# Portfolio Optimization using R and Plotly

April 2, 2016
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In this post we’ll focus on showcasing Plotly’s WebGL capabilities by charting financial portfolios using an R package called PortfolioAnalytics. The package is a generic portfolo optimization framework developed by folks at the University of Washington and Brian Peterson (of the PerformanceAnalytics fame).

You can see the vignette here

Let’s pull in some data first.

```library(PortfolioAnalytics)
library(quantmod)
library(PerformanceAnalytics)
library(zoo)
library(plotly)

# Get data
getSymbols(c("MSFT", "SBUX", "IBM", "AAPL", "^GSPC", "AMZN"))

# Assign to dataframe
prices.data <- merge.zoo(MSFT[,6], SBUX[,6], IBM[,6], AAPL[,6], GSPC[,6], AMZN[,6])

# Calculate returns
returns.data <- CalculateReturns(prices.data)
returns.data <- na.omit(returns.data)

# Set names
colnames(returns.data) <- c("MSFT", "SBUX", "IBM", "AAPL", "^GSPC", "AMZN")

# Save mean return vector and sample covariance matrix
meanReturns <- colMeans(returns.data)
covMat <- cov(returns.data)```

Now that we have some data, let’s get started by creating a portfolio specification. This can be done by using `portfolio.spec()`

```# Start with the names of the assets
port <- portfolio.spec(assets = c("MSFT", "SBUX", "IBM", "AAPL", "^GSPC", "AMZN"))```

Now for some constraints. Let’s use the following:

• Box constraints
• Leverage (weight sum)
```# Box
port <- add.constraint(port, type = "box", min = 0.05, max = 0.8)

# Leverage
port <- add.constraint(portfolio = port, type = "full_investment")```

Let’s use the built-in random solver. This essentially creates a set of feasible portfolios that satisfy all the constraints we have specified. For a full list of supported constraints see here

```# Generate random portfolios
rportfolios <- random_portfolios(port, permutations = 500000, rp_method = "sample")```

Now let’s add some objectives and optimize. For simplicity’s sake let’s do some mean-variance optimization.

```# Get minimum variance portfolio
minvar.port <- add.objective(port, type = "risk", name = "var")

# Optimize
minvar.opt <- optimize.portfolio(returns.data, minvar.port, optimize_method = "random",
rp = rportfolios)

# Generate maximum return portfolio
maxret.port <- add.objective(port, type = "return", name = "mean")

# Optimize
maxret.opt <- optimize.portfolio(returns.data, maxret.port, optimize_method = "random",
rp = rportfolios)

# Generate vector of returns
minret <- 0.06/100
maxret <- maxret.opt\$weights %*% meanReturns

vec <- seq(minret, maxret, length.out = 100)```

Now that we have the minimum variance as well as the maximum return portfolios, we can build out the efficient frontier. Let’s add a weight concentration objective as well to ensure we don’t get highly concentrated portfolios.

Note:

• `random_portfolios()` ignores any diversification constraints. Hence, we didn’t add it previously.
• Using the random solver for each portfolio in the loop below would be very compute intensive. We’ll use the ROI (R Optmization Infrastructure) solver instead.
```eff.frontier <- data.frame(Risk = rep(NA, length(vec)),
Return = rep(NA, length(vec)),
SharpeRatio = rep(NA, length(vec)))

frontier.weights <- mat.or.vec(nr = length(vec), nc = ncol(returns.data))
colnames(frontier.weights) <- colnames(returns.data)

for(i in 1:length(vec)){
eff.port <- add.constraint(port, type = "return", name = "mean", return_target = vec[i])
eff.port <- add.objective(eff.port, type = "risk", name = "var")
# eff.port <- add.objective(eff.port, type = "weight_concentration", name = "HHI",
#                            conc_aversion = 0.001)

eff.port <- optimize.portfolio(returns.data, eff.port, optimize_method = "ROI")

eff.frontier\$Risk[i] <- sqrt(t(eff.port\$weights) %*% covMat %*% eff.port\$weights)

eff.frontier\$Return[i] <- eff.port\$weights %*% meanReturns

eff.frontier\$Sharperatio[i] <- eff.port\$Return[i] / eff.port\$Risk[i]

frontier.weights[i,] = eff.port\$weights

print(paste(round(i/length(vec) * 100, 0), "% done..."))
}```

Now lets plot !

```feasible.sd <- apply(rportfolios, 1, function(x){
return(sqrt(matrix(x, nrow = 1) %*% covMat %*% matrix(x, ncol = 1)))
})

feasible.means <- apply(rportfolios, 1, function(x){
return(x %*% meanReturns)
})

feasible.sr <- feasible.means / feasible.sd

p <- plot_ly(x = feasible.sd, y = feasible.means, color = feasible.sr,
mode = "markers", type = "scattergl", showlegend = F,

marker = list(size = 3, opacity = 0.5,
colorbar = list(title = "Sharpe Ratio"))) %>%

add_trace(data = eff.frontier, x = Risk, y = Return, mode = "markers",
type = "scattergl", showlegend = F,
marker = list(color = "#F7C873", size = 5)) %>%

layout(title = "Random Portfolios with Plotly",
yaxis = list(title = "Mean Returns", tickformat = ".2%"),
xaxis = list(title = "Standard Deviation", tickformat = ".2%"),
plot_bgcolor = "#434343",
paper_bgcolor = "#F8F8F8",
annotations = list(
list(x = 0.4, y = 0.75,
ax = -30, ay = -30,
text = "Efficient frontier",
font = list(color = "#F6E7C1", size = 15),
arrowcolor = "white")
))```

The chart above is plotting 42,749 data points ! Also, you’ll notice that since the portfolios on the frontier(beige dots) have an added weight concentration objective, thefrontier seems sub optimal. Below is a comparison.

Let’s also plot the weights to check how diversified our optimal portfolios are. We’ll use a barchart for this.

```frontier.weights.melt <- reshape2::melt(frontier.weights)

q <- plot_ly(frontier.weights.melt, x = Var1, y = value, group = Var2, type = "bar") %>%
layout(title = "Portfolio weights across frontier", barmode = "stack",
xaxis = list(title = "Index"),
yaxis = list(title = "Weights(%)", tickformat = ".0%"))```

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