Porfolio Optimization using LaGrange methods

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niceseb
Posts: 7
Joined: Thu Nov 03, 2011 8:25 pm

Porfolio Optimization using LaGrange methods

Postby niceseb » Sun Mar 04, 2012 3:27 am

I have a relatively simple question, normally Porfolio Optimization using Lagrange method uses constraints of:

uT w = m
1T w = 1

then Lagrange function differentiate wrt w and solve.

but what if constraint are:

wT 1 = 1
wT u = 0.1

or simply

wT 1 = 1


do I still differentiate wrt w as before, or I now differeniate wrt wT? Also what do these optimization cases perform?

Regards



Sebastian

coalexander
Posts: 815
Joined: Sun Sep 28, 2008 10:30 pm

Re: Porfolio Optimization using LaGrange methods

Postby coalexander » Sun Mar 04, 2012 11:29 am

Hi

Its usually easier to think of the constraints not in vector form, to get the intuition straight. Let w=(w_1,...,w_n)' where I use ' for transpose and all vectors are column vectors. Also set 1 = (1,1,..,1)'. Then w'1 = 1'w = sum w_i. Similarly, if u = (u_1, ..,u_n)' then u'w = w'u - sum u_i*w_i. So, the contraints can be written either way, the problem is exactly the same.

Carol

niceseb
Posts: 7
Joined: Thu Nov 03, 2011 8:25 pm

Re: Porfolio Optimization using LaGrange methods

Postby niceseb » Sun Mar 04, 2012 9:20 pm

Thanks Prof Alexander, but what if now the constraints are simply

wT 1 = 1

this time, no return constraint of uT w=0.1, just a budget equation to say sum of all weights adding up to 1.
Then in this case this is an unconstrained problem? If so then how to get lambda, mu and subsequently A, B and C
coefficients?



Sebastian

coalexander
Posts: 815
Joined: Sun Sep 28, 2008 10:30 pm

Re: Porfolio Optimization using LaGrange methods

Postby coalexander » Mon Mar 05, 2012 10:43 am

Still a constrained problem but there is an analytic solution, see p243-244 and solution is given in excel spread for example I.6.7


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