# How much does "Beta" change depending on time?

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You may often use “Beta” to measure the market exposure of your portfolio because it’s easy to calculate.

Since I have been wondering how much “Beta” change depending on time, more precisely writing, data-set and the period of return time series, I think that I would like to write about that in this article.

First, I get stock price( I selected Mitsubishi UFJ Financial Group, Inc here) and market index(Nikkei225 ) from yahoo Japan by using RFinanceYJ package. These data period are from September, 2010 to September, 2011.

library(RFinanceYJ) #download stock prices of nikkei-225 and MUFJ Financial group mufg <- quoteStockXtsData("8306.T", since="2010-09-30",date.end="2011-09-30")$Close nikkei <- quoteStockXtsData("998407.O", since="2010-09-30",date.end="2011-09-30")$Close(you need to install RFinanceYJ Package to run this code)

I convert this price data into return data, and estimate the "Beta" of Mitsubishi UFJ Financial Group, Inc by using rolling linear regression(every regression have 125 sample data).

#convert to return returns <- merge(mufg / lag(mufg) - 1, nikkei / lag(nikkei) - 1) names(returns) <- c("MUFG","NIKKEI") #estimate beta by rolling regression size.window <- 125 coefs <- rollapplyr(returns, size.window, function(x)coef(lm(x[,1]~x[,2])), by.column = FALSE)

As you know, R language provides us a very easy way to analyze, That is it.

So, Let's visualize these result. The result of simple plot is following that.

plot(as.xts(coefs[, 2]))

This graph shows that how historical beta changes depending on time.

And I create the animation of this result to understand more easily.

And I create the animation of this result to understand more easily.

Here is the code to create this animation.

x <- na.omit(coredata(returns)[ ,2]) y <- na.omit(coredata(returns)[ ,1]) x.max <- c(-max(abs(x)), max(abs(x))) y.max <- c(-max(abs(y)), max(abs(y))) x.lab <- names(returns)[2] y.lab <- names(returns)[1] Snap <- function(val){ val.x <- na.omit(coredata(val)[ ,2]) val.y <- na.omit(coredata(val)[ ,1]) lm.xy <- lm(val.y~val.x) plot(val.x, val.y, xlim = x.max, ylim = y.max, xlab = x.lab, ylab = y.lab) abline(lm.xy) text(x.max[1], y.max[2], paste("Beta :", round(coef(lm.xy)[2],3)), pos = 4) text(x.max[1], y.max[1], as.character(last(index(val))), pos = 4) } library(animation) saveGIF({ for(i in 1:(nrow(returns)-size.window)){ Snap(returns[(i:(i+size.window)),]) } },interval = 0.01)

you need to install "animation" package and ImageMagick (another software) to run this code

Enjoy!

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