Multiple Factor Model – Building 130/30 Index
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Nico brought to my attention the 130/30: The New Long-Only (2008) by A. Lo, P. Patel paper in his comment to the Multiple Factor Model – Building CSFB Factors post. This paper presents a very detailed step by step guide to building 130/30 Index using average CSFB Factors as the alpha model and MSCI Barra Multi-Factor Risk model. Today, I want to adapt this methodology and to show how to build 130/30 Index based on the CSFB Factors we created in the Multiple Factor Model – Building CSFB Factors post and the Risk Model we created in the Multiple Factor Model – Building Risk Model post.
Let’s start by loading the CSFB factors that we saved at the end of the Multiple Factor Model – Building CSFB Factors post. [If you are missing data.factors.Rdata file, please execute fm.all.factor.test() function first to create and save CSFB factors.] Next, let’s load the multiple factor risk model we saved at the end of the Multiple Factor Model – Building Risk Model post. [If you are missing risk.model.Rdata file, please execute fm.risk.model.test() function first to create and save multiple factor risk model.] Next, I will compute betas over a two-year rolling window.
###############################################################################
# Load Systematic Investor Toolbox (SIT)
# http://systematicinvestor.wordpress.com/systematic-investor-toolbox/
###############################################################################
con = gzcon(url('http://www.systematicportfolio.com/sit.gz', 'rb'))
source(con)
close(con)
#*****************************************************************
# Load data
#******************************************************************
load.packages('quantmod')
# Load CSFB factor data that we saved at the end of the fm.all.factor.test function
load(file='data.factors.Rdata')
nperiods = nrow(next.month.ret)
tickers = colnames(next.month.ret)
n = len(tickers)
# Load multiple factor risk model data that we saved at the end of the fm.risk.model.test function
load(file='risk.model.Rdata')
factor.exposures = all.data[,,-1]
factor.names = dimnames(factor.exposures)[[3]]
nfactors = len(factor.names)
#*****************************************************************
# Compute Betas: b = cov(r,m) / var(m)
# The betas are measured on a two-year rolling window
# http://en.wikipedia.org/wiki/Beta_(finance)
#******************************************************************
ret = mlag(next.month.ret)
beta = ret * NA
# 1/n benchmark portfolio
benchmark = ntop(ret, n)
benchmark.ret = rowSums(benchmark * ret, na.rm=T)
# estimate betas
for(t in 24:nperiods) {
t.index = (t-23):t
benchmark.var = var( benchmark.ret[t.index], na.rm=T )
t.count = count(ret[t.index, ])
t.cov = cov( ifna(ret[t.index,], 0), benchmark.ret[t.index], use='complete.obs' )
# require at least 20 months of history
beta[t,] = iif(t.count > 20, t.cov/benchmark.var, NA)
}
To construct efficient portfolio each month, I will solve mean-variance portfolio optimization problem of the following form:
Max weight * return - risk.aversion * weight * Covariance * weight Sum weight = 1 Sum weight * beta = 1 0 <= weight <= 0.1
Please read The Effects of Risk Aversion on Optimization (2010) by S. Liu, R. Xu paper for the detailed discussion of this optimization problem.
Please note that I will use risk.aversion = 0.0075 and portfolio beta = 1 as discussed on the page 22 of the 130/30: The New Long-Only (2008) by A. Lo, P. Patel paper. To model portfolio beta constraint, I will use the fact that portfolio beta is equal to the weighted average of the individual asset betas.
