# Difference between revisions of "Geometric mean"

(Standardise appearance of page) |
(Link with CAGR page and expand page.) |
||

Line 1: | Line 1: | ||

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

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

− | + | ===<span style="color:#4B0082">Example 1: Positive returns or growth</span>=== | |

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

Line 10: | Line 10: | ||

( 1.04 x 1.05 x 1.06 )<sup>(1/3)</sup> - 1 | ( 1.04 x 1.05 x 1.06 )<sup>(1/3)</sup> - 1 | ||

− | = 4.9968%. | + | = '''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'', this relationship is reversed. | ||

+ | |||

+ | |||

+ | ===<span style="color:#4B0082">Example 2: Negative returns or decline</span>=== | ||

+ | |||

+ | 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) )<sup>(1/3)</sup> - 1 | ||

+ | |||

+ | ( 0.96 x 0.95 x 0.94 )<sup>(1/3)</sup> - 1 | ||

+ | |||

+ | = '''-(5.0035)%'''. | ||

+ | |||

+ | |||

+ | ''The negative geometric mean of -(5.0035)% is a bigger negative number 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 == | == See also == | ||

* [[Arithmetic mean]] | * [[Arithmetic mean]] | ||

+ | * [[CAGR]] |

## Revision as of 08:46, 1 December 2015

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

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

## Contents

### 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*, this relationship is reversed.

### 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 bigger negative number 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.)*