2nd letter to friends of Brunei (Measuring Returns over multiple period)

Measuring returns (of investments) over multiple periods builds on the earlier blog entry on total return and relative return. There are 3 common ways of measuing such returns. They are:

1. Cumulative Wealth Index (CWI)

2. Arithmetic Mean (AM)

3. Geometric Mean (GM)

Let's assume you invested into a stock for 5 years as follows:

Year 1, 2, 3, 4, 5

Total Return (TR) 40%, 25%, -20%, -30%, 30%

Relative Return (RR) 1.4, 1.25, 0.8, 0.7, 1.3

Item 1. Cumulative Wealth Index (CWI)

CWI tells you how much you will receive today for every dollar you invested 5 years ago. The formula for CWI is to mulitply all the relative return data. [i.e. (RR1)(RR2)(RR3)(RR4)(RR5)]

In above question, the CWI is (1.4)(1.25)(0.8)(0.7)(1.3) = 1.274.

It means when you invest $1,000 in year 1, you get $1,274 in year 5.

Item 2. Arithmetic Mean (AM)

AM is a measure to help you gauge the return on an average basis. The formula for AM is [sum of total returns / number of total returns].

In above question, the AM is [{40+25+(-20)+(-30)+30} / 5] = 9%

It means if you randomly compare any one-year period within the 5 years of investment, the average return is 9%. (Be careful, sometime AM does not make sense on first look. It merely give you an average return over any 1-year period of a series of years of investment)

Item 3. Geometric Mean (GM)

GM is a measure to help you gauage the return on an annual basis. This is the most common return that appears on professionally managed fund's (e.g. Unit Trusts) factsheet. The formula for GM is {[product of the Relative Returns] }nth root - 1 , where n is the total number of data.

In above question, the GM is

{[1.4 x 1.25 x 0.8 x 0.7 x 1.3]} 1/5 - 1 = 1.0496 - 1 = 0.0496 (or 4.96%)

It means the annual return for this investment is 4.96% p.a.

It is interesting to note that mathematically, AM is either equal or always greater (in value) than GM.