OptimalPortfolio.Rmd

---
title: "Optimal Portfolio"
author: "Pasquale Antonante"
date: "6/5/2018"
output:
  html_document:
    toc: true
    theme: sandstone
---

```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = TRUE)
require(pander)
library(mongolite)
library(quantmod)
library(PortfolioAnalytics)
library(doParallel)

registerDoParallel(cores=4)
etfs_collection <- mongo(collection = "details", db = "etf")
```

## Parameters

```{r params}
# General
q <- 2 # last q years used for evaluate performances (q<k)
k <- 5 # last k-q years used for optimization
risk_free_rate <- .028 / 12 # monthly risk free rate
risk_aversion <-  0.2 # only used if quadratic utility
diversification_target <- 0.1 # lower enforce more diversification
capital <- 800 # in dollars
search_size <- 10000 # number of random generated portfolios

# Asset allocation
group_labels <- c("Stock", "Bond") # do not change
benckmark_weights <- c(0.8,0.2) # how to mix SPY and IEF (iShares 7-10 Year US Bond)
group_min=c(0.6, 0.2)
group_max=c(0.8, 0.4)
```

## Selected ETFs

```{r etfs, echo=FALSE, message=FALSE, warnings=FALSE}
etfs <- etfs_collection$find( 
  fields = '{"_id": false, "ticker": true, "asset_class": true, "category": true,"name": true, "expense_ratio":true}',
  sort = '{"category": 1, "asset_class": 1}')
etfs$expense_ratio <-etfs$expense_ratio*100
colnames(etfs) <- c("Ticker", "Name","Category", "Asset class", "Expense Ratio")
panderOptions('table.split.table', Inf)
pander(etfs, style = 'rmarkdown')
```

```{r symbol-download, include=FALSE, message=FALSE, warnings=FALSE, fig.align='center'}
stock <- etfs_collection$find(query = '{"asset_class":"Equity"}', fields = '{"_id": false, "ticker": true, "expense_ratio":true}')
bond <- etfs_collection$find(query = '{"asset_class":"Bond"}', fields = '{"_id": false, "ticker": true, "expense_ratio":true}')
symbols <- c(stock$ticker, bond$ticker)
expense_ratios <- c(stock$expense_ratio, bond$expense_ratio)
groups <- list( 1:length(stock$ticker),
               (1+length(stock$ticker)):(length(stock$ticker)+length(bond$ticker)))
end_date <- format(Sys.Date() - q*365,"%Y-%m-%d")
start_date <- format(Sys.Date() - k*365, "%Y-%m-%d")
returns_list <- lapply(symbols, function(sym) {
  monthlyReturn(na.omit(Ad(getSymbols(sym, from=start_date, to=end_date, auto.assign=FALSE))))
})
returns <- do.call(merge, returns_list)
colnames(returns) <- symbols
returns <- sweep(returns,2,expense_ratios/12) # removing monthly expense
returns[is.na(returns)] <- 0 # if we can't buy it, we can't profit from it

#chart.Correlation(returns, histogram=FALSE, method = 'kendall')
```

## Portfolio constraint specification

```{r portfolio-spec, message=FALSE, warnings=FALSE}
############## PORTFOLIO INIT ##############

pspec <- portfolio.spec(symbols)

############## CONSTRAINTS ##############

# The long-only, full-investment constraint
pspec <- add.constraint(portfolio=pspec, type="weight_sum", min_sum = 0.99, max_sum = 1.01)

# Add box constraints, each investment should not exceed 50%
pspec <- add.constraint(portfolio=pspec, type="box", min=0.0, max=0.5)

# Add group constraints
pspec <- add.constraint(portfolio=pspec, type="group",
                        groups=groups,
                        group_labels=group_labels,
                        group_min=group_min,
                        group_max=group_max)

# Add position limit constraint such that we have a maximum number of three 
# assets with non-zero weights.
#pspec <- add.constraint(portfolio=pspec, type="position_limit", max_pos=3)

# Add turnover constraint
##pspec <- add.constraint(portfolio=pspec, type="turnover", turnover_target=0.2)

# Add target mean return constraint
#pspec <- add.constraint(portfolio=pspec, type="return", return_target=0.07/12)
```


