Quantitative thoughts

When dealing with transaction cost analysis, a stock’s volume is assumed to be stable or foreseeable.  However, there is different picture, then we are dealing with an illiquid stock.

It is relatively easy to forecast the volume of a liquid stock, because trading volume has high autocorrelation – the volumes at t and t+1 are correlated. For example, let’s take a look at BPN Paribas stock  autocorrelation figure:

X axis shows the lag between the days and Y axis shows percentage of the correlation. For BNP Paribas stock we have 60 % correlation between t and t+1 and 30 % between t and t+2.

Now, let’s look what is autocorrelation  of an illiquid stock:

The figure above shows, that autocorrelation for this emerging market stock is zero. That means, that we can’t forecast tomorrow’s volume, based on today’s volume.

Imagine, that a portfolio manager has to liquidate 100 000 of illiquid stock where the median of daily volume is 4000 and participation rate has to be maximum 20 %. How many days it will take to liquidate the position?

Because the daily volume of the stock is very volatile, we need more randomness in our forecast. To do that, we can use bootstrap – let’s take the last 250 data-points of the volume and generate 10 000 or 100 000 new time series. Once the have a bunch of new time series, let’s check how many days it would take to liquidate 100 000 stock position in each. Finally, we collect the numbers of the days needed for liquidation  and form a new vector. The result is following:

The histogram above shows, that it will take maximum 80 days to liquidate 100 000 of illiquid stock with 95% confidence.

#volume - vector of the stock's volume
#shareNumber - number of the shares to liquidate
#loop - number of the time-series to be created
#participationRate - what is going to be participation rate 
simNumberOfDays<-function(volume,shareNumber,loop=10000,participationRate=0.05)
{
  rez=matrix(nrow=loop)
  for(i in 1:loop)
  {
    x=nasa[sample(NROW(volume))]
    y=cumsum(x*participationRate)
    rez[i,]=which(y>shareNumber)[1]
  }
  return(rez)
}
rez=simNumberOfDays(illiquidStock,100000,10000,0.2)

I came up with an idea to draw correlation network to get a grasp about relationship between a list of stocks. An alternative way to show correlation matrix would be head map, which can have limitations with big matrices (>100).
Unfortunately,  ggplot2 package doesn’t have a easy way to draw the networks, so I was left with igraph or network. I tried both, but somehow chose igraph. If you want to master either package I highly recommend to start from theoretical part – it is very well written and it will save your time trying to understand the package’s conception.

Here is correlation matrix of stocks, which correlation coef. is more than 0.5:

require(xts)
require(quantmod)
require(igraph)
cor_mat<- matrix( runif(100), nr=10 )
cor_mat[ lower.tri(cor_mat, diag=TRUE) ]<- 0
cor_mat[ abs(cor_mat) < 0.5]<- 0
 
graph <- graph.adjacency(cor_mat>0.5, weighted=TRUE, mode="upper")
E(graph)$weight<-t(cor_mat)[abs(t(cor_mat))>0.5]
E(graph)[ weight>0.7 ]$color <- "black" E(graph)[ weight>=0.65 &amp; weight<0.7 ]$color <- "red" E(graph)[ weight>=0.6 &amp;weight<0.65 ]$color <- "green" E(graph)[ weight>=0.55 &amp;weight<0.6 ]$color <- "blue"
E(graph)[ weight<0.55  ]$color <- "yellow"
V(graph)$label<- seq(1:10)#V(graph)$name
graph$layout <- layout.fruchterman.reingold
factor<-as.factor(cut(E(graph)$weight*10,c(4,5,6,7,8),labels=c(1,10,20,30))) png('corr_network.png',width=500) plot(decompose.graph(graph)[[which.max(sapply(decompose.graph(graph), vcount))]],edge.width =as.numeric(factor)*1.5,frame=T) legend("bottomleft", title="Colors", cex=0.75, pch=16, col=c("black", "blue","red", "green","pink"), legend=c(">70%", "65-70","60-65","55-60","50-55"), ncol=2)
dev.off()

The code can be found on github.