library(readxl) library(ggplot2) library(lubridate) library(DT) library(curl) library(tidyquant) library(quantmod) library(purrr) library(SIT)
#load data Betting_Against_Beta_Equity_Factors_Monthly_1_ <- read_excel(“Downloads/Betting Against Beta Equity Factors Monthly (1).xlsx”, sheet = “ME(t-1)”, range = “A19:AD1165”) data <- Betting_Against_Beta_Equity_Factors_Monthly_1_ ########
date <- as.Date(data$DATE, “%m/%d/%Y”)
Betting_Against_Beta_Equity_Factors_Monthly_1_ <- xts(coredata(Betting_Against_Beta_Equity_Factors_Monthly_1_[, -1]), order.by = date)
stockpr = subset(Betting_Against_Beta_Equity_Factors_Monthly_1_, select = c(AUS,CAN,FRA,DEU,JPN,GBR,USA) )
#question 1 stockpr1 <- with(stockpr, stockpr[(date >= “1989-11-30” & date <= “2021-12-31”)])
monthly.return <- na.omit(Return.calculate(stockpr1, method = “discrete”))
head(monthly.return) tail(monthly.return) ######### #question 2
stockpr3 <- with(stockpr, stockpr[(date >= “1993-01-31” & date <= “2021-12-31”)]) data <- new.env() #create 4 required input elements in data data\(prices <- stockpr3 data\)weight <- stockpr3 data\(execution.price <- stockpr3 data\)execution.price[] <- NA data\(symbolnames <- colnames(data\)prices) prices <- data$prices n = ncol(prices)
data$weight = ntop(prices, n)
model <-list() model\(equal.weight <- bt.run(data, trade.summary=T) capital = 100000 data\)weight[] = (capital / prices) * data\(weight #2.1 Compute the equal-weighted portfolio returns EACH month starting from 1993/01 to 2021/12. #Denote this strategy as the Benchmark portfolio and create its backtesting report using SIT package. equal.weight = bt.run(data, type='share') head(equal.weight\)ret) bt.detail.summary(model\(equal.weight) strategy.performance.snapshoot(model, T) data\)weight = ntop(prices, n) head(data$weight)
#Question 2.2: Compute MVP portfolio returns by rebalancing EACH month starting from 1993/01 to 2021/12. #Use in-sample data range of 36 months to compute covariance matrix. #Denote this strategy as the MVP portfolio and create its backtesting report using SIT.
data\(prices <- stockpr3 data\)weight <- stockpr3 data\(execution.price <- stockpr3 data\)execution.price[] <- NA prices <- data$prices
constraints = new.constraints(n, lb = -Inf, ub = +Inf)
constraints = add.constraints(rep(1, n), 1, type = ‘=’, constraints)
ret = prices / mlag(prices) - 1 weight = coredata(prices) weight[] =
NA i = 36 for (i in 36:dim(weight)[1]) { hist = ret[ (i- 36 +1):i, ]
hist = na.omit(hist) ia = create.historical.ia(hist, 12) ia\(cov = cov(coredata(hist)) weight[i,] =
min.risk.portfolio(ia, constraints) } data\)weight[] =
weight
capital = 100000 data\(weight[] = (capital /
prices) * data\)weight model\(min.var.monthly <- bt.run(data, type =
'share', capital = capital)
sum(as.numeric(weight[36,])*as.numeric(ret[37,]))
model\)min.var.monthly$ret[37, ] #Question 2.3: Plot both
strategies side by side and compare their performance and comment.
