three observations. However,
(1.2333) 3 ≠ (1.10) (1.25) (1.35)
Again, the conceptually correct average must be consistent with the time value of money and its “(1 + R) n” format. By modifying the line above and asking what the correct average rate, “R,” should be, we arrive at:
(1 + R) 3 = (1.10) (1.25) (1.35)
(1 + R) = [(1.10) (1.25) (1.35)] 1/3
R = [(1.10) (1.25) (1.35)] 1/3 – 1
R = .228981
Thus, the average multi-period return is 0.2290. This calculation is referred to as the “geometric average” and is consistent with the manner in which we do the time value of money. The general notation for this formula requires the use of product summation notation – “Π” (as opposed to using the usual sigma summation notation, Σ). The notation reads as follows:
Geometric Average = [Π (1 + Ri) 1/n] – 1
Question: What would this average be if the 10% observation were negative?
Answer: 0.1495. How did you get this?
Note: Should there be a negative return in the mix as above, the same method should be used as always. The following should make common sense. For example, should one experience a 50% loss and a 100% gain in consecutive years, the geometric average return would be: [(1 + {-0.50}) × (1 + 1)] ½ – 1 = 0.0.
Had you instead calculated the simple average, you would have gotten: [(-0.50) + (1.0)] ÷ 2 = 25%. That cannot be correct!