#*****************************************************************
# Construct LONG ONLY portfolio using the multiple factor risk model
#******************************************************************
load.packages('quadprog,corpcor,kernlab')
weight = NA * next.month.ret
weights = list()
weights$benchmark = ntop(beta, n)
weights$long.alpha = weight
for(t in 36:nperiods) {
#--------------------------------------------------------------------------
# Create constraints
#--------------------------------------------------------------------------
# set min/max wgts for individual stocks: 0 =< x <= 10/100
constraints = new.constraints(n, lb = 0, ub = 10/100)
# wgts must sum to 1 (fully invested)
constraints = add.constraints(rep(1,n), type = '=', b = 1, constraints)
#--------------------------------------------------------------------------
# beta of portfolio is the weighted average of the individual asset betas
# http://www.duke.edu/~charvey/Classes/ba350/riskman/riskman.htm
#--------------------------------------------------------------------------
constraints = add.constraints(ifna(as.vector(beta[t,]),0), type = '=', b = 1, constraints)
#--------------------------------------------------------------------------
# Create factor exposures constraints
#--------------------------------------------------------------------------
# adjust prior constraints, add factor exposures
constraints = add.variables(nfactors, constraints)
# BX - X1 = 0
constraints = add.constraints(rbind(ifna(factor.exposures[t,,], 0), -diag(nfactors)), rep(0, nfactors), type = '=', constraints)
#--------------------------------------------------------------------------
# Create Covariance matrix
# [Qu 0]
# [ 0 Qf]
#--------------------------------------------------------------------------
temp = diag(n)
diag(temp) = ifna(specific.variance[t,], mean(coredata(specific.variance[t,]), na.rm=T))^2
cov.temp = diag(n + nfactors)
cov.temp[1:n,1:n] = temp
cov.temp[(n+1):(n+nfactors),(n+1):(n+nfactors)] = factor.covariance[t,,]
#--------------------------------------------------------------------------
# create input assumptions
#--------------------------------------------------------------------------
ia = list()
ia$n = nrow(cov.temp)
ia$annual.factor = 12
ia$symbols = c(tickers, factor.names)
ia$cov = cov.temp
#--------------------------------------------------------------------------
# page 9, Risk: We use the Barra default setting, risk aversion value of 0.0075, and
# AS-CF risk aversion ratio of 1.
#
# The Effects of Risk Aversion on Optimization (2010) by S. Liu, R. Xu
# page 4/5
#--------------------------------------------------------------------------
risk.aversion = 0.0075
ia$cov.temp = ia$cov
# set expected return
alpha = factors.avg$AVG[t,] / 5
ia$expected.return = c(ifna(coredata(alpha),0), rep(0, nfactors))
# remove companies that have no beta from optimization
index = which(is.na(beta[t,]))
if( len(index) > 0) {
constraints$ub[index] = 0
constraints$lb[index] = 0
}
# find solution
sol = solve.QP.bounds(Dmat = 2* risk.aversion * ia$cov.temp, dvec = ia$expected.return,
Amat = constraints$A, bvec = constraints$b,
meq = constraints$meq, lb = constraints$lb, ub = constraints$ub)
weights$long.alpha[t,] = sol$solution[1:n]
}
Next, let’s construct 130/30 portfolio. I will restrict portfolio weights to be +/- 10% and will use portfolio construction technique that I documented in the 130/30 Portfolio Construction post.
#*****************************************************************
# Construct Long/Short 130:30 portfolio using the multiple factor risk model
# based on the examples in the aa.long.short.test functions
#******************************************************************
weights$long.short.alpha = weight
for(t in 36:nperiods) {
#--------------------------------------------------------------------------
# Create constraints
#--------------------------------------------------------------------------
# set min/max wgts for individual stocks: -10/100 =< x <= 10/100
constraints = new.constraints(n, lb = -10/100, ub = 10/100)
# wgts must sum to 1 (fully invested)
constraints = add.constraints(rep(1,n), type = '=', b = 1, constraints)
#--------------------------------------------------------------------------
# beta of portfolio is the weighted average of the individual asset betas
# http://www.duke.edu/~charvey/Classes/ba350/riskman/riskman.htm
#--------------------------------------------------------------------------
constraints = add.constraints(ifna(as.vector(beta[t,]),0), type = '=', b = 1, constraints)
#--------------------------------------------------------------------------
# Create factor exposures constraints
#--------------------------------------------------------------------------
# adjust prior constraints, add factor exposures
constraints = add.variables(nfactors, constraints)
