Black-Litterman Model

November 15, 2011
By

(This article was first published on Systematic Investor » R, and kindly contributed to R-bloggers)

The Black-Litterman Model was created by Fisher Black and Robert Litterman in 1992 to resolve shortcomings of traditional Markovitz mean-variance asset allocation model. It addresses following two items:

  • Lack of diversification of portfolios on the mean-variance efficient frontier.
  • Instability of portfolios on the mean-variance efficient frontier: small changes in the input assumptions often lead to very different efficient portfolios.

I recommend a very good non-technical introduction to The Black-Litterman Model, An Introduction for the Practitioner by T. Idzorek (2009).

I will take the country allocation example presented in The Intuition Behind Black-Litterman Model Portfolios by G. He, R. Litterman (1999) paper and update it using current market data.

First, I need market capitalization data for each country to compute equilibrium portfolio. I found following two sources of capitalization data:

I will use market capitalization data from World Databank.

# load Systematic Investor Toolbox
setInternet2(TRUE)
source(gzcon(url('https://github.com/systematicinvestor/SIT/raw/master/sit.gz', 'rb')))

	#--------------------------------------------------------------------------
	# Visualize Market Capitalization History
	#--------------------------------------------------------------------------

	hist.caps = aa.test.hist.capitalization()	
	hist.caps.weight = hist.caps/rowSums(hist.caps)
	
	# Plot Transition of Market Cap Weights in time
	plot.transition.map(hist.caps.weight, index(hist.caps.weight), xlab='', name='Market Capitalization Weight History')

	# Plot History for each Country's Market Cap
	layout( matrix(1:9, nrow = 3, byrow=T) )
	col = plota.colors(ncol(hist.caps))
	for(i in 1:ncol(hist.caps)) {
		plota(hist.caps[,i], type='l', lwd=5, col=col[i], main=colnames(hist.caps)[i])
	}

There is a major shift in weights between Japan and USA from 1988 to 2010. In 1988 Japan represented 47% and USA 33%. In 2010 Japan represents 13% and USA 55%. The shift was driven by inflow of capital to USA, the Japaneses capitalization was pretty stable in time, as can be observed from time series plot for each country.

Second, I need historical prices series for each country to compute covariance matrix. I will use historical data from Yahoo Fiance:

AustraliaEWA
CanadaEWC
FranceEWQ
GermanyEWG
JapanEWJ
U.K.EWU
USASPY

The first step of the Black-Litterman model is to find implied equilibrium returns using reverse optimization.

\Pi = \delta \Sigma w_{eq}

where \Pi are equilibrium returns, \delta is risk aversion, \Sigma is covariance matrix, and w_{eq} are market capitalization weights. The risk aversion parameter can be estimated from historical data by dividing the excess market portfolio return by its variance.

# Use reverse optimization to compute the vector of equilibrium returns
bl.compute.eqret <- function
(
	risk.aversion, 	# Risk Aversion
	cov, 		# Covariance matrix
	cap.weight, 	# Market Capitalization Weights
	risk.free = 0	# Rsik Free Interest Rate
)
{
	return( risk.aversion * cov %*% cap.weight +  risk.free)	
}

	#--------------------------------------------------------------------------
	# Compute Risk Aversion, prepare Black-Litterman input assumptions
	#--------------------------------------------------------------------------
	ia = aa.test.create.ia.country()
	
	# compute Risk Aversion
	risk.aversion = bl.compute.risk.aversion( ia$hist.returns$USA )

	# the latest market capitalization weights
	cap.weight = last(hist.caps.weight)	
		
	# create Black-Litterman input assumptions	
	ia.bl = ia
	ia.bl$expected.return = bl.compute.eqret( risk.aversion, ia$cov, cap.weight )
	
	# Plot market capitalization weights and implied equilibrium returns
	layout( matrix(c(1,1,2,3), nrow=2, byrow=T) )
	pie(coredata(cap.weight), paste(colnames(cap.weight), round(100*cap.weight), '%'), 
		main = paste('Country Market Capitalization Weights for', format(index(cap.weight),'%b %Y'))
		, col=plota.colors(ia$n))
	
	plot.ia(ia.bl, T)

Next, let’s compare the efficient frontier created using historical input assumptions and Black-Litterman input assumptions

	#--------------------------------------------------------------------------
	# Create Efficient Frontier(s)
	#--------------------------------------------------------------------------
	n = ia$n
	
	# -1 <= x.i <= 1
	constraints = new.constraints(n, lb = 0, ub = 1)

	# SUM x.i = 1
	constraints = add.constraints(rep(1, n), 1, type = '=', constraints)		
	
	# create efficient frontier(s)
	ef.risk = portopt(ia, constraints, 50, 'Historical', equally.spaced.risk = T)		
	ef.risk.bl = portopt(ia.bl, constraints, 50, 'Black-Litterman', equally.spaced.risk = T)	

