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st: Return-based style analysis / Shape's Quadratic Programming technique


From   "Christopher Garlich" <cgarlich@mail.uni-mannheim.de>
To   <statalist@hsphsun2.harvard.edu>
Subject   st: Return-based style analysis / Shape's Quadratic Programming technique
Date   Sat, 10 Sep 2011 14:12:04 +0200

Hi everybody,

studying mutual funds, I would like to rebuild Sharpe's (1988/92) Quadratic
Programming technique (which is a return-based style analysis) within STATA.


Basically, it's a simple regression of portfolio returns on the returns of a
set of asset classes (e.g. stocks, bonds, etc.).
So if you had just 2 asset classes, it would look something like this:
R(portfolio) =  b1*R(stocks) + b2*R(bonds)

But in order to determine portfolio weights that actually make sense to an
(long-only) investor, one has to impose 2 restrictions:
b1 + b2 + ... + bn = 1
and 0 <= b <= 1, for all b

In forums I have seen suggestions that using nonlinear least-squares [nl]
will do the trick if you respecify the coefficients to be estimated in the
following way:     
b1 = 1/(1+exp(b'2))
a2 = exp(b'2)/(1+exp(b'2))
...and then recover the betas that you are looking for by using:
nlcom (1/(1+exp(_b[/b2]) ///
exp(_b[/b2])/(1+exp(_b[/b2])


But isn't there a more intuitive way? Additionally, I would definitely need
the R2 from this regression but I wouldn't know how to recover this from the
transformation above.
Has anyone encountered this (or a similar) problem before?


Thanks and regards,
Chris


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