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

Michael Ashton m.ashton at enduringinvestments.com
Tue Dec 15 05:14:17 CET 2015

```All eigenvalues are positive.

From: Krishna Kumar [mailto:kk2250 at optonline.net]
Sent: Monday, December 14, 2015 9:27 PM
To: Michael Ashton
Cc: Michael Weylandt; r-sig-finance at r-project.org
Subject: Re: [R-SIG-Finance] solnp Problem Inverting Hessian

Michael,

Perhaps your covariance matrix is singular (non positive definite) so you can "fix" this matrix by finding the nearest correlation matrix that is non singular.

The simplest recipe is to  set the negative eigenvalues to a small positive number and rescale the  matrix.

Also look at nearPD in the Matrix package.

Best

Krishna

----

On Dec 14, 2015, at 9:06 PM, Michael Ashton <m.ashton at enduringinvestments.com<mailto: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<mailto: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<mailto: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,ineqLB
=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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