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### Expected Portfolio Return Maximization Formulation

Posted: **March 12th, 2018, 9:45 am**

by **GiuseppeAlesii**

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.

max_{w}(w' \mu)

s.t.

w' \Sigma w = \sigma^{2}_{0}

w' i = 1

Could someone please let me know some references about this?

### Re: Expected Portfolio Return Maximization Formulation

Posted: **March 12th, 2018, 11:35 am**

by **bearish**

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.

max_{w}(w' \mu)

s.t.

w' \Sigma w = \sigma^{2}_{0}

w' i = 1

Could someone please let me know some references about this?

The Wikipedia entry on modern portfolio theory is not bad.

### Re: Expected Portfolio Return Maximization Formulation

Posted: **March 12th, 2018, 2:38 pm**

by **GiuseppeAlesii**

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.

max_{w}(w' \mu)

s.t.

w' \Sigma w = \sigma^{2}_{0}

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.

### Re: Expected Portfolio Return Maximization Formulation

Posted: **March 13th, 2018, 10:14 pm**

by **Alan**

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.

### Re: Expected Portfolio Return Maximization Formulation

Posted: **March 14th, 2018, 4:56 pm**

by **GiuseppeAlesii**

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 ... note14.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?

### Re: Expected Portfolio Return Maximization Formulation

Posted: **March 14th, 2018, 5:25 pm**

by **bearish**

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 ... note14.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.

### Re: Expected Portfolio Return Maximization Formulation

Posted: **March 14th, 2018, 10:14 pm**

by **GiuseppeAlesii**

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 ... note14.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

### Re: Expected Portfolio Return Maximization Formulation

Posted: **June 20th, 2019, 3:13 pm**

by **ikicker**

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.

max_{w}(w' \mu)

s.t.

w' \Sigma w = \sigma^{2}_{0}

w' i = 1

Could someone please let me know some references about this?

Good news: This equation exists. They figured out the precise portfolio maximum return with no constraints computation using matrix algebra at NYU and published it in a paper. Check their papers. It will give you negative weights if you use it (the optimal portfolio value includes short positions). There is no practical application because most people don't maximize returns without constraints.

Bad news: I didn't write it down because I use linear programming. I have it saved somewhere in an excel file, but I don't want to look for it that badly. If you really, really need it I can look.

You can easily compute portfolio return using matrix algebra. Corrected - Portfolio variance = W'CW, where W = weight matrix and C = covariance matrix. Your portfolio returns is just weights times position variances. Use linear programming to solve for the weights while minimizing standard deviation given a return. The sum of the weights must be equal to 1. The individual weights may or may not have to be greater than zero, depending up whether or not you have the ability to hold short positions.

### Re: Expected Portfolio Return Maximization Formulation

Posted: **June 20th, 2019, 9:32 pm**

by **GiuseppeAlesii**

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.

max_{w}(w' \mu)

s.t.

w' \Sigma w = \sigma^{2}_{0}

w' i = 1

Could someone please let me know some references about this?

Good news: This equation exists. They figured out the precise portfolio maximum return with no constraints computation using matrix algebra at NYU and published it in a paper. Check their papers. It will give you negative weights if you use it (the optimal portfolio value includes short positions). There is no practical application because most people don't maximize returns without constraints.

Bad news: I didn't write it down because I use linear programming. I have it saved somewhere in an excel file, but I don't want to look for it that badly. If you really, really need it I can look.

You can easily compute portfolio return using matrix algebra. Portfolio return = W'CW, where W = weight matrix and C = covariance matrix. Your portfolio standard deviation is just weights times position variances. Use linear programming to solve for the weights while minimizing standard deviation given a return. The sum of the weights must be equal to 1. The individual weights may or may not have to be greater than zero, depending up whether or not you have the ability to hold short positions.

thanks for your interest in my query. I would be really interested in studying the NYU papers which you suggest. Would you please post some URL or reference so that I can get them.

Also, I would very interested in examining the worksheets you have used applying linear programming to the topic. I wonder how could you solve with linear programming a problem which has weights squared and cross products of them. Finally, why do you write that Portfolio return is equal to W' C W ?

to the best of my modest knowledge to the topic this is the portfolio variance.

Thank in advance for your replies.

### Re: Expected Portfolio Return Maximization Formulation

Posted: **June 20th, 2019, 10:24 pm**

by **ikicker**

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.

max_{w}(w' \mu)

s.t.

w' \Sigma w = \sigma^{2}_{0}

w' i = 1

Could someone please let me know some references about this?

Good news: This equation exists. They figured out the precise portfolio maximum return with no constraints computation using matrix algebra at NYU and published it in a paper. Check their papers. It will give you negative weights if you use it (the optimal portfolio value includes short positions). There is no practical application because most people don't maximize returns without constraints.

Bad news: I didn't write it down because I use linear programming. I have it saved somewhere in an excel file, but I don't want to look for it that badly. If you really, really need it I can look.

You can easily compute portfolio return using matrix algebra. Portfolio return = W'CW, where W = weight matrix and C = covariance matrix. Your portfolio standard deviation is just weights times position variances. Use linear programming to solve for the weights while minimizing standard deviation given a return. The sum of the weights must be equal to 1. The individual weights may or may not have to be greater than zero, depending up whether or not you have the ability to hold short positions.

thanks for your interest in my query. I would be really interested in studying the NYU papers which you suggest. Would you please post some URL or reference so that I can get them.

Also, I would very interested in examining the worksheets you have used applying linear programming to the topic. I wonder how could you solve with linear programming a problem which has weights squared and cross products of them. Finally, why do you write that Portfolio return is equal to W' C W ?

to the best of my modest knowledge to the topic this is the portfolio variance.

Thank in advance for your replies.

Sorry, I might be inverting them. I'm not looking at my spreadsheet. W'CW = portfolio variance; sum(weights x expected return) = expected portfolio return. I amended the correction above.

Also, it's technically called quadratic programming. You can't use simplex method to solve. You have to use a non-linear solver. The squares are a non-issue.