Feasible Space Analysis & Hierarchical 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.

Common 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, Constant Relative Risk Aversion (CRRA), etc.

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

Supports:

• multiple constraint and objective types
• modular constraints and objectives
• an objective function can be any valid R function
• user defined moment functions (e.g. covariance matrix, return projections)
• visualizations
• solver agnostic
• 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, 
##     rolling_window = NULL) 
## NULL

Ledoit and Wolf (2003):

"The central message of this paper is that nobody should be using the sample covariance matrix for the purpose of portfolio optimization."

• Sample
• Shrinkage Estimators
• Factor Model
• Expressing Views
• See Custom Moment and Objective Functions vignette

Here we will analyse the feasible space of allocations to hedge fund strategies

• EDHEC-Risk Alternative Indexes monthly returns from 1/31/1997 to 12/31/2014

# define base portfolio
portf.base <- portfolio.spec(colnames(R))
portf.base <- add.constraint(portf.base, type="weight_sum",
                             min_sum=0.99, max_sum=1.01)
portf.base <- add.constraint(portf.base, type="box", min=0, max=1)
ef <- create.EfficientFrontier(R, portfolio=portf.base, 
                               type="mean-StdDev", n.portfolios=100)
chart.EfficientFrontier(ef, match.col="StdDev", pch=18, col="lightblue")

p <- portfolio.spec(colnames(R))
p <- add.constraint(p, type="weight_sum", min_sum=0.99, max_sum=1.01)
p <- add.constraint(p, type="box", min=0, max=1)
p <- add.objective(p, type="return", name="mean", multiplier=0)
p <- add.objective(p, type="risk", name="StdDev", multiplier=0)
rp <- random_portfolios(p, permutations=5000, rp_method='sample')
opt <- optimize.portfolio(R, p, optimize_method="random", rp=rp, trace=TRUE)
xt <- extractStats(opt)

Consider an equity long short portfolio

• We first consider a 'naive' model and specify the portfolio and constraints

• Extend the model and separate the portfolio into a long portfolio and a short portfolio

• Further extend the model and use a framework that allows one to express a view and reduce allocation/exposure to the short portfolio

Consider an allocation to equity sectors using 9 sector ETFs

portf.naive <- portfolio.spec(colnames(R))
portf.naive <- add.constraint(portf.naive, type="weight_sum",
                              min_sum=-0.05, max_sum=0.05)
portf.naive <- add.constraint(portf.naive, type="box", min=-0.5, max=0.5)
portf.naive <- add.constraint(portf.naive, type = "leverage_exposure", leverage=2)
portf.naive <- add.objective(portf.naive, type="risk", name="StdDev")
portf.naive <- add.objective(portf.naive, type="risk_budget",
                             name="StdDev", max_prisk=0.50)

rp.naive <- random_portfolios(portf.naive, permutations=1000, rp_method='sample')
opt.naive <- optimize.portfolio.rebalancing(R, portf.naive,
                                            optimize_method="random",
                                            rebalance_on="quarters",
                                            training_period=36,
                                            rp=rp.naive, trace=TRUE)
# compute arithmetic portfolio returns because of negative weights
ret.naive <- Return.portfolio(R, extractWeights(opt.naive), geometric = FALSE)
colnames(ret.naive) <- "naive.ls"
charts.PerformanceSummary(ret.naive)

# long portfolio
p.long <- portfolio.spec(assets=colnames(R))
p.long <- add.constraint(p.long, type="weight_sum", min_sum=0.99, max_sum=1.01)
p.long <- add.constraint(p.long, type="box", min=0, max=0.85)
p.long <- add.objective(p.long, type="risk", name="StdDev")
p.long <- add.objective(p.long, type="risk_budget", name="StdDev", max_prisk=0.50)
rp.long <- random_portfolios(p.long, permutations=1000, rp_method='sample')
# short portfolio
p.short <- portfolio.spec(assets=colnames(R))
p.short <- add.constraint(p.short, type="weight_sum", min_sum=-1.01, max_sum=-0.99)
p.short <- add.constraint(p.short, type="box", min=-0.85, max=0)
p.short <- add.objective(p.short, type="risk", name="StdDev")
p.short <- add.objective(p.short, type="risk_budget", name="StdDev", max_prisk=0.50)
rp.short <- random_portfolios(p.short, permutations=1000, rp_method='sample')