	# Plot multiple Efficient Frontiers and Transition Maps
	layout( matrix(1:4, nrow = 2) )
	plot.ef(ia, list(ef.risk), portfolio.risk, T, T)			
	plot.ef(ia.bl, list(ef.risk.bl), portfolio.risk, T, T)			

Comparing the transition maps, the Black-Litterman efficient portfolios are well diversified. Efficient portfolios have allocation to all asset classes at various risk levels. By its construction, the Black-Litterman model is well suited to address the diversification problems.

The Black-Litterman model also introduces a mechanism to incorporate investor’s views into the input assumptions in such a way that small changes in the input assumptions will NOT lead to very different efficient portfolios. The Black-Litterman model adjusts expected returns and covariance:

\bar{\mu} = \left [ (\tau \Sigma)^{-1}+P'\Omega^{-1}P \right ]^{-1}  \left [ (\tau \Sigma)^{-1}\Pi + P'\Omega^{-1}Q  \right ]  \newline\newline  \bar{\Sigma}=\Sigma+\left [ (\tau \Sigma)^{-1}+P'\Omega^{-1}P \right ]^{-1}

where P is Views pick matrix, and Q Views mean vector. The Black-Litterman model assumes that views are N \sim (Q,P).

bl.compute.posterior <- function
(
	mu, 		# Equilibrium returns
	cov, 		# Covariance matrix
	pmat=NULL, 	# Views pick matrix
	qmat=NULL, 	# Views mean vector
	tau=0.025 	# Measure of uncertainty of the prior estimate of the mean returns
)
{
	out = list()	
	omega = diag(c(1,diag(tau * pmat %*% cov %*% t(pmat))))[-1,-1]
		
	temp = solve(solve(tau * cov) + t(pmat) %*% solve(omega) %*% pmat)	
	out$cov = cov + temp

	out$expected.return = temp %*% (solve(tau * cov) %*% mu + t(pmat) %*% solve(omega) %*% qmat)
	return(out)
}

	#--------------------------------------------------------------------------
	# Create Views
	#--------------------------------------------------------------------------
	temp = matrix(rep(0, n), nrow = 1)
		colnames(temp) = ia$symbols
		
	# Relative View
	# Japan will outperform UK by 2%
	temp[,'Japan'] = 1
	temp[,'UK'] = -1

	pmat = temp
	qmat = c(0.02)
	
	# Absolute View
	# Australia's expected return is 12%
	temp[] = 0
	temp[,'Australia'] = 1
	
	pmat = rbind(pmat, temp)	
	qmat = c(qmat, 0.12)

	# compute posterior distribution parameters
	post = bl.compute.posterior(ia.bl$expected.return, ia$cov, pmat, qmat, tau = 0.025 )

	# create Black-Litterman input assumptions with Views	
	ia.bl.view = ia.bl
		ia.bl.view$expected.return = post$expected.return
		ia.bl.view$cov = post$cov
		ia.bl.view$risk = sqrt(diag(ia.bl.view$cov))
		
	# create efficient frontier(s)
	ef.risk.bl.view = portopt(ia.bl.view, constraints, 50, 'Black-Litterman + View(s)', equally.spaced.risk = T)	

	# Plot multiple Efficient Frontiers and Transition Maps
	layout( matrix(1:4, nrow = 2) )
	plot.ef(ia.bl, list(ef.risk.bl), portfolio.risk, T, T)			
	plot.ef(ia.bl.view, list(ef.risk.bl.view), portfolio.risk, T, T)			

Comparing the transition maps, the Black-Litterman + Views efficient portfolios have more allocation to Japan and Australia, as expected. The portfolios are well diversified and are not drastically different from the Black-Litterman efficient portfolios.

The Black-Litterman model provides an elegant way to resolve shortcomings of traditional Markovitz mean-variance asset allocation model based on historical input assumptions. It addresses following two items:

  • Lack of diversification of portfolios on the mean-variance efficient frontier. The Black-Litterman model uses equilibrium returns implied from the current market capitalization weighs to construct well diversified portfolios.
  • Instability of portfolios on the mean-variance efficient frontier. The Black-Litterman model introduces a mechanism to incorporate investor’s views into the input assumptions in such a way that small changes in the input assumptions will NOT lead to very different efficient portfolios.

I highly recommend exploring and reading following articles and websites for better understanding of the Black-Litterman model:

To view the complete source code for this example, please have a look at the aa.black.litterman.test() function in aa.test.r at github.


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