plotbt.custom.report.part1(model\(min.var.monthly, model\)equal.weight)
layout(1:2) plotbt.transition.map(model\(min.var.monthly\)weight) legend(‘topright’, legend = ‘min.var.monthly’, bty = ‘n’) plotbt.transition.map(model\(equal.weight\)weight) legend(‘topright’, legend = ‘equal weight’, bty = ‘n’) strategy.performance.snapshoot(model, T) model <- rev(model) plotbt.custom.report(model)
#question3 ########## stockpr4 <- with(stockpr, stockpr[(date >= “1988-01-31” & date <= “2021-12-31”)])
hist.caps = stockpr4 hist.caps.weight = hist.caps/rowSums(hist.caps)
aa.test.create.ia.country <- function(dates = ‘1990::2021’) { # load.packages(‘quantmod,quadprog’) symbols = spl(‘EWA,EWC,EWQ,EWG,EWJ,EWU,SPY’) symbol.names = spl(‘Australia, Canada, France, Germany, Japan, UK, USA’) getSymbols(symbols, from = ‘1980-01-01’, auto.assign = TRUE) hist.prices = merge(EWA,EWC,EWQ,EWG,EWJ,EWU,SPY) period.ends = endpoints(hist.prices, ‘months’) hist.prices = Ad(hist.prices)[period.ends, ] colnames(hist.prices) = symbol.names annual.factor = 12 hist.prices = na.omit(hist.prices[dates]) hist.returns = na.omit( ROC(hist.prices, type = ‘discrete’) ) ia = create.historical.ia(hist.returns, annual.factor) return(ia) }
plot.ef <- function( ia, efs, portfolio.risk.fn = portfolio.risk,
transition.map = TRUE, layout = NULL ) { risk.label =
as.character(substitute(portfolio.risk.fn)) n = ia\(n x = match.fun(portfolio.risk.fn)(diag(n),
ia) y = ia\)expected.return xlim = range(c(0, x, max(
sapply(efs, function(x) max(match.fun(portfolio.risk.fn)(x\(weight,ia))) ) ), na.rm = T) ylim = range(c(0,
y, min( sapply(efs, function(x)
min(portfolio.return(x\)weight,ia))) ), max( sapply(efs,
function(x) max(portfolio.return(x\(weight,ia))) ) ), na.rm = T) x = 100 * x y =
100 * y xlim = 100 * xlim ylim = 100 * ylim if( !transition.map )
layout = T if( is.null(layout) ) layout(1:2) par(mar = c(4,3,2,1), cex
= 0.8) plot(x, y, xlim = xlim, ylim = ylim, xlab='',
ylab='', main=paste(risk.label, 'vs Return'),
col='black') mtext('Return', side = 2,line = 2, cex =
par('cex')) mtext(risk.label, side = 1,line = 2, cex =
par('cex')) grid(); text(x, y, ia\)symbols, col =
‘blue’, adj = c(1,1), cex = 0.8) for(i in len(efs):1) { ef = efs[[ i ]]
x = 100 * match.fun(portfolio.risk.fn)(ef\(weight, ia) y = 100 * ef\)return lines(x,
y, col=i) } plota.legend(sapply(efs, function(x) x\(name), 1:len(efs)) if(transition.map)
{ plot.transition.map(efs[[i]]\)weight, x, risk.label,
efs[[i]]$name) } } # 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)
} # 3.1 Visualize Market Capitalization History: # a. Plot Transition of
Market Cap Weights in time #b. Plot History for each Country’s Market
Cap stockpr4 <- with(stockpr, stockpr[(date >= “1988-01-31” &
date <= “2021-12-31”)])
hist.caps = stockpr4 hist.caps.weight = hist.caps/rowSums(hist.caps)
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]) } #3.2 Compute Risk Aversion, prepare Black-Litterman input assumptions #————————————————————————– # Compute Risk Aversion, prepare Black-Litterman input assumptions #————————————————————————– ia = aa.test.create.ia.country()
risk.aversion = bl.compute.risk.aversion( ia\(hist.returns\)USA )
risk.aversion # the latest market capitalization weights cap.weight =
last(hist.caps.weight)
ia.bl = ia ia.bl\(expected.return = bl.compute.eqret( risk.aversion, ia\)cov, as.vector(cap.weight) )
#3.2 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$n))
plot.ia(ia.bl, T) #————————————————————————– # Create Efficient Frontier(s) #————————————————————————– n = ia$n
constraints = new.constraints(n, lb = 0, ub = 1)
constraints = add.constraints(rep(1, n), 1, type = ‘=’, constraints)
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)
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) ##
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
temp[,’ Japan’] = 1 temp[,’ UK’] = -1
pmat = temp qmat = c(0.02)
temp[] = 0 # temp[,‘Australia’] = 1 temp[,1] = 1 pmat = rbind(pmat,
temp)
qmat = c(qmat, 0.12)
post = bl.compute.posterior(ia.bl\(expected.return, ia\)cov, pmat, qmat, tau = 0.025 )
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))
ef.risk.bl.view = portopt(ia.bl.view, constraints, 50, ‘Black-Litterman + View(s)’, equally.spaced.risk = T)
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)