Factor Attribution

红太狼 2022-06-13 06:58 160阅读 0赞

I came across a very descriptive visualization of the Factor Attribution that I will replicate today. There is the [Three Factor Rolling Regression Viewer][] at the [mas financial tools][Three Factor Rolling Regression Viewer] web site that performs rolling window Factor Analysis of the [“three-factor model”][three-factor model] of Fama and French. The factor returns are available from the [Kenneth R French: Data Library][Kenneth R French_ Data Library]. I recommend reading the [Efficient Frontier: Rolling Your Own: Three-Factor Analysis by W. Bernstein][Efficient Frontier_ Rolling Your Own_ Three-Factor Analysis by W. Bernstein] for a step by step instructions.

Let’s start by loading historical returns for the [Vanguard Small Cap Value Index (VISVX)][Vanguard Small Cap Value Index _VISVX] and aligning them with Fama/French Monthly Factors. Please [note][] I wrote a helper function, [get.fama.french.data()][get.fama.french.data], to simplify process of loading and analyzing factor data from the [Kenneth R French: Data Library][Kenneth R French_ Data Library].

###############################################################################
    # Load Systematic Investor Toolbox (SIT)
    # http://systematicinvestor.wordpress.com/systematic-investor-toolbox/
    ###############################################################################
    setInternet2(TRUE)
    con = gzcon(url('http://www.systematicportfolio.com/sit.gz', 'rb'))
        source(con)
    close(con)
    
    	#*****************************************************************
    	# Load historical data
    	#****************************************************************** 
    	load.packages('quantmod')	
    	tickers = 'VISVX'
    	
    	data <- new.env()
    	getSymbols(tickers, src = 'yahoo', from = '1980-01-01', env = data, auto.assign = T)
    	for(i in ls(data)) {
    		temp = adjustOHLC(data[[i]], use.Adjusted=T)							
    		
    		period.ends = endpoints(temp, 'months')
    			period.ends = period.ends[period.ends > 0]
    
    		# reformat date to match Fama French Data
    		monthly.dates = as.Date(paste(format(index(temp)[period.ends], '%Y%m'),'01',sep=''), '%Y%m%d')
    		data[[i]] = make.xts(coredata(temp[period.ends,]), monthly.dates)
    	}
    	
    	# Fama/French factors
    	factors = get.fama.french.data('F-F_Research_Data_Factors', periodicity = 'months',download = T, clean = F)
    	
    	# add factors and align
    	data$factors = factors$data / 100
    	bt.prep(data, align='remove.na', dates='1994::')

Next, let’s run Factor Attribution over all available data:

#*****************************************************************
    	# Facto Loadings Regression over whole period
    	#****************************************************************** 
    	prices = data$prices
    		nperiods = nrow(prices)
    		dates = index(data$prices)
    	
    	# compute simple returns	
    	hist.returns = ROC(prices[,tickers], type = 'discrete')
    		hist.returns = hist.returns - data$factors$RF
    		colnames(hist.returns) = 'fund'
    	hist.returns = cbind(hist.returns, data$factors$Mkt.RF,
    					data$factors$SMB, data$factors$HML)
    
    	fit.all = summary(lm(fund~Mkt.RF+SMB+HML, data=hist.returns))
    		estimate.all = c(fit.all$coefficients[,'Estimate'], fit.all$r.squared)
    		std.error.all = c(fit.all$coefficients[,'Std. Error'], NA)

Coefficients:
                  Estimate Std. Error t value Pr(>|t|)
    (Intercept) -0.0006828  0.0012695  -0.538    0.591
    Mkt.RF       0.9973980  0.0262881  37.941   <2e-16 ***
    SMB          0.5478299  0.0364984  15.010   <2e-16 ***
    HML          0.6316528  0.0367979  17.165   <2e-16 ***
    ---
    Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
    
    Multiple R-squared: 0.9276,     Adjusted R-squared: 0.9262

All factor loadings are significant.

