[R-SIG-Finance] Portfolio Min Variance Optimisation with Turnover Constraints

[Graham Ewen]

Thanks for the reply Alexios.

I apologise I can see quite clearly in the documentation now, in Table 1, that the Covariance minrisk problem is not compatible with the NLP solver, so it should have been obvious to me why the NLP constraints were ignored.

I think the lack of convergence in the solnp formulation is potentially due to the initial weights being too far from the solution under the tight turnover constraint. I will take your advice and try to reformulate and solve as a QP problem.

Thank you,

Graham.

-----Original Message-----
From: alexios ghalanos [mailto:alexios at 4dscape.com]
Sent: 10 April 2015 19:10
To: Graham Ewen; r-sig-finance at r-project.org
Cc: alexios at 4dscape.com
Subject: Re: [R-SIG-Finance] Portfolio Min Variance Optimisation with Turnover Constraints

Hi Graham,

The reason it ignores the turnover constraint is because it is an NLP constraint (scenario based), _not_ a QP constraint. From the documentation:

"Provides a number of custom constraints and their jacobians for use with the NLP representation (both minimum risk and the fractional problem)."

As to the lack of convergence of the solnp formulation you describe, I am guessing that it is because of the bounds placed on the ineqUB
(0.2) for the turnover constraint and the presence of the absolute function which is not smooth. To solve this as a QP problem, the abs function would require that you represent it as a second order cone problem and solve it with an SOCP solver. Alternatively, if you can write out the gradients of your functions, you _may_ have better luck with the nloptr solver (see ineqjac.turnover.min for an example).

Regards,

Alexios



On 10/04/2015 19:54, Graham Ewen wrote:
> Hi,
>
>
>
> I am new to using R for optimisation and I am trying to set up a
> simple portfolio minimum variance optimisation with turnover
> constraints, but am having some difficulty setting up the problem
> correctly. I would really appreciate any help. I am trying two
> approaches one using the Rsolnp library and the other using the the
> parma library, but am having trouble with both.
>
>
>
> Using Rsolnp I am having trouble getting the optimisation to converge,
> but this may be that the problem is poorly conditioned but please
> advise if the formulation seems correct.
>
>
>
> ineq.fun.turnover <- function (x, sigma, wInit)
>
> {
>
>   return(sum(abs(x - wInit)))
>
> }
>
>
>
> obj.fun <- function(x, sigma, wInit)
>
> {
>
>   return(as.numeric(x%*%sigma%*%x))
>
> }
>
>
>
> eq.fun <- function(x,sigma,wInit)
>
> {
>
>   return(sum(x))
>
> }
>
>
>
> tst <- solnp(initWeights, fun=obj.fun, eqfun=eq.fun, eqB = 1.0,
> ineqfun = ineq.fun.turnover,
>
>              ineqLB = 0.0, ineqUB = 0.2, LB =
> rep(0,numberOfSecurities), UB = rep(0.1,numberOfSecurities),
>
>              sigma = covarMatx, wInit = initWeights)
>
>
>
>
>
> Where initWeights is a vector of length numberOfSecurities, and the
> covarMatx is a covariance Matrix (generated from a factor model) of
> dimensions numberOfSecurities by numberOfSecurities.
>
>
>
> Using parma, I cannot seem to get parmaspec to recognise the turnover
> constraint inequality functions (using the custom ones in the
> library), so I am guessing I am using incompatible parameters.... ?
>
>
>
> servars = list()
>
> uservars$turnover <- 0.2
>
> uservars$wold <- initWeights
>
>
>
> spec2 <- parmaspec(S = covarMatx,  LB = lowerBounds, UB = upperBounds,
>
>                   risk = "EV", riskType = "minrisk",
>
>                   ineqfun = list(ineqfun.turnover.min), ineqgrad =
> list(ineqjac.turnover.min), uservars = uservars)
>
>
>
>
>
> It simply tries to solve the QP min risk problem ignoring the turnover
> inequality.
>
>
>
>
>
> Thank you,
>
>
>
> Graham.
>
>
>
>
>
>
>
> *Graham Ewen*
> BlueCrest Capital Management (UK) LLP
> 40 Grosvenor Place, London, SW1X 7AW, United Kingdom
> *T*  +44 20 3180 3530
> graham.ewen at bluecrestcapital.com
> <mailto:graham.ewen at bluecrestcapital.com>
>
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