# combined portfolio
p <- portfolio.spec(assets=paste("proxy",1:2, sep="."))
p <- add.constraint(p, type="weight_sum", min_sum=0.99, max_sum=1.01)
p <- add.constraint(p, type="box", min=0.1, max=1)
p <- add.objective(p, type="return", name="mean")
p <- add.objective(p, type="risk", name="StdDev")
rp <- random_portfolios(p, permutations=1000, rp_method='sample')
mult.portf <- mult.portfolio.spec(p)
mult.portf <- add.sub.portfolio(mult.portf, p.long, rp=rp.long, 
                                optimize_method="random", rebalance_on="quarters", 
                                training_period=36)
mult.portf <- add.sub.portfolio(mult.portf, p.short, rp=rp.short, 
                                optimize_method="random", rebalance_on="quarters", 
                                training_period=36)

# run the optimization
opt.m.ls <- optimize.portfolio.rebalancing(R, mult.portf,
                                           optimize_method = "random",
                                           trace = TRUE, rp = rp,
                                           rebalance_on = "quarters",
                                           training_period = 36)
r.ls <- Return.portfolio(opt.m.ls$R, extractWeights(opt.m.ls), geometric = FALSE)
charts.PerformanceSummary(r.ls)

• Define an indicator for overall market direction
  • 12 period exponentially weighted moving average on monthly closing prices of SPY
• Use the trend indicator for a simple regime model
  • Regime 1: Price > EMA (Uptrend)
  • Regime 2: Price <= EMA (Downtrend)
• Define the portfolio optimization problem such that we reduce exposure to the short portfolio when in regime 1

p.short.1 <- portfolio.spec(assets=colnames(R))
p.short.1 <- add.constraint(p.short.1, type="weight_sum",
                            min_sum=-0.26, max_sum=-0.24)
p.short.1 <- add.constraint(portfolio=p.short.1, type="box", min=-0.25, max=0)
p.short.1 <- add.objective(portfolio=p.short.1, type="risk", name="StdDev")
p.short.1 <- add.objective(p.short.1, type="risk_budget",
                           name="StdDev", max_prisk=0.50)
p.short.2 <- p.short
# define the portfolios for the regime model
# regime 1: Price >  EMA ; regime 2: Price <= EMA
trend.ind <- na.omit(cbind(Ad(SPY), TTR::EMA(Ad(SPY), n = 12)))
regime <- ifelse(trend.ind$SPY.Adjusted > trend.ind$EMA, 1, 2)
regime.port <- regime.portfolios(regime, combine.portfolios(list(p.short.1, p.short.2)))

p <- portfolio.spec(assets=paste("proxy",1:2, sep="."))
p <- add.constraint(p, type="weight_sum",min_sum=0.99, max_sum=1.01)
p <- add.constraint(p, type="box", min=0.1, max=1)
p <- add.objective(p, type="return", name="mean")
p <- add.objective(p, type="risk", name="StdDev")
rp <- random_portfolios(p, permutations=1000, rp_method='sample')

# initialize multi-layer (i.e. hierarchical) portfolio specification
mult.portf <- mult.portfolio.spec(p)
mult.portf <- add.sub.portfolio(mult.portf, p.long, rp=rp.long,
                                optimize_method="random", rebalance_on="quarters",
                                training_period=36)
mult.portf <- add.sub.portfolio(mult.portf, regime.port,
                                search_size = 1000, optimize_method="random", 
                                rebalance_on="quarters", training_period=36)

opt.mr.ls <- optimize.portfolio.rebalancing(R, mult.portf,
                                            optimize_method = "random",
                                            trace = TRUE, rp = rp,
                                            rebalance_on = "quarters",
                                            training_period = 36)
r.ls.regime <- Return.portfolio(opt.mr.ls$R, extractWeights(opt.mr.ls), geometric = FALSE)
ret.all <- na.omit(cbind(ret.naive, r.ls, r.ls.regime))
charts.PerformanceSummary(ret.all)

• Introduced the goals and summary of PortfolioAnalytics
• Demonstrated the flexibility through examples
• Plans for continued development
  • Interface to \(parma\)
  • Additional solvers
  • "Gallery" of examples

Acknowledgements

Many thanks to...

• Google: funding Google Summer of Code (GSoC) for 2013 and 2014
• UW CF&RM Program: continued work on PortfolioAnalytics
• GSoC Mentors: Brian Peterson, Peter Carl, Doug Martin, and Guy Yollin
• R/Finance Committee

• PortfolioAnalytics on CRAN

• PortfolioAnalytics on GitHub

Source code for the slides

• https://github.com/rossb34/PortfolioAnalyticsPresentation2016

and view it here

• http://rossb34.github.io/PortfolioAnalyticsPresentation2016/

• ROI
• DEoptim
• pso
• GenSA
• PerformanceAnalytics
• Patrick Burns Random Portfolios
• W.T. Shaw Random Portfolios
• Improved Forecasts of Higher-Order Co-moments and Implications for Portfolio Selection
• Higher Order Comoments of Multifactor Models and Asset Allocation