Final Project IPA

1. Download monthly equity market capitalization of 7 countries

Compute monthly equity market returns (in simple returns) based on market capitalization data.

require(pacman)

## Loading required package: pacman

require(xts)

## Loading required package: xts

## Loading required package: zoo

## 
## Attaching package: 'zoo'

## The following objects are masked from 'package:base':
## 
##     as.Date, as.Date.numeric

require(openxlsx)

## Loading required package: openxlsx

require(quantmod)

## Loading required package: quantmod

## Loading required package: TTR

## Registered S3 method overwritten by 'quantmod':
##   method            from
##   as.zoo.data.frame zoo

require(devtools)

## Loading required package: devtools

## Loading required package: usethis

devtools::install_github('systematicinvestor/SIT.date')

## Skipping install of 'SIT.date' from a github remote, the SHA1 (6263da60) has not changed since last install.
##   Use `force = TRUE` to force installation

require(curl)

## Loading required package: curl

## Using libcurl 7.68.0 with OpenSSL/1.1.1f

curl_download('https://github.com/systematicinvestor/SIT/raw/master/SIT.tar.gz', 'SIT',mode = 'wb',quiet=T)

install.packages('SIT', repos = NULL, type='source')

## Installing package into '/cloud/lib/x86_64-pc-linux-gnu-library/4.2'
## (as 'lib' is unspecified)

require(SIT)

## Loading required package: SIT

## Loading required package: SIT.date

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## Attaching package: 'SIT'

## The following object is masked from 'package:TTR':
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## The following object is masked from 'package:base':
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require(ggplot2)

## Loading required package: ggplot2

p_load(quantmod, quadprog,lpSolve)
p_load(xts)
p_load(TTR)
require(readxl)

## Loading required package: readxl

library(readxl)
library(ggplot2)
library(lubridate)

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## Attaching package: 'lubridate'

## The following objects are masked from 'package:base':
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library(DT)
library(curl)

library(tidyquant)

## Loading required package: PerformanceAnalytics

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## Attaching package: 'PerformanceAnalytics'

## The following object is masked from 'package:graphics':
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library(quantmod)
library(purrr)

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## The following object is masked from 'package:SIT':
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library(SIT)

rm(list=ls())
devtools::install_github('joshuaulrich/xts', force = T)

## Downloading GitHub repo joshuaulrich/xts@HEAD

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## * checking for file ‘/tmp/RtmpVGOl4Q/remotes139d11af2f6c/joshuaulrich-xts-96612b7/DESCRIPTION’ ... OK
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## * building ‘xts_0.12.1.3.tar.gz’
## Warning in sprintf(gettext(fmt, domain = domain, trim = trim), ...) :
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## Warning: invalid uid value replaced by that for user 'nobody'

## Installing package into '/cloud/lib/x86_64-pc-linux-gnu-library/4.2'
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devtools::install_github('joshuaulrich/quantmod', force = T)

## Downloading GitHub repo joshuaulrich/quantmod@HEAD

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## * checking for file ‘/tmp/RtmpVGOl4Q/remotes139d797c0787/joshuaulrich-quantmod-d1b4b17/DESCRIPTION’ ... OK
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## * installing the package to process help pages
## * saving partial Rd database
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#
library(SIT)
library(quantmod)
library(quadprog)

Betting_Against_Beta_Equity_Factors_Monthly_1_ <- read_excel("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) )
stockpr2 <- with(stockpr, stockpr[(date >= "1989-12-31" & date <= "2021-12-31")])

monthly.return <- na.omit(Return.calculate(stockpr2, method = "discrete"))

head(monthly.return)

##                    AUS           CAN          FRA          DEU         JPN
## 1990-01-31  0.03066617  0.0242062712  0.106772375  0.175900236  0.02123232
## 1990-02-28  0.01574424 -0.0631233515 -0.039850578  0.022063077 -0.04670123
## 1990-03-31 -0.06035407  0.0006305838 -0.028605990  0.002080930 -0.08398405
## 1990-04-30 -0.02950981 -0.0104092150  0.067718048  0.095673570 -0.17803619
## 1990-05-31 -0.08247848 -0.0819545270  0.041442220 -0.061484068 -0.01682893
## 1990-06-30  0.09444762  0.0728235944  0.002914819  0.005735468  0.15212730
##                    GBR         USA
## 1990-01-31  0.09773906  0.01799629
## 1990-02-28  0.01176512 -0.07791268
## 1990-03-31 -0.03207884  0.01297222
## 1990-04-30 -0.03012199  0.02484288
## 1990-05-31 -0.06753591 -0.02987430
## 1990-06-30  0.13266871  0.08571573

tail(monthly.return)

