[R-SIG-Finance] [R-sig-finance] portfolio.optim and assets with weigth equals to zero...

[Alberto Santini]

After posting  on R-Help it makes sense posting on r-sig-finance too.

I don't understand a particular output of portfolio.optim (tseries).
I have 4 assets and the portfolio.optim returns an asset with weight equals
to zero.
If I do a portfolio.optim with 3 assets, without the asset with weight
equals to zero,
it returns a completely different result.

That's I would expected the same weights as the run with 4 assets.

Below the code.

Thanks in advance,
Alberto Santini


-------------------------------------

require(tseries)

f.mi <- coredata(get.hist.quote("F.MI", start="2006-09-03", compression="w",
quote="Close"))
eng.mi <- coredata(get.hist.quote("ENG.MI", start="2006-09-03",
compression="w", quote="Close"))
tis.mi <- coredata(get.hist.quote("TIS.MI", start="2006-09-03",
compression="w", quote="Close"))
spmib <- coredata(get.hist.quote("^SPMIB", start="2006-09-03",
compression="w", quote="Close"))

f.mi.rets <- diff(log(f.mi[1:(length(f.mi)-1)]))
eng.mi.rets <- diff(log(eng.mi[1:(length(eng.mi)-1)]))
tis.mi.rets <- diff(log(tis.mi[1:(length(tis.mi)-1)]))
spmib.rets <- diff(log(spmib[1:(length(spmib)-1)]))

x <- cbind(f.mi.rets, eng.mi.rets, tis.mi.rets, spmib.rets)
res <- portfolio.optim(x);
res$pw

x2 <- cbind(f.mi.rets, eng.mi.rets, spmib.rets)
res <- portfolio.optim(x2);
res$pw 

----------------------------------------


I tried with Rmetrics too, but the behaviour is the same. So I am missing
something...
Very interesting to understand why. :)

I think it's a numerical problem: maybe, zero is not zero, it's very near to
zero.

Regards,
Alberto Santini

-----------------------------

require(fPortfolio)
require(tseries)

assets <- c(
    "F.MI",
    "ENG.MI",
    "TIS.MI",
    "^SPMIB"
)

f.mi <- get.hist.quote("F.MI", start="2006-09-03", end="2008-09-03",
    compression="w", quote="Close")
eng.mi <- get.hist.quote("ENG.MI", start="2006-09-03", end="2008-09-03",
    compression="w", quote="Close")
tis.mi <- get.hist.quote("TIS.MI", start="2006-09-03", end="2008-09-03",
    compression="w", quote="Close")  
spmib <- get.hist.quote("^SPMIB", start="2006-09-03", end="2008-09-03",
    compression="w", quote="Close")

X <- cbind(f.mi[1:(length(f.mi)-1)], eng.mi[1:(length(eng.mi)-1)],
tis.mi[1:(length(tis.mi)-1)], spmib[1:(length(spmib)-1)])
colnames(X) <- assets

R <- as.timeSeries(returns(X))

# Spec = portfolioSpec(model = list(type = c("MV"),
#  estimator = c("mean", "cov"), tailRisk = list(), params = list()),
#  portfolio = list(weights = NULL, targetReturn = 0,
#    targetRisk = 0, targetAlpha = 0.05, riskFreeRate = NULL,
#    nFrontierPoints = 50),
#  solver = list(solver = c("quadprog"), trace = FALSE))
Spec = portfolioSpec()
frontier <- portfolioFrontier(R, Spec, c("minW[1:nAssets]=0"))
# weightsSlider(frontier)

ptf <-
frontier at portfolio$weights[which.min(getTargetRisk(frontier)[,1])+1,]*100
ptf

assets2 <- c(
    "ENG.MI",
    "^SPMIB"
)

X2 <- cbind(eng.mi[1:(length(eng.mi)-1)], spmib[1:(length(spmib)-1)])
colnames(X2) <- assets2

R2 <- as.timeSeries(returns(X2))
frontier <- portfolioFrontier(R2, Spec, c("minW[1:nAssets]=0"))

ptf <-
frontier at portfolio$weights[which.min(getTargetRisk(frontier)[,1])+1,]*100
ptf

-----------------------------

The asset allocation (in Rmetrics):


    F.MI   ENG.MI   TIS.MI   ^SPMIB
 0.00000 24.30756  0.00000 75.69244

 ENG.MI   ^SPMIB
24.48980 75.51020


> cov(R)

               F.MI       ENG.MI       TIS.MI       ^SPMIB
F.MI   0.0029905040 0.0008974621 0.0010289887 0.0009954717
ENG.MI 0.0008974621 0.0014935276 0.0004278133 0.0002809398
TIS.MI 0.0010289887 0.0004278133 0.0052708621 0.0007186357
^SPMIB 0.0009954717 0.0002809398 0.0007186357 0.0005617209
> cov(R2)
             ENG.MI       ^SPMIB
ENG.MI 0.0014197889 0.0002578205
^SPMIB 0.0002578205 0.0005791276
 
I would think that if a weight is within epsilon of zero, then removing it
from the computation should have minimal impact on the result.

> eigen(cov(R))
$values
[1] 0.0060129474 0.0030233942 0.0010997423 0.0001805307

$vectors
           [,1]       [,2]        [,3]        [,4]
[1,] -0.4145084  0.7702681  0.36304798 -0.32103879
[2,] -0.1770771  0.3541481 -0.91735850  0.04094150
[3,] -0.8700674 -0.4871272 -0.02330958 -0.07173836
[4,] -0.1995222  0.2096978  0.16157418  0.94345721

> eigen(cov(R2))
$values
[1] 0.0014925599 0.0005063566

$vectors
           [,1]       [,2]
[1,] -0.9623985  0.2716415
[2,] -0.2716415 -0.9623985

The difference of the asset allocation is small. It is not so for
portfolio.optim. I will investigate the difference between Rmetric and
tseries, because in my context the solver is the same (quadprog).

Maybe the measure of the risk used in Rmetrics is not the variance as used
in tseries and it is less sensitive. In fact, the two asset allocations are
different.

I have short constraints and the weights don't change if I change the order
of the assets.


Thanks in advance,
Alberto Santini
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