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GiuseppeAlesii
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Posts: 20
Joined: December 8th, 2013, 3:54 pm

Expected Portfolio Return Maximization Formulation

March 12th, 2018, 9:45 am

hi there,
I am looking for
-- a numerical method 
and
-- a closed form solution 
for the expected return formulation of the classical mean variance portfolio optimization problem.
maxw(w' \mu) 
s.t.
w' \Sigma w = \sigma20
w' i = 1
Could someone please let me know some references about this?
 
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bearish
Posts: 3131
Joined: February 3rd, 2011, 2:19 pm

Re: Expected Portfolio Return Maximization Formulation

March 12th, 2018, 11:35 am

GiuseppeAlesii wrote:
hi there,
I am looking for
-- a numerical method 
and
-- a closed form solution 
for the expected return formulation of the classical mean variance portfolio optimization problem.
maxw(w' \mu) 
s.t.
w' \Sigma w = \sigma20
w' i = 1
Could someone please let me know some references about this?


The Wikipedia entry on modern portfolio theory is not bad.
 
User avatar
GiuseppeAlesii
Topic Author
Posts: 20
Joined: December 8th, 2013, 3:54 pm

Re: Expected Portfolio Return Maximization Formulation

March 12th, 2018, 2:38 pm

bearish wrote:
GiuseppeAlesii wrote:
hi there,
I am looking for
-- a numerical method 
and
-- a closed form solution 
for the expected return formulation of the classical mean variance portfolio optimization problem.
maxw(w' \mu) 
s.t.
w' \Sigma w = \sigma20
w' i = 1
Could someone please let me know some references about this?


The Wikipedia entry on modern portfolio theory is not bad

Thanks a lot for your post, but it does not suit to my query which is much more specific.
In particular, I have a doubt that the usual Lagrangian constrained optimization framework applies to the case of return maximization formulation. To be specific, in the first order condition related to the constraint on volatility weights appear squared and with cross products. And this prevents the derivation of a system of linear equations like in the dual problem, i.e. risk minimization given an expected portfolio return.
 
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Alan
Posts: 9369
Joined: December 19th, 2001, 4:01 am
Location: California
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Re: Expected Portfolio Return Maximization Formulation

March 13th, 2018, 10:14 pm

I am probably missing something, but why isn't every solution to your problem also a solution to what you call the 'dual problem'? If so, just use the latter.
 
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GiuseppeAlesii
Topic Author
Posts: 20
Joined: December 8th, 2013, 3:54 pm

Re: Expected Portfolio Return Maximization Formulation

March 14th, 2018, 4:56 pm

Alan wrote:
I am probably missing something, but why isn't every solution to your problem also a solution to what you call the 'dual problem'? If so, just use the latter.

there are some references which hint to a possible solution with a Lagrangian, for instance on page 7 in 
https://ocw.mit.edu/courses/mathematics/18-s096-topics-in-mathematics-with-applications-in-finance-fall-2013/lecture-notes/MIT18_S096F13_lecnote14.pdf
it looks like that the expression "equivalent Lagrangian" hints to the possible construction of a Lagrangian for the expected return formulation.
Moreover, on page 34 of Fabozzi, Kolm, Pachamanova, Focardi 2007, "the expected return formulation ... is often used by portfolio managers that are required to not take more risk, as measured by the standard deviation of the portfolio return, than a certain pre specified volatility"
They also cite "index tracking" in which, under a given tracking error, the expected excess return of the portfolio over the benchmark is taken as objective function to maximize.
Therefore, it seems that, at least numerically, the problem has been solved. Any hint? 
 
User avatar
bearish
Posts: 3131
Joined: February 3rd, 2011, 2:19 pm

Re: Expected Portfolio Return Maximization Formulation

March 14th, 2018, 5:25 pm

GiuseppeAlesii wrote:
Alan wrote:
I am probably missing something, but why isn't every solution to your problem also a solution to what you call the 'dual problem'? If so, just use the latter.

there are some references which hint to a possible solution with a Lagrangian, for instance on page 7 in 
https://ocw.mit.edu/courses/mathematics/18-s096-topics-in-mathematics-with-applications-in-finance-fall-2013/lecture-notes/MIT18_S096F13_lecnote14.pdf
it looks like that the expression "equivalent Lagrangian" hints to the possible construction of a Lagrangian for the expected return formulation.
Moreover, on page 34 of Fabozzi, Kolm, Pachamanova, Focardi 2007, "the expected return formulation ... is often used by portfolio managers that are required to not take more risk, as measured by the standard deviation of the portfolio return, than a certain pre specified volatility"
They also cite "index tracking" in which, under a given tracking error, the expected excess return of the portfolio over the benchmark is taken as objective function to maximize.
Therefore, it seems that, at least numerically, the problem has been solved. Any hint? 


The approach I have seen people take in practice is to solve the dual problem for a given expected return level and iterate until the resulting volatility (typically tracking error) equals the target.
 
User avatar
GiuseppeAlesii
Topic Author
Posts: 20
Joined: December 8th, 2013, 3:54 pm

Re: Expected Portfolio Return Maximization Formulation

March 14th, 2018, 10:14 pm

bearish wrote:
GiuseppeAlesii wrote:
Alan wrote:
I am probably missing something, but why isn't every solution to your problem also a solution to what you call the 'dual problem'? If so, just use the latter.

there are some references which hint to a possible solution with a Lagrangian, for instance on page 7 in 
https://ocw.mit.edu/courses/mathematics/18-s096-topics-in-mathematics-with-applications-in-finance-fall-2013/lecture-notes/MIT18_S096F13_lecnote14.pdf
it looks like that the expression "equivalent Lagrangian" hints to the possible construction of a Lagrangian for the expected return formulation.
Moreover, on page 34 of Fabozzi, Kolm, Pachamanova, Focardi 2007, "the expected return formulation ... is often used by portfolio managers that are required to not take more risk, as measured by the standard deviation of the portfolio return, than a certain pre specified volatility"
They also cite "index tracking" in which, under a given tracking error, the expected excess return of the portfolio over the benchmark is taken as objective function to maximize.
Therefore, it seems that, at least numerically, the problem has been solved. Any hint? 


The approach I have seen people take in practice is to solve the dual problem for a given expected return level and iterate until the resulting volatility (typically tracking error) equals the target.

thanks
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