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

[alexios ghalanos]

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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