## Portfolio objective

```{r portfolio-obj, message=FALSE, warnings=FALSE}
# Min-Var portfolio
minvar <- add.objective(pspec, type = "return", name = "mean")
minvar <- add.objective(minvar, type = "risk", name = "var", multiplier = 1)

# Min-Expected Shortfall (or Expected Tail Losses - ETL)
minetl <- add.objective(portfolio=pspec, type="risk", name="ES")
minetl <- add.objective(portfolio=minetl, type="return", name="mean")

# Max-Sortino Ration
sortino <- add.objective(portfolio = pspec, type="return", name="SortinoRatio", enabled=TRUE, arguments = list(MAR=risk_free_rate))
# multiplier 0 makes it availble for plotting, but not affect optimization
sortino <- add.objective(portfolio = sortino, type="return", name="mean", enabled=TRUE, multiplier=0) 
```

## Optimization

```{r portfolio-optimization, message=FALSE, warnings=FALSE}
minvar.opt <- optimize.portfolio(R = returns,
                                 optimize_method = "random",
                                 search_size = search_size,
                                 portfolio = minvar,
                                 trace=TRUE, verbose=TRUE)

minetl.opt <- optimize.portfolio(R = returns,
                                 optimize_method = "random",
                                 search_size = search_size,
                                 portfolio = minetl,
                                 trace=TRUE, verbose=TRUE)

sortino.opt <- optimize.portfolio(R = returns,
                                  optimize_method = "random",
                                  search_size = search_size,
                                  portfolio = sortino,
                                  trace=TRUE, verbose=TRUE)
```


```{r portfolio-plots, message=FALSE, warnings=FALSE, fig.align='center'}
plot(minvar.opt, risk.col='StdDev', main="Min-Var")
plot(minetl.opt, risk.col='ES', main="Min-ES")
plot(sortino.opt, risk.col='SortinoRatio', main="Max-Sortino")
```

## Portfolios Weights

```{r portfolio-weights, message=FALSE, warnings=FALSE, fig.align='center'}
minvar.weights <- extractWeights(minvar.opt)
minetl.weights <- extractWeights(minetl.opt)
sortino.weights <- extractWeights(sortino.opt)
weights <- cbind(minvar.weights,minetl.weights,sortino.weights)
colnames(weights) <- c("Min-Var", "Min-ES", "Max-Sortino")
pander(weights, style = 'rmarkdown')
```

## Performances

```{r portfolio-returns, message=FALSE, warnings=FALSE, fig.align='center'}
end_date <- format(Sys.Date(),"%Y-%m-%d")
start_date <- format(Sys.Date() - q*365, "%Y-%m-%d")
# Benckmark
SPY <- monthlyReturn(na.omit(Ad(getSymbols("SPY", from=start_date, to=end_date, auto.assign=FALSE))))
IEF <- monthlyReturn(na.omit(Ad(getSymbols("IEF", from=start_date, to=end_date, auto.assign=FALSE))))
benckmark <- Return.portfolio(merge(SPY,IEF), weights = benckmark_weights, rebalance_on = "quarters")
colnames(benckmark) <- "Index"
colnames(SPY) <- "SPY"

# Getting quotes again returns
returns_list <- lapply(symbols, function(sym) {
  monthlyReturn(na.omit(Ad(getSymbols(sym, from=start_date, to=end_date, auto.assign=FALSE))))
})
returns <- do.call(merge, returns_list)
colnames(returns) <- symbols
returns <- sweep(returns,2,expense_ratios/12) # removing monthly expense
returns[is.na(returns)] <- 0 # if we can't buy it, we can't profit from it

# Portfolio returns
minvar.return <- Return.portfolio(returns, weights = minvar.weights, rebalance_on = "quarters")
colnames(minvar.return) <- "Min-Var"
minetl.return <- Return.portfolio(returns, weights = minetl.weights, rebalance_on = "quarters")
colnames(minetl.return) <- "Min-ES"
sortino.return <- Return.portfolio(returns, weights = sortino.weights, rebalance_on = "quarters")
colnames(sortino.return) <- "Max-Sortino"

all.in.one <- merge(minvar.return, minetl.return, sortino.return, benckmark, SPY)
```

