# [R-SIG-Finance] solnp Problem Inverting Hessian

Michael Ashton m.ashton at enduringinvestments.com
Tue Dec 15 16:17:48 CET 2015

```That is very interesting. Actually some of the projected returns are negative, some are very close to zero. None is exactly zero, which I guess is why it's not singular, but that's why it's ill-conditioned. Very interesting. And hard to solve...if I add 5% to all returns will I still get the same portfolio? Seems to me intuitively I shouldn't since the proportional risk/return tradeoff then is greater for the assets on the low end of the spectrum, but I might be wrong.

-----Original Message-----
From: Paul Gilbert [mailto:pgilbert902 at gmail.com]
Sent: Tuesday, December 15, 2015 9:58 AM
To: Michael Ashton
Cc: Michael Weylandt; r-sig-finance at r-project.org
Subject: Re: [R-SIG-Finance] solnp Problem Inverting Hessian

I have not been paying close attention and I've missed something in this thread, but ...

I think the hessian in the optimization will be the second derivative of this function

>>>>opt.fun <- function(wgt.vect) {-crossprod(wgt.vect,t(Ret.vect))}

WRT  wgt.vect. If there are zero elements in Ret.vect then that hessian will be singular. And if there are orders of magnitude difference between the largest and smallest elements in Ret.vect then the hessian will be ill-conditioned.

The ill-conditioned case is a numerical problem and you might solve it with a different algorithm, or by better scaling. The zero case is a theoretical problem, there is not a unique optimal point but rather whole continuum of optimal points (your parameters corresponding to the zero elements will not make any difference in the function value).

HTH,
Paul

On 12/14/2015 11:39 PM, Michael Weylandt wrote:
> (+list -- I'm not a numerical linear algebra expert or portfolio
> optimization expert, but there are a number on this list who might
> chime in)
>
> That is quite high, and almost certainly problematic.
>
> I'm not sure what an acceptable bound is (will depend on your solver),
> but that's almost certainly above it for any algorithm which will
> require inverting cov.mat.
>
> Two things you could try:
> - use a different (non-Hessian-using) optimization algorithm;
> - use some form of shrinkage/regularization (Ledoit-Wolf is a common
> choice) to tame your covariance matrix.
>
> Michael
>
> On Mon, Dec 14, 2015 at 10:10 PM, Michael Ashton
> <m.ashton at enduringinvestments.com> wrote:
>> Well, not sure whether this makes any sense but kappa(cov.mat) gives me 11148245007.
>>
>> That seems large. But I am not sure what it is supposed to be.
>>
>> -----Original Message-----
>> From: Michael Weylandt [mailto:michael.weylandt at gmail.com]
>> Sent: Monday, December 14, 2015 9:29 PM
>> To: Michael Ashton
>> Cc: r-sig-finance at r-project.org
>> Subject: Re: [R-SIG-Finance] solnp Problem Inverting Hessian
>>
>> What's the condition number of your correlation/covariance matrix?
>> (?kappa)
>>
>> I think I've used quadprog for portfolio optimization with success, but it's been a while.
>>
>> MW
>>
>> On Mon, Dec 14, 2015 at 8:06 PM, Michael Ashton <m.ashton at enduringinvestments.com> wrote:
>>> Do you have a suggestion for such? I have in the past tried fPortfolio, but it would not allow me to specify my own projected return vectors rather than the historical returns of the series (which is exactly backwards).
>>>
>>> As for whether the matrix is comfortably non-singular...I suppose it depends in a Clintonian way on the meaning of "comfortably," but I can create a Cholesky decomposition without it blowing up, which is usually how I can tell if I have done something stupid. Well, stupider than normal.
>>>
>>> -----Original Message-----
>>> From: Michael Weylandt [mailto:michael.weylandt at gmail.com]
>>> Sent: Monday, December 14, 2015 8:32 PM
>>> To: Michael Ashton
>>> Cc: r-sig-finance at r-project.org
>>> Subject: Re: [R-SIG-Finance] solnp Problem Inverting Hessian
>>>
>>> The Hessian is the matrix of second derivatives
>>> (https://en.wikipedia.org/wiki/Hessian_matrix) -- in scalar terms, you're finding a point where the second derivative is zero and then trying to divide by the second derivative to calculate the step size.
>>>
>>> I haven't gone through your code in any detail, but I'd start by checking the covariance matrix since that's proportional to the Hessian of your objective function. Is it (comfortably) non-singular?
>>>
>>> Since you're just solving the standard Markowitz problem, you might try a simpler (quadratic/convex) solver instead of a general non-linear solver. Should behave a bit better.
>>>
>>> Michael
>>>
>>>
>>> On Mon, Dec 14, 2015 at 6:13 PM, Michael Ashton <m.ashton at enduringinvestments.com> wrote:
>>>> I must admit to being flummoxed here, mainly because my linear algebra was 25 years ago and I can't remember what a Hessian is.
>>>>
>>>> I have a matrix of 60 securities' weekly returns, along with 60 projected returns. The returns are in a vector called Ret.vect and the covariance matrix of weekly returns in cov.mat . I have the minConstraints and maxConstraints that the parameters are permitted to take. I cycle through targeted risks and get the same error for each risk targeted...below I have removed the loop to focus on the risk=0.002.
>>>>
>>>> wgt.vect=c(rep(1/60, 60))
>>>> constr.fun <- function(wgt.vect) {;
>>>>                  c1 = sqrt(crossprod(t(wgt.vect %*% cov.mat),wgt.vect));
>>>>                  c2 = sum(wgt.vect);
>>>>                  return(c(c1,c2));
>>>>                  }
>>>> ineqconstr.fun <- function(wgt.vect) {
>>>>                  wgt.vect[1:60];
>>>>                  }
>>>> opt.fun <- function(wgt.vect) {-crossprod(wgt.vect,t(Ret.vect))}
>>>>
>>>> OptimSolution <-
>>>> solnp(wgt.vect,opt.fun,constr.fun,eqB=c(0.002,1),ineqconstr.fun,ine
>>>> qL
>>>> B
>>>> =minConstraints,ineqUB=maxConstraints)
>>>>
>>>> I get the following error:
>>>> solnp--> Solution not reliable....Problem Inverting Hessian.
>>>>
>>>> Well, that doesn't tell me very much! The parameters (weights) that are output for each run, as I cycle through the weights, are very scrambled...lots of little allocations, rather than clumping as you would expect to happen especially at the risky and riskless ends of the spectrum.
>>>>
>>>> Can anyone with more math than me give me a helping hand on the Hessian?
>>>>
>>>> Thanks,
>>>>
>>>>
>>>> Mike
>>>>
>>>> Michael Ashton, CFA
>>>> Managing Principal
>>>>
>>>> Enduring Investments LLC
>>>> W: 973.457.4602
>>>> C: 551.655.8006
>>>>
>>>>
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