Complex Portfolio Optimization with PortfolioAnalytics

• Discuss Portfolio Optimization
• Introduce PortfolioAnalytics
• Demonstrate PortfolioAnalytics with Examples

"Modern" Portfolio Theory (MPT) was introduced by Harry Markowitz in 1952.

In general, MPT states that an investor's objective is to maximize portfolio expected return for a given amount of risk.

General Objectives

• Maximize a measure of gain per unit measure of risk
• Minimize a measure of risk

How do we define risk? What about more complex objectives and constraints?

• Minimize Risk
  • Volatility
  • Tail Loss (VaR, ES)
  • Other Downside Risk Measure
• Maximize Risk Adjusted Return
  • Sharpe Ratio, Modified Sharpe Ratio
  • Several Others
• Risk Budgets
  • Equal Component Contribution to Risk (i.e. Risk Parity)
  • Limits on Component Contribution
• Maximize a Utility Function
  • Quadratic, CRRA, etc.

PortfolioAnalytics is an R package designed to provide numerical solutions and visualizations for portfolio optimization problems with complex constraints and objectives.

• Support for multiple constraint and objective types
• An objective function can be any valid R function
• Modular constraints and objectives
• Support for user defined moment functions
• Visualizations
• Solver agnostic
• Support for parallel computing

Linear and Quadratic Programming Solvers

• R Optimization Infrastructure (ROI)
  • GLPK (Rglpk)
  • Symphony (Rsymphony)
  • Quadprog (quadprog)

Global (stochastic or continuous solvers)

• Random Portfolios
• Differential Evolution (DEoptim)
• Particle Swarm Optimization (pso)
• Generalized Simulated Annealing (GenSA)

PortfolioAnalytics has three methods to generate random portfolios.

1. The sample method to generate random portfolios is based on an idea by Pat Burns.
2. The simplex method to generate random portfolios is based on a paper by W. T. Shaw.
3. The grid method to generate random portfolios is based on the gridSearch function in the NMOF package.

args(portfolio.spec)

## function (assets = NULL, category_labels = NULL, weight_seq = NULL, 
##     message = FALSE) 
## NULL

Initializes the portfolio object that holds portfolio level data, constraints, and objectives

args(add.constraint)

## function (portfolio, type, enabled = TRUE, message = FALSE, ..., 
##     indexnum = NULL) 
## NULL

Supported Constraint Types

• Sum of Weights
• Box
• Group
• Factor Exposure
• Position Limit
• and many more

args(add.objective)

## function (portfolio, constraints = NULL, type, name, arguments = NULL, 
##     enabled = TRUE, ..., indexnum = NULL) 
## NULL

Supported Objective types

• Return
• Risk
• Risk Budget
• Weight Concentration

args(optimize.portfolio)

## function (R, portfolio = NULL, constraints = NULL, objectives = NULL, 
##     optimize_method = c("DEoptim", "random", "ROI", "pso", "GenSA"), 
##     search_size = 20000, trace = FALSE, ..., rp = NULL, momentFUN = "set.portfolio.moments", 
##     message = FALSE) 
## NULL

args(optimize.portfolio.rebalancing)

## function (R, portfolio = NULL, constraints = NULL, objectives = NULL, 
##     optimize_method = c("DEoptim", "random", "ROI"), search_size = 20000, 
##     trace = FALSE, ..., rp = NULL, rebalance_on = NULL, training_period = NULL, 
##     trailing_periods = NULL) 
## NULL

Here we will look at portfolio optimization in the context of stocks.

• Selection of large cap, mid cap, and small cap stocks from CRSP data
• Weekly data from 1/7/1997 to 12/28/2010
• 15 Large Cap
• 15 Mid Cap
• 5 Small Cap

equity.data <- cbind(largecap_weekly[,1:15], 
                     midcap_weekly[,1:15], 
                     smallcap_weekly[,1:5])

Here we consider a portfolio of stocks. Our objective is to maximize portfolio return with a target of 0.0015 and minimize portfolio StdDev with a target of 0.02 subject to dollar neutral, beta, box, and position limit constraints.

portf.dn <- portfolio.spec(stocks)

# Add constraint such that the portfolio weights sum to 0*
portf.dn <- add.constraint(portf.dn, type="weight_sum", 
                                  min_sum=-0.01, max_sum=0.01)

# Add box constraint such that no asset can have a weight of greater than
# 20% or less than -20%
portf.dn <- add.constraint(portf.dn, type="box", min=-0.2, max=0.2)

# Add constraint such that we have at most 20 positions
portf.dn <- add.constraint(portf.dn, type="position_limit", max_pos=20)

# Add constraint such that the portfolio beta is between -0.25 and 0.25
betas <- t(CAPM.beta(equity.data, market, Rf))
portf.dn <- add.constraint(portf.dn, type="factor_exposure", B=betas, 
                           lower=-0.25, upper=0.25)