# BX - X1 = 0
constraints = add.constraints(rbind(ifna(factor.exposures[t,,], 0), -diag(nfactors)), rep(0, nfactors), type = '=', constraints)
#--------------------------------------------------------------------------
# Create 130:30
# -v.i <= x.i <= v.i, v.i>0, SUM(v.i) = 1.6
#--------------------------------------------------------------------------
# adjust prior constraints, add v.i
constraints = add.variables(n, constraints)
# -v.i <= x.i <= v.i
# x.i + v.i >= 0
constraints = add.constraints(rbind(diag(n), matrix(0,nfactors,n) ,diag(n)), rep(0, n), type = '>=', constraints)
# x.i - v.i <= 0
constraints = add.constraints(rbind(diag(n), matrix(0,nfactors,n), -diag(n)), rep(0, n), type = '<=', constraints)
# SUM(v.i) = 1.6
constraints = add.constraints(c(rep(0, n), rep(0, nfactors), rep(1, n)), 1.6, type = '=', constraints)
#--------------------------------------------------------------------------
# Create Covariance matrix
# [Qu 0]
# [ 0 Qf]
#--------------------------------------------------------------------------
temp = diag(n)
diag(temp) = ifna(specific.variance[t,], mean(coredata(specific.variance[t,]), na.rm=T))^2
cov.temp = 0*diag(n + nfactors + n)
cov.temp[1:n,1:n] = temp
cov.temp[(n+1):(n+nfactors),(n+1):(n+nfactors)] = factor.covariance[t,,]
#--------------------------------------------------------------------------
# create input assumptions
#--------------------------------------------------------------------------
ia = list()
ia$n = nrow(cov.temp)
ia$annual.factor = 12
ia$symbols = c(tickers, factor.names, tickers)
ia$cov = cov.temp
#--------------------------------------------------------------------------
# page 9, Risk: We use the Barra default setting, risk aversion value of 0.0075, and
# AS-CF risk aversion ratio of 1.
#
# The Effects of Risk Aversion on Optimization (2010) by S. Liu, R. Xu
# page 4/5
#--------------------------------------------------------------------------
risk.aversion = 0.0075
ia$cov.temp = ia$cov
# set expected return
alpha = factors.avg$AVG[t,] / 5
ia$expected.return = c(ifna(coredata(alpha),0), rep(0, nfactors), rep(0, n))
# remove companies that have no beta from optimization
index = which(is.na(beta[t,]))
if( len(index) > 0) {
constraints$ub[index] = 0
constraints$lb[index] = 0
}
# find solution
sol = solve.QP.bounds(Dmat = 2* risk.aversion * ia$cov.temp, dvec = ia$expected.return,
Amat = constraints$A, bvec = constraints$b,
meq = constraints$meq, lb = constraints$lb, ub = constraints$ub)
weights$long.short.alpha[t,] = sol$solution[1:n]
}
At this point, we created monthly long-only and 130:30 portfolios. Let’s examine their transition maps.
#***************************************************************** # Plot Transition Maps #****************************************************************** layout(1:3) for(i in names(weights)) plotbt.transition.map(weights[[i]], i)
The portfolio weights for the long-only portfolio (long.alpha) sum up to 100% and for the 130:30 portfolio (long.short.alpha) is 130% long and 30% short as expected.
Next let’s create trading strategies based on these portfolios and check their performance.
#*****************************************************************
# Create strategies
#******************************************************************
prices = data$prices
prices = bt.apply.matrix(prices, function(x) ifna.prev(x))
# find month ends
month.ends = endpoints(prices, 'months')
# create strategies that invest in each qutile
models = list()
for(i in names(weights)) {
data$weight[] = NA
data$weight[month.ends,] = weights[[i]]
capital = 100000
data$weight[] = (capital / prices) * (data$weight)
models[[i]] = bt.run(data, type='share', capital=capital)
}
#*****************************************************************
# Create Report
#******************************************************************
models = rev(models)
plotbt.custom.report.part1(models, dates='1998::')
plotbt.custom.report.part2(models)
The 130:30 portfolio works best, producing high returns with smaller drawdowns than long-only and benchmark (1/N) portfolios.
The note of caution: the above results are based on $0 commission rate (i.e. trading is free). Also, I’m using the current Dow Jones index components through out the whole history; hence, introducing survivorship bias (i.e. Dow Jones index changed its composition a few times in the last 20 years).
To see how much of the problem is my assumption of $0 commission rate, let’s have a look at the annual portfolio turnovers.
# Plot Portfolio Turnover for each strategy layout(1) barplot.with.labels(sapply(models, compute.turnover, data), 'Average Annual Portfolio Turnover')
The 130:30 portfolio has a pretty high portfolio turnover; therefore, it will not perform as well in the real life as in our backtest.
To view the complete source code for this example, please have a look at the fm.long.short.test() function in factor.model.test.r at github.
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