Set options, load packages and data

options(digits=3, width=70)
library(IntroCompFinR)

## Loading required package: xts
## Loading required package: zoo
## 
## Attaching package: 'zoo'
## 
## The following objects are masked from 'package:base':
## 
##     as.Date, as.Date.numeric

library(PerformanceAnalytics)

## 
## Attaching package: 'PerformanceAnalytics'
## 
## The following object is masked from 'package:graphics':
## 
##     legend

zoo function rollapply()

Use the zoo function rollapply() to compute functions of data over rolling windows.

args(zoo:::rollapply.zoo)

## function (data, width, FUN, ..., by = 1, by.column = TRUE, fill = if (na.pad) NA, 
##     na.pad = FALSE, partial = FALSE, align = c("center", "left", 
##         "right")) 
## NULL

The most important arguments are:

• data: zoo data
• width: integer window width
• FUN: function to be applied over the rolling windows
• by: integer increment to move windows by
• by.column: logical, if TRUE apply FUN to each column of data; if FALSE apply FUN to all columns of data
• align: character, what part of the window gets the time stamp

Compute 24-month rolling means for SBUX

roll.muhat = rollapply(cerRetC[,"SBUX"], width=24,
                       FUN=mean, align="right")
class(roll.muhat)

## [1] "xts" "zoo"

t(roll.muhat[1:5])

##      Feb 1998 Mar 1998 Apr 1998 May 1998 Jun 1998
## SBUX       NA       NA       NA       NA       NA

t(na.omit(roll.muhat)[1:5])

##      Jan 2000 Feb 2000 Mar 2000 Apr 2000 May 2000
## SBUX   0.0233   0.0239   0.0284  0.00951   0.0145

Plot the rolling means with the return series

plot.zoo(merge(roll.muhat,cerRetC[,"SBUX"]), plot.type="single",
     main="24 month rolling means for SBUX",ylab="returns",
     lwd=c(2,2), col=c("blue","orange"))
abline(h=0)
legend(x="bottomright",legend=c("Rolling mean","Monthly returns"),
       lwd=c(2,2), col=c("blue","orange"))

Notice how the rolling means become negative during the financial crisis. Here, the rolling means are not constant over the sample which suggests that the SBUX returns may not covariance stationary

Compute 24-month rolling standard deviations for SBUX

roll.sigmahat = rollapply(cerRetC[,"SBUX"],width=24,
                          FUN=sd, align="right")
t(na.omit(roll.sigmahat)[1:5])

##      Jan 2000 Feb 2000 Mar 2000 Apr 2000 May 2000
## SBUX    0.172    0.172    0.176    0.196    0.197

plot.zoo(merge(roll.sigmahat,cerRetC[,"SBUX"]), plot.type="single",
     main="24 month rolling SDs for SBUX", ylab="Returns",
     lwd=c(2,2), col=c("blue","orange"))
abline(h=0)
legend(x="bottomright",legend=c("Rolling SD","Monthly returns"),
       lwd=c(2,2), col=c("blue","orange"))

Here, the rolling volatility estimates start high, around 0.172 in early 2000, dip to 0.067 in 2005, and then rise to 0.137 during the financial crisis. There is pretty clear evidence that the volatility of SBUX returns has changed over time.

Show rolling means and volatilities on the same graph

plot.zoo(merge(roll.muhat, roll.sigmahat, cerRetC[,"SBUX"]), 
         plot.type="single",
         main="24 month rolling estimates: SBUX",ylab="returns",
         lwd=2, col=c("blue","orange", "black"))
abline(h=0)
legend(x="bottomright",legend=c("Rolling mean","Rolling sd", "Monthly returns"),
       lwd=2, col=c("blue","orange","black"))

24-Month rolling estimates for MSFT

roll.muhat = rollapply(cerRetC[,"MSFT"], width=24,
                       FUN=mean, align="right")
roll.sigmahat = rollapply(cerRetC[,"MSFT"], width=24,
                          FUN=sd, align="right")
plot.zoo(merge(roll.muhat, roll.sigmahat, cerRetC[,"MSFT"]), 
         plot.type="single",
         main="24 month rolling estimates: MSFT",ylab="returns",
         lwd=2, col=c("blue","orange", "black"))
abline(h=0)
legend(x="bottomright",legend=c("Rolling mean","Rolling sd", "Monthly returns"),
       lwd=2, col=c("blue","orange","black"))

• The rolling means stay pretty close to zero over the whole sample.
• However, the rolling volatilities drop after the dot-com bust and only pick up a little bit during the financial crisis.

24-Month rolling estimates for SP500

roll.muhat = rollapply(cerRetC[,"SP500"], width=24,
                       FUN=mean, align="right")
roll.sigmahat = rollapply(cerRetC[,"SP500"],width=24,
                          FUN=sd, align="right")
plot.zoo(merge(roll.muhat, roll.sigmahat, cerRetC[,"SP500"]), 
         plot.type="single",
         main="24 month rolling estimates: SP500",ylab="returns",
         lwd=2, col=c("blue","orange", "black"))
abline(h=0)
legend(x="bottomright",legend=c("Rolling mean","Rolling sd", "Monthly returns"),
       lwd=2, col=c("blue","orange","black"))

• The rolling means for the S&P500 index move around quite a bit. They dip negative around the dot-com bust, become slightly positive during the mid 2000s, dip negative again during the financial crisis, and become positive again afterwards.
• The rolling volatilities drop during the mid 2000s and then pick up dramatically during the financial crisis and fall a bit afterwards.

24-Month rolling correlations between SP500 and SBUX

First, compute function to compute pairwise correlation between two series:

rhohat = function(x) {
  cor(x)[1,2]
}

Next, call rollapply() with the user-written function

roll.rhohat = rollapply(cerRetC[,c("SP500","SBUX")],
                        width=24,FUN=rhohat, by.column=FALSE,
                        align="right")
t(na.omit(roll.rhohat)[1:5])

##   Jan 2000 Feb 2000 Mar 2000 Apr 2000 May 2000
## x    0.482    0.463    0.495    0.511    0.492

plot.zoo(roll.rhohat, main="Rolling Correlation b/w SP500 and SBUX",
         lwd=2, col="blue", ylab="rho.hat")
abline(h=0)   

The correlations are not stable over the sample period. They start out near 0.5, drop to 0.3 between 2001 and 2004, become negative briefly in 2005 (why?), move back positive mear 0.5 in 2006, and then increase to 0.8 in 2011.

Summary of rolling estimation results

• Rolling window estimates indicate that \(\mu\), \(\sigma\) and \(\rho_{ij}\) are typically not constant over time
  • volatilities and correlations tend to move more than means
• Assumption of covariance stationarity over long samples is suspect