### Performance Summary
```{r portfolio-performance, message=FALSE, warnings=FALSE, fig.align='center'}
charts.PerformanceSummary(all.in.one, risk_free_rate, main="Cumulative returns", legend.loc= "topleft")
```

### Risk/Return Scatter
```{r portfolio-riskreturn, message=FALSE, warnings=FALSE, fig.align='center'}
chart.RiskReturnScatter(all.in.one, Rf = risk_free_rate, main = "Risk/Return Scatter")
```

### Capital Asset Pricing Model (CAPM) parameters
```{r capm, ECHO=FALSE,message=FALSE, warnings=FALSE}
table <- table.CAPM(merge(minvar.return, minetl.return, sortino.return), 
                    merge(benckmark,SPY), Rf = risk_free_rate)
pander(table, style = 'rmarkdown')
```

- **Alpha:** the degree to which the asset's returns are not explained by the return that could be captured from the market.Can be used to directly measure the value added or subtracted by a portfolio manager
- **Beta:** describes the portions of the returns of the asset that could be directly attributed to the returns of a passive investment in the benchmark asset (via linear model regression)
- **TreynorRatio:** ratio of asset's Excess Return to Beta β of the benchmark
- **ActivePremium:** investment's annualized return minus the benchmark's annualized return
- **TrackingError:** a measure of the unexplained portion of performance relative to a benchmark
- **InformationRatio:** ActivePremium/TrackingError. Information Ratio may be used to rank investments in a relative fashion

### Annualized Returns
```{r annual-returns, ECHO=FALSE,message=FALSE, warnings=FALSE}
table <- table.AnnualizedReturns(all.in.one, Rf=risk_free_rate)
pander(table, style = 'rmarkdown')
```

### Sharpe ratios
```{r sharpe, ECHO=FALSE,message=FALSE, warnings=FALSE}
sharpe_95 <- SharpeRatio(all.in.one, Rf = risk_free_rate, p=.95)
sharpe_99 <- SharpeRatio(all.in.one, Rf = risk_free_rate, p=.99)
sharpe <- rbind(sharpe_95,sharpe_99)
pander(sharpe, style = 'rmarkdown')
```

<!-- ### To buy -->

<!-- ```{r byu} -->
<!-- buy<-matrix(0, nrow = length(symbols)+1, ncol = 3) -->
<!-- row.names(buy)<-c(symbols,"CASH") -->
<!-- colnames(buy)<-c("Allocation", "Quote", "Shares") -->
<!-- quotes <- getQuote(symbols) -->
<!-- buy[1:length(symbols),1] <- round(w, digits = 2) -->
<!-- buy[1:length(symbols),2] <- quotes$Last -->
<!-- for(i in 1:length(symbols)){ -->
<!--   buy[i,3] = ifelse(buy[i,1]==0,0,floor(buy[i,1]*capital/buy[i,2])) -->
<!-- } -->
<!-- buy[length(buy[,1]),] <- c(NA,1,round(capital-sum(buy[1:length(symbols),3]*buy[1:length(symbols),2]), digits=2)) -->
<!-- buy[length(buy[,1]),1] <- buy[length(buy[,1]),3] / capital -->
<!-- pander(buy, style = 'rmarkdown') -->
<!-- ``` -->