# Add objective to maximize portfolio return with a target of 0.0015
portf.dn.StdDev <- add.objective(portf.dn, type="return", name="mean",
                                 target=0.0015)

# Add objective to minimize portfolio StdDev with a target of 0.02
portf.dn.StdDev <- add.objective(portf.dn.StdDev, type="risk", name="StdDev",
                                 target=0.02)

# Generate random portfolios
rp <- random_portfolios(portf.dn, 10000, "sample")

# Run the optimization
opt.dn <- optimize.portfolio(equity.data, portf.dn.StdDev, 
                               optimize_method="random", rp=rp,
                               trace=TRUE)

plot(opt.dn, main="Dollar Neutral Portfolio", risk.col="StdDev", neighbors=10)

Here we will look at portfolio optimization in the context of portfolio of hedge funds.

• EDHEC-Risk Alternative Indexes monthly returns from 1/31/1997 to 1/31/2014

R <- edhec[,c("Convertible.Arbitrage", "Equity.Market.Neutral", 
              "Fixed.Income.Arbitrage", 
              "CTA.Global", "Emerging.Markets", "Global.Macro")]
# Abreviate column names for convenience and plotting
colnames(R) <- c("CA", "EMN", "FIA", "CTAG", "EM", "GM")

Consider an allocation to hedge funds using the EDHEC-Risk Alternative Index as a proxy. This will be an extended example starting with an objective to minimize modified expected shortfall, then add risk budget percent contribution limit, and finally add equal risk contribution limit.

• Minimize Modified Expected Shortfall
• Minimize Modified Expected Shortfall with Risk Budget Limit
• Minimize Modified Expected Shortfall with Equal Risk Contribution

# Specify an initial portfolio
funds <- colnames(R)
portf.init <- portfolio.spec(funds)

# Add constraint such that the weights sum to 1*
portf.init <- add.constraint(portf.init, type="weight_sum", 
                             min_sum=0.99, max_sum=1.01)

# Add box constraint such that no asset can have a weight of greater than
# 40% or less than 5%
portf.init <- add.constraint(portf.init, type="box", 
                             min=0.05, max=0.4)

# Add return objective with multiplier=0 such that the portfolio mean
# return is calculated, but does not impact optimization
portf.init <- add.objective(portf.init, type="return", 
                            name="mean", multiplier=0)

# Add objective to minimize expected shortfall
portf.minES <- add.objective(portf.init, type="risk", name="ES")

# Add objective to set upper bound on percentage component contribution
portf.minES.RB <- add.objective(portf.minES, type="risk_budget", 
                                name="ES", max_prisk=0.3)
# Relax box constraints
portf.minES.RB$constraints[[2]]$max <- rep(1,ncol(R))

# Add objective to minimize concentration of modified ES
# component contribution
portf.minES.EqRB <- add.objective(portf.minES, type="risk_budget", 
                                  name="ES", min_concentration=TRUE)
# Relax box constraints
portf.minES.EqRB <- add.constraint(portf.minES.EqRB, type="box", 
                                   min=0.05, max=1, indexnum=2)

# Combine the 3 portfolios
portf <- combine.portfolios(list(minES=portf.minES, 
                                 minES.RB=portf.minES.RB, 
                                 minES.EqRB=portf.minES.EqRB))

# Run the optimization
opt.minES <- optimize.portfolio(R, portf, optimize_method="DEoptim", 
                                search_size=5000, trace=TRUE, traceDE=0)

# Set rebalancing frequency
rebal.freq <- "quarters"

# Training Period
training <- 120

# Trailing Period
trailing <- 72

bt.opt.minES <- optimize.portfolio.rebalancing(R, portf,
                                               optimize_method="DEoptim", 
                                               rebalance_on=rebal.freq, 
                                               training_period=training, 
                                               trailing_periods=trailing,
                                               search_size=5000,
                                               traceDE=0)

ret.bt.opt <- do.call(cbind, lapply(bt.opt.minES, 
                                    function(x) summary(x)$portfolio_returns))
colnames(ret.bt.opt) <- c("min ES", "min ES RB", "min ES Eq RB")

charts.PerformanceSummary(ret.bt.opt)

Consider an allocation to hedge funds using the EDHEC-Risk Alternative Index as a proxy. Our objective to maximize the fourth order expansion of the Constant Relative Risk Aversion (CRRA) expected utility function as in the Boudt paper and Martellini paper. We use the same data as Example 3.

\[ EU_{\lambda}(w) = - \frac{\lambda}{2} m_{(2)}(w) + \frac{\lambda (\lambda + 1)}{6} m_{(3)}(w) - \frac{\lambda (\lambda + 1) (\lambda + 2)}{24} m_{(4)}(w) \]