##                     AUS          CAN           FRA          DEU           JPN
## 2021-07-31 -0.007808452 -0.004923728 -0.0159977640 -0.005650794  6.743379e-05
## 2021-08-31 -0.006427044 -0.005186644  0.0148548942  0.012951049 -1.218586e-02
## 2021-09-30  0.019011675  0.001517633 -0.0004440893  0.015669145  2.783157e-02
## 2021-10-31 -0.024477429 -0.023212521 -0.0501314877 -0.052374557  1.671376e-02
## 2021-11-30  0.041072770  0.068697882  0.0495422303  0.022591338 -3.342947e-02
## 2021-12-31 -0.058206115 -0.046620684 -0.0319030400 -0.047759221 -3.347900e-02
##                     GBR          USA
## 2021-07-31 -0.033224039  0.026827519
## 2021-08-31  0.025475826  0.009958025
## 2021-09-30  0.006104884  0.026998156
## 2021-10-31 -0.036415768 -0.040433011
## 2021-11-30  0.023093370  0.066567980
## 2021-12-31 -0.054600574 -0.016133549

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.

stockpr3 <- with(stockpr, stockpr[(date >= "1993-01-31" & date <= "2021-12-31")])

#convert returns into price (data market value of equity)
#stockpr3 <- cumprod(stockpr3 + 1)*100
#head(stockpr3)
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)

## Latest weights :
##              AUS   CAN   FRA   DEU   JPN   GBR   USA
## 2021-12-31 14.29 14.29 14.29 14.29 14.29 14.29 14.29
## 
## Performance summary :
##  CAGR    Best    Worst   
##  8.2 14.1    -21.7

capital = 100000
data$weight[] = (capital / prices) * data$weight
equal.weight = bt.run(data, type='share')

## Latest weights :
##              AUS   CAN   FRA   DEU   JPN   GBR   USA
## 2021-12-31 14.29 14.29 14.29 14.29 14.29 14.29 14.29
## 
## Performance summary :
##  CAGR    Best    Worst   
##  8.2 14.1    -21.7

head(equal.weight$ret)

##                    AUS
## 1993-01-31 0.000000000
## 1993-02-28 0.003976969
## 1993-03-31 0.044861517
## 1993-04-30 0.065207441
## 1993-05-31 0.035969320
## 1993-06-30 0.024251503

bt.detail.summary(model$equal.weight)

## $System
## $System$Period
## [1] "Jan1993 - Dec2021"
## 
## $System$Cagr
## [1] 8.2
## 
## $System$Sharpe
## [1] 0.57
## 
## $System$DVR
##     [,1]
## AUS 0.51
## 
## $System$Volatility
## [1] 16.02
## 
## $System$MaxDD
## [1] -57.92
## 
## $System$AvgDD
## [1] -7.69
## 
## $System$VaR
##    5% 
## -7.26 
## 
## $System$CVaR
## [1] -11.07
## 
## $System$Exposure
## [1] 99.71
## 
## 
## $Trade
## $Trade$Win.Percent
## [1] 100
## 
## $Trade$Avg.Trade
## [1] 130.2
## 
## $Trade$Avg.Win
## [1] 130.2
## 
## $Trade$Avg.Loss
## [1] NaN
## 
## $Trade$Best.Trade
## [1] 213.68
## 
## $Trade$Worst.Trade
## [1] 25.81
## 
## $Trade$WinLoss.Ratio
## [1] NaN
## 
## $Trade$Avg.Len
## [1] 347
## 
## $Trade$Num.Trades
## [1] 7
## 
## 
## $Period
## $Period$Win.Percent.Day
## [1] 60.3
## 
## $Period$Best.Day
## [1] 14.1
## 
## $Period$Worst.Day
## [1] -21.7
## 
## $Period$Win.Percent.Month
## [1] 60.3
## 
## $Period$Best.Month
## [1] 14.1
## 
## $Period$Worst.Month
## [1] -21.7
## 
## $Period$Win.Percent.Year
## [1] 69
## 
## $Period$Best.Year
## [1] 43
## 
## $Period$Worst.Year
## [1] -48.3

plotbt.monthly.table(model$equal.weight$equity)

