## Monday, October 20, 2014

### Portfolio Optimisation - II

The analysis in the previous post created portfolios that had negative weights, or "short" positions. To overcome we search for options to block short positions and arrived at a couple of other websites that demonstrated portfolio optimization with more complex models. Both the Economist at Large site and the Alphaism site give good inputs on how to address this problem and this leads us to the function portfolio.optim available in the {tseries} that addresses this problem.

In the following program, this portfolio.optim function has been used instead of the custom built function shown in the previous post.

========================================
getwd()
assets <- India5[,3:7]
returns <- log(tail(assets, -1) / head(assets, -1))

library("tseries")

## Create an optimum portfolio with expected means defined and shorts allowed

w2 <- portfolio.optim(as.matrix(returns),pm = 0.005,shorts = TRUE,riskless= FALSE)
w2\$pw
sum(w2\$pw)

## Create an optimum portfolio with expected means defined and shorts NOT ALLOWED

w2 <- portfolio.optim(as.matrix(returns),shorts = FALSE,riskless = FALSE)
w2\$pw
sum(w2\$pw)
w2\$pm

# Create a function to create the frontier using portfolio.optim

frontier2 <- function(return,minRet,maxRet){
rbase <- seq(minRet,maxRet,length=100)
s <- sapply(rbase,function(x){
p2 <- portfolio.optim(as.matrix(returns),pm = x, shorts = TRUE)
p2\$ps^2
})
plot(s,rbase,xlab="Variance",ylab="Return", main="w/Portfolio.Optim")
}

frontier2(returns,-0.0005,0.05)
========================================

We observe that with Short positions allowed ( shorts=true), the portfolio weights are the same as what was obtained in the previous program ( previous post ), and as expected they sum up to 1

> w2\$pw
[1]   4.869750   5.651428   3.375111 -10.711613  -2.184675
> sum(w2\$pw)
[1] 1

however when shorts are not allowed ( shorts = false ) a different portfolio is created

> w2\$pw
[1] 0.22797760 0.19765858 0.03686214 0.16357332 0.37392835
> sum(w2\$pw)
[1] 1
> w2\$pm
[1] 0.0003971317

and they sum up to 1 as well.

However, when short positions are not allowed, it is not possible to specify the portfolio returns before hand ( as it was, in the shorts allowed case, where pm = 0.005). With this constraint, the portfolio returns has dropped to 0.000397.

The efficient frontier remains the same as before

The program given in Alphaism uses portfolio.optim and then extends it further.