CRRA <- function(R, weights, lambda, sigma, m3, m4){
  weights <- matrix(weights, ncol=1)
  M2.w <- t(weights) %*% sigma %*% weights
  M3.w <- t(weights) %*% m3 %*% (weights %x% weights)
  M4.w <- t(weights) %*% m4 %*% (weights %x% weights %x% weights)
  term1 <- (1 / 2) * lambda * M2.w
  term2 <- (1 / 6) * lambda * (lambda + 1) * M3.w
  term3 <- (1 / 24) * lambda * (lambda + 1) * (lambda + 2) * M4.w
  out <- -term1 + term2 - term3
  out
}

The default function for momentFUN is set.portfolio.moments. We need to write our own function to estimate the moments for our objective function.

crra.moments <- function(R, ...) {
    out <- list()
    out$mu <- colMeans(R)
    out$sigma <- cov(R)
    out$m3 <- PerformanceAnalytics:::M3.MM(R)
    out$m4 <- PerformanceAnalytics:::M4.MM(R)
    out
}

# Specify portfolio
portf.crra <- portfolio.spec(funds)

# Add constraint such that the weights sum to 1
portf.crra <- add.constraint(portf.crra, type="weight_sum", 
                             min_sum=0.99, max_sum=1.01)

# Add box constraint such that no asset can have a weight of greater than
# 40% or less than 5% 
portf.crra <- add.constraint(portf.crra, type="box", 
                             min=0.05, max=0.4)

# Add objective to maximize CRRA
portf.crra <- add.objective(portf.crra, type="return", 
                            name="CRRA", arguments=list(lambda=10))

# Dummy objectives for plotting and/or further analysis
portf.crra <- add.objective(portf.crra, type="return", name="mean", multiplier=0)
portf.crra <- add.objective(portf.crra, type="risk", name="ES", multiplier=0)
portf.crra <- add.objective(portf.crra, type="risk", name="StdDev", multiplier=0)

opt.crra <- optimize.portfolio(R, portf.crra, optimize_method="DEoptim", 
                                 search_size=5000, trace=TRUE, traceDE=0,
                                 momentFUN="crra.moments")

head(extractStats(opt.crra),4)

##              CRRA.CRRA     mean      ES  StdDev       out   w.CA  w.EMN
## .DE.portf.1 -0.0009786 0.005906 0.03407 0.01367 0.0009786 0.1667 0.1667
## .DE.portf.2 -0.0007585 0.005716 0.02532 0.01215 0.0007585 0.1560 0.3420
## .DE.portf.3 -0.0019451 0.006380 0.05068 0.01901 0.0019451 0.1180 0.1200
## .DE.portf.4 -0.0007816 0.005536 0.02591 0.01232 0.0007816 0.0900 0.1840
##              w.FIA w.CTAG   w.EM   w.GM
## .DE.portf.1 0.1667 0.1667 0.1667 0.1667
## .DE.portf.2 0.0560 0.2180 0.1280 0.0920
## .DE.portf.3 0.0880 0.1300 0.3900 0.1620
## .DE.portf.4 0.2800 0.2720 0.1160 0.0640

chart.RiskReward(opt.crra, risk.col = "ES")
chart.RiskReward(opt.crra, risk.col = "StdDev")

bt.opt.crra <- optimize.portfolio.rebalancing(R, portf.crra, 
                                              optimize_method="DEoptim",
                                              search_size=5000, trace=TRUE,
                                              traceDE=0,
                                              momentFUN="crra.moments",
                                              rebalance_on=rebal.freq, 
                                              training_period=training, 
                                              trailing_periods=trailing)

ret.crra <- summary(bt.opt.crra)$portfolio_returns
colnames(ret.crra) <- "CRRA"

chart.Weights(bt.opt.crra, main="CRRA Weights", col=bluemono)

charts.PerformanceSummary(cbind(ret.bt.opt, ret.crra), 
                          main="Optimization Performance")

• Introduced the goals and summary of PortfolioAnalytics
• Demonstrated the flexibility through examples
• Exciting plans for GSOC 2014
  • Support for regime switching
  • Support for supervised learning
  • many more

Acknowledgements

Many thanks to...

• Google: funding for Google Summer of Code (GSoC)
• UW CF&RM Program: continued work on PortfolioAnalytics
• GSoC Mentors: Brian Peterson, Peter Carl, Doug Martin, and Guy Yollin
• R/Finance Committee

PortfolioAnalytics is on R-Forge in the ReturnAnalytics project

• PortfolioAnalytics

Source code for the slides

• https://github.com/rossb34/PortfolioAnalyticsPresentation

and view it here

• http://rossb34.github.io/PortfolioAnalyticsPresentation/

• ROI
• DEoptim
• pso
• GenSA
• PerformanceAnalytics
• Patrick Burns Random Portfolios
• W.T. Shaw Random Portfolios
• Martellini paper
• Boudt paper
• Shiny App