##      Jan     Feb     Mar     Apr     May     Jun     Jul     Aug     Sep    
## 1993 "   NA" "  0.4" "  4.5" "  6.5" "  3.6" "  2.4" " -1.1" "  3.1" "  7.0"
## 1994 "  7.4" "  6.5" " -1.3" " -3.5" "  2.4" " -0.3" " -1.7" "  4.3" "  3.8"
## 1995 "  1.1" " -2.6" "  2.0" "  4.4" "  3.1" "  0.4" " -0.4" "  5.9" " -1.9"
## 1996 "  1.5" "  2.8" "  1.3" "  1.0" "  3.3" "  0.6" " -0.4" " -2.3" "  2.1"
## 1997 " -0.6" " -0.8" "  1.8" " -0.7" "  0.8" "  5.2" "  3.9" "  3.8" " -5.4"
## 1998 "  1.0" "  4.1" "  6.2" "  3.5" "  0.8" " -0.5" "  1.1" " -0.7" "-13.4"
## 1999 "  2.4" "  4.4" " -3.4" "  4.2" "  6.6" " -4.3" "  5.8" "  1.6" "  0.5"
## 2000 "  9.4" " -4.1" "  4.3" "  3.6" " -5.7" " -3.6" "  6.1" " -1.8" "  2.7"
## 2001 "  3.3" "  2.4" " -7.5" " -7.4" "  8.0" " -1.6" " -1.6" " -2.5" " -2.9"
## 2002 "  0.4" " -2.6" "  0.3" "  4.9" " -0.1" "  2.1" " -4.3" " -9.1" " -1.9"
## 2003 " -2.4" " -1.4" " -1.9" " -1.0" "  9.6" "  6.5" "  3.3" "  1.6" "  2.4"
## 2004 "  7.1" "  1.5" "  2.3" "  0.9" " -3.9" "  0.5" "  2.6" " -2.4" "  0.1"
## 2005 "  3.9" " -1.4" "  4.2" " -1.9" " -3.3" "  0.6" "  2.1" "  3.8" "  1.4"
## 2006 "  3.9" "  6.7" " -0.4" "  2.7" "  4.7" " -4.0" " -0.6" "  0.2" "  2.5"
## 2007 "  2.7" "  0.4" "  1.3" "  2.5" "  4.4" "  2.9" "  0.0" " -1.9" " -2.2"
## 2008 " -1.5" " -8.7" "  1.7" " -2.6" "  4.8" "  2.3" " -6.8" " -3.4" " -3.7"
## 2009 "  5.3" " -9.9" " -9.4" "  8.1" " 12.5" " 11.6" "  0.0" "  9.6" "  4.4"
## 2010 "  1.7" " -4.4" "  0.7" "  6.2" " -0.5" "-10.9" " -2.8" "  8.9" " -3.7"
## 2011 "  8.5" "  2.2" "  4.3" " -1.7" "  4.8" " -2.9" " -1.2" " -1.6" " -8.3"
## 2012 " -2.1" "  6.5" "  5.0" " -0.6" " -1.4" "-11.3" "  4.6" "  1.5" "  2.6"
## 2013 "  2.4" "  4.8" " -0.6" "  0.7" "  3.4" " -2.0" " -4.1" "  5.4" " -1.1"
## 2014 "  1.6" " -3.7" "  5.0" " -0.2" "  0.8" "  1.5" "  2.0" " -1.9" "  0.8"
## 2015 " -2.0" " -1.4" "  6.1" " -2.3" "  3.5" " -1.3" " -2.5" "  0.9" " -7.0"
## 2016 " -1.9" " -6.3" " -1.0" "  7.6" "  2.7" " -1.3" " -2.7" "  5.4" " -0.5"
## 2017 "  2.9" "  3.0" "  1.6" "  1.9" "  1.5" "  1.7" "  0.8" "  2.9" "  0.0"
## 2018 "  2.5" "  4.5" " -4.8" " -1.6" "  1.7" " -0.7" " -1.1" "  2.0" " -1.1"
## 2019 " -6.0" "  7.4" "  2.5" " -0.4" "  2.7" " -4.8" "  5.6" " -0.7" " -2.6"
## 2020 "  3.2" " -1.7" " -9.7" "-17.0" " 10.1" "  5.2" "  3.6" "  3.8" "  6.0"
## 2021 "  6.0" " -1.0" "  3.2" "  2.6" "  4.4" "  2.8" " -0.6" "  0.6" "  1.4"
## Avg  "  2.2" "  0.3" "  0.6" "  0.7" "  2.9" " -0.1" "  0.3" "  1.3" " -0.6"
##      Oct     Nov     Dec     Year    MaxDD  
## 1993 " -1.4" "  4.6" " -4.1" " 28.0" " -4.1"
## 1994 " -3.2" "  2.2" " -3.7" " 12.6" " -4.7"
## 1995 "  1.2" " -0.9" "  2.8" " 15.6" " -2.6"
## 1996 "  2.7" "  1.9" "  6.9" " 23.3" " -2.7"
## 1997 "  5.5" " -4.9" "  0.1" "  8.4" " -5.4"
## 1998 "  0.7" "  9.3" "  5.9" " 17.4" "-14.0"
## 1999 "  0.9" "  5.0" "  5.3" " 32.2" " -4.3"
## 2000 " -6.5" " -3.7" " -5.1" " -5.8" "-16.9"
## 2001 "-10.4" "  3.4" "  4.6" "-13.2" "-24.1"
## 2002 "-10.4" "  4.9" "  3.6" "-12.8" "-23.4"
## 2003 "  3.0" "  7.3" "  1.7" " 31.8" " -6.6"
## 2004 "  4.2" "  4.0" "  6.4" " 25.3" " -3.9"
## 2005 "  3.8" " -4.0" "  3.5" " 12.8" " -5.1"
## 2006 " -0.5" "  4.7" "  2.6" " 24.3" " -4.5"
## 2007 "  6.1" "  5.0" " -5.2" " 16.5" " -5.2"
## 2008 "-14.4" "-21.7" " -6.8" "-48.3" "-48.3"
## 2009 "  5.5" " -2.1" "  3.6" " 43.0" "-18.4"
## 2010 " 10.4" "  4.0" " -2.7" "  5.0" "-13.8"
## 2011 "-11.0" " 10.8" " -4.5" " -3.0" "-23.0"
## 2012 "  2.7" "  1.3" "  1.2" "  9.1" "-13.1"
## 2013 "  6.5" "  3.6" "  0.3" " 20.2" " -6.0"
## 2014 " -5.2" " -0.5" "  0.8" "  0.7" " -6.7"
## 2015 " -4.5" "  7.3" " -0.5" " -4.5" "-13.8"
## 2016 "  1.0" " -2.3" " -0.7" " -0.7" " -9.0"
## 2017 "  2.5" "  1.4" "  1.2" " 23.4" "  0.0"
## 2018 "  0.1" " -8.5" " -0.6" " -8.0" "-14.1"
## 2019 "  2.2" "  2.7" "  1.8" " 10.1" " -6.0"
## 2020 " -3.2" " -3.0" " 14.1" "  7.4" "-26.3"
## 2021 " -3.0" "  3.4" " -4.1" " 16.2" " -4.1"
## Avg  " -0.5" "  1.2" "  1.0" "  9.9" "-11.4"

