Difference between revisions of "Geometric mean"

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Geometric mean returns are calculated by taking account of compounding.
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Geometric mean returns or growth are calculated by taking account of compounding.
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(Contrasted with the arithmetic mean, which ignores compounding).
 
(Contrasted with the arithmetic mean, which ignores compounding).
  
For example, the geometric mean return calculated from sample returns of 4%, 5% and 6% is given by:
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(1.04 x 1.05 x 1.06)<sup>(1/3)</sup> -1 = 4.9968%.
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===<span style="color:#4B0082">Example 1: Positive returns or growth</span>===
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The geometric mean return calculated from sample returns of 4%, 5% and 6% is given by:
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(1.04 x 1.05 x 1.06)<sup>(1/3)</sup> - 1  
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= '''4.9968%'''.
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===Relationship between geometric mean and arithmetic mean===
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When returns or growth are positive, geometric means are smaller figures than arithmetic means.
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In Example 1 above, the arithmetic mean is:
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(4% + 5% + 6%) / 3 = '''5.0000%'''
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''The geometric mean of +4.9968% is a smaller positive number than the [[arithmetic mean]] of +5.0000%.''
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On the other hand, when returns or growth are ''negative'', the geometric mean is a larger negative number - further away from zero - than the arithmetic mean.
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===<span style="color:#4B0082">Example 2: Negative returns or decline</span>===
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The geometric mean return calculated from three ''negative'' sample returns of -(4)%, -(5)% and -(6)% is given by:
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( (1 - 0.04) x (1 - 0.05) x (1 - 0.06) )<sup>(1/3)</sup> - 1
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(0.96 x 0.95 x 0.94)<sup>(1/3)</sup> - 1
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= '''-(5.0035)%'''.
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The negative geometric mean of -(5.0035)% is a larger negative number - further away from zero - than the arithmetic mean of -(5.0000)%.
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(The arithmetic mean of the negative returns of -(4)%, -(5)% and -(6)% is the three items added together and divided by 3.)
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== See also ==
 
== See also ==
 
* [[Arithmetic mean]]
 
* [[Arithmetic mean]]
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* [[CAGR]]

Latest revision as of 20:31, 15 January 2016

Geometric mean returns or growth are calculated by taking account of compounding.

(Contrasted with the arithmetic mean, which ignores compounding).


Example 1: Positive returns or growth

The geometric mean return calculated from sample returns of 4%, 5% and 6% is given by:

(1.04 x 1.05 x 1.06)(1/3) - 1

= 4.9968%.


Relationship between geometric mean and arithmetic mean

When returns or growth are positive, geometric means are smaller figures than arithmetic means.

In Example 1 above, the arithmetic mean is:

(4% + 5% + 6%) / 3 = 5.0000%


The geometric mean of +4.9968% is a smaller positive number than the arithmetic mean of +5.0000%.


On the other hand, when returns or growth are negative, the geometric mean is a larger negative number - further away from zero - than the arithmetic mean.


Example 2: Negative returns or decline

The geometric mean return calculated from three negative sample returns of -(4)%, -(5)% and -(6)% is given by:

( (1 - 0.04) x (1 - 0.05) x (1 - 0.06) )(1/3) - 1

(0.96 x 0.95 x 0.94)(1/3) - 1

= -(5.0035)%.


The negative geometric mean of -(5.0035)% is a larger negative number - further away from zero - than the arithmetic mean of -(5.0000)%.

(The arithmetic mean of the negative returns of -(4)%, -(5)% and -(6)% is the three items added together and divided by 3.)


See also