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I am trying to find a set of hedge ratios by optimising a portfolio for minimum variance under participation constraints of the assets. Ultimately I would like to optimise the assets weights for other measures of risk such as minimum CVaR, VaR or maximum return/risk.

My portfolio comprises 9 series. 1 domestic asset and 4 international assets which are 0% currency hedged: 

"D_Asset1","F_Asset2","F_Asset3","F_Asset4","F_Asset5"

I have also series for those that are 100% currency hedged:

"H_F_Asset2","H_F_Asset3","H_F_Asset4","H_F_Asset5"

The sum of the portfolio asset weights should be equal to one and long only (i.e no leverage / shorting), the domestic asset will have a weight of 60 %. The other asset will have group constraints so that the sum of the hedged and unhedged asset equal a fix target weight (i.e the weight in my benchmark). The group constraints are defined as:

("F_Asset2"+"H_F_Asset2") = 0.2 & ("F_Asset2"+"H_F_Asset2") >= 0 ("F_Asset3"+"H_F_Asset3") = 0.1 & ("F_Asset3"+"H_F_Asset3") >= 0 ("F_Asset4"+"H_F_Asset4") = 0.05 & ("F_Asset4"+"H_F_Asset4") >= 0 ("F_Asset5"+"H_F_Asset5") = 0.05 & ("F_Asset5"+"H_F_Asset5") >= 0

Therefore I should be able to derive through the weights optimisation the hedge ratio for each of my assets by looking at the proportionality of the optimised weights in each unhedged/hedged groups. I have tried to code this as follows:

opt_data <- as.timeSeries(Asset_returns[,c"D_Asset1","F_Asset2","F_Asset3","F_Asset4","F_Asset5","H_F_Asset2","H_F_Asset3","H_F_Asset4","H_F_Asset5")]) spec <- portfolioSpec() setSolver(spec) <- "solveRquadprog" cons <- c("eqsumW['D_Asset1']=0.6","eqsumW[c('F_Asset2','H_F_Asset2')]=0.2","eqsumW[c('F_Asset3','H_F_Asset3')]=0.1","eqsumW[c('F_Asset4','H_F_Asset4')]=0.05","eqsumW[c('F_Asset5','H_F_Asset5')]=0.05") portfolioConstraints(list(mu=apply(opt_data,2,function(x) mean(x)),sigma=cov(opt_data)), spec,cons) minriskPortfolio(list(mu=apply(opt_data,2,function(x) mean(x)),sigma=cov(opt_data),spec,constraints = cons)

This does not seem to work as the sum of my assets end up being allocated to the hedged asset 4 (0.3809) an 5 (0.6191) only an the fixed weight for the other group of assets is not respected...so I am obviously doing something wrong. Below is the print i get when running my script forr the below data sample. 

Title: MV Minimum Risk Portfolio Estimator: covEstimator Solver: solveRquadprog Optimize: minRisk Constraints: LongOnly Portfolio Weights: D_Asset1 F_Asset2 F_Asset3 F_Asset4 F_Asset5 H_F_Asset2 H_F_Asset3 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 H_F_Asset4 H_F_Asset5 0.3809 0.6191 Covariance Risk Budgets: D_Asset1 F_Asset2 F_Asset3 F_Asset4 F_Asset5 H_F_Asset2 H_F_Asset3 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 H_F_Asset4 H_F_Asset5 0.3809 0.6191 Target Return and Risk: [1] -0.0052 Description: Thu Oct 27 13:53:36 2016 by user: 43951663

My series are as follows:

