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Returns used to compute the alpha and the beta

5 messages · Benoit.Schmid at unige.ch, julien cuisinier, Benoit Schmid

Original
Benoit.Schmid at unige.ch
# 
Hello again,

Quoting julien cuisinier <j_cuisinier at hotmail.com>:
Just to be sure, let's use the following article as a base:
http://www.riskglossary.com/link/return.htm

For time aggregation, they use n*z for logr.
What you are suggesting is to use (1+z)^n-1
instead of n*z.
Am I right?

Thanks for your answer.
julien cuisinier
# 
Hello,
 
Please look at the attached example in the spreadsheet. 
 
The closest I got to "real return" if by using geometric annualization 
 
The link you sent me seems to be correct in the sense that daily returns can be seen as not compounding through the day, but I have harder to consider non compounding of daily return...
 
I guess it depends what is the underlying of the returns...for a stock, one can consider the return as compounding every minute - hence the use of geometric annualization of geometric returns...for an other investment where "return" such as interest are compounded only once a year it might be wise to use arithmetic annualization of arithmetic returns...
 
Personally, the key points is geometric annualization of an average return that make the difference - using arithmetic or geometric returns does not makes much differences...
 
 
Hope that helps
 
Rgds,
Julien
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Benoit Schmid
# 
Good morning,

Thanks for providing the file.

For comparing the averages, you need to to use the same samples.

For the arithmetic averages, you take the values that are in 
the rows 4 up to 42.
For the geometric average, you take the values that are 
in the rows 4 up to 255.

This is why we have very different values.

Your summary is:
	Arithmetic returns	Geometric returns	
arithmetic	-0.13	-0.13	
geometric	-0.07	-0.07	
"real" return	-0.07	-0.07	

If you inverse the sampling selection
(4-42 for geometric average and 4-255 fo arithmetic average)
you get:
	Arithmetic returns	Geometric returns	
arithmetic	-0.07	-0.07	
geometric	-0.12	-0.13	
"real" return	-0.07	-0.07	

Which could be interpreted as arithmetic is better as geometric.

If you use the same samples for both (rows 4-255),
we get a very close value
	Arithmetic returns	Geometric returns
arithmetic	-0.07	-0.07
geometric	-0.07	-0.07		
"real" return	-0.07	-0.07

Basically in you example all means converges to the real values.
This is the case because daily return are small.
Therefore (1+x)*(1+y) ~ 1+x+y because xy is very small.

See you,
julien cuisinier wrote:
Benoit Schmid
# 
Good morning,

Just to add to my previous mail.
This is what you get by amplifying the variation by a factor of 10
in your serial time.

	Arithmetic returns	Geometric returns
arithmetic	1.9	-0.08
geometric	5.67	-0.08
"real" return	-0.08	-0.08

You clearly see that arithmetic returns give bad results, even with
geometric aggregation.

In this example we have the logr that is small (-0.08).
This is why we have logr ~ netr

See you,
julien cuisinier wrote:
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Benoit Schmid
# 
Thanks for the small R code.

I agree with you because the expected value of the exponential
is not equal to the exponential of the expected value.
But the approximation works quite well (E(exp(x)) ~ exp(E(x)).