Next, let’s run a 36 month rolling window Factor Attribution:

#*****************************************************************
    	# Facto Loadings Regression over 36 Month window
    	#****************************************************************** 							
    	window.len = 36
    	
    	colnames = spl('alpha,MKT,SMB,HML,R2')
    	estimate = make.xts(matrix(NA, nr = nperiods, len(colnames)), dates)
    	colnames(estimate) = colnames
    	std.error = estimate
    	
    	# main loop
    	for( i in window.len:nperiods ) {
    		window.index = (i - window.len + 1) : i
    		
    		fit = summary(lm(fund~Mkt.RF+SMB+HML, data=hist.returns[window.index,]))
    		estimate[i,] = c(fit$coefficients[,'Estimate'], fit$r.squared)
    		std.error[i,] = c(fit$coefficients[,'Std. Error'], NA)
    		
    		if( i %% 10 == 0) cat(i, '\n')
    	}

Finally, let’s re-create the timeseries and histogram charts as presented by [Three Factor Rolling Regression Viewer][].

#*****************************************************************
    	# Reports
    	#****************************************************************** 
    	layout(matrix(1:10,nc=2,byrow=T))
    	
    	for(i in 1:5) {	
    		#-------------------------------------------------------------------------
    		# Time plot
    		#-------------------------------------------------------------------------
    		est = estimate[,i]
    		est.std.error = ifna(std.error[,i], 0)
    		
    		plota(est, 
    			ylim = range( c(
    				range(est + est.std.error, na.rm=T),
    				range(est - est.std.error, na.rm=T)		
    				)))
    	
    		polygon(c(dates,rev(dates)), 
    			c(coredata(est + est.std.error), 
    			rev(coredata(est - est.std.error))), 
    			border=NA, col=col.add.alpha('red',50))
    	
    		est = estimate.all[i]
    		est.std.error = std.error.all[i]
    	
    		polygon(c(range(dates),rev(range(dates))), 
    			c(rep(est + est.std.error,2),
    			rep(est - est.std.error,2)),
    			border=NA, col=col.add.alpha('blue',50))
    		
    		abline(h=0, col='blue', lty='dashed')
    			
    		abline(h=est, col='blue')
    	
    		plota.lines(estimate[,i], type='l', col='red')
    		
    		#-------------------------------------------------------------------------
    		# Histogram
    		#-------------------------------------------------------------------------
    		par(mar = c(4,3,2,1))
    		hist(estimate[,i], col='red', border='gray', las=1,
    			xlab='', ylab='', main=colnames(estimate)[i])
    			abline(v=estimate.all[i], col='blue', lwd=2)
    	}

[![plot1][]][plot1 1]

In the next post, I plan to replicate the Style charts and provide more examples.

To view the complete source code for this example, please have a look at the [three.factor.rolling.regression() function in bt.test.r at github][three.factor.rolling.regression_ function in bt.test.r at github].

[Three Factor Rolling Regression Viewer]: http://mas.xtreemhost.com/
[three-factor model]: http://en.wikipedia.org/wiki/Fama%E2%80%93French_three-factor_model
[Kenneth R French_ Data Library]: http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html
[Efficient Frontier_ Rolling Your Own_ Three-Factor Analysis by W. Bernstein]: http://www.efficientfrontier.com/ef/101/roll101.htm
[Vanguard Small Cap Value Index _VISVX]: http://www.google.com/finance?q=VISVX
[note]: http://www.ebay.com/sch/i.html?_nkw=note
[get.fama.french.data]: https://github.com/systematicinvestor/SIT/blob/master/R/data.r
[plot1]: http://systematicinvestor.files.wordpress.com/2012/06/plot1.png?w=900#038;h=1200
[plot1 1]: http://systematicinvestor.files.wordpress.com/2012/06/plot1.png
[three.factor.rolling.regression_ function in bt.test.r at github]: https://github.com/systematicinvestor/SIT/blob/master/R/bt.test.r