How to Derive Formula for Geometric Mean

[arthurwhite]

Ok we all know that the formula for geometric mean is , given a set with elements A1... An.

The Geometric mean = (A1xA2x...An)^1/n [1]( Geometric mean - Wikipedia, the free encyclopedia ) .

But there is an alternate formula that is often used in textbooks (finance) which I did not know about it is :

Geometric average return = [(1+R1)x (1+R2) x (1 + R3) ...(1+Rt)]^1/t - 1 [2].

Now I don't see how the two formulae are the same. ? Could someone please show me a simple example and derivation as to how the two are same? The reason why I ask is because Geometric mean's classic formula (see [1]) cannot handle negative values but formula [2] can handle negative values. Could someone be nice enough to explain how [2] is the same as [1] ? I notice someone has given an example here http://www.mrexcel.com/forum/excel-...ic-mean-some-negative-values.html#post1299339 Example (modified)
You invest $50
Year 1: return $50 or 100% (total: $100)
Year 2: return -$50 or -50% (total: $50)

The geometric mean is calculated relative to the total (100%)

=((1+1)*(1-.5))^(1/2)-1= 0%
The question is how do you justify just adding and 1 to every element in the set, take the square root and then subtract 1 from it? and how is that the same as never adding any 1s or subtracting any 1s.

 

[JoeMo]

The question is how do you justify just adding and 1 to every element in the set, take the square root and then subtract 1 from it?

The addition of 1 converts the return as a percentage to return as a factor (return factor = 1 + return%/100). Since your % return can never be less than -100% this ensures that all return factors are >=0. The Excel GEOMEAN function only accepts non-negative numbers, so you need to use the return factor in that function to calculate the average annual return. Here's an example for a ten year period of an investment of $1000.

Excel Workbook
	A	B	C	D	E	F	G
1	Derive the average annual return of a portfolio over 10 years
2	Suppose start with a portfolio value of $1000
3		Yr	Ret (%)	Start	End	Ret as a factor
4		1	10.91	$1,000.00	$1,109.10	1.11
5		2	3.21	$1,109.10	$1,144.70	1.03
6		3	2.21	$1,144.70	$1,170.00	1.02
7		4	28.28	$1,170.00	$1,500.88	1.28
8		5	19.72	$1,500.88	$1,796.85	1.20
9		6	12.93	$1,796.85	$2,029.18	1.13
10		7	16.37	$2,029.18	$2,361.36	1.16
11		8	13.33	$2,361.36	$2,676.13	1.13
12		9	(29.84)	$2,676.13	$1,877.57	0.70
13		10	32.13	$1,877.57	$2,480.83	1.32
14
23					Average Annual Return	9.51	percent
Sheet1

 

[arthurwhite]

Oh this is not what I was asking for. I was looking for a justification of why +1 can be added to each element and then -1 added to the square root (or nth root) of the product of the elements (with 1 added). I know how +1 can make negative values into positive values. What I wished to know was why it is the same as the classical formula (nth root of multiple of elements) etc. Not looking for a demonstration or example. I already provided a simple example . Now if someone could show me why adding +1 to each element and then subtracting 1 from the square root of the products is the same as simply multiplying the elements and taking a square root?

 

[JackBean]

Take an example with two numbers: 1.2 and 0.8

The Geometric Mean (GM) = (1.2*0.8)^(1/2) = 0.979796 = GEOMEAN(1.2,0.8)

The Geometric Mean Return = ((1+1.2)*(1+0.8))^(1/2)-1 = 0.989975 = GEOMEAN((1+{1.2,0.8}))-1

Note that GEOMEAN is the EXCEL Function for GM.

The Geometric Mean Return (GMR) is the same as the Geometric Average Return.

It is time to examine your statement that the GM and GMR are the same.

The GMR is the equivalent monthly or annual compounded change per time interval.

For this example 1.2 = 120% and 0.8 = 80%.

Start with 1.
One year 120% and one year 80% interest compounded, end up with 3.96.
Two years 98.9975% interest compounded, end up with 3.96.

So how is the GMR related to the GM?
There is the equation for the example above:
GMR = GEOMEAN((1+{1.2,0.8}))-1

The proportional changes are 2.20 and 1.8
The GM of the proportional changes is (2.2*1.8)^0.5 = 1.989975
The proportional change is then converted to a fractional change by subtracting 1.

Geometrically this is the square (1.989975x1.989975) for 2 years that is equivalent to:
2.20x1.8
It would be a cube for 3 years and so on.
Note that the order of the returns in each year does not matter,
and this makes sense geometrically.

So the GMR allows the use of fractional changes directly instead of the proportional changes that GM uses.

 

[JoeMo]

Oh this is not what I was asking for. I was looking for a justification of why +1 can be added to each element and then -1 added to the square root (or nth root) of the product of the elements (with 1 added). I know how +1 can make negative values into positive values. What I wished to know was why it is the same as the classical formula (nth root of multiple of elements) etc. Not looking for a demonstration or example. I already provided a simple example . Now if someone could show me why adding +1 to each element and then subtracting 1 from the square root of the products is the same as simply multiplying the elements and taking a square root?

Say you start with an investment of S and after n years that investment is worth E. The average annual rate of return r can be determined from:

E = S (1+r)^n = S (1+R1)(1+R2) .....(1+Rn) where Ri is the rate of return for year i.

Solve for r:
r= (E/S)^(1/n) -1 =[(1+R1)(1+R2) .....(1+Rn)]^(1/n) - 1

r is the geometric mean of the terms (1+Ri) (the bold part of the equation above) minus 1. It is not the geometirc mean, but the geometric mean of the terms (1+Ri) is used in its calculation as I showed by example in my initial post.