[R-SIG-Finance] comparing solve.pq and nloptr for min variance portfolio: nloptr needs resampling

[Alec Schmidt]

Hi  Enrico,
Indeed I had a bug in defining gradient. I attach the corrected script. But nloptr still needs resampling:

Say when I use the initial condition   x0<-rep(1/n, n), I get
        XLY XLF       XLV     Std.Dev  Exp.Return    sharpe
0.4957209   0 0.5042791 0.007334353 0.001318627 0.1797878

For x0<-c(0,1,0), I get
        XLY XLF       XLV     Std.Dev  Exp.Return    sharpe
0.4352796   0 0.5647204 0.007322604 0.001316634 0.1798041

For x0<-c(1,0,0), I get
       XLY XLF      XLV     Std.Dev  Exp.Return    sharpe
0.594686   0 0.405314 0.007378273 0.001321891 0.1791599

And finally for x0<-c(0,0,1), the  nloptr result 
        XLY          XLF       XLV     Std.Dev  Exp.Return    sharpe
0.4752062 3.665469e-19 0.5247938 0.007329077 0.001317951 0.1798249

practically coincides with that for solve.QP (with default initial condition) with max Sharpe value:
        XLY          XLF       XLV     Std.Dev  Exp.Return    sharpe
0.4752112 4.308351e-18 0.5247888 0.007329079 0.001317951 0.1798249

For other data samples (say for 2011-2013),  x0<-rep(1/n, n) may give the same result as solve.qp.

I wonder what is default initial condiiton in solve.qp and if nloptr sensitivity to initial conditions was ever discussed.

Best, Alec 
________________________________________
From: Enrico Schumann <es at enricoschumann.net>
Sent: Saturday, March 19, 2016 4:14 PM
To: Alec Schmidt
Cc: R-SIG-Finance at r-project.org
Subject: Re: [R-SIG-Finance] comparing solve.pq and nloptr for min variance portfolio

On Fri, 18 Mar 2016, Alec Schmidt <aschmid1 at stevens.edu> writes:

> Enrico,
> Here we're. I attach two scripts: one for solve.pq, another for
> nloptr. Both run with the same input file and print output on the
> screen.
>
> Thanks again, Alec

Hi Alec,

but these are still long programmes, which makes it hard to
figure out what exactly is wrong. For a start, I would not
compute the whole frontier, but concentrate on one point.

I computed the minimum-variance portfolio (without short
sales) with an alternative method, and it matches your
output for 'solve.QP'. So, one thing to check is your
implementation of the objective function for 'nloptr'.

Kind regards (and good luck)
     Enrico


> ________________________________________
> From: Enrico Schumann <es at enricoschumann.net>
> Sent: Friday, March 18, 2016 11:08 AM
> To: Alec Schmidt
> Cc: R-SIG-Finance at r-project.org
> Subject: Re: [R-SIG-Finance] comparing solve.pq and nloptr for min variance portfolio
>
> On Fri, 18 Mar 2016, Alec Schmidt <aschmid1 at stevens.edu> writes:
>
>> Hi Enrico,
>> Many thanks for your interest. I attach my script and input file with
>> asset tickers. Sorry for lots of unrelated stuff - it's a working
>> draft.
>>
>> Alec
>
> Thanks for sending the script, Alec. But you will need to
> simplify it if people are to help you. [My bad: I should have
> said _minimal_ reproducible example:
> https://stackoverflow.com/questions/5963269/how-to-make-a-great-r-reproducible-example
> ]
>
>
>
>> ________________________________________
>> From: Enrico Schumann <es at enricoschumann.net>
>> Sent: Friday, March 18, 2016 10:25 AM
>> To: Alec Schmidt
>> Cc: R-SIG-Finance at r-project.org
>> Subject: Re: [R-SIG-Finance] comparing solve.pq and nloptr for min variance portfolio
>>
>> On Fri, 18 Mar 2016, Alec Schmidt <aschmid1 at stevens.edu> writes:
>>
>>> I'm puzzled that I cannot reproduce results for asset weights using
>>> solve.pq and nloptr even in the case of just three assets.  E.g. if I
>>> use NLOPT_LD_SLSQP and start with initial weights of 1/3, I may obtain
>>> (0.47, 0, 0.53) vs (0.52, 0, 0.47).  If I start with (0.52, 0, 0.47),
>>> I do get (0.52, 0, 0.47)...
>>>
>>> When I use NLOPT_GN_ISRES or other nloptr solvers that permit equality
>>> constraint sum(weights)=1 with initial weights of 1/3, I obtain
>>> (almost) the same initial weights after 20000 iterations with
>>> xtol_rel=1.0e-8...
>>>
>>> I remember from my MC simulations of protein structures (20 years ago)
>>> that sampling is key due to multiple local minimums but is it so bad
>>> for a simple portfolio?
>>>
>>>
>>> I'll greatly appreciate relevant comments.
>>>
>>> Alec
>>
>> [...]
>>
>> Unless your covariance matrix is 'broken' in some way, a
>> minimum-variance portfolio with only a budget constraint should be
>> fairly easy to compute (no multiple local minima, smooth objective
>> function, ...). Please provide a reproducible example.
>>
>> Kind regards,
>>         Enrico

--
Enrico Schumann
Lucerne, Switzerland
http://enricoschumann.net
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