 GMT D_Asset1 F_Asset2 F_Asset3 F_Asset4 F_Asset5 H_F_Asset2 H_F_Asset3 H_F_Asset4 H_F_Asset5 2002-07-05 0.0060728958 -0.013756357 -0.011423593 0.0134417810 -0.0003016946 -0.0009105038 -0.0018985396 0.014808963 -0.003882658 2002-07-12 -0.0897779000 -0.085188382 -0.089815019 -0.0138095353 -0.0762572363 -0.0685047064 -0.0832575916 -0.022109799 -0.047944944 2002-07-19 -0.0943660524 -0.046920616 -0.030421831 -0.0382901308 -0.0624865993 -0.0794080445 -0.0524289818 -0.031577514 -0.041396426 2002-07-26 0.0133063390 -0.057651106 -0.020437497 -0.0728233620 -0.0456981099 0.0064529146 -0.0396719802 -0.054443226 -0.024872592 2002-08-02 0.0112493491 0.009167815 0.016499805 0.0103302726 0.0252948946 0.0143938534 0.0118099642 0.014237543 0.026552984 2002-08-09 0.0802254120 0.067077593 0.057022734 0.0469759343 0.0558895137 0.0513020851 0.0548215098 0.029355000 0.019311812 2002-08-16 0.0121210802 0.004739185 0.002730157 -0.0071254354 -0.0147625074 0.0220800644 -0.0005414763 -0.017939892 -0.013482516 2002-08-23 0.0254279436 0.028461524 0.013045475 0.0007546919 0.0223311149 0.0133616906 0.0279480422 0.005884930 0.008278473 2002-08-30 -0.0478436889 -0.049203575 -0.039813657 -0.0356601715 -0.0256056009 -0.0276201640 -0.0386866322 -0.025089402 -0.004296057 2002-09-06 -0.0320257540 -0.038044390 -0.028008660 -0.0577045132 -0.0281928317 -0.0252990267 -0.0312053440 -0.049959243 -0.021289249 2002-09-13 -0.0005754118 -0.031013743 -0.024805745 -0.0045048220 -0.0035485481 -0.0055489175 -0.0259398560 0.017170857 0.004670684 2002-09-20 -0.0544596063 -0.049806038 -0.037835532 0.0040695629 -0.0467206777 -0.0523013152 -0.0589901499 0.020442551 -0.049443302 2002-09-27 -0.0264640157 0.003081610 0.011115685 0.0119000112 -0.0229705875 -0.0220045767 0.0091850943 0.009544600 -0.014769134 2002-10-04 -0.0350683237 -0.044400596 -0.026363396 -0.0591064104 -0.0404089273 -0.0317751096 -0.0394601664 -0.049204737 -0.028352867 2002-10-11 0.0455968470 0.047623114 0.034835901 -0.0516194024 0.0202454036 0.0435196284 0.0364044803 -0.047211423 0.013827130 2002-10-18 0.0682138514 0.050215807 0.044584318 0.0531850741 0.0789798029 0.0584691831 0.0566157920 0.055481410 0.060939407 2002-10-25 0.0112759198 -0.016565486 -0.019723747 -0.0234843100 0.0107604217 0.0149227748 -0.0177273307 -0.028938189 0.010213889 2002-11-01 -0.0040060015 0.012830724 -0.013284706 0.0043773949 -0.0193135586 0.0053397013 0.0007975884 -0.003459391 -0.015845981 2002-11-08 -0.0235192733 -0.004405182 0.008119140 -0.0033849145 -0.0101840271 -0.0059125661 -0.0036278513 -0.004716045 0.011824785 2002-11-15 0.0227169708 0.027997661 0.013501476 -0.0219131086 0.0096590849 0.0160309555 0.0242988333 -0.022932122 0.014512210 2002-11-22 0.0237869242 0.018522337 0.020965808 0.0062667869 0.0233331986 0.0242851881 0.0316120790 0.026096681 0.019139845

I would be quite happy to use another optimiser as long it is in R and allows me to handle other measure of risk such as CVaR, VaR etc...Any idea on how to solve for this problem would be really helpful to me and probably others on this forum.

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There are typos in your code. Review the following lines?

opt_data <- as.timeSeries(Asset_returns[,c("D_Asset1","F_Asset2","F_Asset3","F_Asset4","F_Asset5","H_F_Asset2","H_F_Asset3","H_F_Asset4","H_F_Asset5")])

Whit upper cons:

cons <- c("eqsumW['D_Asset1']=0.6","maxsumW[c('F_Asset2','H_F_Asset2')]=0.2","maxsumW[c('F_Asset3','H_F_Asset3')]=0.1","maxsumW[c('F_Asset4','H_F_Asset4')]=0.05","maxsumW[c('F_Asset5','H_F_Asset5')]=0.05")

I got:

minriskPortfolio(list(mu=apply(opt_data,2,function(x) mean(x)),sigma=cov(opt_data)),spec,constraints = cons) Title: MV Minimum Risk Portfolio Estimator: covEstimator Solver: solveRquadprog Optimize: minRisk Constraints: Portfolio Weights: D_Asset1 F_Asset2 F_Asset3 F_Asset4 F_Asset5 H_F_Asset2 H_F_Asset3 0.6000 0.0308 0.1000 0.0000 0.0000 0.1692 0.0000 H_F_Asset4 H_F_Asset5 0.0500 0.0500 Covariance Risk Budgets: D_Asset1 F_Asset2 F_Asset3 F_Asset4 F_Asset5 H_F_Asset2 H_F_Asset3 0.6932 0.0283 0.0762 0.0000 0.0000 0.1557 0.0000 H_F_Asset4 H_F_Asset5 0.0172 0.0294 Target Return and Risk: [1] -0.0043
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3 Comments

Thank you for your reply Robert, this is really useful. May I ask why you use maxsumW in the box constraints ? I thought that if you used EqsumW this would impose that asset group would have to sum up to the desired weight whilst leaving either of the asset in the group to vary between zero and the target weight. I understood that if you used minsumW or maxsumW this would insure that the assets in the group would have a minimum or maximum imposed weight nbut that the group could vary above / below this weight? however the above seem to do what I am after so I am very thankful.
I used maxsumW just to have more possibilities, but if necessary you could continue using eqsumW
Thank you Robert, I think eqsumW is more appropriate for what I am trying to solve. Thank you for your help much appreciated

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