strategy.performance.snapshoot(model, T)

## NULL

2.2 Compute MVP portfolio returns by rebalancing EACH month starting from 1993/01 to 2021/12.

Use in-sample data range of previous 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)

# SUM x.i = 1
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
min.var = bt.run(data, type='share', capital=capital)

## Latest weights :
##               AUS   CAN   FRA   DEU    JPN   GBR  USA
## 2021-12-31 -33.98 21.66 -8.95 15.53 107.47 -4.46 2.72
## 
## Performance summary :
##  CAGR    Best    Worst   
##  4.5 11.4    -15.9

model$min.var.monthly <- bt.run(data, trade.summary = T)

## Latest weights :
##             AUS  CAN   FRA  DEU  JPN   GBR  USA
## 2021-12-31 -1.8 0.72 -0.26 0.53 1.56 -0.12 0.01
## 
## Performance summary :
##  CAGR    Best    Worst   
##  -0.1    1.1 -1.5

sum(as.numeric(weight[36,])*as.numeric(ret[37,]))

## [1] 0.0318602

model$min.var.monthly$ret[37, ]

##                    AUS
## 1996-01-31 0.005077245

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)

## NULL

model <- rev(model)
plotbt.custom.report(model)

Similar to homework 9 (reference file: Black_Literman_model_SIT.R and website: (https://systematicinvestor.wordpress.com/2011/11/16/black-litterman-model/), update the annual data from 1988-2021 based on the enclosed excel file from AQR. By updating required parameter inputs conditional on the new data, show your answers to the following questions:

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 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) )

3.2 Compute Risk Aversion, prepare Black-Litterman input assumptions

a. compute Risk Aversion

b. comoute implied equilibrium returns

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])
}

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 = '1991-01-01', 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)
}

# 3. Load up efficient frontier plotting function:
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)    
}

#--------------------------------------------------------------------------
# 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, as.vector(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)