# Portfolio Optimization

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Changing tracks, I want to now look at portfolio optimization. Although this is very different from developing trading strategies, it is useful to know how to construct minimum-variance portfolios and the like, if only for curiosity’s sake. Also, just a -I hope unnecessary- note, portfolio optimization and parameter optimization (which I covered in the last post) are two completely different things.

Minimum-variance portfolio optimization has a lot of problems associated with it, but it makes for a good starting point as it is the most commonly discussed optimization technique in classroom-finance. One of my biggest issues is with the measurement of risk via volatility. Security out-performance contributes as much to volatility -hence risk- as security under-performance, which ideally shouldn’t be the case.

First, install the package tseries:

**install.packages(‘tseries’)**

The function of interest is **portfolio.optim()**. I decided to write my own function to enter in a vector of tickers, start and end dates for the dataset, min and max weight constraints and short-selling constraints. This function first processes the data and then passes it to portfolio.optim to determine the minimum variance portfolio for a given level of return. It then cycles through increasingly higher returns to check how high the Sharpe ratio can go.

Here is the code with comments:

minVarPortfolio= function(tickers,start='2000-01-01',end=Sys.Date(), riskfree=0,short=TRUE,lowestWeight=-1,highestWeight=1){ # Load up the package require(tseries) #Initialize all the variables we will be using. returnMatrix is #initailized as a vector,with length equal to one of the input #ticker vectors (dependent on the start and end dates). #Sharpe is set to 0. The weights vector is set equal in #length to the number of tickers. The portfolio is set to #NULL. A 'constraint' variable is created to pass on the #short parameter to the portfolio.optim function. And vectors #are created with the low and high weight restrictions, which #are then passed to the portfolio.optim function as well. ## returnMatrix=vector(length=length(getSymbols(tickers[1], auto.assign=FALSE,from=start,to=end))) sharpe=0 weights=vector(,length(tickers)) port=NULL constraint=short lowVec=rep(lowestWeight,length(tickers)) hiVec=rep(highestWeight,length(tickers)) #This is a for-loop which cycles through the tickers, calculates #their return, and stores the returns in a matrix, adding #the return vector for each ticker to the matrix for(i in 1:length(tickers)){ temp=getSymbols(tickers[i],auto.assign=FALSE,from=start,to=end) if(i==1){ returnMatrix=diff(log(Ad(temp))) } else{ returnMatrix=cbind(returnMatrix,diff(log(Ad(temp)))) } } returnMatrix[is.na(returnMatrix)]=0 it #This for-loop cycles through returns to test the portfolio.optim function #for the highest Sharpe ratio. for(j in 1:100){ #Stores the log of the return in retcalc retcalc=log((1+j/100)) retcalc=retcalc/252 print(paste("Ret Calc:",retcalc)) #Tries to see if the specified return from retcalc can result #in an efficient portfolio try(port<-portfolio.optim(returnMatrix,pm=retcalc,shorts=constraint, reslow=lowVec,reshigh=hiVec,riskfree=riskfree),silent=T) #If the portfolio exists, it is compared against previous portfolios #for different returns using the #Sharpe ratio. If it has the highest #Sharpe ratio, it is stored and the old one is discarded. if(!is.null(port)){ print('Not Null') sd=port$ps tSharpe=((retcalc-riskfree)/sd) print(paste("Sharpe",tSharpe)) if(tSharpe>sharpe){ sharpe=tSharpe weights=port$pw }} } print(paste('Sharpe:', sharpe)) print(rbind(tickers,weights)) return(returnMatrix) }

This code works fine except for when the restrictions are too strict, the portfolio.optim function can’t find a minimum variance portfolio. This happens if the optimum portfolio has negative returns, which my code doesn’t test for. For this reason, I wanted to try out other ways of finding the highest Sharpe portfolio. There are numerous tutorials out there on how to do this. Some of them are:

1-http://blog.streeteye.com/blog/2012/01/portfolio-optimization-and-efficient-frontiers-in-r/

2-http://quantivity.wordpress.com/2011/04/17/minimum-variance-portfolios/

3-http://www.rinfinance.com/RinFinance2009/presentations/yollin_slides.pdf

4-http://systematicinvestor.wordpress.com/2013/03/22/maximum-sharpe-portfolio/

5-http://alphaism.wordpress.com/2012/05/04/finding-efficient-frontier-and-optimal-portfolio-in-r/

After I run my function, with the following tickers and constraints:

**matrix=minVarPortfolio(c(‘NVDA’, ‘YHOO’, ‘GOOG’, ‘CAT’, ‘BNS’, ‘POT’, ‘STO’, ‘MBT’ ,’SNE’),lowestWeight=0,highestWeight=0.2,start=’2000-01-01′, end=’2013-06-01′)**

This is the output I get:

[1] “Sharpe: 0.177751547083007″

tickers ”NVDA” “YHOO” ”GOOG”

weights “-1.58276161084957e-19″ ”2.02785605793095e-17″ “0.2″

tickers “CAT” “BNS” “POT”

weights “0.104269676769825″ “0.2″ “0.2″

tickers “STO” “MBT”

weights “0.189985091184918″ “0.105745232045257″

tickers “SNE”

weights “-2.85654465380669e-17″

The ‘e-XX’ weights basically indicate a weighting of zero on that particular security (NVDA, YHOO and SNE above). In the next post I will look at how all this can be done using a package called ‘fPortfolio’. Happy trading!

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